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Invariants ​

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julia
is_snf(G::FinGenAbGroup) -> Bool
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julia
number_of_generators(G::FinGenAbGroup) -> Int
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julia
number_of_relations(G::FinGenAbGroup) -> Int
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julia
rels(A::FinGenAbGroup) -> ZZMatrix
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julia
isfinite(A::FinGenAbGroup) -> Bool
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julia
torsion_free_rank(A::FinGenAbGroup) -> Int
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julia
order(A::FinGenAbGroup) -> ZZRingElem
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julia
exponent(A::FinGenAbGroup) -> ZZRingElem
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julia
is_trivial(A::FinGenAbGroup) -> Bool
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julia
is_torsion(G::FinGenAbGroup) -> Bool

Return whether G is a torsion group.

source

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julia
is_cyclic(G::FinGenAbGroup) -> Bool
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julia
elementary_divisors(G::FinGenAbGroup) -> Vector{ZZRingElem}
',1)),t("p",null,[s[183]||(s[183]=e("Given ")),t("mjx-container",F1,[(a(),l("svg",Z1,s[171]||(s[171]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 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Invariants ​

`,2)),t("details",U,[t("summary",null,[s[73]||(s[73]=t("a",{id:"is_snf-Tuple{FinGenAbGroup}",href:"#is_snf-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_snf")],-1)),s[74]||(s[74]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[79]||(s[79]=i('
julia
is_snf(G::FinGenAbGroup) -> Bool
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julia
number_of_generators(G::FinGenAbGroup) -> Int
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julia
number_of_relations(G::FinGenAbGroup) -> Int
',1)),t("p",null,[s[93]||(s[93]=e("Return the number of relations of ")),t("mjx-container",Y,[(a(),l("svg",_,s[91]||(s[91]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[92]||(s[92]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[94]||(s[94]=e(" in the current representation."))]),s[96]||(s[96]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",t1,[t("summary",null,[s[97]||(s[97]=t("a",{id:"rels-Tuple{FinGenAbGroup}",href:"#rels-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"rels")],-1)),s[98]||(s[98]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[103]||(s[103]=i('
julia
rels(A::FinGenAbGroup) -> ZZMatrix
',1)),t("p",null,[s[101]||(s[101]=e("Return the currently used relations of ")),t("mjx-container",s1,[(a(),l("svg",e1,s[99]||(s[99]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[100]||(s[100]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[102]||(s[102]=e(" as a single matrix."))]),s[104]||(s[104]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",i1,[t("summary",null,[s[105]||(s[105]=t("a",{id:"is_finite-Tuple{FinGenAbGroup}",href:"#is_finite-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_finite")],-1)),s[106]||(s[106]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[111]||(s[111]=i('
julia
isfinite(A::FinGenAbGroup) -> Bool
',1)),t("p",null,[s[109]||(s[109]=e("Return whether ")),t("mjx-container",l1,[(a(),l("svg",a1,s[107]||(s[107]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),s[108]||(s[108]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"A")])],-1))]),s[110]||(s[110]=e(" is finite."))]),s[112]||(s[112]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",n1,[t("summary",null,[s[113]||(s[113]=t("a",{id:"torsion_free_rank-Tuple{FinGenAbGroup}",href:"#torsion_free_rank-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"torsion_free_rank")],-1)),s[114]||(s[114]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[125]||(s[125]=i('
julia
torsion_free_rank(A::FinGenAbGroup) -> Int
',1)),t("p",null,[s[121]||(s[121]=e("Return the torsion free rank of ")),t("mjx-container",o1,[(a(),l("svg",r1,s[115]||(s[115]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),s[116]||(s[116]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"A")])],-1))]),s[122]||(s[122]=e(", that is, the dimension of the ")),t("mjx-container",Q1,[(a(),l("svg",p1,s[117]||(s[117]=[i('',1)]))),s[118]||(s[118]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"Q")])])],-1))]),s[123]||(s[123]=e("-vectorspace ")),t("mjx-container",d1,[(a(),l("svg",T1,s[119]||(s[119]=[i('',1)]))),s[120]||(s[120]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"A"),t("msub",null,[t("mo",null,"⊗"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"Z")])])]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"Q")])])],-1))]),s[124]||(s[124]=e("."))]),s[126]||(s[126]=t("p",null,[e("See also "),t("a",{href:"/v0.34.8/manual/quad_forms/Zgenera#rank-Tuple{ZZGenus}"},[t("code",null,"rank")]),e(".")],-1)),s[127]||(s[127]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",h1,[t("summary",null,[s[128]||(s[128]=t("a",{id:"order-Tuple{FinGenAbGroup}",href:"#order-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"order")],-1)),s[129]||(s[129]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[137]||(s[137]=i('
julia
order(A::FinGenAbGroup) -> ZZRingElem
',1)),t("p",null,[s[134]||(s[134]=e("Return the order of ")),t("mjx-container",m1,[(a(),l("svg",g1,s[130]||(s[130]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),s[131]||(s[131]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"A")])],-1))]),s[135]||(s[135]=e(". It is assumed that ")),t("mjx-container",u1,[(a(),l("svg",k1,s[132]||(s[132]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),s[133]||(s[133]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"A")])],-1))]),s[136]||(s[136]=e(" is finite."))]),s[138]||(s[138]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",x1,[t("summary",null,[s[139]||(s[139]=t("a",{id:"exponent-Tuple{FinGenAbGroup}",href:"#exponent-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"exponent")],-1)),s[140]||(s[140]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[148]||(s[148]=i('
julia
exponent(A::FinGenAbGroup) -> ZZRingElem
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julia
is_trivial(A::FinGenAbGroup) -> Bool
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julia
is_torsion(G::FinGenAbGroup) -> Bool

Return whether G is a torsion group.

source

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julia
is_cyclic(G::FinGenAbGroup) -> Bool
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julia
elementary_divisors(G::FinGenAbGroup) -> Vector{ZZRingElem}
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a/v0.34.8/assets/manual_abelian_introduction.md.B8MmCMfI.lean.js b/v0.34.8/assets/manual_abelian_introduction.md.B8MmCMfI.lean.js index b937086597..52417611b2 100644 --- a/v0.34.8/assets/manual_abelian_introduction.md.B8MmCMfI.lean.js +++ b/v0.34.8/assets/manual_abelian_introduction.md.B8MmCMfI.lean.js @@ -6,4 +6,4 @@ import{_ as r,c as l,j as t,a as e,a5 as i,G as o,B as Q,o as a}from"./chunks/fr 3-element Vector{FinGenAbGroup}: (Z/2)^3 Z/2 x Z/4 - Z/8

Invariants ​

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julia
is_snf(G::FinGenAbGroup) -> Bool
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julia
number_of_generators(G::FinGenAbGroup) -> Int
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julia
number_of_relations(G::FinGenAbGroup) -> Int
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julia
rels(A::FinGenAbGroup) -> ZZMatrix
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julia
isfinite(A::FinGenAbGroup) -> Bool
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julia
torsion_free_rank(A::FinGenAbGroup) -> Int
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julia
order(A::FinGenAbGroup) -> ZZRingElem
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julia
exponent(A::FinGenAbGroup) -> ZZRingElem
',1)),t("p",null,[s[145]||(s[145]=e("Return the exponent of ")),t("mjx-container",w1,[(a(),l("svg",b1,s[141]||(s[141]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),s[142]||(s[142]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"A")])],-1))]),s[146]||(s[146]=e(". It is assumed that ")),t("mjx-container",c1,[(a(),l("svg",y1,s[143]||(s[143]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),s[144]||(s[144]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"A")])],-1))]),s[147]||(s[147]=e(" is finite."))]),s[149]||(s[149]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",f1,[t("summary",null,[s[150]||(s[150]=t("a",{id:"is_trivial-Tuple{FinGenAbGroup}",href:"#is_trivial-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_trivial")],-1)),s[151]||(s[151]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[156]||(s[156]=i('
julia
is_trivial(A::FinGenAbGroup) -> Bool
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julia
is_torsion(G::FinGenAbGroup) -> Bool

Return whether G is a torsion group.

source

',3))]),t("details",H1,[t("summary",null,[s[161]||(s[161]=t("a",{id:"is_cyclic-Tuple{FinGenAbGroup}",href:"#is_cyclic-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_cyclic")],-1)),s[162]||(s[162]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[167]||(s[167]=i('
julia
is_cyclic(G::FinGenAbGroup) -> Bool
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julia
elementary_divisors(G::FinGenAbGroup) -> Vector{ZZRingElem}
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Invariants ​

`,2)),t("details",U,[t("summary",null,[s[73]||(s[73]=t("a",{id:"is_snf-Tuple{FinGenAbGroup}",href:"#is_snf-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_snf")],-1)),s[74]||(s[74]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[79]||(s[79]=i('
julia
is_snf(G::FinGenAbGroup) -> Bool
',1)),t("p",null,[s[77]||(s[77]=e("Return whether the current relation matrix of the group ")),t("mjx-container",P,[(a(),l("svg",N,s[75]||(s[75]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[76]||(s[76]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[78]||(s[78]=e(" is in Smith normal form."))]),s[80]||(s[80]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",q,[t("summary",null,[s[81]||(s[81]=t("a",{id:"number_of_generators-Tuple{FinGenAbGroup}",href:"#number_of_generators-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"number_of_generators")],-1)),s[82]||(s[82]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[87]||(s[87]=i('
julia
number_of_generators(G::FinGenAbGroup) -> Int
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julia
number_of_relations(G::FinGenAbGroup) -> Int
',1)),t("p",null,[s[93]||(s[93]=e("Return the number of relations of ")),t("mjx-container",Y,[(a(),l("svg",_,s[91]||(s[91]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[92]||(s[92]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[94]||(s[94]=e(" in the current representation."))]),s[96]||(s[96]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",t1,[t("summary",null,[s[97]||(s[97]=t("a",{id:"rels-Tuple{FinGenAbGroup}",href:"#rels-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"rels")],-1)),s[98]||(s[98]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[103]||(s[103]=i('
julia
rels(A::FinGenAbGroup) -> ZZMatrix
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julia
isfinite(A::FinGenAbGroup) -> Bool
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julia
torsion_free_rank(A::FinGenAbGroup) -> Int
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julia
order(A::FinGenAbGroup) -> ZZRingElem
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julia
exponent(A::FinGenAbGroup) -> ZZRingElem
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julia
is_trivial(A::FinGenAbGroup) -> Bool
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julia
is_torsion(G::FinGenAbGroup) -> Bool

Return whether G is a torsion group.

source

',3))]),t("details",H1,[t("summary",null,[s[161]||(s[161]=t("a",{id:"is_cyclic-Tuple{FinGenAbGroup}",href:"#is_cyclic-Tuple{FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_cyclic")],-1)),s[162]||(s[162]=e()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[167]||(s[167]=i('
julia
is_cyclic(G::FinGenAbGroup) -> Bool
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julia
elementary_divisors(G::FinGenAbGroup) -> Vector{ZZRingElem}
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586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[172]||(s[172]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[184]||(s[184]=e(", return the elementary divisors of ")),t("mjx-container",D1,[(a(),l("svg",A1,s[173]||(s[173]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 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a/v0.34.8/assets/manual_abelian_structural.md.BDMufbJO.js b/v0.34.8/assets/manual_abelian_structural.md.BDMufbJO.js index 42ea28a819..dde9f67c70 100644 --- a/v0.34.8/assets/manual_abelian_structural.md.BDMufbJO.js +++ b/v0.34.8/assets/manual_abelian_structural.md.BDMufbJO.js @@ -8,7 +8,7 @@ import{_ as Q,c as e,a5 as a,j as t,a as i,G as o,B as r,o as l}from"./chunks/fr julia> divexact(order(A), order(H)) 45

source

`,3))]),t("details",M2,[t("summary",null,[s[159]||(s[159]=t("a",{id:"has_quotient-Tuple{FinGenAbGroup, Vector{Int64}}",href:"#has_quotient-Tuple{FinGenAbGroup, Vector{Int64}}"},[t("span",{class:"jlbinding"},"has_quotient")],-1)),s[160]||(s[160]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=a('
julia
has_quotient(G::FinGenAbGroup, invariant::Vector{Int}) -> Bool
',1)),t("p",null,[s[163]||(s[163]=i("Given an abelian group ")),t("mjx-container",C2,[(l(),e("svg",L2,s[161]||(s[161]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[162]||(s[162]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[164]||(s[164]=i(", return true if it has a quotient with given elementary divisors and false otherwise."))]),s[166]||(s[166]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",j2,[t("summary",null,[s[167]||(s[167]=t("a",{id:"has_complement-Tuple{FinGenAbGroupHom}",href:"#has_complement-Tuple{FinGenAbGroupHom}"},[t("span",{class:"jlbinding"},"has_complement")],-1)),s[168]||(s[168]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[180]||(s[180]=a(`
julia
has_complement(f::FinGenAbGroupHom) -> Bool, FinGenAbGroupHom
-has_complement(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool, FinGenAbGroupHom
`,1)),t("p",null,[s[173]||(s[173]=i("Given a map representing a subgroup of a group ")),t("mjx-container",G2,[(l(),e("svg",A2,s[169]||(s[169]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[170]||(s[170]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[174]||(s[174]=i(", or a subgroup ")),s[175]||(s[175]=t("code",null,"U",-1)),s[176]||(s[176]=i(" of a group ")),s[177]||(s[177]=t("code",null,"G",-1)),s[178]||(s[178]=i(", return either true and an injection of a complement in ")),t("mjx-container",Z2,[(l(),e("svg",B2,s[171]||(s[171]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[172]||(s[172]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[179]||(s[179]=i(", or false."))]),s[181]||(s[181]=t("p",null,[i("See also: "),t("a",{href:"/Hecke.jl/v0.34.8/manual/abelian/structural#is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("code",null,"is_pure")])],-1)),s[182]||(s[182]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",D2,[t("summary",null,[s[183]||(s[183]=t("a",{id:"is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_pure")],-1)),s[184]||(s[184]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[185]||(s[185]=a(`
julia
is_pure(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool

A subgroup U of G is called pure if for all n an element in U that is in the image of the multiplication by n map of G is also a multiple of an element in U.

For finite abelian groups this is equivalent to U having a complement in G. They are also know as isolated subgroups and serving subgroups.

See also: is_neat, has_complement

EXAMPLES

julia
julia> G = abelian_group([2, 8]);
+has_complement(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool, FinGenAbGroupHom
`,1)),t("p",null,[s[173]||(s[173]=i("Given a map representing a subgroup of a group ")),t("mjx-container",G2,[(l(),e("svg",A2,s[169]||(s[169]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[170]||(s[170]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[174]||(s[174]=i(", or a subgroup ")),s[175]||(s[175]=t("code",null,"U",-1)),s[176]||(s[176]=i(" of a group ")),s[177]||(s[177]=t("code",null,"G",-1)),s[178]||(s[178]=i(", return either true and an injection of a complement in ")),t("mjx-container",Z2,[(l(),e("svg",B2,s[171]||(s[171]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[172]||(s[172]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[179]||(s[179]=i(", or false."))]),s[181]||(s[181]=t("p",null,[i("See also: "),t("a",{href:"/v0.34.8/manual/abelian/structural#is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("code",null,"is_pure")])],-1)),s[182]||(s[182]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",D2,[t("summary",null,[s[183]||(s[183]=t("a",{id:"is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_pure")],-1)),s[184]||(s[184]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[185]||(s[185]=a(`
julia
is_pure(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool

A subgroup U of G is called pure if for all n an element in U that is in the image of the multiplication by n map of G is also a multiple of an element in U.

For finite abelian groups this is equivalent to U having a complement in G. They are also know as isolated subgroups and serving subgroups.

See also: is_neat, has_complement

EXAMPLES

julia
julia> G = abelian_group([2, 8]);
 
 julia> U, _ = sub(G, [G[1]+2*G[2]]);
 
@@ -21,7 +21,7 @@ import{_ as Q,c as e,a5 as a,j as t,a as i,G as o,B as r,o as l}from"./chunks/fr
 true
 
 julia> has_complement(U, G)[1]
-true

source

`,7))]),t("details",V2,[t("summary",null,[s[186]||(s[186]=t("a",{id:"is_neat-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#is_neat-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_neat")],-1)),s[187]||(s[187]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[188]||(s[188]=a(`
julia
is_neat(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool

A subgroup U of G is called neat if for all primes p an element in U that is in the image of the multiplication by p map of G is also a multiple of an element in U.

See also: is_pure

EXAMPLES

julia
julia> G = abelian_group([2, 8]);
+true

source

`,7))]),t("details",V2,[t("summary",null,[s[186]||(s[186]=t("a",{id:"is_neat-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#is_neat-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_neat")],-1)),s[187]||(s[187]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[188]||(s[188]=a(`
julia
is_neat(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool

A subgroup U of G is called neat if for all primes p an element in U that is in the image of the multiplication by p map of G is also a multiple of an element in U.

See also: is_pure

EXAMPLES

julia
julia> G = abelian_group([2, 8]);
 
 julia> U, _ = sub(G, [G[1] + 2*G[2]]);
 
@@ -29,7 +29,7 @@ import{_ as Q,c as e,a5 as a,j as t,a as i,G as o,B as r,o as l}from"./chunks/fr
 true
 
 julia> is_pure(U, G)
-false

source

`,6))]),t("details",S2,[t("summary",null,[s[189]||(s[189]=t("a",{id:"saturate-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#saturate-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"saturate")],-1)),s[190]||(s[190]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[191]||(s[191]=a('
julia
saturate(U::FinGenAbGroup, G::FinGenAbGroup) -> FinGenAbGroup

For a subgroup U of G find a minimal overgroup that is pure, and thus has a complement.

See also: is_pure, has_complement

source

',4))]),s[468]||(s[468]=t("p",null,"A sophisticated algorithm for the enumeration of all (or selected) subgroups of a finite abelian group is available.",-1)),t("details",J2,[t("summary",null,[s[192]||(s[192]=t("a",{id:"psubgroups-Tuple{FinGenAbGroup, Integer}",href:"#psubgroups-Tuple{FinGenAbGroup, Integer}"},[t("span",{class:"jlbinding"},"psubgroups")],-1)),s[193]||(s[193]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[205]||(s[205]=a(`
julia
psubgroups(g::FinGenAbGroup, p::Integer;
+false

source

`,6))]),t("details",S2,[t("summary",null,[s[189]||(s[189]=t("a",{id:"saturate-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#saturate-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"saturate")],-1)),s[190]||(s[190]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[191]||(s[191]=a('
julia
saturate(U::FinGenAbGroup, G::FinGenAbGroup) -> FinGenAbGroup

For a subgroup U of G find a minimal overgroup that is pure, and thus has a complement.

See also: is_pure, has_complement

source

',4))]),s[468]||(s[468]=t("p",null,"A sophisticated algorithm for the enumeration of all (or selected) subgroups of a finite abelian group is available.",-1)),t("details",J2,[t("summary",null,[s[192]||(s[192]=t("a",{id:"psubgroups-Tuple{FinGenAbGroup, Integer}",href:"#psubgroups-Tuple{FinGenAbGroup, Integer}"},[t("span",{class:"jlbinding"},"psubgroups")],-1)),s[193]||(s[193]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[205]||(s[205]=a(`
julia
psubgroups(g::FinGenAbGroup, p::Integer;
            subtype = :all,
            quotype = :all,
            index = -1,
diff --git a/v0.34.8/assets/manual_abelian_structural.md.BDMufbJO.lean.js b/v0.34.8/assets/manual_abelian_structural.md.BDMufbJO.lean.js
index 42ea28a819..dde9f67c70 100644
--- a/v0.34.8/assets/manual_abelian_structural.md.BDMufbJO.lean.js
+++ b/v0.34.8/assets/manual_abelian_structural.md.BDMufbJO.lean.js
@@ -8,7 +8,7 @@ import{_ as Q,c as e,a5 as a,j as t,a as i,G as o,B as r,o as l}from"./chunks/fr
 
 julia> divexact(order(A), order(H))
 45

source

`,3))]),t("details",M2,[t("summary",null,[s[159]||(s[159]=t("a",{id:"has_quotient-Tuple{FinGenAbGroup, Vector{Int64}}",href:"#has_quotient-Tuple{FinGenAbGroup, Vector{Int64}}"},[t("span",{class:"jlbinding"},"has_quotient")],-1)),s[160]||(s[160]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=a('
julia
has_quotient(G::FinGenAbGroup, invariant::Vector{Int}) -> Bool
',1)),t("p",null,[s[163]||(s[163]=i("Given an abelian group ")),t("mjx-container",C2,[(l(),e("svg",L2,s[161]||(s[161]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[162]||(s[162]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[164]||(s[164]=i(", return true if it has a quotient with given elementary divisors and false otherwise."))]),s[166]||(s[166]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",j2,[t("summary",null,[s[167]||(s[167]=t("a",{id:"has_complement-Tuple{FinGenAbGroupHom}",href:"#has_complement-Tuple{FinGenAbGroupHom}"},[t("span",{class:"jlbinding"},"has_complement")],-1)),s[168]||(s[168]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[180]||(s[180]=a(`
julia
has_complement(f::FinGenAbGroupHom) -> Bool, FinGenAbGroupHom
-has_complement(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool, FinGenAbGroupHom
`,1)),t("p",null,[s[173]||(s[173]=i("Given a map representing a subgroup of a group ")),t("mjx-container",G2,[(l(),e("svg",A2,s[169]||(s[169]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[170]||(s[170]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[174]||(s[174]=i(", or a subgroup ")),s[175]||(s[175]=t("code",null,"U",-1)),s[176]||(s[176]=i(" of a group ")),s[177]||(s[177]=t("code",null,"G",-1)),s[178]||(s[178]=i(", return either true and an injection of a complement in ")),t("mjx-container",Z2,[(l(),e("svg",B2,s[171]||(s[171]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[172]||(s[172]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[179]||(s[179]=i(", or false."))]),s[181]||(s[181]=t("p",null,[i("See also: "),t("a",{href:"/Hecke.jl/v0.34.8/manual/abelian/structural#is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("code",null,"is_pure")])],-1)),s[182]||(s[182]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",D2,[t("summary",null,[s[183]||(s[183]=t("a",{id:"is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_pure")],-1)),s[184]||(s[184]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[185]||(s[185]=a(`
julia
is_pure(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool

A subgroup U of G is called pure if for all n an element in U that is in the image of the multiplication by n map of G is also a multiple of an element in U.

For finite abelian groups this is equivalent to U having a complement in G. They are also know as isolated subgroups and serving subgroups.

See also: is_neat, has_complement

EXAMPLES

julia
julia> G = abelian_group([2, 8]);
+has_complement(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool, FinGenAbGroupHom
`,1)),t("p",null,[s[173]||(s[173]=i("Given a map representing a subgroup of a group ")),t("mjx-container",G2,[(l(),e("svg",A2,s[169]||(s[169]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[170]||(s[170]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[174]||(s[174]=i(", or a subgroup ")),s[175]||(s[175]=t("code",null,"U",-1)),s[176]||(s[176]=i(" of a group ")),s[177]||(s[177]=t("code",null,"G",-1)),s[178]||(s[178]=i(", return either true and an injection of a complement in ")),t("mjx-container",Z2,[(l(),e("svg",B2,s[171]||(s[171]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[172]||(s[172]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"G")])],-1))]),s[179]||(s[179]=i(", or false."))]),s[181]||(s[181]=t("p",null,[i("See also: "),t("a",{href:"/v0.34.8/manual/abelian/structural#is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("code",null,"is_pure")])],-1)),s[182]||(s[182]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",D2,[t("summary",null,[s[183]||(s[183]=t("a",{id:"is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#is_pure-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_pure")],-1)),s[184]||(s[184]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[185]||(s[185]=a(`
julia
is_pure(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool

A subgroup U of G is called pure if for all n an element in U that is in the image of the multiplication by n map of G is also a multiple of an element in U.

For finite abelian groups this is equivalent to U having a complement in G. They are also know as isolated subgroups and serving subgroups.

See also: is_neat, has_complement

EXAMPLES

julia
julia> G = abelian_group([2, 8]);
 
 julia> U, _ = sub(G, [G[1]+2*G[2]]);
 
@@ -21,7 +21,7 @@ import{_ as Q,c as e,a5 as a,j as t,a as i,G as o,B as r,o as l}from"./chunks/fr
 true
 
 julia> has_complement(U, G)[1]
-true

source

`,7))]),t("details",V2,[t("summary",null,[s[186]||(s[186]=t("a",{id:"is_neat-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#is_neat-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_neat")],-1)),s[187]||(s[187]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[188]||(s[188]=a(`
julia
is_neat(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool

A subgroup U of G is called neat if for all primes p an element in U that is in the image of the multiplication by p map of G is also a multiple of an element in U.

See also: is_pure

EXAMPLES

julia
julia> G = abelian_group([2, 8]);
+true

source

`,7))]),t("details",V2,[t("summary",null,[s[186]||(s[186]=t("a",{id:"is_neat-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#is_neat-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"is_neat")],-1)),s[187]||(s[187]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[188]||(s[188]=a(`
julia
is_neat(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool

A subgroup U of G is called neat if for all primes p an element in U that is in the image of the multiplication by p map of G is also a multiple of an element in U.

See also: is_pure

EXAMPLES

julia
julia> G = abelian_group([2, 8]);
 
 julia> U, _ = sub(G, [G[1] + 2*G[2]]);
 
@@ -29,7 +29,7 @@ import{_ as Q,c as e,a5 as a,j as t,a as i,G as o,B as r,o as l}from"./chunks/fr
 true
 
 julia> is_pure(U, G)
-false

source

`,6))]),t("details",S2,[t("summary",null,[s[189]||(s[189]=t("a",{id:"saturate-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#saturate-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"saturate")],-1)),s[190]||(s[190]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[191]||(s[191]=a('
julia
saturate(U::FinGenAbGroup, G::FinGenAbGroup) -> FinGenAbGroup

For a subgroup U of G find a minimal overgroup that is pure, and thus has a complement.

See also: is_pure, has_complement

source

',4))]),s[468]||(s[468]=t("p",null,"A sophisticated algorithm for the enumeration of all (or selected) subgroups of a finite abelian group is available.",-1)),t("details",J2,[t("summary",null,[s[192]||(s[192]=t("a",{id:"psubgroups-Tuple{FinGenAbGroup, Integer}",href:"#psubgroups-Tuple{FinGenAbGroup, Integer}"},[t("span",{class:"jlbinding"},"psubgroups")],-1)),s[193]||(s[193]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[205]||(s[205]=a(`
julia
psubgroups(g::FinGenAbGroup, p::Integer;
+false

source

`,6))]),t("details",S2,[t("summary",null,[s[189]||(s[189]=t("a",{id:"saturate-Tuple{FinGenAbGroup, FinGenAbGroup}",href:"#saturate-Tuple{FinGenAbGroup, FinGenAbGroup}"},[t("span",{class:"jlbinding"},"saturate")],-1)),s[190]||(s[190]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[191]||(s[191]=a('
julia
saturate(U::FinGenAbGroup, G::FinGenAbGroup) -> FinGenAbGroup

For a subgroup U of G find a minimal overgroup that is pure, and thus has a complement.

See also: is_pure, has_complement

source

',4))]),s[468]||(s[468]=t("p",null,"A sophisticated algorithm for the enumeration of all (or selected) subgroups of a finite abelian group is available.",-1)),t("details",J2,[t("summary",null,[s[192]||(s[192]=t("a",{id:"psubgroups-Tuple{FinGenAbGroup, Integer}",href:"#psubgroups-Tuple{FinGenAbGroup, Integer}"},[t("span",{class:"jlbinding"},"psubgroups")],-1)),s[193]||(s[193]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[205]||(s[205]=a(`
julia
psubgroups(g::FinGenAbGroup, p::Integer;
            subtype = :all,
            quotype = :all,
            index = -1,
diff --git a/v0.34.8/assets/manual_algebras_basics.md.CCIsU1L-.js b/v0.34.8/assets/manual_algebras_basics.md.CCIsU1L-.js
index 05f25ef047..e34f8f296c 100644
--- a/v0.34.8/assets/manual_algebras_basics.md.CCIsU1L-.js
+++ b/v0.34.8/assets/manual_algebras_basics.md.CCIsU1L-.js
@@ -1,8 +1,8 @@
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Basics ​
Creation of algebras ​
See the corresponding sections on structure constant algebras.
',3)),s("details",d,[s("summary",null,[t[0]||(t[0]=s("a",{id:"zero_algebra-Tuple{Field}",href:"#zero_algebra-Tuple{Field}"},[s("span",{class:"jlbinding"},"zero_algebra")],-1)),t[1]||(t[1]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[6]||(t[6]=a('
julia
zero_algebra([T, ] K::Field) -> AbstractAssociativeAlgebra
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The optional first argument determines the type of the algebra, and can be StructureConstantAlgebra (default) or MatrixAlgebra.
Examples
julia
julia> A = zero_algebra(QQ)
-Structure constant algebra of dimension 0 over QQ
source
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julia
base_ring(A::AbstractAssociativeAlgebra) -> Field
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julia
basis(A::AbstractAssociativeAlgebra) -> Vector
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0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[35]||(t[35]=i("-basis of ")),s("mjx-container",L,[(l(),e("svg",H,t[30]||(t[30]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 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See also ")),t[37]||(t[37]=s("a",{href:"/Hecke.jl/v0.34.8/manual/number_fields/elements#coordinates-Tuple{NumFieldElem}"},[s("code",null,"coordinates")],-1)),t[38]||(t[38]=i(" to get the the coordinates of an element with respect to the bases."))]),t[40]||(t[40]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t[177]||(t[177]=s("h2",{id:"predicates",tabindex:"-1"},[i("Predicates "),s("a",{class:"header-anchor",href:"#predicates","aria-label":'Permalink to "Predicates"'},"​")],-1)),s("details",j,[s("summary",null,[t[41]||(t[41]=s("a",{id:"is_zero-Tuple{Hecke.AbstractAssociativeAlgebra}",href:"#is_zero-Tuple{Hecke.AbstractAssociativeAlgebra}"},[s("span",{class:"jlbinding"},"is_zero")],-1)),t[42]||(t[42]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[47]||(t[47]=a('
julia
is_zero(A::AbstractAssociativeAlgebra) -> Bool
',1)),s("p",null,[t[45]||(t[45]=i("Return whether ")),s("mjx-container",M,[(l(),e("svg",A,t[43]||(t[43]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[44]||(t[44]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[46]||(t[46]=i(" is the zero algebra."))]),t[48]||(t[48]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",B,[s("summary",null,[t[49]||(t[49]=s("a",{id:"is_commutative-Tuple{Hecke.AbstractAssociativeAlgebra}",href:"#is_commutative-Tuple{Hecke.AbstractAssociativeAlgebra}"},[s("span",{class:"jlbinding"},"is_commutative")],-1)),t[50]||(t[50]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[55]||(t[55]=a('
julia
is_commutative(A::AbstractAssociativeAlgebra) -> Bool
',1)),s("p",null,[t[53]||(t[53]=i("Return whether ")),s("mjx-container",D,[(l(),e("svg",Z,t[51]||(t[51]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[52]||(t[52]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[54]||(t[54]=i(" is commutative."))]),t[56]||(t[56]=a(`
Examples
julia
julia> A = matrix_algebra(QQ, 2);
+import{_ as r,c as e,a5 as a,j as s,a as i,G as o,B as p,o as l}from"./chunks/framework.DbhH0_mT.js";const qs=JSON.parse('{"title":"Basics","description":"","frontmatter":{},"headers":[],"relativePath":"manual/algebras/basics.md","filePath":"manual/algebras/basics.md","lastUpdated":null}'),h={name:"manual/algebras/basics.md"},d={class:"jldocstring custom-block",open:""},Q={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},k={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.011ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 889 683","aria-hidden":"true"},g={class:"jldocstring custom-block",open:""},m={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},u={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.011ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 889 683","aria-hidden":"true"},x={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},T={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"1.697ex",height:"1.62ex",role:"img",focusable:"false",viewBox:"0 -716 750 716","aria-hidden":"true"},w={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},c={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.011ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 889 683","aria-hidden":"true"},b={class:"jldocstring custom-block",open:""},y={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},v={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.011ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 889 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Basics ​
Creation of algebras ​
See the corresponding sections on structure constant algebras.
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julia
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The optional first argument determines the type of the algebra, and can be StructureConstantAlgebra (default) or MatrixAlgebra.
Examples
julia
julia> A = zero_algebra(QQ)
+Structure constant algebra of dimension 0 over QQ
source
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julia
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julia
is_zero(A::AbstractAssociativeAlgebra) -> Bool
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julia
is_commutative(A::AbstractAssociativeAlgebra) -> Bool
',1)),s("p",null,[t[53]||(t[53]=i("Return whether ")),s("mjx-container",D,[(l(),e("svg",Z,t[51]||(t[51]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[52]||(t[52]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[54]||(t[54]=i(" is commutative."))]),t[56]||(t[56]=a(`
Examples
julia
julia> A = matrix_algebra(QQ, 2);
 
 julia> is_commutative(A)
-false
source
`,3))]),s("details",V,[s("summary",null,[t[57]||(t[57]=s("a",{id:"is_central-Tuple{Hecke.AbstractAssociativeAlgebra}",href:"#is_central-Tuple{Hecke.AbstractAssociativeAlgebra}"},[s("span",{class:"jlbinding"},"is_central")],-1)),t[58]||(t[58]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[72]||(t[72]=a('
julia
is_central(A::AbstractAssociativeAlgebra)
',1)),s("p",null,[t[67]||(t[67]=i("Return whether the ")),s("mjx-container",G,[(l(),e("svg",z,t[59]||(t[59]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[60]||(t[60]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[68]||(t[68]=i("-algebra 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1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[69]||(t[69]=i(" is central, that is, whether ")),s("mjx-container",K,[(l(),e("svg",O,t[63]||(t[63]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[64]||(t[64]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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julia
gens(A::AbstractAssociativeAlgebra; thorough_search::Bool = false) -> Vector
',1)),s("p",null,[t[84]||(t[84]=i("Given a ")),s("mjx-container",R,[(l(),e("svg",q,t[76]||(t[76]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 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718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[81]||(t[81]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[89]||(t[89]=i(" as an algebra over ")),s("mjx-container",Y,[(l(),e("svg",_,t[82]||(t[82]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[83]||(t[83]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[90]||(t[90]=i("."))]),t[92]||(t[92]=a(`
If thorough_search is true, the number of returned generators is possibly smaller. This will in general increase the runtime. It is not guaranteed that the number of generators is minimal in any case.
The gens_with_data function computes additional data for expressing a basis as words in the generators.
Examples
julia
julia> A = matrix_algebra(QQ, 3);
+false
source
`,3))]),s("details",V,[s("summary",null,[t[57]||(t[57]=s("a",{id:"is_central-Tuple{Hecke.AbstractAssociativeAlgebra}",href:"#is_central-Tuple{Hecke.AbstractAssociativeAlgebra}"},[s("span",{class:"jlbinding"},"is_central")],-1)),t[58]||(t[58]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[72]||(t[72]=a('
julia
is_central(A::AbstractAssociativeAlgebra)
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"Generators"'},"​")],-1)),s("details",N,[s("summary",null,[t[74]||(t[74]=s("a",{id:"gens-Tuple{Hecke.AbstractAssociativeAlgebra}",href:"#gens-Tuple{Hecke.AbstractAssociativeAlgebra}"},[s("span",{class:"jlbinding"},"gens")],-1)),t[75]||(t[75]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[91]||(t[91]=a('
julia
gens(A::AbstractAssociativeAlgebra; thorough_search::Bool = false) -> Vector
',1)),s("p",null,[t[84]||(t[84]=i("Given a ")),s("mjx-container",R,[(l(),e("svg",q,t[76]||(t[76]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 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")),s("mjx-container",Y,[(l(),e("svg",_,t[82]||(t[82]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[83]||(t[83]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[90]||(t[90]=i("."))]),t[92]||(t[92]=a(`
If thorough_search is true, the number of returned generators is possibly smaller. This will in general increase the runtime. It is not guaranteed that the number of generators is minimal in any case.
The gens_with_data function computes additional data for expressing a basis as words in the generators.
Examples
julia
julia> A = matrix_algebra(QQ, 3);
 
 julia> gens(A; thorough_search = true)
 5-element Vector{MatAlgebraElem{QQFieldElem, QQMatrix}}:
diff --git a/v0.34.8/assets/manual_algebras_basics.md.CCIsU1L-.lean.js b/v0.34.8/assets/manual_algebras_basics.md.CCIsU1L-.lean.js
index 05f25ef047..e34f8f296c 100644
--- a/v0.34.8/assets/manual_algebras_basics.md.CCIsU1L-.lean.js
+++ b/v0.34.8/assets/manual_algebras_basics.md.CCIsU1L-.lean.js
@@ -1,8 +1,8 @@
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Basics ​
Creation of algebras ​
See the corresponding sections on structure constant algebras.
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The optional first argument determines the type of the algebra, and can be StructureConstantAlgebra (default) or MatrixAlgebra.
Examples
julia
julia> A = zero_algebra(QQ)
-Structure constant algebra of dimension 0 over QQ
source
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See also ")),t[37]||(t[37]=s("a",{href:"/Hecke.jl/v0.34.8/manual/number_fields/elements#coordinates-Tuple{NumFieldElem}"},[s("code",null,"coordinates")],-1)),t[38]||(t[38]=i(" to get the the coordinates of an element with respect to the bases."))]),t[40]||(t[40]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t[177]||(t[177]=s("h2",{id:"predicates",tabindex:"-1"},[i("Predicates "),s("a",{class:"header-anchor",href:"#predicates","aria-label":'Permalink to "Predicates"'},"​")],-1)),s("details",j,[s("summary",null,[t[41]||(t[41]=s("a",{id:"is_zero-Tuple{Hecke.AbstractAssociativeAlgebra}",href:"#is_zero-Tuple{Hecke.AbstractAssociativeAlgebra}"},[s("span",{class:"jlbinding"},"is_zero")],-1)),t[42]||(t[42]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[47]||(t[47]=a('
julia
is_zero(A::AbstractAssociativeAlgebra) -> Bool
',1)),s("p",null,[t[45]||(t[45]=i("Return whether ")),s("mjx-container",M,[(l(),e("svg",A,t[43]||(t[43]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[44]||(t[44]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[46]||(t[46]=i(" is the zero algebra."))]),t[48]||(t[48]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",B,[s("summary",null,[t[49]||(t[49]=s("a",{id:"is_commutative-Tuple{Hecke.AbstractAssociativeAlgebra}",href:"#is_commutative-Tuple{Hecke.AbstractAssociativeAlgebra}"},[s("span",{class:"jlbinding"},"is_commutative")],-1)),t[50]||(t[50]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[55]||(t[55]=a('
julia
is_commutative(A::AbstractAssociativeAlgebra) -> Bool
',1)),s("p",null,[t[53]||(t[53]=i("Return whether ")),s("mjx-container",D,[(l(),e("svg",Z,t[51]||(t[51]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[52]||(t[52]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[54]||(t[54]=i(" is commutative."))]),t[56]||(t[56]=a(`
Examples
julia
julia> A = matrix_algebra(QQ, 2);
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Basics ​
Creation of algebras ​
See the corresponding sections on structure constant algebras.
',3)),s("details",d,[s("summary",null,[t[0]||(t[0]=s("a",{id:"zero_algebra-Tuple{Field}",href:"#zero_algebra-Tuple{Field}"},[s("span",{class:"jlbinding"},"zero_algebra")],-1)),t[1]||(t[1]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[6]||(t[6]=a('
julia
zero_algebra([T, ] K::Field) -> AbstractAssociativeAlgebra
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The optional first argument determines the type of the algebra, and can be StructureConstantAlgebra (default) or MatrixAlgebra.
Examples
julia
julia> A = zero_algebra(QQ)
+Structure constant algebra of dimension 0 over QQ
source
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julia
base_ring(A::AbstractAssociativeAlgebra) -> Field
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julia
basis(A::AbstractAssociativeAlgebra) -> Vector
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See also ")),t[37]||(t[37]=s("a",{href:"/v0.34.8/manual/number_fields/elements#coordinates-Tuple{NumFieldElem}"},[s("code",null,"coordinates")],-1)),t[38]||(t[38]=i(" to get the the coordinates of an element with respect to the bases."))]),t[40]||(t[40]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t[177]||(t[177]=s("h2",{id:"predicates",tabindex:"-1"},[i("Predicates "),s("a",{class:"header-anchor",href:"#predicates","aria-label":'Permalink to "Predicates"'},"​")],-1)),s("details",j,[s("summary",null,[t[41]||(t[41]=s("a",{id:"is_zero-Tuple{Hecke.AbstractAssociativeAlgebra}",href:"#is_zero-Tuple{Hecke.AbstractAssociativeAlgebra}"},[s("span",{class:"jlbinding"},"is_zero")],-1)),t[42]||(t[42]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[47]||(t[47]=a('
julia
is_zero(A::AbstractAssociativeAlgebra) -> Bool
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julia
is_commutative(A::AbstractAssociativeAlgebra) -> Bool
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Examples
julia
julia> A = matrix_algebra(QQ, 2);
 
 julia> is_commutative(A)
-false
source
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julia
is_central(A::AbstractAssociativeAlgebra)
',1)),s("p",null,[t[67]||(t[67]=i("Return whether the ")),s("mjx-container",G,[(l(),e("svg",z,t[59]||(t[59]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[60]||(t[60]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[68]||(t[68]=i("-algebra ")),s("mjx-container",S,[(l(),e("svg",J,t[61]||(t[61]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[62]||(t[62]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[69]||(t[69]=i(" is central, that is, whether ")),s("mjx-container",K,[(l(),e("svg",O,t[63]||(t[63]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[64]||(t[64]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[70]||(t[70]=i(" is the center of ")),s("mjx-container",P,[(l(),e("svg",I,t[65]||(t[65]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[66]||(t[66]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[71]||(t[71]=i("."))]),t[73]||(t[73]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t[178]||(t[178]=s("h2",{id:"generators",tabindex:"-1"},[i("Generators "),s("a",{class:"header-anchor",href:"#generators","aria-label":'Permalink to "Generators"'},"​")],-1)),s("details",N,[s("summary",null,[t[74]||(t[74]=s("a",{id:"gens-Tuple{Hecke.AbstractAssociativeAlgebra}",href:"#gens-Tuple{Hecke.AbstractAssociativeAlgebra}"},[s("span",{class:"jlbinding"},"gens")],-1)),t[75]||(t[75]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[91]||(t[91]=a('
julia
gens(A::AbstractAssociativeAlgebra; thorough_search::Bool = false) -> Vector
',1)),s("p",null,[t[84]||(t[84]=i("Given a ")),s("mjx-container",R,[(l(),e("svg",q,t[76]||(t[76]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[77]||(t[77]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[85]||(t[85]=i("-algebra ")),s("mjx-container",$,[(l(),e("svg",U,t[78]||(t[78]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[79]||(t[79]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[86]||(t[86]=i(", return a subset of ")),t[87]||(t[87]=s("code",null,"basis(A)",-1)),t[88]||(t[88]=i(", which generates ")),s("mjx-container",W,[(l(),e("svg",X,t[80]||(t[80]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[81]||(t[81]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[89]||(t[89]=i(" as an algebra over ")),s("mjx-container",Y,[(l(),e("svg",_,t[82]||(t[82]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[83]||(t[83]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[90]||(t[90]=i("."))]),t[92]||(t[92]=a(`
If thorough_search is true, the number of returned generators is possibly smaller. This will in general increase the runtime. It is not guaranteed that the number of generators is minimal in any case.
The gens_with_data function computes additional data for expressing a basis as words in the generators.
Examples
julia
julia> A = matrix_algebra(QQ, 3);
+false
source
`,3))]),s("details",V,[s("summary",null,[t[57]||(t[57]=s("a",{id:"is_central-Tuple{Hecke.AbstractAssociativeAlgebra}",href:"#is_central-Tuple{Hecke.AbstractAssociativeAlgebra}"},[s("span",{class:"jlbinding"},"is_central")],-1)),t[58]||(t[58]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[72]||(t[72]=a('
julia
is_central(A::AbstractAssociativeAlgebra)
',1)),s("p",null,[t[67]||(t[67]=i("Return whether the ")),s("mjx-container",G,[(l(),e("svg",z,t[59]||(t[59]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[60]||(t[60]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[68]||(t[68]=i("-algebra ")),s("mjx-container",S,[(l(),e("svg",J,t[61]||(t[61]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[62]||(t[62]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[69]||(t[69]=i(" is central, that is, whether ")),s("mjx-container",K,[(l(),e("svg",O,t[63]||(t[63]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[64]||(t[64]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[66]||(t[66]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[71]||(t[71]=i("."))]),t[73]||(t[73]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t[178]||(t[178]=s("h2",{id:"generators",tabindex:"-1"},[i("Generators "),s("a",{class:"header-anchor",href:"#generators","aria-label":'Permalink to "Generators"'},"​")],-1)),s("details",N,[s("summary",null,[t[74]||(t[74]=s("a",{id:"gens-Tuple{Hecke.AbstractAssociativeAlgebra}",href:"#gens-Tuple{Hecke.AbstractAssociativeAlgebra}"},[s("span",{class:"jlbinding"},"gens")],-1)),t[75]||(t[75]=i()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[91]||(t[91]=a('
julia
gens(A::AbstractAssociativeAlgebra; thorough_search::Bool = false) -> Vector
',1)),s("p",null,[t[84]||(t[84]=i("Given a ")),s("mjx-container",R,[(l(),e("svg",q,t[76]||(t[76]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[77]||(t[77]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[85]||(t[85]=i("-algebra ")),s("mjx-container",$,[(l(),e("svg",U,t[78]||(t[78]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[79]||(t[79]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[86]||(t[86]=i(", return a subset of ")),t[87]||(t[87]=s("code",null,"basis(A)",-1)),t[88]||(t[88]=i(", which generates ")),s("mjx-container",W,[(l(),e("svg",X,t[80]||(t[80]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[81]||(t[81]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[89]||(t[89]=i(" as an algebra over ")),s("mjx-container",Y,[(l(),e("svg",_,t[82]||(t[82]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[83]||(t[83]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[90]||(t[90]=i("."))]),t[92]||(t[92]=a(`
If thorough_search is true, the number of returned generators is possibly smaller. This will in general increase the runtime. It is not guaranteed that the number of generators is minimal in any case.
The gens_with_data function computes additional data for expressing a basis as words in the generators.
Examples
julia
julia> A = matrix_algebra(QQ, 3);
 
 julia> gens(A; thorough_search = true)
 5-element Vector{MatAlgebraElem{QQFieldElem, QQMatrix}}:
diff --git a/v0.34.8/assets/manual_algebras_structureconstant.md.DVA5t00J.js b/v0.34.8/assets/manual_algebras_structureconstant.md.DVA5t00J.js
index e2afe1a5fd..6c42c101ad 100644
--- a/v0.34.8/assets/manual_algebras_structureconstant.md.DVA5t00J.js
+++ b/v0.34.8/assets/manual_algebras_structureconstant.md.DVA5t00J.js
@@ -3,7 +3,7 @@ import{_ as o,c as e,j as s,a as i,G as r,a5 as l,B as p,o as a}from"./chunks/fr
 Structure constant algebra of dimension 2 over QQ
source
`,4))]),s("details",w,[s("summary",null,[t[24]||(t[24]=s("a",{id:"structure_constant_algebra-Tuple{SimpleNumField}",href:"#structure_constant_algebra-Tuple{SimpleNumField}"},[s("span",{class:"jlbinding"},"structure_constant_algebra")],-1)),t[25]||(t[25]=i()),r(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[45]||(t[45]=l('
julia
structure_constant_algebra(K::SimpleNumField) -> StructureConstantAlgebra, Map
',1)),s("p",null,[t[38]||(t[38]=i("Given a number field ")),s("mjx-container",b,[(a(),e("svg",C,t[26]||(t[26]=[l('',1)]))),t[27]||(t[27]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"L"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"K")])],-1))]),t[39]||(t[39]=i(", return ")),s("mjx-container",v,[(a(),e("svg",F,t[28]||(t[28]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),t[29]||(t[29]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"L")])],-1))]),t[40]||(t[40]=i(" as a ")),s("mjx-container",f,[(a(),e("svg",H,t[30]||(t[30]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[31]||(t[31]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[41]||(t[41]=i("-algebra ")),s("mjx-container",M,[(a(),e("svg",L,t[32]||(t[32]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[33]||(t[33]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[42]||(t[42]=i(" together with a ")),s("mjx-container",B,[(a(),e("svg",j,t[34]||(t[34]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),t[35]||(t[35]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"K")])],-1))]),t[43]||(t[43]=i("-linear map ")),s("mjx-container",A,[(a(),e("svg",D,t[36]||(t[36]=[l('',1)]))),t[37]||(t[37]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"L")])],-1))]),t[44]||(t[44]=i("."))]),t[46]||(t[46]=l(`
Examples
julia
julia> L, = quadratic_field(2);
 
 julia> structure_constant_algebra(L)
-(Structure constant algebra of dimension 2 over QQ, Map: structure constant algebra -> real quadratic field)
source
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julia
structure_constant_table(A::StructureConstantAlgebra; copy::Bool = true) -> Array{_, 3}
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Examples
julia
julia> A = associative_algebra(QQ, reshape([1, 0, 0, 2, 0, 1, 1, 0], (2, 2, 2)));
+(Structure constant algebra of dimension 2 over QQ, Map: structure constant algebra -> real quadratic field)
source
`,3))]),t[62]||(t[62]=s("h2",{id:"Structure-constant-table",tabindex:"-1"},[i("Structure constant table "),s("a",{class:"header-anchor",href:"#Structure-constant-table","aria-label":'Permalink to "Structure constant table {#Structure-constant-table}"'},"​")],-1)),s("details",S,[s("summary",null,[t[47]||(t[47]=s("a",{id:"structure_constant_table-Tuple{StructureConstantAlgebra}",href:"#structure_constant_table-Tuple{StructureConstantAlgebra}"},[s("span",{class:"jlbinding"},"structure_constant_table")],-1)),t[48]||(t[48]=i()),r(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[58]||(t[58]=l('
julia
structure_constant_table(A::StructureConstantAlgebra; copy::Bool = true) -> Array{_, 3}
',1)),s("p",null,[t[53]||(t[53]=i("Given an algebra ")),s("mjx-container",Z,[(a(),e("svg",V,t[49]||(t[49]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[50]||(t[50]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[54]||(t[54]=i(", return the structure constant table of ")),s("mjx-container",G,[(a(),e("svg",J,t[51]||(t[51]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[52]||(t[52]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[55]||(t[55]=i(". See ")),t[56]||(t[56]=s("a",{href:"/v0.34.8/manual/algebras/structureconstant#structure_constant_algebra-Tuple{Ring, Array{<:Any, 3}}"},[s("code",null,"structure_constant_algebra")],-1)),t[57]||(t[57]=i(" for the defining property."))]),t[59]||(t[59]=l(`
Examples
julia
julia> A = associative_algebra(QQ, reshape([1, 0, 0, 2, 0, 1, 1, 0], (2, 2, 2)));
 
 julia> structure_constant_table(A)
 2×2×2 Array{QQFieldElem, 3}:
diff --git a/v0.34.8/assets/manual_algebras_structureconstant.md.DVA5t00J.lean.js b/v0.34.8/assets/manual_algebras_structureconstant.md.DVA5t00J.lean.js
index e2afe1a5fd..6c42c101ad 100644
--- a/v0.34.8/assets/manual_algebras_structureconstant.md.DVA5t00J.lean.js
+++ b/v0.34.8/assets/manual_algebras_structureconstant.md.DVA5t00J.lean.js
@@ -3,7 +3,7 @@ import{_ as o,c as e,j as s,a as i,G as r,a5 as l,B as p,o as a}from"./chunks/fr
 Structure constant algebra of dimension 2 over QQ
source
`,4))]),s("details",w,[s("summary",null,[t[24]||(t[24]=s("a",{id:"structure_constant_algebra-Tuple{SimpleNumField}",href:"#structure_constant_algebra-Tuple{SimpleNumField}"},[s("span",{class:"jlbinding"},"structure_constant_algebra")],-1)),t[25]||(t[25]=i()),r(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[45]||(t[45]=l('
julia
structure_constant_algebra(K::SimpleNumField) -> StructureConstantAlgebra, Map
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Examples
julia
julia> L, = quadratic_field(2);
 
 julia> structure_constant_algebra(L)
-(Structure constant algebra of dimension 2 over QQ, Map: structure constant algebra -> real quadratic field)
source
`,3))]),t[62]||(t[62]=s("h2",{id:"Structure-constant-table",tabindex:"-1"},[i("Structure constant table "),s("a",{class:"header-anchor",href:"#Structure-constant-table","aria-label":'Permalink to "Structure constant table {#Structure-constant-table}"'},"​")],-1)),s("details",S,[s("summary",null,[t[47]||(t[47]=s("a",{id:"structure_constant_table-Tuple{StructureConstantAlgebra}",href:"#structure_constant_table-Tuple{StructureConstantAlgebra}"},[s("span",{class:"jlbinding"},"structure_constant_table")],-1)),t[48]||(t[48]=i()),r(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[58]||(t[58]=l('
julia
structure_constant_table(A::StructureConstantAlgebra; copy::Bool = true) -> Array{_, 3}
',1)),s("p",null,[t[53]||(t[53]=i("Given an algebra ")),s("mjx-container",Z,[(a(),e("svg",V,t[49]||(t[49]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[50]||(t[50]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[54]||(t[54]=i(", return the structure constant table of ")),s("mjx-container",G,[(a(),e("svg",J,t[51]||(t[51]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[52]||(t[52]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[55]||(t[55]=i(". See ")),t[56]||(t[56]=s("a",{href:"/Hecke.jl/v0.34.8/manual/algebras/structureconstant#structure_constant_algebra-Tuple{Ring, Array{<:Any, 3}}"},[s("code",null,"structure_constant_algebra")],-1)),t[57]||(t[57]=i(" for the defining property."))]),t[59]||(t[59]=l(`
Examples
julia
julia> A = associative_algebra(QQ, reshape([1, 0, 0, 2, 0, 1, 1, 0], (2, 2, 2)));
+(Structure constant algebra of dimension 2 over QQ, Map: structure constant algebra -> real quadratic field)
source
`,3))]),t[62]||(t[62]=s("h2",{id:"Structure-constant-table",tabindex:"-1"},[i("Structure constant table "),s("a",{class:"header-anchor",href:"#Structure-constant-table","aria-label":'Permalink to "Structure constant table {#Structure-constant-table}"'},"​")],-1)),s("details",S,[s("summary",null,[t[47]||(t[47]=s("a",{id:"structure_constant_table-Tuple{StructureConstantAlgebra}",href:"#structure_constant_table-Tuple{StructureConstantAlgebra}"},[s("span",{class:"jlbinding"},"structure_constant_table")],-1)),t[48]||(t[48]=i()),r(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[58]||(t[58]=l('
julia
structure_constant_table(A::StructureConstantAlgebra; copy::Bool = true) -> Array{_, 3}
',1)),s("p",null,[t[53]||(t[53]=i("Given an algebra ")),s("mjx-container",Z,[(a(),e("svg",V,t[49]||(t[49]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[50]||(t[50]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[54]||(t[54]=i(", return the structure constant table of ")),s("mjx-container",G,[(a(),e("svg",J,t[51]||(t[51]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[52]||(t[52]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"A")])],-1))]),t[55]||(t[55]=i(". See ")),t[56]||(t[56]=s("a",{href:"/v0.34.8/manual/algebras/structureconstant#structure_constant_algebra-Tuple{Ring, Array{<:Any, 3}}"},[s("code",null,"structure_constant_algebra")],-1)),t[57]||(t[57]=i(" for the defining property."))]),t[59]||(t[59]=l(`
Examples
julia
julia> A = associative_algebra(QQ, reshape([1, 0, 0, 2, 0, 1, 1, 0], (2, 2, 2)));
 
 julia> structure_constant_table(A)
 2×2×2 Array{QQFieldElem, 3}:
diff --git a/v0.34.8/assets/manual_misc_FacElem.md.D_L3aOnQ.js b/v0.34.8/assets/manual_misc_FacElem.md.D_L3aOnQ.js
index 4a712c872a..dae18b558c 100644
--- a/v0.34.8/assets/manual_misc_FacElem.md.D_L3aOnQ.js
+++ b/v0.34.8/assets/manual_misc_FacElem.md.D_L3aOnQ.js
@@ -5,4 +5,4 @@ import{_ as r,c as i,j as e,a as s,a5 as a,G as o,B as d,o as l}from"./chunks/fr
 simplify(x::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> FacElem
`,1)),e("p",null,[t[93]||(t[93]=s("Uses ")),t[94]||(t[94]=e("code",null,"coprime_base",-1)),t[95]||(t[95]=s(" to obtain a simplified version of ")),e("mjx-container",W,[(l(),i("svg",Y,t[91]||(t[91]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),t[92]||(t[92]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"x")])],-1))]),t[96]||(t[96]=s(", ie. in the simplified version all base ideals will be pariwise coprime but not necessarily prime!."))]),t[98]||(t[98]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",_,[e("summary",null,[t[99]||(t[99]=e("a",{id:"simplify-Tuple{FacElem{QQFieldElem}}",href:"#simplify-Tuple{FacElem{QQFieldElem}}"},[e("span",{class:"jlbinding"},"simplify")],-1)),t[100]||(t[100]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[101]||(t[101]=a(`
julia
simplify(x::FacElem{QQFieldElem}) -> FacElem{QQFieldElem}
 simplify(x::FacElem{ZZRingElem}) -> FacElem{ZZRingElem}
Simplfies the factored element, i.e. arranges for the base to be coprime.
source
`,3))]),t[295]||(t[295]=e("p",null,"The simplified version can then be used further:",-1)),e("details",e1,[e("summary",null,[t[102]||(t[102]=e("a",{id:"isone-Tuple{FacElem{QQFieldElem}}",href:"#isone-Tuple{FacElem{QQFieldElem}}"},[e("span",{class:"jlbinding"},"isone")],-1)),t[103]||(t[103]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[111]||(t[111]=a(`
julia
isone(x::FacElem{QQFieldElem}) -> Bool
 isone(x::FacElem{ZZRingElem}) -> Bool
`,1)),e("p",null,[t[108]||(t[108]=s("Tests if ")),e("mjx-container",t1,[(l(),i("svg",s1,t[104]||(t[104]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),t[105]||(t[105]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"x")])],-1))]),t[109]||(t[109]=s(" represents ")),e("mjx-container",i1,[(l(),i("svg",l1,t[106]||(t[106]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mn"},[e("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),t[107]||(t[107]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mn",null,"1")])],-1))]),t[110]||(t[110]=s(" without an evaluation."))]),t[112]||(t[112]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",a1,[e("summary",null,[t[113]||(t[113]=e("a",{id:"factor_coprime-Tuple{FacElem{ZZRingElem}}",href:"#factor_coprime-Tuple{FacElem{ZZRingElem}}"},[e("span",{class:"jlbinding"},"factor_coprime")],-1)),t[114]||(t[114]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[122]||(t[122]=a('
julia
factor_coprime(x::FacElem{ZZRingElem}) -> Fac{ZZRingElem}
',1)),e("p",null,[t[119]||(t[119]=s("Computed a partial factorisation of ")),e("mjx-container",n1,[(l(),i("svg",o1,t[115]||(t[115]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),t[116]||(t[116]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"x")])],-1))]),t[120]||(t[120]=s(", ie. writes ")),e("mjx-container",r1,[(l(),i("svg",d1,t[117]||(t[117]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),t[118]||(t[118]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"x")])],-1))]),t[121]||(t[121]=s(" as a product of pariwise coprime integers."))]),t[123]||(t[123]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",Q1,[e("summary",null,[t[124]||(t[124]=e("a",{id:"factor_coprime-Tuple{FacElem{AbsSimpleNumFieldOrderIdeal, Hecke.AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}}",href:"#factor_coprime-Tuple{FacElem{AbsSimpleNumFieldOrderIdeal, Hecke.AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}}"},[e("span",{class:"jlbinding"},"factor_coprime")],-1)),t[125]||(t[125]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[133]||(t[133]=a('
julia
factor_coprime(x::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}
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julia
factor_coprime(Q::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}
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julia
factor_coprime(I::AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}, a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}
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julia
 factor(Q::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}
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julia
factor(I::AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}, a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}
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julia
compact_presentation(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, n::Int = 2; decom, arb_prec = 100, short_prec = 1000) -> FacElem
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As the algorithm needs the factorisation of the principal ideal generated by ")),e("mjx-container",R1,[(l(),i("svg",P1,t[187]||(t[187]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),t[188]||(t[188]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"a")])],-1))]),t[196]||(t[196]=s(", it can be passed in in \\code{decom}."))]),t[198]||(t[198]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",q1,[e("summary",null,[t[199]||(t[199]=e("a",{id:"valuation-Tuple{FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, AbsSimpleNumFieldOrderIdeal}",href:"#valuation-Tuple{FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, AbsSimpleNumFieldOrderIdeal}"},[e("span",{class:"jlbinding"},"valuation")],-1)),t[200]||(t[200]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[208]||(t[208]=a('
julia
valuation(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
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julia
valuation(A::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem})
-valuation(A::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem})
`,1)),e("p",null,[t[216]||(t[216]=s("The valuation of ")),e("mjx-container",Y1,[(l(),i("svg",_1,t[212]||(t[212]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[213]||(t[213]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"A")])],-1))]),t[217]||(t[217]=s(" at ")),e("mjx-container",e2,[(l(),i("svg",t2,t[214]||(t[214]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D443",d:"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z",style:{"stroke-width":"3"}})])])],-1)]))),t[215]||(t[215]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"P")])],-1))]),t[218]||(t[218]=s("."))]),t[220]||(t[220]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",s2,[e("summary",null,[t[221]||(t[221]=e("a",{id:"evaluate_mod-Tuple{FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, AbsSimpleNumFieldOrderFractionalIdeal}",href:"#evaluate_mod-Tuple{FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, AbsSimpleNumFieldOrderFractionalIdeal}"},[e("span",{class:"jlbinding"},"evaluate_mod")],-1)),t[222]||(t[222]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[236]||(t[236]=a('
julia
evaluate_mod(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, B::AbsSimpleNumFieldOrderFractionalIdeal) -> AbsSimpleNumFieldElem
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Useful in cases where ")),e("mjx-container",d2,[(l(),i("svg",Q2,t[229]||(t[229]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),t[230]||(t[230]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"a")])],-1))]),t[235]||(t[235]=s(' has huge exponents, but the evaluated element is actually "small".'))]),t[237]||(t[237]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",p2,[e("summary",null,[t[238]||(t[238]=e("a",{id:"reduce_ideal-Tuple{FacElem{AbsSimpleNumFieldOrderIdeal, Hecke.AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}}",href:"#reduce_ideal-Tuple{FacElem{AbsSimpleNumFieldOrderIdeal, Hecke.AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}}"},[e("span",{class:"jlbinding"},"reduce_ideal")],-1)),t[239]||(t[239]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[250]||(t[250]=a('
julia
reduce_ideal(A::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, FacElem{AbsSimpleNumFieldElem}
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julia
modular_proj(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, me::modular_env) -> Vector{fqPolyRepFieldElem}
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Positivity & Signs ​
Factored elements can be used instead of number field elements for the functions sign, signs, is_positive, is_negative and is_totally_positive, see Positivity & Signs
Miscellaneous ​
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julia
max_exp(a::FacElem)
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julia
min_exp(a::FacElem)
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julia
maxabs_exp(a::FacElem)
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+valuation(A::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem})
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julia
evaluate_mod(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, B::AbsSimpleNumFieldOrderFractionalIdeal) -> AbsSimpleNumFieldElem
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julia
reduce_ideal(A::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, FacElem{AbsSimpleNumFieldElem}
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julia
modular_proj(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, me::modular_env) -> Vector{fqPolyRepFieldElem}
',1)),e("p",null,[t[258]||(t[258]=s("Given an algebraic number ")),e("mjx-container",c2,[(l(),i("svg",w2,t[254]||(t[254]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),t[255]||(t[255]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"a")])],-1))]),t[259]||(t[259]=s(" in factored form and data \\code{me} as computed by \\code{modular_init}, project ")),e("mjx-container",b2,[(l(),i("svg",y2,t[256]||(t[256]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),t[257]||(t[257]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"a")])],-1))]),t[260]||(t[260]=s(" onto the residue class fields."))]),t[262]||(t[262]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t[297]||(t[297]=a('
Positivity & Signs ​
Factored elements can be used instead of number field elements for the functions sign, signs, is_positive, is_negative and is_totally_positive, see Positivity & Signs
Miscellaneous ​
',3)),e("details",f2,[e("summary",null,[t[263]||(t[263]=e("a",{id:"max_exp-Tuple{FacElem}",href:"#max_exp-Tuple{FacElem}"},[e("span",{class:"jlbinding"},"max_exp")],-1)),t[264]||(t[264]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[269]||(t[269]=a('
julia
max_exp(a::FacElem)
',1)),e("p",null,[t[267]||(t[267]=s("Finds the largest exponent in the factored element ")),e("mjx-container",v2,[(l(),i("svg",E2,t[265]||(t[265]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),t[266]||(t[266]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"a")])],-1))]),t[268]||(t[268]=s("."))]),t[270]||(t[270]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",F2,[e("summary",null,[t[271]||(t[271]=e("a",{id:"min_exp-Tuple{FacElem}",href:"#min_exp-Tuple{FacElem}"},[e("span",{class:"jlbinding"},"min_exp")],-1)),t[272]||(t[272]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[277]||(t[277]=a('
julia
min_exp(a::FacElem)
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julia
maxabs_exp(a::FacElem)
',1)),e("p",null,[t[283]||(t[283]=s("Finds the largest exponent by absolute value in the factored element ")),e("mjx-container",C2,[(l(),i("svg",L2,t[281]||(t[281]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),t[282]||(t[282]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"a")])],-1))]),t[284]||(t[284]=s("."))]),t[286]||(t[286]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))])])}const I2=r(Q,[["render",A2]]);export{O2 as __pageData,I2 as default};
diff --git a/v0.34.8/assets/manual_misc_FacElem.md.D_L3aOnQ.lean.js b/v0.34.8/assets/manual_misc_FacElem.md.D_L3aOnQ.lean.js
index 4a712c872a..dae18b558c 100644
--- a/v0.34.8/assets/manual_misc_FacElem.md.D_L3aOnQ.lean.js
+++ b/v0.34.8/assets/manual_misc_FacElem.md.D_L3aOnQ.lean.js
@@ -5,4 +5,4 @@ import{_ as r,c as i,j as e,a as s,a5 as a,G as o,B as d,o as l}from"./chunks/fr
 simplify(x::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> FacElem
`,1)),e("p",null,[t[93]||(t[93]=s("Uses ")),t[94]||(t[94]=e("code",null,"coprime_base",-1)),t[95]||(t[95]=s(" to obtain a simplified version of ")),e("mjx-container",W,[(l(),i("svg",Y,t[91]||(t[91]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),t[92]||(t[92]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"x")])],-1))]),t[96]||(t[96]=s(", ie. in the simplified version all base ideals will be pariwise coprime but not necessarily prime!."))]),t[98]||(t[98]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",_,[e("summary",null,[t[99]||(t[99]=e("a",{id:"simplify-Tuple{FacElem{QQFieldElem}}",href:"#simplify-Tuple{FacElem{QQFieldElem}}"},[e("span",{class:"jlbinding"},"simplify")],-1)),t[100]||(t[100]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[101]||(t[101]=a(`
julia
simplify(x::FacElem{QQFieldElem}) -> FacElem{QQFieldElem}
 simplify(x::FacElem{ZZRingElem}) -> FacElem{ZZRingElem}
Simplfies the factored element, i.e. arranges for the base to be coprime.
source
`,3))]),t[295]||(t[295]=e("p",null,"The simplified version can then be used further:",-1)),e("details",e1,[e("summary",null,[t[102]||(t[102]=e("a",{id:"isone-Tuple{FacElem{QQFieldElem}}",href:"#isone-Tuple{FacElem{QQFieldElem}}"},[e("span",{class:"jlbinding"},"isone")],-1)),t[103]||(t[103]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[111]||(t[111]=a(`
julia
isone(x::FacElem{QQFieldElem}) -> Bool
 isone(x::FacElem{ZZRingElem}) -> Bool
`,1)),e("p",null,[t[108]||(t[108]=s("Tests if ")),e("mjx-container",t1,[(l(),i("svg",s1,t[104]||(t[104]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),t[105]||(t[105]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"x")])],-1))]),t[109]||(t[109]=s(" represents ")),e("mjx-container",i1,[(l(),i("svg",l1,t[106]||(t[106]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mn"},[e("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),t[107]||(t[107]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mn",null,"1")])],-1))]),t[110]||(t[110]=s(" without an evaluation."))]),t[112]||(t[112]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",a1,[e("summary",null,[t[113]||(t[113]=e("a",{id:"factor_coprime-Tuple{FacElem{ZZRingElem}}",href:"#factor_coprime-Tuple{FacElem{ZZRingElem}}"},[e("span",{class:"jlbinding"},"factor_coprime")],-1)),t[114]||(t[114]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[122]||(t[122]=a('
julia
factor_coprime(x::FacElem{ZZRingElem}) -> Fac{ZZRingElem}
',1)),e("p",null,[t[119]||(t[119]=s("Computed a partial factorisation of ")),e("mjx-container",n1,[(l(),i("svg",o1,t[115]||(t[115]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),t[116]||(t[116]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"x")])],-1))]),t[120]||(t[120]=s(", ie. writes ")),e("mjx-container",r1,[(l(),i("svg",d1,t[117]||(t[117]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),t[118]||(t[118]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"x")])],-1))]),t[121]||(t[121]=s(" as a product of pariwise coprime integers."))]),t[123]||(t[123]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",Q1,[e("summary",null,[t[124]||(t[124]=e("a",{id:"factor_coprime-Tuple{FacElem{AbsSimpleNumFieldOrderIdeal, Hecke.AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}}",href:"#factor_coprime-Tuple{FacElem{AbsSimpleNumFieldOrderIdeal, Hecke.AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}}"},[e("span",{class:"jlbinding"},"factor_coprime")],-1)),t[125]||(t[125]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[133]||(t[133]=a('
julia
factor_coprime(x::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}
',1)),e("p",null,[t[130]||(t[130]=s("Computed a partial factorisation of ")),e("mjx-container",p1,[(l(),i("svg",m1,t[126]||(t[126]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),t[127]||(t[127]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"x")])],-1))]),t[131]||(t[131]=s(", ie. writes ")),e("mjx-container",T1,[(l(),i("svg",h1,t[128]||(t[128]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),t[129]||(t[129]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"x")])],-1))]),t[132]||(t[132]=s(" as a product of pariwise coprime integral ideals."))]),t[134]||(t[134]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",u1,[e("summary",null,[t[135]||(t[135]=e("a",{id:"factor_coprime-Tuple{FacElem{AbsSimpleNumFieldOrderFractionalIdeal, Hecke.AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}}",href:"#factor_coprime-Tuple{FacElem{AbsSimpleNumFieldOrderFractionalIdeal, Hecke.AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}}"},[e("span",{class:"jlbinding"},"factor_coprime")],-1)),t[136]||(t[136]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[144]||(t[144]=a('
julia
factor_coprime(Q::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}
',1)),e("p",null,[t[141]||(t[141]=s("A coprime factorisation of ")),e("mjx-container",g1,[(l(),i("svg",k1,t[137]||(t[137]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D444",d:"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z",style:{"stroke-width":"3"}})])])],-1)]))),t[138]||(t[138]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"Q")])],-1))]),t[142]||(t[142]=s(": each ideal in ")),e("mjx-container",x1,[(l(),i("svg",c1,t[139]||(t[139]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D444",d:"M399 -80Q399 -47 400 -30T402 -11V-7L387 -11Q341 -22 303 -22Q208 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435Q740 255 592 107Q529 47 461 16L444 8V3Q444 2 449 -24T470 -66T516 -82Q551 -82 583 -60T625 -3Q631 11 638 11Q647 11 649 2Q649 -6 639 -34T611 -100T557 -165T481 -194Q399 -194 399 -87V-80ZM636 468Q636 523 621 564T580 625T530 655T477 665Q429 665 379 640Q277 591 215 464T153 216Q153 110 207 59Q231 38 236 38V46Q236 86 269 120T347 155Q372 155 390 144T417 114T429 82T435 55L448 64Q512 108 557 185T619 334T636 468ZM314 18Q362 18 404 39L403 49Q399 104 366 115Q354 117 347 117Q344 117 341 117T337 118Q317 118 296 98T274 52Q274 18 314 18Z",style:{"stroke-width":"3"}})])])],-1)]))),t[140]||(t[140]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"Q")])],-1))]),t[143]||(t[143]=s(" is split using \\code{integral_split} and then a coprime basis is computed. This does {\\bf not} use any factorisation."))]),t[145]||(t[145]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",w1,[e("summary",null,[t[146]||(t[146]=e("a",{id:"factor_coprime-Tuple{Hecke.AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}, FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}}",href:"#factor_coprime-Tuple{Hecke.AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}, FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}}"},[e("span",{class:"jlbinding"},"factor_coprime")],-1)),t[147]||(t[147]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[155]||(t[155]=a('
julia
factor_coprime(I::AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}, a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}
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julia
 factor(Q::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}
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julia
factor(I::AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}, a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}
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julia
compact_presentation(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, n::Int = 2; decom, arb_prec = 100, short_prec = 1000) -> FacElem
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As the algorithm needs the factorisation of the principal ideal generated by ")),e("mjx-container",R1,[(l(),i("svg",P1,t[187]||(t[187]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),t[188]||(t[188]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"a")])],-1))]),t[196]||(t[196]=s(", it can be passed in in \\code{decom}."))]),t[198]||(t[198]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",q1,[e("summary",null,[t[199]||(t[199]=e("a",{id:"valuation-Tuple{FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, AbsSimpleNumFieldOrderIdeal}",href:"#valuation-Tuple{FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, AbsSimpleNumFieldOrderIdeal}"},[e("span",{class:"jlbinding"},"valuation")],-1)),t[200]||(t[200]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[208]||(t[208]=a('
julia
valuation(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
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julia
valuation(A::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem})
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julia
evaluate_mod(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, B::AbsSimpleNumFieldOrderFractionalIdeal) -> AbsSimpleNumFieldElem
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julia
reduce_ideal(A::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, FacElem{AbsSimpleNumFieldElem}
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julia
modular_proj(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, me::modular_env) -> Vector{fqPolyRepFieldElem}
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Positivity & Signs ​
Factored elements can be used instead of number field elements for the functions sign, signs, is_positive, is_negative and is_totally_positive, see Positivity & Signs
Miscellaneous ​
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julia
max_exp(a::FacElem)
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julia
min_exp(a::FacElem)
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julia
maxabs_exp(a::FacElem)
',1)),e("p",null,[t[283]||(t[283]=s("Finds the largest exponent by absolute value in the factored element ")),e("mjx-container",C2,[(l(),i("svg",L2,t[281]||(t[281]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),t[282]||(t[282]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"a")])],-1))]),t[284]||(t[284]=s("."))]),t[286]||(t[286]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))])])}const I2=r(Q,[["render",A2]]);export{O2 as __pageData,I2 as default};
+valuation(A::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem})
`,1)),e("p",null,[t[216]||(t[216]=s("The valuation of ")),e("mjx-container",Y1,[(l(),i("svg",_1,t[212]||(t[212]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D434",d:"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z",style:{"stroke-width":"3"}})])])],-1)]))),t[213]||(t[213]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"A")])],-1))]),t[217]||(t[217]=s(" at ")),e("mjx-container",e2,[(l(),i("svg",t2,t[214]||(t[214]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D443",d:"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z",style:{"stroke-width":"3"}})])])],-1)]))),t[215]||(t[215]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"P")])],-1))]),t[218]||(t[218]=s("."))]),t[220]||(t[220]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",s2,[e("summary",null,[t[221]||(t[221]=e("a",{id:"evaluate_mod-Tuple{FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, AbsSimpleNumFieldOrderFractionalIdeal}",href:"#evaluate_mod-Tuple{FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, AbsSimpleNumFieldOrderFractionalIdeal}"},[e("span",{class:"jlbinding"},"evaluate_mod")],-1)),t[222]||(t[222]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[236]||(t[236]=a('
julia
evaluate_mod(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, B::AbsSimpleNumFieldOrderFractionalIdeal) -> AbsSimpleNumFieldElem
',1)),e("p",null,[t[231]||(t[231]=s("Evaluates ")),e("mjx-container",i2,[(l(),i("svg",l2,t[223]||(t[223]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),t[224]||(t[224]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"a")])],-1))]),t[232]||(t[232]=s(" using CRT and small primes. Assumes that the ideal generated by ")),e("mjx-container",a2,[(l(),i("svg",n2,t[225]||(t[225]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),t[226]||(t[226]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"a")])],-1))]),t[233]||(t[233]=s(" is in fact ")),e("mjx-container",o2,[(l(),i("svg",r2,t[227]||(t[227]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),t[228]||(t[228]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"B")])],-1))]),t[234]||(t[234]=s(". Useful in cases where ")),e("mjx-container",d2,[(l(),i("svg",Q2,t[229]||(t[229]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),t[230]||(t[230]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"a")])],-1))]),t[235]||(t[235]=s(' has huge exponents, but the evaluated element is actually "small".'))]),t[237]||(t[237]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",p2,[e("summary",null,[t[238]||(t[238]=e("a",{id:"reduce_ideal-Tuple{FacElem{AbsSimpleNumFieldOrderIdeal, Hecke.AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}}",href:"#reduce_ideal-Tuple{FacElem{AbsSimpleNumFieldOrderIdeal, Hecke.AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}}"},[e("span",{class:"jlbinding"},"reduce_ideal")],-1)),t[239]||(t[239]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[250]||(t[250]=a('
julia
reduce_ideal(A::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, FacElem{AbsSimpleNumFieldElem}
',1)),e("p",null,[t[246]||(t[246]=s("Computes ")),e("mjx-container",m2,[(l(),i("svg",T2,t[240]||(t[240]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),t[241]||(t[241]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"B")])],-1))]),t[247]||(t[247]=s(" and ")),e("mjx-container",h2,[(l(),i("svg",u2,t[242]||(t[242]=[e("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[e("g",{"data-mml-node":"math"},[e("g",{"data-mml-node":"mi"},[e("path",{"data-c":"1D6FC",d:"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z",style:{"stroke-width":"3"}})])])],-1)]))),t[243]||(t[243]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"α")])],-1))]),t[248]||(t[248]=s(" in factored form, such that ")),e("mjx-container",g2,[(l(),i("svg",k2,t[244]||(t[244]=[a('',1)]))),t[245]||(t[245]=e("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[e("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[e("mi",null,"α"),e("mi",null,"B"),e("mo",null,"="),e("mi",null,"A")])],-1))]),t[249]||(t[249]=s("."))]),t[251]||(t[251]=e("p",null,[e("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e("details",x2,[e("summary",null,[t[252]||(t[252]=e("a",{id:"modular_proj-Tuple{FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, Hecke.modular_env}",href:"#modular_proj-Tuple{FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, Hecke.modular_env}"},[e("span",{class:"jlbinding"},"modular_proj")],-1)),t[253]||(t[253]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),t[261]||(t[261]=a('
julia
modular_proj(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, me::modular_env) -> Vector{fqPolyRepFieldElem}
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Positivity & Signs ​
Factored elements can be used instead of number field elements for the functions sign, signs, is_positive, is_negative and is_totally_positive, see Positivity & Signs
Miscellaneous ​
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julia
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julia
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julia
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diff --git a/v0.34.8/assets/manual_misc_pmat.md.DCq1lfbi.js b/v0.34.8/assets/manual_misc_pmat.md.DCq1lfbi.js
index adbcd8c8a2..5a1de95d80 100644
--- a/v0.34.8/assets/manual_misc_pmat.md.DCq1lfbi.js
+++ b/v0.34.8/assets/manual_misc_pmat.md.DCq1lfbi.js
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julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
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julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
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julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
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julia
coefficient_ideals(M::PMat)
Returns the vector of coefficient ideals.
source
',3))]),t("details",T2,[t("summary",null,[e[182]||(e[182]=t("a",{id:"matrix-Tuple{Hecke.PMat}",href:"#matrix-Tuple{Hecke.PMat}"},[t("span",{class:"jlbinding"},"matrix")],-1)),e[183]||(e[183]=l()),a(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[184]||(e[184]=Q('
julia
matrix(M::PMat)
Returns the matrix part of the PMat.
source
',3))]),t("details",r2,[t("summary",null,[e[185]||(e[185]=t("a",{id:"base_ring-Tuple{Hecke.PMat}",href:"#base_ring-Tuple{Hecke.PMat}"},[t("span",{class:"jlbinding"},"base_ring")],-1)),e[186]||(e[186]=l()),a(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[205]||(e[205]=Q('
julia
base_ring(M::PMat)
',1)),t("p",null,[e[197]||(e[197]=l("The ")),e[198]||(e[198]=t("code",null,"PMat",-1)),e[199]||(e[199]=l()),t("mjx-container",d2,[(i(),s("svg",m2,e[187]||(e[187]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),e[188]||(e[188]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"M")])],-1))]),e[200]||(e[200]=l(" defines an ")),t("mjx-container",p2,[(i(),s("svg",h2,e[189]||(e[189]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D445",d:"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 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658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z",style:{"stroke-width":"3"}})])])],-1)]))),e[192]||(e[192]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"R")])],-1))]),e[202]||(e[202]=l(". This function returns the ")),t("mjx-container",w2,[(i(),s("svg",g2,e[193]||(e[193]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D445",d:"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z",style:{"stroke-width":"3"}})])])],-1)]))),e[194]||(e[194]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"R")])],-1))]),e[203]||(e[203]=l(" that was used to defined ")),t("mjx-container",c2,[(i(),s("svg",k2,e[195]||(e[195]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),e[196]||(e[196]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"M")])],-1))]),e[204]||(e[204]=l("."))]),e[206]||(e[206]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",b2,[t("summary",null,[e[207]||(e[207]=t("a",{id:"pseudo_hnf-Tuple{Hecke.PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}}",href:"#pseudo_hnf-Tuple{Hecke.PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}}"},[t("span",{class:"jlbinding"},"pseudo_hnf")],-1)),e[208]||(e[208]=l()),a(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[216]||(e[216]=Q('
julia
pseudo_hnf(P::PMat)
',1)),t("p",null,[e[213]||(e[213]=l("Transforms ")),t("mjx-container",f2,[(i(),s("svg",v2,e[209]||(e[209]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D443",d:"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z",style:{"stroke-width":"3"}})])])],-1)]))),e[210]||(e[210]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"P")])],-1))]),e[214]||(e[214]=l(" into pseudo-Hermite form as defined by Cohen. Essentially the matrix part of ")),t("mjx-container",y2,[(i(),s("svg",L2,e[211]||(e[211]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D443",d:"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z",style:{"stroke-width":"3"}})])])],-1)]))),e[212]||(e[212]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"P")])],-1))]),e[215]||(e[215]=l(" will be upper triangular with some technical normalisation for the off-diagonal elements. This operation preserves the module."))]),e[217]||(e[217]=t("p",null,[l("A optional second argument can be specified as a symbols, indicating the desired shape of the echelon form. Possible are "),t("code",null,":upperright"),l(" (the default) and "),t("code",null,":lowerleft")],-1)),e[218]||(e[218]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",M2,[t("summary",null,[e[219]||(e[219]=t("a",{id:"pseudo_hnf_with_transform-Tuple{Hecke.PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}}",href:"#pseudo_hnf_with_transform-Tuple{Hecke.PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}}"},[t("span",{class:"jlbinding"},"pseudo_hnf_with_transform")],-1)),e[220]||(e[220]=l()),a(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[228]||(e[228]=Q('
julia
pseudo_hnf_with_transform(P::PMat)
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julia
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julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
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julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
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julia
coefficient_ideals(M::PMat)
Returns the vector of coefficient ideals.
source
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julia
matrix(M::PMat)
Returns the matrix part of the PMat.
source
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julia
base_ring(M::PMat)
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julia
pseudo_hnf(P::PMat)
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julia
pseudo_hnf_with_transform(P::PMat)
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diff --git a/v0.34.8/assets/manual_misc_pmat.md.DCq1lfbi.lean.js b/v0.34.8/assets/manual_misc_pmat.md.DCq1lfbi.lean.js
index adbcd8c8a2..5a1de95d80 100644
--- a/v0.34.8/assets/manual_misc_pmat.md.DCq1lfbi.lean.js
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julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
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julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
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julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
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julia
coefficient_ideals(M::PMat)
Returns the vector of coefficient ideals.
source
',3))]),t("details",T2,[t("summary",null,[e[182]||(e[182]=t("a",{id:"matrix-Tuple{Hecke.PMat}",href:"#matrix-Tuple{Hecke.PMat}"},[t("span",{class:"jlbinding"},"matrix")],-1)),e[183]||(e[183]=l()),a(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[184]||(e[184]=Q('
julia
matrix(M::PMat)
Returns the matrix part of the PMat.
source
',3))]),t("details",r2,[t("summary",null,[e[185]||(e[185]=t("a",{id:"base_ring-Tuple{Hecke.PMat}",href:"#base_ring-Tuple{Hecke.PMat}"},[t("span",{class:"jlbinding"},"base_ring")],-1)),e[186]||(e[186]=l()),a(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[205]||(e[205]=Q('
julia
base_ring(M::PMat)
',1)),t("p",null,[e[197]||(e[197]=l("The ")),e[198]||(e[198]=t("code",null,"PMat",-1)),e[199]||(e[199]=l()),t("mjx-container",d2,[(i(),s("svg",m2,e[187]||(e[187]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),e[188]||(e[188]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"M")])],-1))]),e[200]||(e[200]=l(" defines an ")),t("mjx-container",p2,[(i(),s("svg",h2,e[189]||(e[189]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D445",d:"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 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658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z",style:{"stroke-width":"3"}})])])],-1)]))),e[192]||(e[192]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"R")])],-1))]),e[202]||(e[202]=l(". This function returns the ")),t("mjx-container",w2,[(i(),s("svg",g2,e[193]||(e[193]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D445",d:"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z",style:{"stroke-width":"3"}})])])],-1)]))),e[194]||(e[194]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"R")])],-1))]),e[203]||(e[203]=l(" that was used to defined ")),t("mjx-container",c2,[(i(),s("svg",k2,e[195]||(e[195]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),e[196]||(e[196]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"M")])],-1))]),e[204]||(e[204]=l("."))]),e[206]||(e[206]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",b2,[t("summary",null,[e[207]||(e[207]=t("a",{id:"pseudo_hnf-Tuple{Hecke.PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}}",href:"#pseudo_hnf-Tuple{Hecke.PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}}"},[t("span",{class:"jlbinding"},"pseudo_hnf")],-1)),e[208]||(e[208]=l()),a(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[216]||(e[216]=Q('
julia
pseudo_hnf(P::PMat)
',1)),t("p",null,[e[213]||(e[213]=l("Transforms ")),t("mjx-container",f2,[(i(),s("svg",v2,e[209]||(e[209]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D443",d:"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z",style:{"stroke-width":"3"}})])])],-1)]))),e[210]||(e[210]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"P")])],-1))]),e[214]||(e[214]=l(" into pseudo-Hermite form as defined by Cohen. Essentially the matrix part of ")),t("mjx-container",y2,[(i(),s("svg",L2,e[211]||(e[211]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D443",d:"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z",style:{"stroke-width":"3"}})])])],-1)]))),e[212]||(e[212]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"P")])],-1))]),e[215]||(e[215]=l(" will be upper triangular with some technical normalisation for the off-diagonal elements. This operation preserves the module."))]),e[217]||(e[217]=t("p",null,[l("A optional second argument can be specified as a symbols, indicating the desired shape of the echelon form. Possible are "),t("code",null,":upperright"),l(" (the default) and "),t("code",null,":lowerleft")],-1)),e[218]||(e[218]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",M2,[t("summary",null,[e[219]||(e[219]=t("a",{id:"pseudo_hnf_with_transform-Tuple{Hecke.PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}}",href:"#pseudo_hnf_with_transform-Tuple{Hecke.PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}}"},[t("span",{class:"jlbinding"},"pseudo_hnf_with_transform")],-1)),e[220]||(e[220]=l()),a(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[228]||(e[228]=Q('
julia
pseudo_hnf_with_transform(P::PMat)
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julia
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julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
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julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
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julia
coefficient_ideals(M::PMat)
Returns the vector of coefficient ideals.
source
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julia
matrix(M::PMat)
Returns the matrix part of the PMat.
source
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julia
base_ring(M::PMat)
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diff --git a/v0.34.8/assets/manual_number_fields_elements.md.BTwKw5-v.js b/v0.34.8/assets/manual_number_fields_elements.md.BTwKw5-v.js
index 0ef36772e5..c8f2b840ff 100644
--- a/v0.34.8/assets/manual_number_fields_elements.md.BTwKw5-v.js
+++ b/v0.34.8/assets/manual_number_fields_elements.md.BTwKw5-v.js
@@ -9,5 +9,5 @@ import{_ as Q,c as l,j as t,a as s,G as n,a5 as a,B as r,o as i}from"./chunks/fr
 (Relative number field of degree 3 over number field, b)
 
 julia> L([a, 1, 1//2])
-1//2*b^2 + b + a
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julia
quadratic_defect(a::Union{NumFieldElem,Rational,QQFieldElem}, p) -> Union{Inf, PosInf}
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julia
hilbert_symbol(a::NumFieldElem, b::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Int
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julia
representation_matrix(a::NumFieldElem) -> MatElem
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julia
basis_matrix(v::Vector{NumFieldElem}) -> Mat
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129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z",style:{"stroke-width":"3"}})])])],-1)]))),e[95]||(e[95]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"d")])],-1))]),e[110]||(e[110]=s(", this function returns an ")),t("mjx-container",n1,[(i(),l("svg",Q1,e[96]||(e[96]=[a('',1)]))),e[97]||(e[97]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"n"),t("mo",null,"×"),t("mi",null,"d")])],-1))]),e[111]||(e[111]=s(" matrix with entries in the base field of ")),t("mjx-container",r1,[(i(),l("svg",T1,e[98]||(e[98]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[99]||(e[99]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[112]||(e[112]=s(", where row ")),t("mjx-container",d1,[(i(),l("svg",m1,e[100]||(e[100]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D456",d:"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z",style:{"stroke-width":"3"}})])])],-1)]))),e[101]||(e[101]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"i")])],-1))]),e[113]||(e[113]=s(" contains the coefficients of ")),t("mjx-container",p1,[(i(),l("svg",h1,e[102]||(e[102]=[a('',1)]))),e[103]||(e[103]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"v"),t("mo",{stretchy:"false"},"["),t("mi",null,"i"),t("mo",{stretchy:"false"},"]")])],-1))]),e[114]||(e[114]=s(" with respect of the canonical basis of ")),t("mjx-container",g1,[(i(),l("svg",x1,e[104]||(e[104]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[105]||(e[105]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[115]||(e[115]=s("."))]),e[117]||(e[117]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",u1,[t("summary",null,[e[118]||(e[118]=t("a",{id:"coefficients-Tuple{SimpleNumFieldElem}",href:"#coefficients-Tuple{SimpleNumFieldElem}"},[t("span",{class:"jlbinding"},"coefficients")],-1)),e[119]||(e[119]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[120]||(e[120]=a('
julia
coefficients(a::SimpleNumFieldElem, i::Int) -> Vector{FieldElem}

Given a number field element a of a simple number field extension L/K, this function returns the coefficients of a, when expanded in the canonical power basis of L.

source

',3))]),t("details",k1,[t("summary",null,[e[121]||(e[121]=t("a",{id:"coordinates-Tuple{NumFieldElem}",href:"#coordinates-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"coordinates")],-1)),e[122]||(e[122]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[136]||(e[136]=a('
julia
coordinates(x::NumFieldElem{T}) -> Vector{T}
',1)),t("p",null,[e[131]||(e[131]=s("Given an element ")),t("mjx-container",w1,[(i(),l("svg",c1,e[123]||(e[123]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),e[124]||(e[124]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x")])],-1))]),e[132]||(e[132]=s(" in a number field ")),t("mjx-container",y1,[(i(),l("svg",f1,e[125]||(e[125]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[126]||(e[126]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),e[128]||(e[128]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x")])],-1))]),e[134]||(e[134]=s(" with respect to the basis of 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julia
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julia
coeff(a::SimpleNumFieldElem, i::Int) -> FieldElem

Given a number field element a of a simple number field extension L/K, this function returns the i-th coefficient of a, when expanded in the canonical power basis of L. The result is an element of K.

source

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julia
valuation(a::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
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julia
torsion_unit_order(x::AbsSimpleNumFieldElem, n::Int)
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julia
algebraic_split(a::AbsSimpleNumFieldElem) -> AbsSimpleNumFieldElem, AbsSimpleNumFieldElem

Writes the input as a quotient of two "small" algebraic integers.

source

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julia
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julia
conjugates(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
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julia
conjugates_arb_log(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
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julia
conjugates_arb_real(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
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julia
conjugates_complex(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
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julia
conjugates_arb_log_normalise(x::AbsSimpleNumFieldElem, p::Int = 10)
+1//2*b^2 + b + a
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julia
quadratic_defect(a::Union{NumFieldElem,Rational,QQFieldElem}, p) -> Union{Inf, PosInf}
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julia
hilbert_symbol(a::NumFieldElem, b::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Int
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julia
representation_matrix(a::NumFieldElem) -> MatElem
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julia
basis_matrix(v::Vector{NumFieldElem}) -> Mat
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julia
coefficients(a::SimpleNumFieldElem, i::Int) -> Vector{FieldElem}

Given a number field element a of a simple number field extension L/K, this function returns the coefficients of a, when expanded in the canonical power basis of L.

source

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julia
coordinates(x::NumFieldElem{T}) -> Vector{T}
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julia
absolute_coordinates(x::NumFieldElem{T}) -> Vector{T}
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julia
coeff(a::SimpleNumFieldElem, i::Int) -> FieldElem

Given a number field element a of a simple number field extension L/K, this function returns the i-th coefficient of a, when expanded in the canonical power basis of L. The result is an element of K.

source

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julia
valuation(a::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
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julia
torsion_unit_order(x::AbsSimpleNumFieldElem, n::Int)
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julia
algebraic_split(a::AbsSimpleNumFieldElem) -> AbsSimpleNumFieldElem, AbsSimpleNumFieldElem

Writes the input as a quotient of two "small" algebraic integers.

source

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julia
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julia
conjugates(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
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julia
conjugates_arb_log(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
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julia
conjugates_arb_real(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
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julia
conjugates_complex(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
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julia
conjugates_arb_log_normalise(x::AbsSimpleNumFieldElem, p::Int = 10)
 conjugates_arb_log_normalise(x::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, p::Int = 10)
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julia
minkowski_map(a::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
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julia
is_integral(a::NumFieldElem) -> Bool
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julia
is_torsion_unit(x::AbsSimpleNumFieldElem, checkisunit::Bool = false) -> Bool
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julia
is_local_norm(L::NumField, a::NumFieldElem, P)
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julia
is_norm_divisible(a::AbsSimpleNumFieldElem, n::ZZRingElem) -> Bool
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julia
is_norm(K::AbsSimpleNumField, a::ZZRingElem; extra::Vector{ZZRingElem}) -> Bool, AbsSimpleNumFieldElem
',1)),t("p",null,[e[452]||(e[452]=s("For a ZZRingElem ")),t("mjx-container",Y4,[(i(),l("svg",_4,e[444]||(e[444]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[445]||(e[445]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[453]||(e[453]=s(", try to find ")),t("mjx-container",t3,[(i(),l("svg",e3,e[446]||(e[446]=[a('',1)]))),e[447]||(e[447]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"T"),t("mo",null,"∈"),t("mi",null,"K")])],-1))]),e[454]||(e[454]=s(" s.th. ")),t("mjx-container",s3,[(i(),l("svg",l3,e[448]||(e[448]=[a('',1)]))),e[449]||(e[449]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"N"),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"a")])],-1))]),e[455]||(e[455]=s(" holds. If successful, return true and ")),t("mjx-container",i3,[(i(),l("svg",a3,e[450]||(e[450]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D447",d:"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z",style:{"stroke-width":"3"}})])])],-1)]))),e[451]||(e[451]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"T")])],-1))]),e[456]||(e[456]=s(", otherwise false and some element. In \\testtt{extra} one can pass in additional prime numbers that are allowed to occur in the solution. This will then be supplemented. The element will be returned in factored form."))]),e[458]||(e[458]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e[578]||(e[578]=t("h3",{id:"invariants",tabindex:"-1"},[s("Invariants "),t("a",{class:"header-anchor",href:"#invariants","aria-label":'Permalink to "Invariants"'},"​")],-1)),t("details",o3,[t("summary",null,[e[459]||(e[459]=t("a",{id:"norm-Tuple{NumFieldElem}",href:"#norm-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"norm")],-1)),e[460]||(e[460]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[471]||(e[471]=a('
julia
norm(a::NumFieldElem) -> NumFieldElem
',1)),t("p",null,[e[467]||(e[467]=s("Returns the norm of an element ")),t("mjx-container",n3,[(i(),l("svg",Q3,e[461]||(e[461]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[462]||(e[462]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[468]||(e[468]=s(" of a number field extension ")),t("mjx-container",r3,[(i(),l("svg",T3,e[463]||(e[463]=[a('',1)]))),e[464]||(e[464]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"L"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("mi",null,"K")])],-1))]),e[469]||(e[469]=s(". This will be an element of ")),t("mjx-container",d3,[(i(),l("svg",m3,e[465]||(e[465]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[466]||(e[466]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[470]||(e[470]=s("."))]),e[472]||(e[472]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",p3,[t("summary",null,[e[473]||(e[473]=t("a",{id:"absolute_norm-Tuple{NumFieldElem}",href:"#absolute_norm-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"absolute_norm")],-1)),e[474]||(e[474]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[482]||(e[482]=a('
julia
absolute_norm(a::NumFieldElem) -> QQFieldElem
',1)),t("p",null,[e[479]||(e[479]=s("Given a number field element ")),t("mjx-container",h3,[(i(),l("svg",g3,e[475]||(e[475]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[476]||(e[476]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[480]||(e[480]=s(", returns the absolute norm of ")),t("mjx-container",x3,[(i(),l("svg",u3,e[477]||(e[477]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[478]||(e[478]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[481]||(e[481]=s("."))]),e[483]||(e[483]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",k3,[t("summary",null,[e[484]||(e[484]=t("a",{id:"minpoly-Tuple{NumFieldElem}",href:"#minpoly-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"minpoly")],-1)),e[485]||(e[485]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[499]||(e[499]=a('
julia
minpoly(a::NumFieldElem) -> PolyRingElem
',1)),t("p",null,[e[494]||(e[494]=s("Given a number field element ")),t("mjx-container",w3,[(i(),l("svg",c3,e[486]||(e[486]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[487]||(e[487]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[495]||(e[495]=s(" of a number field ")),t("mjx-container",y3,[(i(),l("svg",f3,e[488]||(e[488]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[489]||(e[489]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[496]||(e[496]=s(", this function returns the minimal polynomial of ")),t("mjx-container",b3,[(i(),l("svg",v3,e[490]||(e[490]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[491]||(e[491]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[497]||(e[497]=s(" over the base field of ")),t("mjx-container",H3,[(i(),l("svg",M3,e[492]||(e[492]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[493]||(e[493]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[498]||(e[498]=s("."))]),e[500]||(e[500]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",E3,[t("summary",null,[e[501]||(e[501]=t("a",{id:"absolute_minpoly-Tuple{NumFieldElem}",href:"#absolute_minpoly-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"absolute_minpoly")],-1)),e[502]||(e[502]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[516]||(e[516]=a('
julia
absolute_minpoly(a::NumFieldElem) -> PolyRingElem
',1)),t("p",null,[e[511]||(e[511]=s("Given a number field element ")),t("mjx-container",L3,[(i(),l("svg",j3,e[503]||(e[503]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[504]||(e[504]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[512]||(e[512]=s(" of a number field ")),t("mjx-container",C3,[(i(),l("svg",Z3,e[505]||(e[505]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[506]||(e[506]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[513]||(e[513]=s(", this function returns the minimal polynomial of ")),t("mjx-container",F3,[(i(),l("svg",V3,e[507]||(e[507]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[508]||(e[508]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[514]||(e[514]=s(" over the rationals ")),t("mjx-container",D3,[(i(),l("svg",B3,e[509]||(e[509]=[a('',1)]))),e[510]||(e[510]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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julia
charpoly(a::NumFieldElem) -> PolyRingElem
',1)),t("p",null,[e[528]||(e[528]=s("Given a number field element ")),t("mjx-container",S3,[(i(),l("svg",N3,e[520]||(e[520]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[521]||(e[521]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[529]||(e[529]=s(" of a number field ")),t("mjx-container",G3,[(i(),l("svg",O3,e[522]||(e[522]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[523]||(e[523]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[530]||(e[530]=s(", this function returns the characteristic polynomial of ")),t("mjx-container",z3,[(i(),l("svg",J3,e[524]||(e[524]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[525]||(e[525]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[531]||(e[531]=s(" over the base field of ")),t("mjx-container",R3,[(i(),l("svg",K3,e[526]||(e[526]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[527]||(e[527]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[532]||(e[532]=s("."))]),e[534]||(e[534]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",I3,[t("summary",null,[e[535]||(e[535]=t("a",{id:"absolute_charpoly-Tuple{NumFieldElem}",href:"#absolute_charpoly-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"absolute_charpoly")],-1)),e[536]||(e[536]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[550]||(e[550]=a('
julia
absolute_charpoly(a::NumFieldElem) -> PolyRingElem
',1)),t("p",null,[e[545]||(e[545]=s("Given a number field element ")),t("mjx-container",X3,[(i(),l("svg",P3,e[537]||(e[537]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[538]||(e[538]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[546]||(e[546]=s(" of a number field ")),t("mjx-container",q3,[(i(),l("svg",U3,e[539]||(e[539]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[540]||(e[540]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[547]||(e[547]=s(", this function returns the characteristic polynomial of ")),t("mjx-container",$3,[(i(),l("svg",W3,e[541]||(e[541]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[542]||(e[542]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[548]||(e[548]=s(" over the rationals ")),t("mjx-container",Y3,[(i(),l("svg",_3,e[543]||(e[543]=[a('',1)]))),e[544]||(e[544]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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julia
norm(a::NumFieldElem, k::NumField) -> NumFieldElem
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380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),e[561]||(e[561]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"L")])],-1))]),e[568]||(e[568]=s(". 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julia
quadratic_defect(a::Union{NumFieldElem,Rational,QQFieldElem}, p) -> Union{Inf, PosInf}
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julia
hilbert_symbol(a::NumFieldElem, b::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Int
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julia
representation_matrix(a::NumFieldElem) -> MatElem
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julia
basis_matrix(v::Vector{NumFieldElem}) -> Mat
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129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z",style:{"stroke-width":"3"}})])])],-1)]))),e[95]||(e[95]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"d")])],-1))]),e[110]||(e[110]=s(", this function returns an ")),t("mjx-container",n1,[(i(),l("svg",Q1,e[96]||(e[96]=[a('',1)]))),e[97]||(e[97]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"n"),t("mo",null,"×"),t("mi",null,"d")])],-1))]),e[111]||(e[111]=s(" matrix with entries in the base field of ")),t("mjx-container",r1,[(i(),l("svg",T1,e[98]||(e[98]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[99]||(e[99]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[112]||(e[112]=s(", where row ")),t("mjx-container",d1,[(i(),l("svg",m1,e[100]||(e[100]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D456",d:"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z",style:{"stroke-width":"3"}})])])],-1)]))),e[101]||(e[101]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"i")])],-1))]),e[113]||(e[113]=s(" contains the coefficients of ")),t("mjx-container",p1,[(i(),l("svg",h1,e[102]||(e[102]=[a('',1)]))),e[103]||(e[103]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"v"),t("mo",{stretchy:"false"},"["),t("mi",null,"i"),t("mo",{stretchy:"false"},"]")])],-1))]),e[114]||(e[114]=s(" with respect of the canonical basis of ")),t("mjx-container",g1,[(i(),l("svg",x1,e[104]||(e[104]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[105]||(e[105]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[115]||(e[115]=s("."))]),e[117]||(e[117]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",u1,[t("summary",null,[e[118]||(e[118]=t("a",{id:"coefficients-Tuple{SimpleNumFieldElem}",href:"#coefficients-Tuple{SimpleNumFieldElem}"},[t("span",{class:"jlbinding"},"coefficients")],-1)),e[119]||(e[119]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[120]||(e[120]=a('
julia
coefficients(a::SimpleNumFieldElem, i::Int) -> Vector{FieldElem}

Given a number field element a of a simple number field extension L/K, this function returns the coefficients of a, when expanded in the canonical power basis of L.

source

',3))]),t("details",k1,[t("summary",null,[e[121]||(e[121]=t("a",{id:"coordinates-Tuple{NumFieldElem}",href:"#coordinates-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"coordinates")],-1)),e[122]||(e[122]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[136]||(e[136]=a('
julia
coordinates(x::NumFieldElem{T}) -> Vector{T}
',1)),t("p",null,[e[131]||(e[131]=s("Given an element ")),t("mjx-container",w1,[(i(),l("svg",c1,e[123]||(e[123]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D465",d:"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),e[124]||(e[124]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x")])],-1))]),e[132]||(e[132]=s(" in a number field ")),t("mjx-container",y1,[(i(),l("svg",f1,e[125]||(e[125]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[126]||(e[126]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z",style:{"stroke-width":"3"}})])])],-1)]))),e[128]||(e[128]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"x")])],-1))]),e[134]||(e[134]=s(" with respect to the basis of 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julia
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julia
coeff(a::SimpleNumFieldElem, i::Int) -> FieldElem

Given a number field element a of a simple number field extension L/K, this function returns the i-th coefficient of a, when expanded in the canonical power basis of L. The result is an element of K.

source

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julia
valuation(a::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
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julia
torsion_unit_order(x::AbsSimpleNumFieldElem, n::Int)
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julia
algebraic_split(a::AbsSimpleNumFieldElem) -> AbsSimpleNumFieldElem, AbsSimpleNumFieldElem

Writes the input as a quotient of two "small" algebraic integers.

source

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julia
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julia
conjugates(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
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julia
conjugates_arb_log(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
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julia
conjugates_arb_real(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
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julia
conjugates_complex(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
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julia
conjugates_arb_log_normalise(x::AbsSimpleNumFieldElem, p::Int = 10)
+1//2*b^2 + b + a
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julia
quadratic_defect(a::Union{NumFieldElem,Rational,QQFieldElem}, p) -> Union{Inf, PosInf}
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julia
hilbert_symbol(a::NumFieldElem, b::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Int
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julia
representation_matrix(a::NumFieldElem) -> MatElem
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julia
basis_matrix(v::Vector{NumFieldElem}) -> Mat
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julia
coefficients(a::SimpleNumFieldElem, i::Int) -> Vector{FieldElem}

Given a number field element a of a simple number field extension L/K, this function returns the coefficients of a, when expanded in the canonical power basis of L.

source

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julia
coordinates(x::NumFieldElem{T}) -> Vector{T}
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julia
absolute_coordinates(x::NumFieldElem{T}) -> Vector{T}
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julia
coeff(a::SimpleNumFieldElem, i::Int) -> FieldElem

Given a number field element a of a simple number field extension L/K, this function returns the i-th coefficient of a, when expanded in the canonical power basis of L. The result is an element of K.

source

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julia
valuation(a::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
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julia
torsion_unit_order(x::AbsSimpleNumFieldElem, n::Int)
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julia
algebraic_split(a::AbsSimpleNumFieldElem) -> AbsSimpleNumFieldElem, AbsSimpleNumFieldElem

Writes the input as a quotient of two "small" algebraic integers.

source

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julia
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julia
conjugates(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
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julia
conjugates_arb_log(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
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julia
conjugates_arb_real(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
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julia
conjugates_complex(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
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julia
conjugates_arb_log_normalise(x::AbsSimpleNumFieldElem, p::Int = 10)
 conjugates_arb_log_normalise(x::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, p::Int = 10)
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julia
minkowski_map(a::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
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julia
is_integral(a::NumFieldElem) -> Bool
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julia
is_torsion_unit(x::AbsSimpleNumFieldElem, checkisunit::Bool = false) -> Bool
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julia
is_local_norm(L::NumField, a::NumFieldElem, P)
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julia
is_norm_divisible(a::AbsSimpleNumFieldElem, n::ZZRingElem) -> Bool
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julia
is_norm(K::AbsSimpleNumField, a::ZZRingElem; extra::Vector{ZZRingElem}) -> Bool, AbsSimpleNumFieldElem
',1)),t("p",null,[e[452]||(e[452]=s("For a ZZRingElem ")),t("mjx-container",Y4,[(i(),l("svg",_4,e[444]||(e[444]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[445]||(e[445]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[453]||(e[453]=s(", try to find ")),t("mjx-container",t3,[(i(),l("svg",e3,e[446]||(e[446]=[a('',1)]))),e[447]||(e[447]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"T"),t("mo",null,"∈"),t("mi",null,"K")])],-1))]),e[454]||(e[454]=s(" s.th. ")),t("mjx-container",s3,[(i(),l("svg",l3,e[448]||(e[448]=[a('',1)]))),e[449]||(e[449]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"N"),t("mo",{stretchy:"false"},"("),t("mi",null,"T"),t("mo",{stretchy:"false"},")"),t("mo",null,"="),t("mi",null,"a")])],-1))]),e[455]||(e[455]=s(" holds. If successful, return true and ")),t("mjx-container",i3,[(i(),l("svg",a3,e[450]||(e[450]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D447",d:"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z",style:{"stroke-width":"3"}})])])],-1)]))),e[451]||(e[451]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"T")])],-1))]),e[456]||(e[456]=s(", otherwise false and some element. In \\testtt{extra} one can pass in additional prime numbers that are allowed to occur in the solution. This will then be supplemented. The element will be returned in factored form."))]),e[458]||(e[458]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e[578]||(e[578]=t("h3",{id:"invariants",tabindex:"-1"},[s("Invariants "),t("a",{class:"header-anchor",href:"#invariants","aria-label":'Permalink to "Invariants"'},"​")],-1)),t("details",o3,[t("summary",null,[e[459]||(e[459]=t("a",{id:"norm-Tuple{NumFieldElem}",href:"#norm-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"norm")],-1)),e[460]||(e[460]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[471]||(e[471]=a('
julia
norm(a::NumFieldElem) -> NumFieldElem
',1)),t("p",null,[e[467]||(e[467]=s("Returns the norm of an element ")),t("mjx-container",n3,[(i(),l("svg",Q3,e[461]||(e[461]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[462]||(e[462]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[468]||(e[468]=s(" of a number field extension ")),t("mjx-container",r3,[(i(),l("svg",T3,e[463]||(e[463]=[a('',1)]))),e[464]||(e[464]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"L"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("mi",null,"K")])],-1))]),e[469]||(e[469]=s(". This will be an element of ")),t("mjx-container",d3,[(i(),l("svg",m3,e[465]||(e[465]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[466]||(e[466]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[470]||(e[470]=s("."))]),e[472]||(e[472]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",p3,[t("summary",null,[e[473]||(e[473]=t("a",{id:"absolute_norm-Tuple{NumFieldElem}",href:"#absolute_norm-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"absolute_norm")],-1)),e[474]||(e[474]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[482]||(e[482]=a('
julia
absolute_norm(a::NumFieldElem) -> QQFieldElem
',1)),t("p",null,[e[479]||(e[479]=s("Given a number field element ")),t("mjx-container",h3,[(i(),l("svg",g3,e[475]||(e[475]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[476]||(e[476]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[480]||(e[480]=s(", returns the absolute norm of ")),t("mjx-container",x3,[(i(),l("svg",u3,e[477]||(e[477]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[478]||(e[478]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[481]||(e[481]=s("."))]),e[483]||(e[483]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",k3,[t("summary",null,[e[484]||(e[484]=t("a",{id:"minpoly-Tuple{NumFieldElem}",href:"#minpoly-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"minpoly")],-1)),e[485]||(e[485]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[499]||(e[499]=a('
julia
minpoly(a::NumFieldElem) -> PolyRingElem
',1)),t("p",null,[e[494]||(e[494]=s("Given a number field element ")),t("mjx-container",w3,[(i(),l("svg",c3,e[486]||(e[486]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[487]||(e[487]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[495]||(e[495]=s(" of a number field ")),t("mjx-container",y3,[(i(),l("svg",f3,e[488]||(e[488]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[489]||(e[489]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[496]||(e[496]=s(", this function returns the minimal polynomial of ")),t("mjx-container",b3,[(i(),l("svg",v3,e[490]||(e[490]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[491]||(e[491]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[497]||(e[497]=s(" over the base field of ")),t("mjx-container",H3,[(i(),l("svg",M3,e[492]||(e[492]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[493]||(e[493]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[498]||(e[498]=s("."))]),e[500]||(e[500]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",E3,[t("summary",null,[e[501]||(e[501]=t("a",{id:"absolute_minpoly-Tuple{NumFieldElem}",href:"#absolute_minpoly-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"absolute_minpoly")],-1)),e[502]||(e[502]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[516]||(e[516]=a('
julia
absolute_minpoly(a::NumFieldElem) -> PolyRingElem
',1)),t("p",null,[e[511]||(e[511]=s("Given a number field element ")),t("mjx-container",L3,[(i(),l("svg",j3,e[503]||(e[503]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[504]||(e[504]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[512]||(e[512]=s(" of a number field ")),t("mjx-container",C3,[(i(),l("svg",Z3,e[505]||(e[505]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[506]||(e[506]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[513]||(e[513]=s(", this function returns the minimal polynomial of ")),t("mjx-container",F3,[(i(),l("svg",V3,e[507]||(e[507]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[508]||(e[508]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[514]||(e[514]=s(" over the rationals ")),t("mjx-container",D3,[(i(),l("svg",B3,e[509]||(e[509]=[a('',1)]))),e[510]||(e[510]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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julia
charpoly(a::NumFieldElem) -> PolyRingElem
',1)),t("p",null,[e[528]||(e[528]=s("Given a number field element ")),t("mjx-container",S3,[(i(),l("svg",N3,e[520]||(e[520]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[521]||(e[521]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[529]||(e[529]=s(" of a number field ")),t("mjx-container",G3,[(i(),l("svg",O3,e[522]||(e[522]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[523]||(e[523]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[530]||(e[530]=s(", this function returns the characteristic polynomial of ")),t("mjx-container",z3,[(i(),l("svg",J3,e[524]||(e[524]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[525]||(e[525]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[531]||(e[531]=s(" over the base field of ")),t("mjx-container",R3,[(i(),l("svg",K3,e[526]||(e[526]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[527]||(e[527]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[532]||(e[532]=s("."))]),e[534]||(e[534]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",I3,[t("summary",null,[e[535]||(e[535]=t("a",{id:"absolute_charpoly-Tuple{NumFieldElem}",href:"#absolute_charpoly-Tuple{NumFieldElem}"},[t("span",{class:"jlbinding"},"absolute_charpoly")],-1)),e[536]||(e[536]=s()),n(o,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[550]||(e[550]=a('
julia
absolute_charpoly(a::NumFieldElem) -> PolyRingElem
',1)),t("p",null,[e[545]||(e[545]=s("Given a number field element ")),t("mjx-container",X3,[(i(),l("svg",P3,e[537]||(e[537]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[538]||(e[538]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[546]||(e[546]=s(" of a number field ")),t("mjx-container",q3,[(i(),l("svg",U3,e[539]||(e[539]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[540]||(e[540]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[547]||(e[547]=s(", this function returns the characteristic polynomial of ")),t("mjx-container",$3,[(i(),l("svg",W3,e[541]||(e[541]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[542]||(e[542]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[548]||(e[548]=s(" over the rationals ")),t("mjx-container",Y3,[(i(),l("svg",_3,e[543]||(e[543]=[a('',1)]))),e[544]||(e[544]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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julia
norm(a::NumFieldElem, k::NumField) -> NumFieldElem
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Introduction ​

This chapter deals with number fields and orders there of. We follow the common terminology and conventions as e.g. used in [2], [1], [3] or [4].

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As far as possible, the interaction and the interface for orders of absolute number fields and of relative number fields is the same."))]),e[140]||(e[140]=t("h2",{id:"Orders-of-absolute-number-fields",tabindex:"-1"},[s("Orders of absolute number fields "),t("a",{class:"header-anchor",href:"#Orders-of-absolute-number-fields","aria-label":'Permalink to "Orders of absolute number fields {#Orders-of-absolute-number-fields}"'},"​")],-1)),t("p",null,[e[45]||(e[45]=s("Assume that ")),t("mjx-container",v,[(i(),a("svg",L,e[27]||(e[27]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[28]||(e[28]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[46]||(e[46]=s(" is defined as an absolute field. 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As a ring, the order ")),t("mjx-container",h1,[(i(),a("svg",g1,e[93]||(e[93]=[l('',1)]))),e[94]||(e[94]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[107]||(e[107]=s(" is unitary and has ")),t("mjx-container",x1,[(i(),a("svg",k1,e[95]||(e[95]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),e[96]||(e[96]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"L")])],-1))]),e[108]||(e[108]=s(" as a fraction field. Due to ")),t("mjx-container",w1,[(i(),a("svg",u1,e[97]||(e[97]=[l('',1)]))),e[98]||(e[98]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")]),t("mi",null,"K")])])],-1))]),e[109]||(e[109]=s(" in general not being a principal ideal domain, the module structure is more complicated and requires so called pseudo-matrices. See ")),e[110]||(e[110]=t("a",{href:"/Hecke.jl/v0.34.8/manual/misc/pmat#PMatLink"},"here",-1)),e[111]||(e[111]=s(" for details on pseudo-matrices, or [")),e[112]||(e[112]=t("a",{href:"/Hecke.jl/v0.34.8/references#Coh00"},"1",-1)),e[113]||(e[113]=s("], Chapter 1 for an introduction."))]),t("p",null,[e[130]||(e[130]=s("In short, ")),t("mjx-container",c1,[(i(),a("svg",y1,e[114]||(e[114]=[l('',1)]))),e[115]||(e[115]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[131]||(e[131]=s(" is represented as ")),t("mjx-container",f1,[(i(),a("svg",b1,e[116]||(e[116]=[l('',1)]))),e[117]||(e[117]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{"data-mjx-texclass":"OP"},"∑"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"a")]),t("mi",null,"i")]),t("msub",null,[t("mi",null,"ω"),t("mi",null,"i")])])],-1))]),e[132]||(e[132]=s(" with fractional ")),t("mjx-container",v1,[(i(),a("svg",L1,e[118]||(e[118]=[l('',1)]))),e[119]||(e[119]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")]),t("mi",null,"K")])])],-1))]),e[133]||(e[133]=s(" ideals ")),t("mjx-container",H1,[(i(),a("svg",E1,e[120]||(e[120]=[l('',1)]))),e[121]||(e[121]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"a")]),t("mi",null,"i")]),t("mo",null,"⊂"),t("mi",null,"K")])],-1))]),e[134]||(e[134]=s(" and ")),t("mjx-container",M1,[(i(),a("svg",j1,e[122]||(e[122]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 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In general it is impossible to have both ")),t("mjx-container",V1,[(i(),a("svg",D1,e[126]||(e[126]=[l('',1)]))),e[127]||(e[127]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"a")]),t("mi",null,"i")])])],-1))]),e[137]||(e[137]=s(" integral and ")),t("mjx-container",F1,[(i(),a("svg",B1,e[128]||(e[128]=[l('',1)]))),e[129]||(e[129]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"i")]),t("mo",null,"∈"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[138]||(e[138]=s(", thus coefficients will not be integral and/or generators not in the structure."))]),e[142]||(e[142]=l(`

Examples ​

Usually, to create an order, one starts with a field (or a polynomial):

julia

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Introduction ​
This chapter deals with number fields and orders there of. We follow the common terminology and conventions as e.g. used in [2], [1], [3] or [4].
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Due to ")),t("mjx-container",w1,[(i(),a("svg",u1,e[97]||(e[97]=[l('',1)]))),e[98]||(e[98]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")]),t("mi",null,"K")])])],-1))]),e[109]||(e[109]=s(" in general not being a principal ideal domain, the module structure is more complicated and requires so called pseudo-matrices. See ")),e[110]||(e[110]=t("a",{href:"/v0.34.8/manual/misc/pmat#PMatLink"},"here",-1)),e[111]||(e[111]=s(" for details on pseudo-matrices, or [")),e[112]||(e[112]=t("a",{href:"/v0.34.8/references#Coh00"},"1",-1)),e[113]||(e[113]=s("], Chapter 1 for an introduction."))]),t("p",null,[e[130]||(e[130]=s("In short, ")),t("mjx-container",c1,[(i(),a("svg",y1,e[114]||(e[114]=[l('',1)]))),e[115]||(e[115]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[131]||(e[131]=s(" is represented 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In general it is impossible to have both ")),t("mjx-container",V1,[(i(),a("svg",D1,e[126]||(e[126]=[l('',1)]))),e[127]||(e[127]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"a")]),t("mi",null,"i")])])],-1))]),e[137]||(e[137]=s(" integral and ")),t("mjx-container",F1,[(i(),a("svg",B1,e[128]||(e[128]=[l('',1)]))),e[129]||(e[129]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"i")]),t("mo",null,"∈"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[138]||(e[138]=s(", thus coefficients will not be integral and/or generators not in the structure."))]),e[142]||(e[142]=l(`
Examples ​
Usually, to create an order, one starts with a field (or a polynomial):
julia

 julia> Qx, x = polynomial_ring(QQ, "x");
 
 julia> K, a = number_field(x^2 - 10, "a");
diff --git a/v0.34.8/assets/manual_orders_introduction.md.BTbzdtxm.lean.js b/v0.34.8/assets/manual_orders_introduction.md.BTbzdtxm.lean.js
index c28d8279d6..b3f7b1b12b 100644
--- a/v0.34.8/assets/manual_orders_introduction.md.BTbzdtxm.lean.js
+++ b/v0.34.8/assets/manual_orders_introduction.md.BTbzdtxm.lean.js
@@ -1,4 +1,4 @@
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Introduction ​
This chapter deals with number fields and orders there of. We follow the common terminology and conventions as e.g. used in [2], [1], [3] or [4].
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As a ring, the order ")),t("mjx-container",h1,[(i(),a("svg",g1,e[93]||(e[93]=[l('',1)]))),e[94]||(e[94]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[107]||(e[107]=s(" is unitary and has ")),t("mjx-container",x1,[(i(),a("svg",k1,e[95]||(e[95]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),e[96]||(e[96]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"L")])],-1))]),e[108]||(e[108]=s(" as a fraction field. Due to ")),t("mjx-container",w1,[(i(),a("svg",u1,e[97]||(e[97]=[l('',1)]))),e[98]||(e[98]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")]),t("mi",null,"K")])])],-1))]),e[109]||(e[109]=s(" in general not being a principal ideal domain, the module structure is more complicated and requires so called pseudo-matrices. See ")),e[110]||(e[110]=t("a",{href:"/Hecke.jl/v0.34.8/manual/misc/pmat#PMatLink"},"here",-1)),e[111]||(e[111]=s(" for details on pseudo-matrices, or [")),e[112]||(e[112]=t("a",{href:"/Hecke.jl/v0.34.8/references#Coh00"},"1",-1)),e[113]||(e[113]=s("], Chapter 1 for an introduction."))]),t("p",null,[e[130]||(e[130]=s("In short, ")),t("mjx-container",c1,[(i(),a("svg",y1,e[114]||(e[114]=[l('',1)]))),e[115]||(e[115]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[131]||(e[131]=s(" is represented as ")),t("mjx-container",f1,[(i(),a("svg",b1,e[116]||(e[116]=[l('',1)]))),e[117]||(e[117]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{"data-mjx-texclass":"OP"},"∑"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"a")]),t("mi",null,"i")]),t("msub",null,[t("mi",null,"ω"),t("mi",null,"i")])])],-1))]),e[132]||(e[132]=s(" with fractional ")),t("mjx-container",v1,[(i(),a("svg",L1,e[118]||(e[118]=[l('',1)]))),e[119]||(e[119]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")]),t("mi",null,"K")])])],-1))]),e[133]||(e[133]=s(" ideals ")),t("mjx-container",H1,[(i(),a("svg",E1,e[120]||(e[120]=[l('',1)]))),e[121]||(e[121]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"a")]),t("mi",null,"i")]),t("mo",null,"⊂"),t("mi",null,"K")])],-1))]),e[134]||(e[134]=s(" and ")),t("mjx-container",M1,[(i(),a("svg",j1,e[122]||(e[122]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 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In general it is impossible to have both ")),t("mjx-container",V1,[(i(),a("svg",D1,e[126]||(e[126]=[l('',1)]))),e[127]||(e[127]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"a")]),t("mi",null,"i")])])],-1))]),e[137]||(e[137]=s(" integral and ")),t("mjx-container",F1,[(i(),a("svg",B1,e[128]||(e[128]=[l('',1)]))),e[129]||(e[129]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"i")]),t("mo",null,"∈"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[138]||(e[138]=s(", thus coefficients will not be integral and/or generators not in the structure."))]),e[142]||(e[142]=l(`
Examples ​
Usually, to create an order, one starts with a field (or a polynomial):
julia

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Introduction ​
This chapter deals with number fields and orders there of. We follow the common terminology and conventions as e.g. used in [2], [1], [3] or [4].
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Due to ")),t("mjx-container",w1,[(i(),a("svg",u1,e[97]||(e[97]=[l('',1)]))),e[98]||(e[98]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")]),t("mi",null,"K")])])],-1))]),e[109]||(e[109]=s(" in general not being a principal ideal domain, the module structure is more complicated and requires so called pseudo-matrices. See ")),e[110]||(e[110]=t("a",{href:"/v0.34.8/manual/misc/pmat#PMatLink"},"here",-1)),e[111]||(e[111]=s(" for details on pseudo-matrices, or [")),e[112]||(e[112]=t("a",{href:"/v0.34.8/references#Coh00"},"1",-1)),e[113]||(e[113]=s("], Chapter 1 for an introduction."))]),t("p",null,[e[130]||(e[130]=s("In short, ")),t("mjx-container",c1,[(i(),a("svg",y1,e[114]||(e[114]=[l('',1)]))),e[115]||(e[115]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[131]||(e[131]=s(" is represented 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1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")]),t("mi",null,"K")])])],-1))]),e[133]||(e[133]=s(" ideals ")),t("mjx-container",H1,[(i(),a("svg",E1,e[120]||(e[120]=[l('',1)]))),e[121]||(e[121]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"a")]),t("mi",null,"i")]),t("mo",null,"⊂"),t("mi",null,"K")])],-1))]),e[134]||(e[134]=s(" and ")),t("mjx-container",M1,[(i(),a("svg",j1,e[122]||(e[122]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[123]||(e[123]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[135]||(e[135]=s("-linear independent elements ")),t("mjx-container",C1,[(i(),a("svg",Z1,e[124]||(e[124]=[l('',1)]))),e[125]||(e[125]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"i")]),t("mo",null,"∈"),t("mi",null,"L")])],-1))]),e[136]||(e[136]=s(". In general it is impossible to have both ")),t("mjx-container",V1,[(i(),a("svg",D1,e[126]||(e[126]=[l('',1)]))),e[127]||(e[127]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"a")]),t("mi",null,"i")])])],-1))]),e[137]||(e[137]=s(" integral and ")),t("mjx-container",F1,[(i(),a("svg",B1,e[128]||(e[128]=[l('',1)]))),e[129]||(e[129]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msub",null,[t("mi",null,"ω"),t("mi",null,"i")]),t("mo",null,"∈"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[138]||(e[138]=s(", thus coefficients will not be integral and/or generators not in the structure."))]),e[142]||(e[142]=l(`
Examples ​
Usually, to create an order, one starts with a field (or a polynomial):
julia

 julia> Qx, x = polynomial_ring(QQ, "x");
 
 julia> K, a = number_field(x^2 - 10, "a");
diff --git a/v0.34.8/assets/manual_orders_orders.md.Bk8W8VLT.js b/v0.34.8/assets/manual_orders_orders.md.Bk8W8VLT.js
index cff9d10581..2edd1383af 100644
--- a/v0.34.8/assets/manual_orders_orders.md.Bk8W8VLT.js
+++ b/v0.34.8/assets/manual_orders_orders.md.Bk8W8VLT.js
@@ -13,4 +13,4 @@ import{_ as r,c as i,j as t,a as s,a5 as a,G as o,B as d,o as l}from"./chunks/fr
 with Z-basis AbsSimpleNumFieldOrderElem[1, a]
`,2)),t("details",w1,[t("summary",null,[e[145]||(e[145]=t("a",{id:"parent-Tuple{AbsSimpleNumFieldOrder}",href:"#parent-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"parent")],-1)),e[146]||(e[146]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[151]||(e[151]=a('
julia
parent(O::AbsNumFieldOrder) -> AbsNumFieldOrderSet
',1)),t("p",null,[e[149]||(e[149]=s("Returns the parent of ")),t("mjx-container",c1,[(l(),i("svg",b1,e[147]||(e[147]=[a('',1)]))),e[148]||(e[148]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[150]||(e[150]=s(", that is, the set of orders of the ambient number field."))]),e[152]||(e[152]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",y1,[t("summary",null,[e[153]||(e[153]=t("a",{id:"signature-Tuple{AbsSimpleNumFieldOrder}",href:"#signature-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"signature")],-1)),e[154]||(e[154]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[159]||(e[159]=a('
julia
signature(O::NumFieldOrder) -> Tuple{Int, Int}
',1)),t("p",null,[e[157]||(e[157]=s("Returns the signature of the ambient number field of ")),t("mjx-container",f1,[(l(),i("svg",v1,e[155]||(e[155]=[a('',1)]))),e[156]||(e[156]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[158]||(e[158]=s("."))]),e[160]||(e[160]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",E1,[t("summary",null,[e[161]||(e[161]=t("a",{id:"nf-Tuple{AbsSimpleNumFieldOrder}",href:"#nf-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"nf")],-1)),e[162]||(e[162]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[167]||(e[167]=a('
julia
nf(O::NumFieldOrder) -> NumField
',1)),t("p",null,[e[165]||(e[165]=s("Returns the ambient number field of ")),t("mjx-container",H1,[(l(),i("svg",M1,e[163]||(e[163]=[a('',1)]))),e[164]||(e[164]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[166]||(e[166]=s("."))]),e[168]||(e[168]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",j1,[t("summary",null,[e[169]||(e[169]=t("a",{id:"basis-Tuple{AbsSimpleNumFieldOrder}",href:"#basis-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"basis")],-1)),e[170]||(e[170]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[185]||(e[185]=a('
julia
basis(O::AbsNumFieldOrder) -> Vector{AbsNumFieldOrderElem}
',1)),t("p",null,[e[175]||(e[175]=s("Returns the ")),t("mjx-container",F1,[(l(),i("svg",L1,e[171]||(e[171]=[a('',1)]))),e[172]||(e[172]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"Z")])])],-1))]),e[176]||(e[176]=s("-basis of ")),t("mjx-container",C1,[(l(),i("svg",O1,e[173]||(e[173]=[a('',1)]))),e[174]||(e[174]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[177]||(e[177]=s("."))]),e[186]||(e[186]=a('
source
julia
basis(I::AbsNumFieldOrderFractionalIdeal) -> Vector{AbsSimpleNumFieldElem}
',2)),t("p",null,[e[182]||(e[182]=s("Returns the ")),t("mjx-container",A1,[(l(),i("svg",D1,e[178]||(e[178]=[a('',1)]))),e[179]||(e[179]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"Z")])])],-1))]),e[183]||(e[183]=s("-basis of ")),t("mjx-container",Z1,[(l(),i("svg",V1,e[180]||(e[180]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43C",d:"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z",style:{"stroke-width":"3"}})])])],-1)]))),e[181]||(e[181]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"I")])],-1))]),e[184]||(e[184]=s("."))]),e[187]||(e[187]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",B1,[t("summary",null,[e[188]||(e[188]=t("a",{id:"lll_basis-Tuple{AbsSimpleNumFieldOrder}",href:"#lll_basis-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"lll_basis")],-1)),e[189]||(e[189]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[194]||(e[194]=a('
julia
lll_basis(M::NumFieldOrder) -> Vector{NumFieldElem}
',1)),t("p",null,[e[192]||(e[192]=s("A basis for ")),t("mjx-container",S1,[(l(),i("svg",R1,e[190]||(e[190]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),e[191]||(e[191]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"M")])],-1))]),e[193]||(e[193]=s(" that is reduced using the LLL algorithm for the Minkowski metric."))]),e[195]||(e[195]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",N1,[t("summary",null,[e[196]||(e[196]=t("a",{id:"basis-Tuple{AbsSimpleNumFieldOrder, AbsSimpleNumField}",href:"#basis-Tuple{AbsSimpleNumFieldOrder, AbsSimpleNumField}"},[t("span",{class:"jlbinding"},"basis")],-1)),e[197]||(e[197]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[205]||(e[205]=a('
julia
basis(O::AbsSimpleNumFieldOrder, K::AbsSimpleNumField) -> Vector{AbsSimpleNumFieldElem}
',1)),t("p",null,[e[202]||(e[202]=s("Returns the ")),t("mjx-container",G1,[(l(),i("svg",z1,e[198]||(e[198]=[a('',1)]))),e[199]||(e[199]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"Z")])])],-1))]),e[203]||(e[203]=s("-basis elements of ")),t("mjx-container",J1,[(l(),i("svg",X1,e[200]||(e[200]=[a('',1)]))),e[201]||(e[201]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[204]||(e[204]=s(" as elements of the ambient number field."))]),e[206]||(e[206]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",I1,[t("summary",null,[e[207]||(e[207]=t("a",{id:"pseudo_basis-Tuple{Hecke.RelNumFieldOrder}",href:"#pseudo_basis-Tuple{Hecke.RelNumFieldOrder}"},[t("span",{class:"jlbinding"},"pseudo_basis")],-1)),e[208]||(e[208]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[213]||(e[213]=a('
julia
  pseudo_basis(O::RelNumFieldOrder{T, S}) -> Vector{Tuple{NumFieldElem{T}, S}}
',1)),t("p",null,[e[211]||(e[211]=s("Returns the pseudo-basis of ")),t("mjx-container",q1,[(l(),i("svg",K1,e[209]||(e[209]=[a('',1)]))),e[210]||(e[210]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[212]||(e[212]=s("."))]),e[214]||(e[214]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",P1,[t("summary",null,[e[215]||(e[215]=t("a",{id:"basis_pmatrix-Tuple{Hecke.RelNumFieldOrder}",href:"#basis_pmatrix-Tuple{Hecke.RelNumFieldOrder}"},[t("span",{class:"jlbinding"},"basis_pmatrix")],-1)),e[216]||(e[216]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[221]||(e[221]=a('
julia
  basis_pmatrix(O::RelNumFieldOrder) -> PMat
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julia
  basis_nf(O::RelNumFieldOrder) -> Vector{NumFieldElem}
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julia
  inv_coeff_ideals(O::RelNumFieldOrder{T, S}) -> Vector{S}
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julia
basis_matrix(O::AbsNumFieldOrder) -> QQMatrix
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julia
basis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat
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julia
gen_index(O::AbsSimpleNumFieldOrder) -> QQFieldElem
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julia
is_index_divisor(O::AbsSimpleNumFieldOrder, d::ZZRingElem) -> Bool
 is_index_divisor(O::AbsSimpleNumFieldOrder, d::Int) -> Bool
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julia
minkowski_matrix(O::AbsNumFieldOrder, abs_tol::Int = 64) -> ArbMatrix
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julia
in(a::NumFieldElem, O::NumFieldOrder) -> Bool
',1)),t("p",null,[e[304]||(e[304]=s("Checks whether ")),t("mjx-container",F3,[(l(),i("svg",L3,e[300]||(e[300]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),e[301]||(e[301]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"a")])],-1))]),e[305]||(e[305]=s(" lies in ")),t("mjx-container",C3,[(l(),i("svg",O3,e[302]||(e[302]=[a('',1)]))),e[303]||(e[303]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[306]||(e[306]=s("."))]),e[308]||(e[308]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",A3,[t("summary",null,[e[309]||(e[309]=t("a",{id:"norm_change_const-Tuple{AbsSimpleNumFieldOrder}",href:"#norm_change_const-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"norm_change_const")],-1)),e[310]||(e[310]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[333]||(e[333]=a('
julia
norm_change_const(O::AbsSimpleNumFieldOrder) -> (Float64, Float64)
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0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[332]||(e[332]=s("."))]),e[334]||(e[334]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",P3,[t("summary",null,[e[335]||(e[335]=t("a",{id:"trace_matrix-Tuple{AbsSimpleNumFieldOrder}",href:"#trace_matrix-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"trace_matrix")],-1)),e[336]||(e[336]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[344]||(e[344]=a('
julia
trace_matrix(O::AbsNumFieldOrder) -> ZZMatrix
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julia
+(R::AbsSimpleNumFieldOrder, S::AbsSimpleNumFieldOrder) -> AbsSimpleNumFieldOrder
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julia
poverorder(O::AbsSimpleNumFieldOrder, p::ZZRingElem) -> AbsSimpleNumFieldOrder
 poverorder(O::AbsSimpleNumFieldOrder, p::Integer) -> AbsSimpleNumFieldOrder
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julia
poverorders(O, p) -> Vector{Ord}
Returns all p-overorders of O, that is all overorders M, such that the index of O in M is a p-power.
source
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julia
pmaximal_overorder(O::AbsSimpleNumFieldOrder, p::ZZRingElem) -> AbsSimpleNumFieldOrder
-pmaximal_overorder(O::AbsSimpleNumFieldOrder, p::Integer) -> AbsSimpleNumFieldOrder
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julia
pradical(O::AbsSimpleNumFieldOrder, p::{ZZRingElem|Integer}) -> AbsNumFieldOrderIdeal
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julia
  pradical(O::RelNumFieldOrder, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> RelNumFieldOrderIdeal
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julia
ring_of_multipliers(I::AbsNumFieldOrderIdeal) -> AbsNumFieldOrder
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julia
discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
reduced_discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
degree(O::NumFieldOrder) -> Int
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julia
index(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
different(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal
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julia
codifferent(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
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julia
is_gorenstein(O::AbsSimpleNumFieldOrder) -> Bool
Return whether the order \\mathcal{O} is Gorenstein.
source
',3))]),t("details",K2,[t("summary",null,[e[544]||(e[544]=t("a",{id:"is_bass-Tuple{AbsSimpleNumFieldOrder}",href:"#is_bass-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"is_bass")],-1)),e[545]||(e[545]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[546]||(e[546]=a('
julia
is_bass(O::AbsSimpleNumFieldOrder) -> Bool
Return whether the order \\mathcal{O} is Bass.
source
',3))]),t("details",P2,[t("summary",null,[e[547]||(e[547]=t("a",{id:"is_equation_order-Tuple{AbsSimpleNumFieldOrder}",href:"#is_equation_order-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"is_equation_order")],-1)),e[548]||(e[548]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[556]||(e[556]=a('
julia
is_equation_order(O::NumFieldOrder) -> Bool
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julia
zeta_log_residue(O::AbsSimpleNumFieldOrder, error::Float64) -> ArbFieldElem
',1)),t("p",null,[e[564]||(e[564]=s("Computes the residue of the zeta function of ")),t("mjx-container",tt,[(l(),i("svg",et,e[560]||(e[560]=[a('',1)]))),e[561]||(e[561]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[565]||(e[565]=s(" at ")),t("mjx-container",st,[(l(),i("svg",it,e[562]||(e[562]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mn"},[t("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),e[563]||(e[563]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"1")])],-1))]),e[566]||(e[566]=s(". The output will be an element of type ")),e[567]||(e[567]=t("code",null,"ArbFieldElem",-1)),e[568]||(e[568]=s(" with radius less then ")),e[569]||(e[569]=t("code",null,"error",-1)),e[570]||(e[570]=s("."))]),e[572]||(e[572]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",lt,[t("summary",null,[e[573]||(e[573]=t("a",{id:"ramified_primes-Tuple{AbsSimpleNumFieldOrder}",href:"#ramified_primes-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"ramified_primes")],-1)),e[574]||(e[574]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[579]||(e[579]=a('
julia
ramified_primes(O::AbsNumFieldOrder) -> Vector{ZZRingElem}
',1)),t("p",null,[e[577]||(e[577]=s("Returns the list of prime numbers that divide ")),t("mjx-container",at,[(l(),i("svg",nt,e[575]||(e[575]=[a('',1)]))),e[576]||(e[576]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"disc"),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")]),t("mo",{stretchy:"false"},")")])],-1))]),e[578]||(e[578]=s("."))]),e[580]||(e[580]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e[607]||(e[607]=a('
Arithmetic ​
Progress and intermediate results of the functions mentioned here can be obtained via verbose_level, supported are
  • ClassGroup
  • UnitGroup
All of the functions have a very similar interface: they return an abelian group and a map converting elements of the group into the objects required. The maps also allow a point-wise inverse to server as the discrete logarithm map. For more information on abelian groups, see here, for ideals, here.
For the processing of units, there are a couple of helper functions also available:
',6)),t("details",ot,[t("summary",null,[e[581]||(e[581]=t("a",{id:"is_independent",href:"#is_independent"},[t("span",{class:"jlbinding"},"is_independent")],-1)),e[582]||(e[582]=s()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[583]||(e[583]=a('
julia
is_independent{T}(x::Vector{T})
Given an array of non-zero units in a number field, returns whether they are multiplicatively independent.
source
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julia
is_contained(R::AbsNumFieldOrder, S::AbsNumFieldOrder) -> Bool
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julia
is_maximal(R::AbsNumFieldOrder) -> Bool
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+pmaximal_overorder(O::AbsSimpleNumFieldOrder, p::Integer) -> AbsSimpleNumFieldOrder
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julia
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julia
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julia
ring_of_multipliers(I::AbsNumFieldOrderIdeal) -> AbsNumFieldOrder
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julia
discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
reduced_discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
degree(O::NumFieldOrder) -> Int
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julia
index(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
different(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal
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julia
codifferent(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
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julia
is_gorenstein(O::AbsSimpleNumFieldOrder) -> Bool
Return whether the order \\mathcal{O} is Gorenstein.
source
',3))]),t("details",K2,[t("summary",null,[e[544]||(e[544]=t("a",{id:"is_bass-Tuple{AbsSimpleNumFieldOrder}",href:"#is_bass-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"is_bass")],-1)),e[545]||(e[545]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[546]||(e[546]=a('
julia
is_bass(O::AbsSimpleNumFieldOrder) -> Bool
Return whether the order \\mathcal{O} is Bass.
source
',3))]),t("details",P2,[t("summary",null,[e[547]||(e[547]=t("a",{id:"is_equation_order-Tuple{AbsSimpleNumFieldOrder}",href:"#is_equation_order-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"is_equation_order")],-1)),e[548]||(e[548]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[556]||(e[556]=a('
julia
is_equation_order(O::NumFieldOrder) -> Bool
',1)),t("p",null,[e[553]||(e[553]=s("Returns whether ")),t("mjx-container",U2,[(l(),i("svg",$2,e[549]||(e[549]=[a('',1)]))),e[550]||(e[550]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[554]||(e[554]=s(" is the equation order of the ambient number field ")),t("mjx-container",W2,[(l(),i("svg",Y2,e[551]||(e[551]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43E",d:"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z",style:{"stroke-width":"3"}})])])],-1)]))),e[552]||(e[552]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"K")])],-1))]),e[555]||(e[555]=s("."))]),e[557]||(e[557]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",_2,[t("summary",null,[e[558]||(e[558]=t("a",{id:"zeta_log_residue-Tuple{AbsSimpleNumFieldOrder, Float64}",href:"#zeta_log_residue-Tuple{AbsSimpleNumFieldOrder, Float64}"},[t("span",{class:"jlbinding"},"zeta_log_residue")],-1)),e[559]||(e[559]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[571]||(e[571]=a('
julia
zeta_log_residue(O::AbsSimpleNumFieldOrder, error::Float64) -> ArbFieldElem
',1)),t("p",null,[e[564]||(e[564]=s("Computes the residue of the zeta function of ")),t("mjx-container",tt,[(l(),i("svg",et,e[560]||(e[560]=[a('',1)]))),e[561]||(e[561]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[565]||(e[565]=s(" at ")),t("mjx-container",st,[(l(),i("svg",it,e[562]||(e[562]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mn"},[t("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),e[563]||(e[563]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"1")])],-1))]),e[566]||(e[566]=s(". The output will be an element of type ")),e[567]||(e[567]=t("code",null,"ArbFieldElem",-1)),e[568]||(e[568]=s(" with radius less then ")),e[569]||(e[569]=t("code",null,"error",-1)),e[570]||(e[570]=s("."))]),e[572]||(e[572]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",lt,[t("summary",null,[e[573]||(e[573]=t("a",{id:"ramified_primes-Tuple{AbsSimpleNumFieldOrder}",href:"#ramified_primes-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"ramified_primes")],-1)),e[574]||(e[574]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[579]||(e[579]=a('
julia
ramified_primes(O::AbsNumFieldOrder) -> Vector{ZZRingElem}
',1)),t("p",null,[e[577]||(e[577]=s("Returns the list of prime numbers that divide ")),t("mjx-container",at,[(l(),i("svg",nt,e[575]||(e[575]=[a('',1)]))),e[576]||(e[576]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"disc"),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")]),t("mo",{stretchy:"false"},")")])],-1))]),e[578]||(e[578]=s("."))]),e[580]||(e[580]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e[607]||(e[607]=a('
Arithmetic ​
Progress and intermediate results of the functions mentioned here can be obtained via verbose_level, supported are
  • ClassGroup
  • UnitGroup
All of the functions have a very similar interface: they return an abelian group and a map converting elements of the group into the objects required. The maps also allow a point-wise inverse to server as the discrete logarithm map. For more information on abelian groups, see here, for ideals, here.
For the processing of units, there are a couple of helper functions also available:
',6)),t("details",ot,[t("summary",null,[e[581]||(e[581]=t("a",{id:"is_independent",href:"#is_independent"},[t("span",{class:"jlbinding"},"is_independent")],-1)),e[582]||(e[582]=s()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[583]||(e[583]=a('
julia
is_independent{T}(x::Vector{T})
Given an array of non-zero units in a number field, returns whether they are multiplicatively independent.
source
',3))]),e[608]||(e[608]=t("h2",{id:"predicates",tabindex:"-1"},[s("Predicates "),t("a",{class:"header-anchor",href:"#predicates","aria-label":'Permalink to "Predicates"'},"​")],-1)),t("details",rt,[t("summary",null,[e[584]||(e[584]=t("a",{id:"is_contained-Tuple{AbsNumFieldOrder, AbsNumFieldOrder}",href:"#is_contained-Tuple{AbsNumFieldOrder, AbsNumFieldOrder}"},[t("span",{class:"jlbinding"},"is_contained")],-1)),e[585]||(e[585]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[593]||(e[593]=a('
julia
is_contained(R::AbsNumFieldOrder, S::AbsNumFieldOrder) -> Bool
',1)),t("p",null,[e[590]||(e[590]=s("Checks if ")),t("mjx-container",dt,[(l(),i("svg",Qt,e[586]||(e[586]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D445",d:"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z",style:{"stroke-width":"3"}})])])],-1)]))),e[587]||(e[587]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"R")])],-1))]),e[591]||(e[591]=s(" is contained in ")),t("mjx-container",pt,[(l(),i("svg",mt,e[588]||(e[588]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D446",d:"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z",style:{"stroke-width":"3"}})])])],-1)]))),e[589]||(e[589]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"S")])],-1))]),e[592]||(e[592]=s("."))]),e[594]||(e[594]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Tt,[t("summary",null,[e[595]||(e[595]=t("a",{id:"is_maximal-Tuple{AbsNumFieldOrder}",href:"#is_maximal-Tuple{AbsNumFieldOrder}"},[t("span",{class:"jlbinding"},"is_maximal")],-1)),e[596]||(e[596]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[601]||(e[601]=a('
julia
is_maximal(R::AbsNumFieldOrder) -> Bool
',1)),t("p",null,[e[599]||(e[599]=s("Tests if the order ")),t("mjx-container",ht,[(l(),i("svg",gt,e[597]||(e[597]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D445",d:"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z",style:{"stroke-width":"3"}})])])],-1)]))),e[598]||(e[598]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"R")])],-1))]),e[600]||(e[600]=s(" is maximal. This might trigger the computation of the maximal order."))]),e[602]||(e[602]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))])])}const vt=r(Q,[["render",ut]]);export{ft as __pageData,vt as default};
diff --git a/v0.34.8/assets/manual_orders_orders.md.Bk8W8VLT.lean.js b/v0.34.8/assets/manual_orders_orders.md.Bk8W8VLT.lean.js
index cff9d10581..2edd1383af 100644
--- a/v0.34.8/assets/manual_orders_orders.md.Bk8W8VLT.lean.js
+++ b/v0.34.8/assets/manual_orders_orders.md.Bk8W8VLT.lean.js
@@ -13,4 +13,4 @@ import{_ as r,c as i,j as t,a as s,a5 as a,G as o,B as d,o as l}from"./chunks/fr
 with Z-basis AbsSimpleNumFieldOrderElem[1, a]
`,2)),t("details",w1,[t("summary",null,[e[145]||(e[145]=t("a",{id:"parent-Tuple{AbsSimpleNumFieldOrder}",href:"#parent-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"parent")],-1)),e[146]||(e[146]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[151]||(e[151]=a('
julia
parent(O::AbsNumFieldOrder) -> AbsNumFieldOrderSet
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julia
signature(O::NumFieldOrder) -> Tuple{Int, Int}
',1)),t("p",null,[e[157]||(e[157]=s("Returns the signature of the ambient number field of ")),t("mjx-container",f1,[(l(),i("svg",v1,e[155]||(e[155]=[a('',1)]))),e[156]||(e[156]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[158]||(e[158]=s("."))]),e[160]||(e[160]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",E1,[t("summary",null,[e[161]||(e[161]=t("a",{id:"nf-Tuple{AbsSimpleNumFieldOrder}",href:"#nf-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"nf")],-1)),e[162]||(e[162]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[167]||(e[167]=a('
julia
nf(O::NumFieldOrder) -> NumField
',1)),t("p",null,[e[165]||(e[165]=s("Returns the ambient number field of ")),t("mjx-container",H1,[(l(),i("svg",M1,e[163]||(e[163]=[a('',1)]))),e[164]||(e[164]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[166]||(e[166]=s("."))]),e[168]||(e[168]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",j1,[t("summary",null,[e[169]||(e[169]=t("a",{id:"basis-Tuple{AbsSimpleNumFieldOrder}",href:"#basis-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"basis")],-1)),e[170]||(e[170]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[185]||(e[185]=a('
julia
basis(O::AbsNumFieldOrder) -> Vector{AbsNumFieldOrderElem}
',1)),t("p",null,[e[175]||(e[175]=s("Returns the ")),t("mjx-container",F1,[(l(),i("svg",L1,e[171]||(e[171]=[a('',1)]))),e[172]||(e[172]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"Z")])])],-1))]),e[176]||(e[176]=s("-basis of ")),t("mjx-container",C1,[(l(),i("svg",O1,e[173]||(e[173]=[a('',1)]))),e[174]||(e[174]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[177]||(e[177]=s("."))]),e[186]||(e[186]=a('
source
julia
basis(I::AbsNumFieldOrderFractionalIdeal) -> Vector{AbsSimpleNumFieldElem}
',2)),t("p",null,[e[182]||(e[182]=s("Returns the ")),t("mjx-container",A1,[(l(),i("svg",D1,e[178]||(e[178]=[a('',1)]))),e[179]||(e[179]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"Z")])])],-1))]),e[183]||(e[183]=s("-basis of ")),t("mjx-container",Z1,[(l(),i("svg",V1,e[180]||(e[180]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D43C",d:"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z",style:{"stroke-width":"3"}})])])],-1)]))),e[181]||(e[181]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"I")])],-1))]),e[184]||(e[184]=s("."))]),e[187]||(e[187]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",B1,[t("summary",null,[e[188]||(e[188]=t("a",{id:"lll_basis-Tuple{AbsSimpleNumFieldOrder}",href:"#lll_basis-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"lll_basis")],-1)),e[189]||(e[189]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[194]||(e[194]=a('
julia
lll_basis(M::NumFieldOrder) -> Vector{NumFieldElem}
',1)),t("p",null,[e[192]||(e[192]=s("A basis for ")),t("mjx-container",S1,[(l(),i("svg",R1,e[190]||(e[190]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),e[191]||(e[191]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"M")])],-1))]),e[193]||(e[193]=s(" that is reduced using the LLL algorithm for the Minkowski metric."))]),e[195]||(e[195]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",N1,[t("summary",null,[e[196]||(e[196]=t("a",{id:"basis-Tuple{AbsSimpleNumFieldOrder, AbsSimpleNumField}",href:"#basis-Tuple{AbsSimpleNumFieldOrder, AbsSimpleNumField}"},[t("span",{class:"jlbinding"},"basis")],-1)),e[197]||(e[197]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[205]||(e[205]=a('
julia
basis(O::AbsSimpleNumFieldOrder, K::AbsSimpleNumField) -> Vector{AbsSimpleNumFieldElem}
',1)),t("p",null,[e[202]||(e[202]=s("Returns the ")),t("mjx-container",G1,[(l(),i("svg",z1,e[198]||(e[198]=[a('',1)]))),e[199]||(e[199]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"bold"},"Z")])])],-1))]),e[203]||(e[203]=s("-basis elements of ")),t("mjx-container",J1,[(l(),i("svg",X1,e[200]||(e[200]=[a('',1)]))),e[201]||(e[201]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[204]||(e[204]=s(" as elements of the ambient number field."))]),e[206]||(e[206]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",I1,[t("summary",null,[e[207]||(e[207]=t("a",{id:"pseudo_basis-Tuple{Hecke.RelNumFieldOrder}",href:"#pseudo_basis-Tuple{Hecke.RelNumFieldOrder}"},[t("span",{class:"jlbinding"},"pseudo_basis")],-1)),e[208]||(e[208]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[213]||(e[213]=a('
julia
  pseudo_basis(O::RelNumFieldOrder{T, S}) -> Vector{Tuple{NumFieldElem{T}, S}}
',1)),t("p",null,[e[211]||(e[211]=s("Returns the pseudo-basis of ")),t("mjx-container",q1,[(l(),i("svg",K1,e[209]||(e[209]=[a('',1)]))),e[210]||(e[210]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[212]||(e[212]=s("."))]),e[214]||(e[214]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",P1,[t("summary",null,[e[215]||(e[215]=t("a",{id:"basis_pmatrix-Tuple{Hecke.RelNumFieldOrder}",href:"#basis_pmatrix-Tuple{Hecke.RelNumFieldOrder}"},[t("span",{class:"jlbinding"},"basis_pmatrix")],-1)),e[216]||(e[216]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[221]||(e[221]=a('
julia
  basis_pmatrix(O::RelNumFieldOrder) -> PMat
',1)),t("p",null,[e[219]||(e[219]=s("Returns the basis pseudo-matrix of ")),t("mjx-container",U1,[(l(),i("svg",$1,e[217]||(e[217]=[a('',1)]))),e[218]||(e[218]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[220]||(e[220]=s(" with respect to the power basis of the ambient number field."))]),e[222]||(e[222]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",W1,[t("summary",null,[e[223]||(e[223]=t("a",{id:"basis_nf-Tuple{Hecke.RelNumFieldOrder}",href:"#basis_nf-Tuple{Hecke.RelNumFieldOrder}"},[t("span",{class:"jlbinding"},"basis_nf")],-1)),e[224]||(e[224]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[229]||(e[229]=a('
julia
  basis_nf(O::RelNumFieldOrder) -> Vector{NumFieldElem}
',1)),t("p",null,[e[227]||(e[227]=s("Returns the elements of the pseudo-basis of ")),t("mjx-container",Y1,[(l(),i("svg",_1,e[225]||(e[225]=[a('',1)]))),e[226]||(e[226]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[228]||(e[228]=s(" as elements of the ambient number field."))]),e[230]||(e[230]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",t3,[t("summary",null,[e[231]||(e[231]=t("a",{id:"inv_coeff_ideals-Tuple{Hecke.RelNumFieldOrder}",href:"#inv_coeff_ideals-Tuple{Hecke.RelNumFieldOrder}"},[t("span",{class:"jlbinding"},"inv_coeff_ideals")],-1)),e[232]||(e[232]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[237]||(e[237]=a('
julia
  inv_coeff_ideals(O::RelNumFieldOrder{T, S}) -> Vector{S}
',1)),t("p",null,[e[235]||(e[235]=s("Returns the inverses of the coefficient ideals of the pseudo basis of ")),t("mjx-container",e3,[(l(),i("svg",s3,e[233]||(e[233]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D442",d:"M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435ZM637 476Q637 565 591 615T476 665Q396 665 322 605Q242 542 200 428T157 216Q157 126 200 73T314 19Q404 19 485 98T608 313Q637 408 637 476Z",style:{"stroke-width":"3"}})])])],-1)]))),e[234]||(e[234]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"O")])],-1))]),e[236]||(e[236]=s("."))]),e[238]||(e[238]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",i3,[t("summary",null,[e[239]||(e[239]=t("a",{id:"basis_matrix-Tuple{AbsNumFieldOrder}",href:"#basis_matrix-Tuple{AbsNumFieldOrder}"},[t("span",{class:"jlbinding"},"basis_matrix")],-1)),e[240]||(e[240]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[245]||(e[245]=a('
julia
basis_matrix(O::AbsNumFieldOrder) -> QQMatrix
',1)),t("p",null,[e[243]||(e[243]=s("Returns the basis matrix of ")),t("mjx-container",l3,[(l(),i("svg",a3,e[241]||(e[241]=[a('',1)]))),e[242]||(e[242]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[244]||(e[244]=s(" with respect to the basis of the ambient number field."))]),e[246]||(e[246]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",n3,[t("summary",null,[e[247]||(e[247]=t("a",{id:"basis_mat_inv-Tuple{AbsSimpleNumFieldOrder}",href:"#basis_mat_inv-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"basis_mat_inv")],-1)),e[248]||(e[248]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[253]||(e[253]=a('
julia
basis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat
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julia
gen_index(O::AbsSimpleNumFieldOrder) -> QQFieldElem
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julia
is_index_divisor(O::AbsSimpleNumFieldOrder, d::ZZRingElem) -> Bool
 is_index_divisor(O::AbsSimpleNumFieldOrder, d::Int) -> Bool
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julia
minkowski_matrix(O::AbsNumFieldOrder, abs_tol::Int = 64) -> ArbMatrix
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julia
in(a::NumFieldElem, O::NumFieldOrder) -> Bool
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julia
norm_change_const(O::AbsSimpleNumFieldOrder) -> (Float64, Float64)
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trace_matrix(O::AbsNumFieldOrder) -> ZZMatrix
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julia
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julia
poverorder(O::AbsSimpleNumFieldOrder, p::ZZRingElem) -> AbsSimpleNumFieldOrder
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julia
poverorders(O, p) -> Vector{Ord}
Returns all p-overorders of O, that is all overorders M, such that the index of O in M is a p-power.
source
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julia
pmaximal_overorder(O::AbsSimpleNumFieldOrder, p::ZZRingElem) -> AbsSimpleNumFieldOrder
-pmaximal_overorder(O::AbsSimpleNumFieldOrder, p::Integer) -> AbsSimpleNumFieldOrder
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julia
pradical(O::AbsSimpleNumFieldOrder, p::{ZZRingElem|Integer}) -> AbsNumFieldOrderIdeal
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julia
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julia
ring_of_multipliers(I::AbsNumFieldOrderIdeal) -> AbsNumFieldOrder
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julia
discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
reduced_discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
degree(O::NumFieldOrder) -> Int
',1)),t("p",null,[e[501]||(e[501]=s("Returns the degree of ")),t("mjx-container",M2,[(l(),i("svg",j2,e[499]||(e[499]=[a('',1)]))),e[500]||(e[500]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[502]||(e[502]=s("."))]),e[504]||(e[504]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",F2,[t("summary",null,[e[505]||(e[505]=t("a",{id:"index-Tuple{AbsSimpleNumFieldOrder}",href:"#index-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"index")],-1)),e[506]||(e[506]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[517]||(e[517]=a('
julia
index(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
different(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal
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julia
codifferent(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
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julia
is_gorenstein(O::AbsSimpleNumFieldOrder) -> Bool
Return whether the order \\mathcal{O} is Gorenstein.
source
',3))]),t("details",K2,[t("summary",null,[e[544]||(e[544]=t("a",{id:"is_bass-Tuple{AbsSimpleNumFieldOrder}",href:"#is_bass-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"is_bass")],-1)),e[545]||(e[545]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[546]||(e[546]=a('
julia
is_bass(O::AbsSimpleNumFieldOrder) -> Bool
Return whether the order \\mathcal{O} is Bass.
source
',3))]),t("details",P2,[t("summary",null,[e[547]||(e[547]=t("a",{id:"is_equation_order-Tuple{AbsSimpleNumFieldOrder}",href:"#is_equation_order-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"is_equation_order")],-1)),e[548]||(e[548]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[556]||(e[556]=a('
julia
is_equation_order(O::NumFieldOrder) -> Bool
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julia
zeta_log_residue(O::AbsSimpleNumFieldOrder, error::Float64) -> ArbFieldElem
',1)),t("p",null,[e[564]||(e[564]=s("Computes the residue of the zeta function of ")),t("mjx-container",tt,[(l(),i("svg",et,e[560]||(e[560]=[a('',1)]))),e[561]||(e[561]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[565]||(e[565]=s(" at ")),t("mjx-container",st,[(l(),i("svg",it,e[562]||(e[562]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mn"},[t("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),e[563]||(e[563]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"1")])],-1))]),e[566]||(e[566]=s(". The output will be an element of type ")),e[567]||(e[567]=t("code",null,"ArbFieldElem",-1)),e[568]||(e[568]=s(" with radius less then ")),e[569]||(e[569]=t("code",null,"error",-1)),e[570]||(e[570]=s("."))]),e[572]||(e[572]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",lt,[t("summary",null,[e[573]||(e[573]=t("a",{id:"ramified_primes-Tuple{AbsSimpleNumFieldOrder}",href:"#ramified_primes-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"ramified_primes")],-1)),e[574]||(e[574]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[579]||(e[579]=a('
julia
ramified_primes(O::AbsNumFieldOrder) -> Vector{ZZRingElem}
',1)),t("p",null,[e[577]||(e[577]=s("Returns the list of prime numbers that divide ")),t("mjx-container",at,[(l(),i("svg",nt,e[575]||(e[575]=[a('',1)]))),e[576]||(e[576]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"disc"),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")]),t("mo",{stretchy:"false"},")")])],-1))]),e[578]||(e[578]=s("."))]),e[580]||(e[580]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e[607]||(e[607]=a('
Arithmetic ​
Progress and intermediate results of the functions mentioned here can be obtained via verbose_level, supported are
  • ClassGroup
  • UnitGroup
All of the functions have a very similar interface: they return an abelian group and a map converting elements of the group into the objects required. The maps also allow a point-wise inverse to server as the discrete logarithm map. For more information on abelian groups, see here, for ideals, here.
For the processing of units, there are a couple of helper functions also available:
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julia
is_independent{T}(x::Vector{T})
Given an array of non-zero units in a number field, returns whether they are multiplicatively independent.
source
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julia
is_contained(R::AbsNumFieldOrder, S::AbsNumFieldOrder) -> Bool
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julia
is_maximal(R::AbsNumFieldOrder) -> Bool
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julia
pradical(O::AbsSimpleNumFieldOrder, p::{ZZRingElem|Integer}) -> AbsNumFieldOrderIdeal
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julia
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julia
ring_of_multipliers(I::AbsNumFieldOrderIdeal) -> AbsNumFieldOrder
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julia
discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
reduced_discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
degree(O::NumFieldOrder) -> Int
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julia
index(O::AbsSimpleNumFieldOrder) -> ZZRingElem
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julia
different(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal
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julia
codifferent(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
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julia
is_gorenstein(O::AbsSimpleNumFieldOrder) -> Bool

Return whether the order \\mathcal{O} is Gorenstein.

source

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julia
is_bass(O::AbsSimpleNumFieldOrder) -> Bool

Return whether the order \\mathcal{O} is Bass.

source

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julia
is_equation_order(O::NumFieldOrder) -> Bool
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julia
zeta_log_residue(O::AbsSimpleNumFieldOrder, error::Float64) -> ArbFieldElem
',1)),t("p",null,[e[564]||(e[564]=s("Computes the residue of the zeta function of ")),t("mjx-container",tt,[(l(),i("svg",et,e[560]||(e[560]=[a('',1)]))),e[561]||(e[561]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")])])],-1))]),e[565]||(e[565]=s(" at ")),t("mjx-container",st,[(l(),i("svg",it,e[562]||(e[562]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mn"},[t("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),e[563]||(e[563]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"1")])],-1))]),e[566]||(e[566]=s(". The output will be an element of type ")),e[567]||(e[567]=t("code",null,"ArbFieldElem",-1)),e[568]||(e[568]=s(" with radius less then ")),e[569]||(e[569]=t("code",null,"error",-1)),e[570]||(e[570]=s("."))]),e[572]||(e[572]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",lt,[t("summary",null,[e[573]||(e[573]=t("a",{id:"ramified_primes-Tuple{AbsSimpleNumFieldOrder}",href:"#ramified_primes-Tuple{AbsSimpleNumFieldOrder}"},[t("span",{class:"jlbinding"},"ramified_primes")],-1)),e[574]||(e[574]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[579]||(e[579]=a('
julia
ramified_primes(O::AbsNumFieldOrder) -> Vector{ZZRingElem}
',1)),t("p",null,[e[577]||(e[577]=s("Returns the list of prime numbers that divide ")),t("mjx-container",at,[(l(),i("svg",nt,e[575]||(e[575]=[a('',1)]))),e[576]||(e[576]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"disc"),t("mo",{stretchy:"false"},"("),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{"data-mjx-variant":"-tex-calligraphic",mathvariant:"script"},"O")]),t("mo",{stretchy:"false"},")")])],-1))]),e[578]||(e[578]=s("."))]),e[580]||(e[580]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),e[607]||(e[607]=a('

Arithmetic ​

Progress and intermediate results of the functions mentioned here can be obtained via verbose_level, supported are

  • ClassGroup

  • UnitGroup

All of the functions have a very similar interface: they return an abelian group and a map converting elements of the group into the objects required. The maps also allow a point-wise inverse to server as the discrete logarithm map. For more information on abelian groups, see here, for ideals, here.

For the processing of units, there are a couple of helper functions also available:

',6)),t("details",ot,[t("summary",null,[e[581]||(e[581]=t("a",{id:"is_independent",href:"#is_independent"},[t("span",{class:"jlbinding"},"is_independent")],-1)),e[582]||(e[582]=s()),o(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),e[583]||(e[583]=a('
julia
is_independent{T}(x::Vector{T})

Given an array of non-zero units in a number field, returns whether they are multiplicatively independent.

source

',3))]),e[608]||(e[608]=t("h2",{id:"predicates",tabindex:"-1"},[s("Predicates "),t("a",{class:"header-anchor",href:"#predicates","aria-label":'Permalink to "Predicates"'},"​")],-1)),t("details",rt,[t("summary",null,[e[584]||(e[584]=t("a",{id:"is_contained-Tuple{AbsNumFieldOrder, AbsNumFieldOrder}",href:"#is_contained-Tuple{AbsNumFieldOrder, AbsNumFieldOrder}"},[t("span",{class:"jlbinding"},"is_contained")],-1)),e[585]||(e[585]=s()),o(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),e[593]||(e[593]=a('
julia
is_contained(R::AbsNumFieldOrder, S::AbsNumFieldOrder) -> Bool
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julia
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")),t("mjx-container",E,[(o(),n("svg",v,s[12]||(s[12]=[i('',1)]))),s[13]||(s[13]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"M"),t("mo",null,"⊗"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"R")]),t("mo",null,"≅"),t("mi",null,"N"),t("mo",null,"⊗"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"R")])])],-1))]),s[21]||(s[21]=e("."))]),t("p",null,[s[24]||(s[24]=e("The genus of a ")),t("mjx-container",w,[(o(),n("svg",Z,s[22]||(s[22]=[i('',1)]))),s[23]||(s[23]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[25]||(s[25]=e("-lattice is encoded in its Conway-Sloane genus symbol. 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julia
ZZGenus

A collection of local genus symbols (at primes) and a signature pair. Together they represent the genus of a non-degenerate integer_lattice.

source

',3))]),s[318]||(s[318]=t("h2",{id:"Creation-of-Genera",tabindex:"-1"},[e("Creation of Genera "),t("a",{class:"header-anchor",href:"#Creation-of-Genera","aria-label":'Permalink to "Creation of Genera {#Creation-of-Genera}"'},"​")],-1)),s[319]||(s[319]=t("h3",{id:"From-an-integral-Lattice",tabindex:"-1"},[e("From an integral Lattice "),t("a",{class:"header-anchor",href:"#From-an-integral-Lattice","aria-label":'Permalink to "From an integral Lattice {#From-an-integral-Lattice}"'},"​")],-1)),t("details",M,[t("summary",null,[s[37]||(s[37]=t("a",{id:"genus-Tuple{ZZLat}",href:"#genus-Tuple{ZZLat}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[38]||(s[38]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[39]||(s[39]=i('
julia
genus(L::ZZLat) -> ZZGenus

Return the genus of the lattice L.

source

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julia
genus(A::MatElem) -> ZZGenus
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julia
integer_genera(sig_pair::Vector{Int}, determinant::RationalUnion;
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")),t("mjx-container",E,[(o(),n("svg",v,s[12]||(s[12]=[i('',1)]))),s[13]||(s[13]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"M"),t("mo",null,"⊗"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"R")]),t("mo",null,"≅"),t("mi",null,"N"),t("mo",null,"⊗"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"R")])])],-1))]),s[21]||(s[21]=e("."))]),t("p",null,[s[24]||(s[24]=e("The genus of a ")),t("mjx-container",w,[(o(),n("svg",Z,s[22]||(s[22]=[i('',1)]))),s[23]||(s[23]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[25]||(s[25]=e("-lattice is encoded in its Conway-Sloane genus symbol. The genus symbol itself is a collection of its local genus symbols. See [")),s[26]||(s[26]=t("a",{href:"/v0.34.8/references#CS99"},"5",-1)),s[27]||(s[27]=e("] Chapter 15 for the definitions. Note that genera for non-integral lattices are supported."))]),t("p",null,[s[30]||(s[30]=e("The class ")),s[31]||(s[31]=t("code",null,"ZZGenus",-1)),s[32]||(s[32]=e(" supports genera of ")),t("mjx-container",j,[(o(),n("svg",L,s[28]||(s[28]=[i('',1)]))),s[29]||(s[29]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[33]||(s[33]=e("-lattices."))]),t("details",C,[t("summary",null,[s[34]||(s[34]=t("a",{id:"ZZGenus",href:"#ZZGenus"},[t("span",{class:"jlbinding"},"ZZGenus")],-1)),s[35]||(s[35]=e()),l(a,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[36]||(s[36]=i('
julia
ZZGenus
A collection of local genus symbols (at primes) and a signature pair. Together they represent the genus of a non-degenerate integer_lattice.
source
',3))]),s[318]||(s[318]=t("h2",{id:"Creation-of-Genera",tabindex:"-1"},[e("Creation of Genera "),t("a",{class:"header-anchor",href:"#Creation-of-Genera","aria-label":'Permalink to "Creation of Genera {#Creation-of-Genera}"'},"​")],-1)),s[319]||(s[319]=t("h3",{id:"From-an-integral-Lattice",tabindex:"-1"},[e("From an integral Lattice "),t("a",{class:"header-anchor",href:"#From-an-integral-Lattice","aria-label":'Permalink to "From an integral Lattice {#From-an-integral-Lattice}"'},"​")],-1)),t("details",M,[t("summary",null,[s[37]||(s[37]=t("a",{id:"genus-Tuple{ZZLat}",href:"#genus-Tuple{ZZLat}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[38]||(s[38]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[39]||(s[39]=i('
julia
genus(L::ZZLat) -> ZZGenus
Return the genus of the lattice L.
source
',3))]),s[320]||(s[320]=t("h3",{id:"From-a-gram-matrix",tabindex:"-1"},[e("From a gram matrix "),t("a",{class:"header-anchor",href:"#From-a-gram-matrix","aria-label":'Permalink to "From a gram matrix {#From-a-gram-matrix}"'},"​")],-1)),t("details",H,[t("summary",null,[s[40]||(s[40]=t("a",{id:"genus-Tuple{MatElem}",href:"#genus-Tuple{MatElem}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[41]||(s[41]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[48]||(s[48]=i('
julia
genus(A::MatElem) -> ZZGenus
',1)),t("p",null,[s[44]||(s[44]=e("Return the genus of a ")),t("mjx-container",F,[(o(),n("svg",G,s[42]||(s[42]=[i('',1)]))),s[43]||(s[43]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[45]||(s[45]=e("-lattice with gram matrix ")),s[46]||(s[46]=t("code",null,"A",-1)),s[47]||(s[47]=e("."))]),s[49]||(s[49]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[321]||(s[321]=t("h3",{id:"Enumeration-of-genus-symbols",tabindex:"-1"},[e("Enumeration of genus symbols "),t("a",{class:"header-anchor",href:"#Enumeration-of-genus-symbols","aria-label":'Permalink to "Enumeration of genus symbols {#Enumeration-of-genus-symbols}"'},"​")],-1)),t("details",D,[t("summary",null,[s[50]||(s[50]=t("a",{id:"integer_genera-Tuple{Tuple{Int64, Int64}, Union{Int64, ZZRingElem}}",href:"#integer_genera-Tuple{Tuple{Int64, Int64}, Union{Int64, ZZRingElem}}"},[t("span",{class:"jlbinding"},"integer_genera")],-1)),s[51]||(s[51]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[56]||(s[56]=i(`
julia
integer_genera(sig_pair::Vector{Int}, determinant::RationalUnion;
        min_scale::RationalUnion = min(one(QQ), QQ(abs(determinant))),
        max_scale::RationalUnion = max(one(QQ), QQ(abs(determinant))),
-       even=false)                                         -> Vector{ZZGenus}
`,1)),t("p",null,[s[54]||(s[54]=e("Return a list of all genera with the given conditions. Genera of non-integral ")),t("mjx-container",B,[(o(),n("svg",A,s[52]||(s[52]=[i('',1)]))),s[53]||(s[53]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[55]||(s[55]=e("-lattices are also supported."))]),s[57]||(s[57]=i('
Arguments
  • sig_pair: a pair of non-negative integers giving the signature
  • determinant: a rational number; the sign is ignored
  • min_scale: a rational number; return only genera whose scale is an integer multiple of min_scale (default: min(one(QQ), QQ(abs(determinant))))
  • max_scale: a rational number; return only genera such that max_scale is an integer multiple of the scale (default: max(one(QQ), QQ(abs(determinant))))
  • even: boolean; if set to true, return only the even genera (default: false)
source
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julia
direct_sum(G1::ZZGenus, G2::ZZGenus) -> ZZGenus
Return the genus of the direct sum of G1 and G2.
The direct sum is defined via representatives.
source
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julia
dim(G::ZZGenus) -> Int
Return the dimension of this genus.
source
',3))]),t("details",O,[t("summary",null,[s[64]||(s[64]=t("a",{id:"rank-Tuple{ZZGenus}",href:"#rank-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"rank")],-1)),s[65]||(s[65]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[66]||(s[66]=i('
julia
rank(G::ZZGenus) -> Int
Return the rank of a (representative of) the genus G.
source
',3))]),t("details",S,[t("summary",null,[s[67]||(s[67]=t("a",{id:"signature-Tuple{ZZGenus}",href:"#signature-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"signature")],-1)),s[68]||(s[68]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[69]||(s[69]=i('
julia
signature(G::ZZGenus) -> Int
Return the signature of this genus.
The signature is p - n where p is the number of positive eigenvalues and n the number of negative eigenvalues.
source
',4))]),t("details",I,[t("summary",null,[s[70]||(s[70]=t("a",{id:"det-Tuple{ZZGenus}",href:"#det-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"det")],-1)),s[71]||(s[71]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[72]||(s[72]=i('
julia
det(G::ZZGenus) -> QQFieldElem
Return the determinant of this genus.
source
',3))]),t("details",z,[t("summary",null,[s[73]||(s[73]=t("a",{id:"iseven-Tuple{ZZGenus}",href:"#iseven-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"iseven")],-1)),s[74]||(s[74]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[75]||(s[75]=i('
julia
iseven(G::ZZGenus) -> Bool
Return if this genus is even.
source
',3))]),t("details",J,[t("summary",null,[s[76]||(s[76]=t("a",{id:"is_definite-Tuple{ZZGenus}",href:"#is_definite-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_definite")],-1)),s[77]||(s[77]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[78]||(s[78]=i('
julia
is_definite(G::ZZGenus) -> Bool
Return if this genus is definite.
source
',3))]),t("details",P,[t("summary",null,[s[79]||(s[79]=t("a",{id:"level-Tuple{ZZGenus}",href:"#level-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"level")],-1)),s[80]||(s[80]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[81]||(s[81]=i('
julia
level(G::ZZGenus) -> QQFieldElem
Return the level of this genus.
This is the denominator of the inverse gram matrix of a representative.
source
',4))]),t("details",U,[t("summary",null,[s[82]||(s[82]=t("a",{id:"scale-Tuple{ZZGenus}",href:"#scale-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"scale")],-1)),s[83]||(s[83]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[84]||(s[84]=i('
julia
scale(G::ZZGenus) -> QQFieldElem
Return the scale of this genus.
Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).
source
',4))]),t("details",X,[t("summary",null,[s[85]||(s[85]=t("a",{id:"norm-Tuple{ZZGenus}",href:"#norm-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"norm")],-1)),s[86]||(s[86]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[97]||(s[97]=i('
julia
norm(G::ZZGenus) -> QQFieldElem
Return the norm of this genus.
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julia
primes(G::ZZGenus) -> Vector{ZZRingElem}
Return the list of primes of the local symbols of G.
Note that 2 is always in the output since the 2-adic symbol of a ZZGenus is, by convention, always defined.
source
',4))]),t("details",W,[t("summary",null,[s[102]||(s[102]=t("a",{id:"is_integral-Tuple{ZZGenus}",href:"#is_integral-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_integral")],-1)),s[103]||(s[103]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[110]||(s[110]=i('
julia
is_integral(G::ZZGenus) -> Bool
',1)),t("p",null,[s[106]||(s[106]=e("Return whether ")),s[107]||(s[107]=t("code",null,"G",-1)),s[108]||(s[108]=e(" is a genus of integral ")),t("mjx-container",K,[(o(),n("svg",Y,s[104]||(s[104]=[i('',1)]))),s[105]||(s[105]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[109]||(s[109]=e("-lattices."))]),s[111]||(s[111]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[324]||(s[324]=i('
Discriminant group ​
discriminant_group(::ZZGenus)
Primary genera ​
',3)),t("details",_,[t("summary",null,[s[112]||(s[112]=t("a",{id:"is_primary_with_prime-Tuple{ZZGenus}",href:"#is_primary_with_prime-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),s[113]||(s[113]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[128]||(s[128]=i('
julia
is_primary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
',1)),t("p",null,[s[116]||(s[116]=e("Given a genus of ")),t("mjx-container",ss,[(o(),n("svg",ts,s[114]||(s[114]=[i('',1)]))),s[115]||(s[115]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[117]||(s[117]=e("-lattices ")),s[118]||(s[118]=t("code",null,"G",-1)),s[119]||(s[119]=e(", return whether it is primary, that is whether the bilinear form is integral and the associated discriminant form (see ")),s[120]||(s[120]=t("a",{href:"/Hecke.jl/v0.34.8/manual/quad_forms/discriminant_group#discriminant_group-Tuple{ZZLat}"},[t("code",null,"discriminant_group")],-1)),s[121]||(s[121]=e(") is a ")),s[122]||(s[122]=t("code",null,"p",-1)),s[123]||(s[123]=e("-group for some prime number ")),s[124]||(s[124]=t("code",null,"p",-1)),s[125]||(s[125]=e(". In case it is, ")),s[126]||(s[126]=t("code",null,"p",-1)),s[127]||(s[127]=e(" is also returned as second output."))]),s[129]||(s[129]=t("p",null,[e("Note that for unimodular genera, this function returns "),t("code",null,"(true, 1)"),e(". If the genus is not primary, the second return value is "),t("code",null,"-1"),e(" by default.")],-1)),s[130]||(s[130]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",es,[t("summary",null,[s[131]||(s[131]=t("a",{id:"is_primary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}",href:"#is_primary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"is_primary")],-1)),s[132]||(s[132]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[137]||(s[137]=i('
julia
is_primary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
',1)),t("p",null,[s[135]||(s[135]=e("Given a genus of integral ")),t("mjx-container",is,[(o(),n("svg",as,s[133]||(s[133]=[i('',1)]))),s[134]||(s[134]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[136]||(s[136]=i('-lattices G and a prime number p, return whether G is p-primary, that is whether the associated discriminant form (see discriminant_group) is a p-group.',13))]),s[138]||(s[138]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",ls,[t("summary",null,[s[139]||(s[139]=t("a",{id:"is_elementary_with_prime-Tuple{ZZGenus}",href:"#is_elementary_with_prime-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_elementary_with_prime")],-1)),s[140]||(s[140]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[155]||(s[155]=i('
julia
is_elementary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
',1)),t("p",null,[s[143]||(s[143]=e("Given a genus of ")),t("mjx-container",ns,[(o(),n("svg",os,s[141]||(s[141]=[i('',1)]))),s[142]||(s[142]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[144]||(s[144]=e("-lattices ")),s[145]||(s[145]=t("code",null,"G",-1)),s[146]||(s[146]=e(", return whether it is elementary, that is whether the bilinear form is inegtral and the associated discriminant form (see ")),s[147]||(s[147]=t("a",{href:"/Hecke.jl/v0.34.8/manual/quad_forms/discriminant_group#discriminant_group-Tuple{ZZLat}"},[t("code",null,"discriminant_group")],-1)),s[148]||(s[148]=e(") is an elementary ")),s[149]||(s[149]=t("code",null,"p",-1)),s[150]||(s[150]=e("-group for some prime number ")),s[151]||(s[151]=t("code",null,"p",-1)),s[152]||(s[152]=e(". In case it is, ")),s[153]||(s[153]=t("code",null,"p",-1)),s[154]||(s[154]=e(" is also returned as second output."))]),s[156]||(s[156]=t("p",null,[e("Note that for unimodular genera, this function returns "),t("code",null,"(true, 1)"),e(". If the genus is not elementary, the second return value is "),t("code",null,"-1"),e(" by default.")],-1)),s[157]||(s[157]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",rs,[t("summary",null,[s[158]||(s[158]=t("a",{id:"is_elementary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}",href:"#is_elementary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"is_elementary")],-1)),s[159]||(s[159]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[164]||(s[164]=i('
julia
is_elementary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
',1)),t("p",null,[s[162]||(s[162]=e("Given a genus of integral ")),t("mjx-container",ps,[(o(),n("svg",ds,s[160]||(s[160]=[i('',1)]))),s[161]||(s[161]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[163]||(s[163]=i('-lattices G and a prime number p, return whether G is p-elementary, that is whether its associated discriminant form (see discriminant_group) is an elementary p-group.',13))]),s[165]||(s[165]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[325]||(s[325]=t("h3",{id:"local-Symbol",tabindex:"-1"},[e("local Symbol "),t("a",{class:"header-anchor",href:"#local-Symbol","aria-label":'Permalink to "local Symbol {#local-Symbol}"'},"​")],-1)),t("details",hs,[t("summary",null,[s[166]||(s[166]=t("a",{id:"local_symbol-Tuple{ZZGenus, Any}",href:"#local_symbol-Tuple{ZZGenus, Any}"},[t("span",{class:"jlbinding"},"local_symbol")],-1)),s[167]||(s[167]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=i('
julia
local_symbol(G::ZZGenus, p) -> ZZLocalGenus
Return the local symbol at p.
source
',3))]),s[326]||(s[326]=t("h2",{id:"representative-s",tabindex:"-1"},[e("Representative(s) "),t("a",{class:"header-anchor",href:"#representative-s","aria-label":'Permalink to "Representative(s)"'},"​")],-1)),t("details",Qs,[t("summary",null,[s[169]||(s[169]=t("a",{id:"quadratic_space-Tuple{ZZGenus}",href:"#quadratic_space-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"quadratic_space")],-1)),s[170]||(s[170]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[171]||(s[171]=i('
julia
quadratic_space(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}
Return the quadratic space defined by this genus.
source
',3))]),t("details",ms,[t("summary",null,[s[172]||(s[172]=t("a",{id:"rational_representative-Tuple{ZZGenus}",href:"#rational_representative-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"rational_representative")],-1)),s[173]||(s[173]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[174]||(s[174]=i('
julia
rational_representative(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}
Return the quadratic space defined by this genus.
source
',3))]),t("details",gs,[t("summary",null,[s[175]||(s[175]=t("a",{id:"representative-Tuple{ZZGenus}",href:"#representative-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"representative")],-1)),s[176]||(s[176]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[177]||(s[177]=i('
julia
representative(G::ZZGenus) -> ZZLat
Compute a representative of this genus && cache it.
source
',3))]),t("details",ks,[t("summary",null,[s[178]||(s[178]=t("a",{id:"representatives-Tuple{ZZGenus}",href:"#representatives-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"representatives")],-1)),s[179]||(s[179]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[180]||(s[180]=i('
julia
representatives(G::ZZGenus) -> Vector{ZZLat}
Return a list of representatives of the isometry classes in this genus.
source
',3))]),t("details",Ts,[t("summary",null,[s[181]||(s[181]=t("a",{id:"mass-Tuple{ZZGenus}",href:"#mass-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"mass")],-1)),s[182]||(s[182]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[189]||(s[189]=i('
julia
mass(G::ZZGenus) -> QQFieldElem
Return the mass of this genus.
',2)),t("p",null,[s[185]||(s[185]=e("The genus must be definite. Let ")),s[186]||(s[186]=t("code",null,"L_1, ... L_n",-1)),s[187]||(s[187]=e(" be a complete list of representatives of the isometry classes in this genus. Its mass is defined as ")),t("mjx-container",us,[(o(),n("svg",ys,s[183]||(s[183]=[i('',1)]))),s[184]||(s[184]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("munderover",null,[t("mo",{"data-mjx-texclass":"OP"},"∑"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mo",null,"="),t("mn",null,"1")]),t("mi",null,"n")]),t("mfrac",null,[t("mn",null,"1"),t("mrow",null,[t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mi",null,"O"),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"L"),t("mi",null,"i")]),t("mo",{stretchy:"false"},")"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])])])],-1))]),s[188]||(s[188]=e("."))]),s[190]||(s[190]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",cs,[t("summary",null,[s[191]||(s[191]=t("a",{id:"rescale-Tuple{ZZGenus, Union{Integer, QQFieldElem, ZZRingElem, Rational}}",href:"#rescale-Tuple{ZZGenus, Union{Integer, QQFieldElem, ZZRingElem, Rational}}"},[t("span",{class:"jlbinding"},"rescale")],-1)),s[192]||(s[192]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[203]||(s[203]=i('
julia
rescale(G::ZZGenus, a::RationalUnion) -> ZZGenus
',1)),t("p",null,[s[195]||(s[195]=e("Given a genus symbol ")),s[196]||(s[196]=t("code",null,"G",-1)),s[197]||(s[197]=e(" of ")),t("mjx-container",bs,[(o(),n("svg",xs,s[193]||(s[193]=[i('',1)]))),s[194]||(s[194]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[198]||(s[198]=e("-lattices, return the genus symbol of any representative of ")),s[199]||(s[199]=t("code",null,"G",-1)),s[200]||(s[200]=e(" rescaled by ")),s[201]||(s[201]=t("code",null,"a",-1)),s[202]||(s[202]=e("."))]),s[204]||(s[204]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[327]||(s[327]=t("h2",{id:"Embeddings-and-Representations",tabindex:"-1"},[e("Embeddings and Representations "),t("a",{class:"header-anchor",href:"#Embeddings-and-Representations","aria-label":'Permalink to "Embeddings and Representations {#Embeddings-and-Representations}"'},"​")],-1)),t("details",fs,[t("summary",null,[s[205]||(s[205]=t("a",{id:"represents-Tuple{ZZGenus, ZZGenus}",href:"#represents-Tuple{ZZGenus, ZZGenus}"},[t("span",{class:"jlbinding"},"represents")],-1)),s[206]||(s[206]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[207]||(s[207]=i('
julia
represents(G1::ZZGenus, G2::ZZGenus) -> Bool
Return if G1 represents G2. That is if some element in the genus of G1 represents some element in the genus of G2.
source
',3))]),s[328]||(s[328]=t("h2",{id:"Local-genus-Symbols",tabindex:"-1"},[e("Local genus Symbols "),t("a",{class:"header-anchor",href:"#Local-genus-Symbols","aria-label":'Permalink to "Local genus Symbols {#Local-genus-Symbols}"'},"​")],-1)),t("details",Es,[t("summary",null,[s[208]||(s[208]=t("a",{id:"ZZLocalGenus",href:"#ZZLocalGenus"},[t("span",{class:"jlbinding"},"ZZLocalGenus")],-1)),s[209]||(s[209]=e()),l(a,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[210]||(s[210]=i('
julia
ZZLocalGenus
Local genus symbol over a p-adic ring.
The genus symbol of a component p^m A for odd prime = p is of the form (m,n,d), where
  • m = valuation of the component
  • n = rank of A
  • d = det(A) \\in \\{1,u\\} for a normalized quadratic non-residue u.
The genus symbol of a component 2^m A is of the form (m, n, s, d, o), where
  • m = valuation of the component
  • n = rank of A
  • d = det(A) in {1,3,5,7}
  • s = 0 (or 1) if even (or odd)
  • o = oddity of A (= 0 if s = 0) in Z/8Z = the trace of the diagonalization of A
The genus symbol is a list of such symbols (ordered by m) for each of the Jordan blocks A_1,...,A_t.
Reference: [5] Chapter 15, Section 7.
Arguments
  • prime: a prime number
  • symbol: the list of invariants for Jordan blocks A_t,...,A_t given as a list of lists of integers
source
',11))]),s[329]||(s[329]=t("h3",{id:"creation",tabindex:"-1"},[e("Creation "),t("a",{class:"header-anchor",href:"#creation","aria-label":'Permalink to "Creation"'},"​")],-1)),t("details",vs,[t("summary",null,[s[211]||(s[211]=t("a",{id:"genus-Tuple{ZZLat, Union{Integer, ZZRingElem}}",href:"#genus-Tuple{ZZLat, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[212]||(s[212]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[213]||(s[213]=i('
julia
genus(L::ZZLat, p::IntegerUnion) -> ZZLocalGenus
Return the local genus symbol of L at the prime p.
source
',3))]),t("details",ws,[t("summary",null,[s[214]||(s[214]=t("a",{id:"genus-Tuple{QQMatrix, Union{Integer, ZZRingElem}}",href:"#genus-Tuple{QQMatrix, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[215]||(s[215]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[216]||(s[216]=i('
julia
genus(A::QQMatrix, p::IntegerUnion) -> ZZLocalGenus
Return the local genus symbol of a Z-lattice with gram matrix A at the prime p.
source
',3))]),s[330]||(s[330]=t("h3",{id:"attributes",tabindex:"-1"},[e("Attributes "),t("a",{class:"header-anchor",href:"#attributes","aria-label":'Permalink to "Attributes"'},"​")],-1)),t("details",Zs,[t("summary",null,[s[217]||(s[217]=t("a",{id:"prime-Tuple{ZZLocalGenus}",href:"#prime-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"prime")],-1)),s[218]||(s[218]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[219]||(s[219]=i('
julia
prime(S::ZZLocalGenus) -> ZZRingElem
Return the prime p of this p-adic genus.
source
',3))]),t("details",js,[t("summary",null,[s[220]||(s[220]=t("a",{id:"iseven-Tuple{ZZLocalGenus}",href:"#iseven-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"iseven")],-1)),s[221]||(s[221]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[222]||(s[222]=i('
julia
iseven(S::ZZLocalGenus) -> Bool
Return if the underlying p-adic lattice is even.
If p is odd, every lattice is even.
source
',4))]),t("details",Ls,[t("summary",null,[s[223]||(s[223]=t("a",{id:"symbol-Tuple{ZZLocalGenus, Int64}",href:"#symbol-Tuple{ZZLocalGenus, Int64}"},[t("span",{class:"jlbinding"},"symbol")],-1)),s[224]||(s[224]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[225]||(s[225]=i('
julia
symbol(S::ZZLocalGenus, scale::Int) -> Vector{Int}
Return the underlying lists of integers for the Jordan block of the given scale
source
',3))]),t("details",Cs,[t("summary",null,[s[226]||(s[226]=t("a",{id:"hasse_invariant-Tuple{ZZLocalGenus}",href:"#hasse_invariant-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"hasse_invariant")],-1)),s[227]||(s[227]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[230]||(s[230]=i('
julia
hasse_invariant(S::ZZLocalGenus) -> Int
Return the Hasse invariant of a representative. If the representative is diagonal (a_1, ... , a_n) Then the Hasse invariant is
',2)),t("mjx-container",Ms,[(o(),n("svg",Hs,s[228]||(s[228]=[i('',1)]))),s[229]||(s[229]=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("munder",null,[t("mo",{"data-mjx-texclass":"OP"},"∏"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mo",null,"<"),t("mi",null,"j")])]),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"a"),t("mi",null,"i")]),t("mo",null,","),t("msub",null,[t("mi",null,"a"),t("mi",null,"j")]),t("msub",null,[t("mo",{stretchy:"false"},")"),t("mi",null,"p")])])],-1))]),s[231]||(s[231]=t("p",null,".",-1)),s[232]||(s[232]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Fs,[t("summary",null,[s[233]||(s[233]=t("a",{id:"det-Tuple{ZZLocalGenus}",href:"#det-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"det")],-1)),s[234]||(s[234]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[235]||(s[235]=i('
julia
det(S::ZZLocalGenus) -> QQFieldElem
Return an rational representing the determinant of this genus.
source
',3))]),t("details",Gs,[t("summary",null,[s[236]||(s[236]=t("a",{id:"dim-Tuple{ZZLocalGenus}",href:"#dim-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"dim")],-1)),s[237]||(s[237]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[238]||(s[238]=i('
julia
dim(S::ZZLocalGenus) -> Int
Return the dimension of this genus.
source
',3))]),t("details",Ds,[t("summary",null,[s[239]||(s[239]=t("a",{id:"rank-Tuple{ZZLocalGenus}",href:"#rank-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"rank")],-1)),s[240]||(s[240]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[241]||(s[241]=i('
julia
rank(S::ZZLocalGenus) -> Int
Return the rank of (a representative of) S.
source
',3))]),t("details",Bs,[t("summary",null,[s[242]||(s[242]=t("a",{id:"excess-Tuple{ZZLocalGenus}",href:"#excess-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"excess")],-1)),s[243]||(s[243]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[244]||(s[244]=i('
julia
excess(S::ZZLocalGenus) -> zzModRingElem
Return the p-excess of the quadratic form whose Hessian matrix is the symmetric matrix A.
When p = 2 the p-excess is called the oddity. The p-excess is always even && is divisible by 4 if p is congruent 1 mod 4.
Reference
[5] pp 370-371.
source
',6))]),t("details",As,[t("summary",null,[s[245]||(s[245]=t("a",{id:"signature-Tuple{ZZLocalGenus}",href:"#signature-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"signature")],-1)),s[246]||(s[246]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[254]||(s[254]=i('
julia
signature(S::ZZLocalGenus) -> zzModRingElem
',1)),t("p",null,[s[251]||(s[251]=e("Return the ")),t("mjx-container",Vs,[(o(),n("svg",Rs,s[247]||(s[247]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z",style:{"stroke-width":"3"}})])])],-1)]))),s[248]||(s[248]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"p")])],-1))]),s[252]||(s[252]=e("-signature of this ")),t("mjx-container",Os,[(o(),n("svg",Ss,s[249]||(s[249]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z",style:{"stroke-width":"3"}})])])],-1)]))),s[250]||(s[250]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"p")])],-1))]),s[253]||(s[253]=e("-adic form."))]),s[255]||(s[255]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Is,[t("summary",null,[s[256]||(s[256]=t("a",{id:"oddity-Tuple{ZZLocalGenus}",href:"#oddity-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"oddity")],-1)),s[257]||(s[257]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[262]||(s[262]=i('
julia
oddity(S::ZZLocalGenus) -> zzModRingElem
',1)),t("p",null,[s[260]||(s[260]=e("Return the oddity of this even form. The oddity is also called the ")),t("mjx-container",zs,[(o(),n("svg",Js,s[258]||(s[258]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mn"},[t("path",{"data-c":"32",d:"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z",style:{"stroke-width":"3"}})])])],-1)]))),s[259]||(s[259]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"2")])],-1))]),s[261]||(s[261]=e("-signature"))]),s[263]||(s[263]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Ps,[t("summary",null,[s[264]||(s[264]=t("a",{id:"scale-Tuple{ZZLocalGenus}",href:"#scale-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"scale")],-1)),s[265]||(s[265]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[266]||(s[266]=i('
julia
scale(S::ZZLocalGenus) -> QQFieldElem
Return the scale of this local genus.
Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).
source
',4))]),t("details",Us,[t("summary",null,[s[267]||(s[267]=t("a",{id:"norm-Tuple{ZZLocalGenus}",href:"#norm-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"norm")],-1)),s[268]||(s[268]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[279]||(s[279]=i('
julia
norm(S::ZZLocalGenus) -> QQFieldElem
Return the norm of this local genus.
',2)),t("p",null,[s[271]||(s[271]=e("Let ")),s[272]||(s[272]=t("code",null,"L",-1)),s[273]||(s[273]=e(" be a lattice with bilinear form ")),s[274]||(s[274]=t("code",null,"b",-1)),s[275]||(s[275]=e(". The norm of ")),s[276]||(s[276]=t("code",null,"(L,b)",-1)),s[277]||(s[277]=e(" is defined as the ideal generated by ")),t("mjx-container",Xs,[(o(),n("svg",qs,s[269]||(s[269]=[i('',1)]))),s[270]||(s[270]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{fence:"false",stretchy:"false"},"{"),t("mi",null,"b"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",null,","),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mi",null,"x"),t("mo",null,"∈"),t("mi",null,"L"),t("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[278]||(s[278]=e("."))]),s[280]||(s[280]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Ns,[t("summary",null,[s[281]||(s[281]=t("a",{id:"level-Tuple{ZZLocalGenus}",href:"#level-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"level")],-1)),s[282]||(s[282]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[283]||(s[283]=i('
julia
level(S::ZZLocalGenus) -> QQFieldElem
Return the maximal scale of a jordan component.
source
',3))]),s[331]||(s[331]=t("h3",{id:"representative",tabindex:"-1"},[e("Representative "),t("a",{class:"header-anchor",href:"#representative","aria-label":'Permalink to "Representative"'},"​")],-1)),t("details",$s,[t("summary",null,[s[284]||(s[284]=t("a",{id:"representative-Tuple{ZZLocalGenus}",href:"#representative-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"representative")],-1)),s[285]||(s[285]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[286]||(s[286]=i('
julia
representative(S::ZZLocalGenus) -> ZZLat
Return an integer lattice which represents this local genus.
source
',3))]),t("details",Ws,[t("summary",null,[s[287]||(s[287]=t("a",{id:"gram_matrix-Tuple{ZZLocalGenus}",href:"#gram_matrix-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"gram_matrix")],-1)),s[288]||(s[288]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[289]||(s[289]=i('
julia
gram_matrix(S::ZZLocalGenus) -> MatElem
Return a gram matrix of some representative of this local genus.
source
',3))]),t("details",Ks,[t("summary",null,[s[290]||(s[290]=t("a",{id:"rescale-Tuple{ZZLocalGenus, Union{Integer, QQFieldElem, ZZRingElem, Rational}}",href:"#rescale-Tuple{ZZLocalGenus, Union{Integer, QQFieldElem, ZZRingElem, Rational}}"},[t("span",{class:"jlbinding"},"rescale")],-1)),s[291]||(s[291]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[302]||(s[302]=i('
julia
rescale(G::ZZLocalGenus, a::RationalUnion) -> ZZLocalGenus
',1)),t("p",null,[s[294]||(s[294]=e("Given a local genus symbol ")),s[295]||(s[295]=t("code",null,"G",-1)),s[296]||(s[296]=e(" of ")),t("mjx-container",Ys,[(o(),n("svg",_s,s[292]||(s[292]=[i('',1)]))),s[293]||(s[293]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[297]||(s[297]=e("-lattices, return the local genus symbol of any representative of ")),s[298]||(s[298]=t("code",null,"G",-1)),s[299]||(s[299]=e(" rescaled by ")),s[300]||(s[300]=t("code",null,"a",-1)),s[301]||(s[301]=e("."))]),s[303]||(s[303]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[332]||(s[332]=t("h3",{id:"Direct-sums",tabindex:"-1"},[e("Direct sums "),t("a",{class:"header-anchor",href:"#Direct-sums","aria-label":'Permalink to "Direct sums {#Direct-sums}"'},"​")],-1)),t("details",s1,[t("summary",null,[s[304]||(s[304]=t("a",{id:"direct_sum-Tuple{ZZLocalGenus, ZZLocalGenus}",href:"#direct_sum-Tuple{ZZLocalGenus, ZZLocalGenus}"},[t("span",{class:"jlbinding"},"direct_sum")],-1)),s[305]||(s[305]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[306]||(s[306]=i('
julia
direct_sum(S1::ZZLocalGenus, S2::ZZLocalGenus) -> ZZLocalGenus
Return the local genus of the direct sum of two representatives.
source
',3))]),s[333]||(s[333]=t("h3",{id:"embeddings-representations",tabindex:"-1"},[e("Embeddings/Representations "),t("a",{class:"header-anchor",href:"#embeddings-representations","aria-label":'Permalink to "Embeddings/Representations"'},"​")],-1)),t("details",t1,[t("summary",null,[s[307]||(s[307]=t("a",{id:"represents-Tuple{ZZLocalGenus, ZZLocalGenus}",href:"#represents-Tuple{ZZLocalGenus, ZZLocalGenus}"},[t("span",{class:"jlbinding"},"represents")],-1)),s[308]||(s[308]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[315]||(s[315]=i('
julia
represents(g1::ZZLocalGenus, g2::ZZLocalGenus) -> Bool
Return whether g1 represents g2.
',2)),t("p",null,[s[311]||(s[311]=e("Based on O'Meara Integral Representations of Quadratic Forms Over Local Fields Note that for ")),s[312]||(s[312]=t("code",null,"p == 2",-1)),s[313]||(s[313]=e(" there is a typo in O'Meara Theorem 3 (V). The correct statement is (V) ")),t("mjx-container",e1,[(o(),n("svg",i1,s[309]||(s[309]=[i('',1)]))),s[310]||(s[310]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mn",null,"2"),t("mi",null,"i")]),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",null,"+"),t("mn",null,"4"),t("mi",null,"ω"),t("mo",{stretchy:"false"},")"),t("mo",{accent:"false",stretchy:"false"},"→"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"L")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mo",null,"+"),t("mn",null,"1")])]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"l")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",{stretchy:"false"},"["),t("mi",null,"i"),t("mo",{stretchy:"false"},"]")])])])],-1))]),s[314]||(s[314]=e("."))]),s[316]||(s[316]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))])])}const Q1=r(d,[["render",a1]]);export{h1 as __pageData,Q1 as default};
+       even=false)                                         -> Vector{ZZGenus}
`,1)),t("p",null,[s[54]||(s[54]=e("Return a list of all genera with the given conditions. Genera of non-integral ")),t("mjx-container",B,[(o(),n("svg",A,s[52]||(s[52]=[i('',1)]))),s[53]||(s[53]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[55]||(s[55]=e("-lattices are also supported."))]),s[57]||(s[57]=i('

Arguments

  • sig_pair: a pair of non-negative integers giving the signature

  • determinant: a rational number; the sign is ignored

  • min_scale: a rational number; return only genera whose scale is an integer multiple of min_scale (default: min(one(QQ), QQ(abs(determinant))))

  • max_scale: a rational number; return only genera such that max_scale is an integer multiple of the scale (default: max(one(QQ), QQ(abs(determinant))))

  • even: boolean; if set to true, return only the even genera (default: false)

source

',3))]),s[322]||(s[322]=t("h3",{id:"From-other-genus-symbols",tabindex:"-1"},[e("From other genus symbols "),t("a",{class:"header-anchor",href:"#From-other-genus-symbols","aria-label":'Permalink to "From other genus symbols {#From-other-genus-symbols}"'},"​")],-1)),t("details",V,[t("summary",null,[s[58]||(s[58]=t("a",{id:"direct_sum-Tuple{ZZGenus, ZZGenus}",href:"#direct_sum-Tuple{ZZGenus, ZZGenus}"},[t("span",{class:"jlbinding"},"direct_sum")],-1)),s[59]||(s[59]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[60]||(s[60]=i('
julia
direct_sum(G1::ZZGenus, G2::ZZGenus) -> ZZGenus

Return the genus of the direct sum of G1 and G2.

The direct sum is defined via representatives.

source

',4))]),s[323]||(s[323]=t("h2",{id:"Attributes-of-the-genus",tabindex:"-1"},[e("Attributes of the genus "),t("a",{class:"header-anchor",href:"#Attributes-of-the-genus","aria-label":'Permalink to "Attributes of the genus {#Attributes-of-the-genus}"'},"​")],-1)),t("details",R,[t("summary",null,[s[61]||(s[61]=t("a",{id:"dim-Tuple{ZZGenus}",href:"#dim-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"dim")],-1)),s[62]||(s[62]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[63]||(s[63]=i('
julia
dim(G::ZZGenus) -> Int

Return the dimension of this genus.

source

',3))]),t("details",O,[t("summary",null,[s[64]||(s[64]=t("a",{id:"rank-Tuple{ZZGenus}",href:"#rank-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"rank")],-1)),s[65]||(s[65]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[66]||(s[66]=i('
julia
rank(G::ZZGenus) -> Int

Return the rank of a (representative of) the genus G.

source

',3))]),t("details",S,[t("summary",null,[s[67]||(s[67]=t("a",{id:"signature-Tuple{ZZGenus}",href:"#signature-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"signature")],-1)),s[68]||(s[68]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[69]||(s[69]=i('
julia
signature(G::ZZGenus) -> Int

Return the signature of this genus.

The signature is p - n where p is the number of positive eigenvalues and n the number of negative eigenvalues.

source

',4))]),t("details",I,[t("summary",null,[s[70]||(s[70]=t("a",{id:"det-Tuple{ZZGenus}",href:"#det-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"det")],-1)),s[71]||(s[71]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[72]||(s[72]=i('
julia
det(G::ZZGenus) -> QQFieldElem

Return the determinant of this genus.

source

',3))]),t("details",z,[t("summary",null,[s[73]||(s[73]=t("a",{id:"iseven-Tuple{ZZGenus}",href:"#iseven-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"iseven")],-1)),s[74]||(s[74]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[75]||(s[75]=i('
julia
iseven(G::ZZGenus) -> Bool

Return if this genus is even.

source

',3))]),t("details",J,[t("summary",null,[s[76]||(s[76]=t("a",{id:"is_definite-Tuple{ZZGenus}",href:"#is_definite-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_definite")],-1)),s[77]||(s[77]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[78]||(s[78]=i('
julia
is_definite(G::ZZGenus) -> Bool

Return if this genus is definite.

source

',3))]),t("details",P,[t("summary",null,[s[79]||(s[79]=t("a",{id:"level-Tuple{ZZGenus}",href:"#level-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"level")],-1)),s[80]||(s[80]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[81]||(s[81]=i('
julia
level(G::ZZGenus) -> QQFieldElem

Return the level of this genus.

This is the denominator of the inverse gram matrix of a representative.

source

',4))]),t("details",U,[t("summary",null,[s[82]||(s[82]=t("a",{id:"scale-Tuple{ZZGenus}",href:"#scale-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"scale")],-1)),s[83]||(s[83]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[84]||(s[84]=i('
julia
scale(G::ZZGenus) -> QQFieldElem

Return the scale of this genus.

Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).

source

',4))]),t("details",X,[t("summary",null,[s[85]||(s[85]=t("a",{id:"norm-Tuple{ZZGenus}",href:"#norm-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"norm")],-1)),s[86]||(s[86]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[97]||(s[97]=i('
julia
norm(G::ZZGenus) -> QQFieldElem

Return the norm of this genus.

',2)),t("p",null,[s[89]||(s[89]=e("Let ")),s[90]||(s[90]=t("code",null,"L",-1)),s[91]||(s[91]=e(" be a lattice with bilinear form ")),s[92]||(s[92]=t("code",null,"b",-1)),s[93]||(s[93]=e(". The norm of ")),s[94]||(s[94]=t("code",null,"(L,b)",-1)),s[95]||(s[95]=e(" is defined as the ideal generated by ")),t("mjx-container",q,[(o(),n("svg",N,s[87]||(s[87]=[i('',1)]))),s[88]||(s[88]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{fence:"false",stretchy:"false"},"{"),t("mi",null,"b"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",null,","),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mi",null,"x"),t("mo",null,"∈"),t("mi",null,"L"),t("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[96]||(s[96]=e("."))]),s[98]||(s[98]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",$,[t("summary",null,[s[99]||(s[99]=t("a",{id:"primes-Tuple{ZZGenus}",href:"#primes-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"primes")],-1)),s[100]||(s[100]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[101]||(s[101]=i('
julia
primes(G::ZZGenus) -> Vector{ZZRingElem}

Return the list of primes of the local symbols of G.

Note that 2 is always in the output since the 2-adic symbol of a ZZGenus is, by convention, always defined.

source

',4))]),t("details",W,[t("summary",null,[s[102]||(s[102]=t("a",{id:"is_integral-Tuple{ZZGenus}",href:"#is_integral-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_integral")],-1)),s[103]||(s[103]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[110]||(s[110]=i('
julia
is_integral(G::ZZGenus) -> Bool
',1)),t("p",null,[s[106]||(s[106]=e("Return whether ")),s[107]||(s[107]=t("code",null,"G",-1)),s[108]||(s[108]=e(" is a genus of integral ")),t("mjx-container",K,[(o(),n("svg",Y,s[104]||(s[104]=[i('',1)]))),s[105]||(s[105]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[109]||(s[109]=e("-lattices."))]),s[111]||(s[111]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[324]||(s[324]=i('

Discriminant group ​

discriminant_group(::ZZGenus)

Primary genera ​

',3)),t("details",_,[t("summary",null,[s[112]||(s[112]=t("a",{id:"is_primary_with_prime-Tuple{ZZGenus}",href:"#is_primary_with_prime-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),s[113]||(s[113]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[128]||(s[128]=i('
julia
is_primary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
',1)),t("p",null,[s[116]||(s[116]=e("Given a genus of ")),t("mjx-container",ss,[(o(),n("svg",ts,s[114]||(s[114]=[i('',1)]))),s[115]||(s[115]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[117]||(s[117]=e("-lattices ")),s[118]||(s[118]=t("code",null,"G",-1)),s[119]||(s[119]=e(", return whether it is primary, that is whether the bilinear form is integral and the associated discriminant form (see ")),s[120]||(s[120]=t("a",{href:"/v0.34.8/manual/quad_forms/discriminant_group#discriminant_group-Tuple{ZZLat}"},[t("code",null,"discriminant_group")],-1)),s[121]||(s[121]=e(") is a ")),s[122]||(s[122]=t("code",null,"p",-1)),s[123]||(s[123]=e("-group for some prime number ")),s[124]||(s[124]=t("code",null,"p",-1)),s[125]||(s[125]=e(". In case it is, ")),s[126]||(s[126]=t("code",null,"p",-1)),s[127]||(s[127]=e(" is also returned as second output."))]),s[129]||(s[129]=t("p",null,[e("Note that for unimodular genera, this function returns "),t("code",null,"(true, 1)"),e(". If the genus is not primary, the second return value is "),t("code",null,"-1"),e(" by default.")],-1)),s[130]||(s[130]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",es,[t("summary",null,[s[131]||(s[131]=t("a",{id:"is_primary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}",href:"#is_primary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"is_primary")],-1)),s[132]||(s[132]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[137]||(s[137]=i('
julia
is_primary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
',1)),t("p",null,[s[135]||(s[135]=e("Given a genus of integral ")),t("mjx-container",is,[(o(),n("svg",as,s[133]||(s[133]=[i('',1)]))),s[134]||(s[134]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[136]||(s[136]=i('-lattices G and a prime number p, return whether G is p-primary, that is whether the associated discriminant form (see discriminant_group) is a p-group.',13))]),s[138]||(s[138]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",ls,[t("summary",null,[s[139]||(s[139]=t("a",{id:"is_elementary_with_prime-Tuple{ZZGenus}",href:"#is_elementary_with_prime-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_elementary_with_prime")],-1)),s[140]||(s[140]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[155]||(s[155]=i('
julia
is_elementary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
',1)),t("p",null,[s[143]||(s[143]=e("Given a genus of ")),t("mjx-container",ns,[(o(),n("svg",os,s[141]||(s[141]=[i('',1)]))),s[142]||(s[142]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[144]||(s[144]=e("-lattices ")),s[145]||(s[145]=t("code",null,"G",-1)),s[146]||(s[146]=e(", return whether it is elementary, that is whether the bilinear form is inegtral and the associated discriminant form (see ")),s[147]||(s[147]=t("a",{href:"/v0.34.8/manual/quad_forms/discriminant_group#discriminant_group-Tuple{ZZLat}"},[t("code",null,"discriminant_group")],-1)),s[148]||(s[148]=e(") is an elementary ")),s[149]||(s[149]=t("code",null,"p",-1)),s[150]||(s[150]=e("-group for some prime number ")),s[151]||(s[151]=t("code",null,"p",-1)),s[152]||(s[152]=e(". In case it is, ")),s[153]||(s[153]=t("code",null,"p",-1)),s[154]||(s[154]=e(" is also returned as second output."))]),s[156]||(s[156]=t("p",null,[e("Note that for unimodular genera, this function returns "),t("code",null,"(true, 1)"),e(". If the genus is not elementary, the second return value is "),t("code",null,"-1"),e(" by default.")],-1)),s[157]||(s[157]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",rs,[t("summary",null,[s[158]||(s[158]=t("a",{id:"is_elementary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}",href:"#is_elementary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"is_elementary")],-1)),s[159]||(s[159]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[164]||(s[164]=i('
julia
is_elementary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
',1)),t("p",null,[s[162]||(s[162]=e("Given a genus of integral ")),t("mjx-container",ps,[(o(),n("svg",ds,s[160]||(s[160]=[i('',1)]))),s[161]||(s[161]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[163]||(s[163]=i('-lattices G and a prime number p, return whether G is p-elementary, that is whether its associated discriminant form (see discriminant_group) is an elementary p-group.',13))]),s[165]||(s[165]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[325]||(s[325]=t("h3",{id:"local-Symbol",tabindex:"-1"},[e("local Symbol "),t("a",{class:"header-anchor",href:"#local-Symbol","aria-label":'Permalink to "local Symbol {#local-Symbol}"'},"​")],-1)),t("details",hs,[t("summary",null,[s[166]||(s[166]=t("a",{id:"local_symbol-Tuple{ZZGenus, Any}",href:"#local_symbol-Tuple{ZZGenus, Any}"},[t("span",{class:"jlbinding"},"local_symbol")],-1)),s[167]||(s[167]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=i('
julia
local_symbol(G::ZZGenus, p) -> ZZLocalGenus

Return the local symbol at p.

source

',3))]),s[326]||(s[326]=t("h2",{id:"representative-s",tabindex:"-1"},[e("Representative(s) "),t("a",{class:"header-anchor",href:"#representative-s","aria-label":'Permalink to "Representative(s)"'},"​")],-1)),t("details",Qs,[t("summary",null,[s[169]||(s[169]=t("a",{id:"quadratic_space-Tuple{ZZGenus}",href:"#quadratic_space-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"quadratic_space")],-1)),s[170]||(s[170]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[171]||(s[171]=i('
julia
quadratic_space(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}

Return the quadratic space defined by this genus.

source

',3))]),t("details",ms,[t("summary",null,[s[172]||(s[172]=t("a",{id:"rational_representative-Tuple{ZZGenus}",href:"#rational_representative-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"rational_representative")],-1)),s[173]||(s[173]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[174]||(s[174]=i('
julia
rational_representative(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}

Return the quadratic space defined by this genus.

source

',3))]),t("details",gs,[t("summary",null,[s[175]||(s[175]=t("a",{id:"representative-Tuple{ZZGenus}",href:"#representative-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"representative")],-1)),s[176]||(s[176]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[177]||(s[177]=i('
julia
representative(G::ZZGenus) -> ZZLat

Compute a representative of this genus && cache it.

source

',3))]),t("details",ks,[t("summary",null,[s[178]||(s[178]=t("a",{id:"representatives-Tuple{ZZGenus}",href:"#representatives-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"representatives")],-1)),s[179]||(s[179]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[180]||(s[180]=i('
julia
representatives(G::ZZGenus) -> Vector{ZZLat}

Return a list of representatives of the isometry classes in this genus.

source

',3))]),t("details",Ts,[t("summary",null,[s[181]||(s[181]=t("a",{id:"mass-Tuple{ZZGenus}",href:"#mass-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"mass")],-1)),s[182]||(s[182]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[189]||(s[189]=i('
julia
mass(G::ZZGenus) -> QQFieldElem

Return the mass of this genus.

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julia
rescale(G::ZZGenus, a::RationalUnion) -> ZZGenus
',1)),t("p",null,[s[195]||(s[195]=e("Given a genus symbol ")),s[196]||(s[196]=t("code",null,"G",-1)),s[197]||(s[197]=e(" of ")),t("mjx-container",bs,[(o(),n("svg",xs,s[193]||(s[193]=[i('',1)]))),s[194]||(s[194]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[198]||(s[198]=e("-lattices, return the genus symbol of any representative of ")),s[199]||(s[199]=t("code",null,"G",-1)),s[200]||(s[200]=e(" rescaled by ")),s[201]||(s[201]=t("code",null,"a",-1)),s[202]||(s[202]=e("."))]),s[204]||(s[204]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[327]||(s[327]=t("h2",{id:"Embeddings-and-Representations",tabindex:"-1"},[e("Embeddings and Representations "),t("a",{class:"header-anchor",href:"#Embeddings-and-Representations","aria-label":'Permalink to "Embeddings and Representations {#Embeddings-and-Representations}"'},"​")],-1)),t("details",fs,[t("summary",null,[s[205]||(s[205]=t("a",{id:"represents-Tuple{ZZGenus, ZZGenus}",href:"#represents-Tuple{ZZGenus, ZZGenus}"},[t("span",{class:"jlbinding"},"represents")],-1)),s[206]||(s[206]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[207]||(s[207]=i('
julia
represents(G1::ZZGenus, G2::ZZGenus) -> Bool

Return if G1 represents G2. That is if some element in the genus of G1 represents some element in the genus of G2.

source

',3))]),s[328]||(s[328]=t("h2",{id:"Local-genus-Symbols",tabindex:"-1"},[e("Local genus Symbols "),t("a",{class:"header-anchor",href:"#Local-genus-Symbols","aria-label":'Permalink to "Local genus Symbols {#Local-genus-Symbols}"'},"​")],-1)),t("details",Es,[t("summary",null,[s[208]||(s[208]=t("a",{id:"ZZLocalGenus",href:"#ZZLocalGenus"},[t("span",{class:"jlbinding"},"ZZLocalGenus")],-1)),s[209]||(s[209]=e()),l(a,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[210]||(s[210]=i('
julia
ZZLocalGenus

Local genus symbol over a p-adic ring.

The genus symbol of a component p^m A for odd prime = p is of the form (m,n,d), where

  • m = valuation of the component

  • n = rank of A

  • d = det(A) \\in \\{1,u\\} for a normalized quadratic non-residue u.

The genus symbol of a component 2^m A is of the form (m, n, s, d, o), where

  • m = valuation of the component

  • n = rank of A

  • d = det(A) in {1,3,5,7}

  • s = 0 (or 1) if even (or odd)

  • o = oddity of A (= 0 if s = 0) in Z/8Z = the trace of the diagonalization of A

The genus symbol is a list of such symbols (ordered by m) for each of the Jordan blocks A_1,...,A_t.

Reference: [5] Chapter 15, Section 7.

Arguments

  • prime: a prime number

  • symbol: the list of invariants for Jordan blocks A_t,...,A_t given as a list of lists of integers

source

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julia
genus(L::ZZLat, p::IntegerUnion) -> ZZLocalGenus

Return the local genus symbol of L at the prime p.

source

',3))]),t("details",ws,[t("summary",null,[s[214]||(s[214]=t("a",{id:"genus-Tuple{QQMatrix, Union{Integer, ZZRingElem}}",href:"#genus-Tuple{QQMatrix, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[215]||(s[215]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[216]||(s[216]=i('
julia
genus(A::QQMatrix, p::IntegerUnion) -> ZZLocalGenus

Return the local genus symbol of a Z-lattice with gram matrix A at the prime p.

source

',3))]),s[330]||(s[330]=t("h3",{id:"attributes",tabindex:"-1"},[e("Attributes "),t("a",{class:"header-anchor",href:"#attributes","aria-label":'Permalink to "Attributes"'},"​")],-1)),t("details",Zs,[t("summary",null,[s[217]||(s[217]=t("a",{id:"prime-Tuple{ZZLocalGenus}",href:"#prime-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"prime")],-1)),s[218]||(s[218]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[219]||(s[219]=i('
julia
prime(S::ZZLocalGenus) -> ZZRingElem

Return the prime p of this p-adic genus.

source

',3))]),t("details",js,[t("summary",null,[s[220]||(s[220]=t("a",{id:"iseven-Tuple{ZZLocalGenus}",href:"#iseven-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"iseven")],-1)),s[221]||(s[221]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[222]||(s[222]=i('
julia
iseven(S::ZZLocalGenus) -> Bool

Return if the underlying p-adic lattice is even.

If p is odd, every lattice is even.

source

',4))]),t("details",Ls,[t("summary",null,[s[223]||(s[223]=t("a",{id:"symbol-Tuple{ZZLocalGenus, Int64}",href:"#symbol-Tuple{ZZLocalGenus, Int64}"},[t("span",{class:"jlbinding"},"symbol")],-1)),s[224]||(s[224]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[225]||(s[225]=i('
julia
symbol(S::ZZLocalGenus, scale::Int) -> Vector{Int}

Return the underlying lists of integers for the Jordan block of the given scale

source

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julia
hasse_invariant(S::ZZLocalGenus) -> Int

Return the Hasse invariant of a representative. If the representative is diagonal (a_1, ... , a_n) Then the Hasse invariant is

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julia
det(S::ZZLocalGenus) -> QQFieldElem

Return an rational representing the determinant of this genus.

source

',3))]),t("details",Gs,[t("summary",null,[s[236]||(s[236]=t("a",{id:"dim-Tuple{ZZLocalGenus}",href:"#dim-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"dim")],-1)),s[237]||(s[237]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[238]||(s[238]=i('
julia
dim(S::ZZLocalGenus) -> Int

Return the dimension of this genus.

source

',3))]),t("details",Ds,[t("summary",null,[s[239]||(s[239]=t("a",{id:"rank-Tuple{ZZLocalGenus}",href:"#rank-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"rank")],-1)),s[240]||(s[240]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[241]||(s[241]=i('
julia
rank(S::ZZLocalGenus) -> Int

Return the rank of (a representative of) S.

source

',3))]),t("details",Bs,[t("summary",null,[s[242]||(s[242]=t("a",{id:"excess-Tuple{ZZLocalGenus}",href:"#excess-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"excess")],-1)),s[243]||(s[243]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[244]||(s[244]=i('
julia
excess(S::ZZLocalGenus) -> zzModRingElem

Return the p-excess of the quadratic form whose Hessian matrix is the symmetric matrix A.

When p = 2 the p-excess is called the oddity. The p-excess is always even && is divisible by 4 if p is congruent 1 mod 4.

Reference

[5] pp 370-371.

source

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julia
signature(S::ZZLocalGenus) -> zzModRingElem
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julia
oddity(S::ZZLocalGenus) -> zzModRingElem
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julia
scale(S::ZZLocalGenus) -> QQFieldElem

Return the scale of this local genus.

Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).

source

',4))]),t("details",Us,[t("summary",null,[s[267]||(s[267]=t("a",{id:"norm-Tuple{ZZLocalGenus}",href:"#norm-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"norm")],-1)),s[268]||(s[268]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[279]||(s[279]=i('
julia
norm(S::ZZLocalGenus) -> QQFieldElem

Return the norm of this local genus.

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julia
level(S::ZZLocalGenus) -> QQFieldElem

Return the maximal scale of a jordan component.

source

',3))]),s[331]||(s[331]=t("h3",{id:"representative",tabindex:"-1"},[e("Representative "),t("a",{class:"header-anchor",href:"#representative","aria-label":'Permalink to "Representative"'},"​")],-1)),t("details",$s,[t("summary",null,[s[284]||(s[284]=t("a",{id:"representative-Tuple{ZZLocalGenus}",href:"#representative-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"representative")],-1)),s[285]||(s[285]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[286]||(s[286]=i('
julia
representative(S::ZZLocalGenus) -> ZZLat

Return an integer lattice which represents this local genus.

source

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julia
gram_matrix(S::ZZLocalGenus) -> MatElem

Return a gram matrix of some representative of this local genus.

source

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julia
rescale(G::ZZLocalGenus, a::RationalUnion) -> ZZLocalGenus
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julia
direct_sum(S1::ZZLocalGenus, S2::ZZLocalGenus) -> ZZLocalGenus

Return the local genus of the direct sum of two representatives.

source

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julia
represents(g1::ZZLocalGenus, g2::ZZLocalGenus) -> Bool

Return whether g1 represents g2.

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")),t("mjx-container",w,[(o(),n("svg",Z,s[22]||(s[22]=[i('',1)]))),s[23]||(s[23]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[25]||(s[25]=e("-lattice is encoded in its Conway-Sloane genus symbol. 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julia
ZZGenus

A collection of local genus symbols (at primes) and a signature pair. Together they represent the genus of a non-degenerate integer_lattice.

source

',3))]),s[318]||(s[318]=t("h2",{id:"Creation-of-Genera",tabindex:"-1"},[e("Creation of Genera "),t("a",{class:"header-anchor",href:"#Creation-of-Genera","aria-label":'Permalink to "Creation of Genera {#Creation-of-Genera}"'},"​")],-1)),s[319]||(s[319]=t("h3",{id:"From-an-integral-Lattice",tabindex:"-1"},[e("From an integral Lattice "),t("a",{class:"header-anchor",href:"#From-an-integral-Lattice","aria-label":'Permalink to "From an integral Lattice {#From-an-integral-Lattice}"'},"​")],-1)),t("details",M,[t("summary",null,[s[37]||(s[37]=t("a",{id:"genus-Tuple{ZZLat}",href:"#genus-Tuple{ZZLat}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[38]||(s[38]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[39]||(s[39]=i('
julia
genus(L::ZZLat) -> ZZGenus

Return the genus of the lattice L.

source

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julia
genus(A::MatElem) -> ZZGenus
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julia
integer_genera(sig_pair::Vector{Int}, determinant::RationalUnion;
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")),t("mjx-container",E,[(o(),n("svg",v,s[12]||(s[12]=[i('',1)]))),s[13]||(s[13]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"M"),t("mo",null,"⊗"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"R")]),t("mo",null,"≅"),t("mi",null,"N"),t("mo",null,"⊗"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"R")])])],-1))]),s[21]||(s[21]=e("."))]),t("p",null,[s[24]||(s[24]=e("The genus of a ")),t("mjx-container",w,[(o(),n("svg",Z,s[22]||(s[22]=[i('',1)]))),s[23]||(s[23]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[25]||(s[25]=e("-lattice is encoded in its Conway-Sloane genus symbol. The genus symbol itself is a collection of its local genus symbols. See [")),s[26]||(s[26]=t("a",{href:"/v0.34.8/references#CS99"},"5",-1)),s[27]||(s[27]=e("] Chapter 15 for the definitions. Note that genera for non-integral lattices are supported."))]),t("p",null,[s[30]||(s[30]=e("The class ")),s[31]||(s[31]=t("code",null,"ZZGenus",-1)),s[32]||(s[32]=e(" supports genera of ")),t("mjx-container",j,[(o(),n("svg",L,s[28]||(s[28]=[i('',1)]))),s[29]||(s[29]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[33]||(s[33]=e("-lattices."))]),t("details",C,[t("summary",null,[s[34]||(s[34]=t("a",{id:"ZZGenus",href:"#ZZGenus"},[t("span",{class:"jlbinding"},"ZZGenus")],-1)),s[35]||(s[35]=e()),l(a,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[36]||(s[36]=i('
julia
ZZGenus
A collection of local genus symbols (at primes) and a signature pair. Together they represent the genus of a non-degenerate integer_lattice.
source
',3))]),s[318]||(s[318]=t("h2",{id:"Creation-of-Genera",tabindex:"-1"},[e("Creation of Genera "),t("a",{class:"header-anchor",href:"#Creation-of-Genera","aria-label":'Permalink to "Creation of Genera {#Creation-of-Genera}"'},"​")],-1)),s[319]||(s[319]=t("h3",{id:"From-an-integral-Lattice",tabindex:"-1"},[e("From an integral Lattice "),t("a",{class:"header-anchor",href:"#From-an-integral-Lattice","aria-label":'Permalink to "From an integral Lattice {#From-an-integral-Lattice}"'},"​")],-1)),t("details",M,[t("summary",null,[s[37]||(s[37]=t("a",{id:"genus-Tuple{ZZLat}",href:"#genus-Tuple{ZZLat}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[38]||(s[38]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[39]||(s[39]=i('
julia
genus(L::ZZLat) -> ZZGenus
Return the genus of the lattice L.
source
',3))]),s[320]||(s[320]=t("h3",{id:"From-a-gram-matrix",tabindex:"-1"},[e("From a gram matrix "),t("a",{class:"header-anchor",href:"#From-a-gram-matrix","aria-label":'Permalink to "From a gram matrix {#From-a-gram-matrix}"'},"​")],-1)),t("details",H,[t("summary",null,[s[40]||(s[40]=t("a",{id:"genus-Tuple{MatElem}",href:"#genus-Tuple{MatElem}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[41]||(s[41]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[48]||(s[48]=i('
julia
genus(A::MatElem) -> ZZGenus
',1)),t("p",null,[s[44]||(s[44]=e("Return the genus of a ")),t("mjx-container",F,[(o(),n("svg",G,s[42]||(s[42]=[i('',1)]))),s[43]||(s[43]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[45]||(s[45]=e("-lattice with gram matrix ")),s[46]||(s[46]=t("code",null,"A",-1)),s[47]||(s[47]=e("."))]),s[49]||(s[49]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[321]||(s[321]=t("h3",{id:"Enumeration-of-genus-symbols",tabindex:"-1"},[e("Enumeration of genus symbols "),t("a",{class:"header-anchor",href:"#Enumeration-of-genus-symbols","aria-label":'Permalink to "Enumeration of genus symbols {#Enumeration-of-genus-symbols}"'},"​")],-1)),t("details",D,[t("summary",null,[s[50]||(s[50]=t("a",{id:"integer_genera-Tuple{Tuple{Int64, Int64}, Union{Int64, ZZRingElem}}",href:"#integer_genera-Tuple{Tuple{Int64, Int64}, Union{Int64, ZZRingElem}}"},[t("span",{class:"jlbinding"},"integer_genera")],-1)),s[51]||(s[51]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[56]||(s[56]=i(`
julia
integer_genera(sig_pair::Vector{Int}, determinant::RationalUnion;
        min_scale::RationalUnion = min(one(QQ), QQ(abs(determinant))),
        max_scale::RationalUnion = max(one(QQ), QQ(abs(determinant))),
-       even=false)                                         -> Vector{ZZGenus}
`,1)),t("p",null,[s[54]||(s[54]=e("Return a list of all genera with the given conditions. Genera of non-integral ")),t("mjx-container",B,[(o(),n("svg",A,s[52]||(s[52]=[i('',1)]))),s[53]||(s[53]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[55]||(s[55]=e("-lattices are also supported."))]),s[57]||(s[57]=i('
Arguments
  • sig_pair: a pair of non-negative integers giving the signature
  • determinant: a rational number; the sign is ignored
  • min_scale: a rational number; return only genera whose scale is an integer multiple of min_scale (default: min(one(QQ), QQ(abs(determinant))))
  • max_scale: a rational number; return only genera such that max_scale is an integer multiple of the scale (default: max(one(QQ), QQ(abs(determinant))))
  • even: boolean; if set to true, return only the even genera (default: false)
source
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julia
direct_sum(G1::ZZGenus, G2::ZZGenus) -> ZZGenus
Return the genus of the direct sum of G1 and G2.
The direct sum is defined via representatives.
source
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julia
dim(G::ZZGenus) -> Int
Return the dimension of this genus.
source
',3))]),t("details",O,[t("summary",null,[s[64]||(s[64]=t("a",{id:"rank-Tuple{ZZGenus}",href:"#rank-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"rank")],-1)),s[65]||(s[65]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[66]||(s[66]=i('
julia
rank(G::ZZGenus) -> Int
Return the rank of a (representative of) the genus G.
source
',3))]),t("details",S,[t("summary",null,[s[67]||(s[67]=t("a",{id:"signature-Tuple{ZZGenus}",href:"#signature-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"signature")],-1)),s[68]||(s[68]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[69]||(s[69]=i('
julia
signature(G::ZZGenus) -> Int
Return the signature of this genus.
The signature is p - n where p is the number of positive eigenvalues and n the number of negative eigenvalues.
source
',4))]),t("details",I,[t("summary",null,[s[70]||(s[70]=t("a",{id:"det-Tuple{ZZGenus}",href:"#det-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"det")],-1)),s[71]||(s[71]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[72]||(s[72]=i('
julia
det(G::ZZGenus) -> QQFieldElem
Return the determinant of this genus.
source
',3))]),t("details",z,[t("summary",null,[s[73]||(s[73]=t("a",{id:"iseven-Tuple{ZZGenus}",href:"#iseven-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"iseven")],-1)),s[74]||(s[74]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[75]||(s[75]=i('
julia
iseven(G::ZZGenus) -> Bool
Return if this genus is even.
source
',3))]),t("details",J,[t("summary",null,[s[76]||(s[76]=t("a",{id:"is_definite-Tuple{ZZGenus}",href:"#is_definite-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_definite")],-1)),s[77]||(s[77]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[78]||(s[78]=i('
julia
is_definite(G::ZZGenus) -> Bool
Return if this genus is definite.
source
',3))]),t("details",P,[t("summary",null,[s[79]||(s[79]=t("a",{id:"level-Tuple{ZZGenus}",href:"#level-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"level")],-1)),s[80]||(s[80]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[81]||(s[81]=i('
julia
level(G::ZZGenus) -> QQFieldElem
Return the level of this genus.
This is the denominator of the inverse gram matrix of a representative.
source
',4))]),t("details",U,[t("summary",null,[s[82]||(s[82]=t("a",{id:"scale-Tuple{ZZGenus}",href:"#scale-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"scale")],-1)),s[83]||(s[83]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[84]||(s[84]=i('
julia
scale(G::ZZGenus) -> QQFieldElem
Return the scale of this genus.
Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).
source
',4))]),t("details",X,[t("summary",null,[s[85]||(s[85]=t("a",{id:"norm-Tuple{ZZGenus}",href:"#norm-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"norm")],-1)),s[86]||(s[86]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[97]||(s[97]=i('
julia
norm(G::ZZGenus) -> QQFieldElem
Return the norm of this genus.
',2)),t("p",null,[s[89]||(s[89]=e("Let ")),s[90]||(s[90]=t("code",null,"L",-1)),s[91]||(s[91]=e(" be a lattice with bilinear form ")),s[92]||(s[92]=t("code",null,"b",-1)),s[93]||(s[93]=e(". The norm of ")),s[94]||(s[94]=t("code",null,"(L,b)",-1)),s[95]||(s[95]=e(" is defined as the ideal generated by ")),t("mjx-container",q,[(o(),n("svg",N,s[87]||(s[87]=[i('',1)]))),s[88]||(s[88]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{fence:"false",stretchy:"false"},"{"),t("mi",null,"b"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",null,","),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mi",null,"x"),t("mo",null,"∈"),t("mi",null,"L"),t("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[96]||(s[96]=e("."))]),s[98]||(s[98]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",$,[t("summary",null,[s[99]||(s[99]=t("a",{id:"primes-Tuple{ZZGenus}",href:"#primes-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"primes")],-1)),s[100]||(s[100]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[101]||(s[101]=i('
julia
primes(G::ZZGenus) -> Vector{ZZRingElem}
Return the list of primes of the local symbols of G.
Note that 2 is always in the output since the 2-adic symbol of a ZZGenus is, by convention, always defined.
source
',4))]),t("details",W,[t("summary",null,[s[102]||(s[102]=t("a",{id:"is_integral-Tuple{ZZGenus}",href:"#is_integral-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_integral")],-1)),s[103]||(s[103]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[110]||(s[110]=i('
julia
is_integral(G::ZZGenus) -> Bool
',1)),t("p",null,[s[106]||(s[106]=e("Return whether ")),s[107]||(s[107]=t("code",null,"G",-1)),s[108]||(s[108]=e(" is a genus of integral ")),t("mjx-container",K,[(o(),n("svg",Y,s[104]||(s[104]=[i('',1)]))),s[105]||(s[105]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[109]||(s[109]=e("-lattices."))]),s[111]||(s[111]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[324]||(s[324]=i('
Discriminant group ​
discriminant_group(::ZZGenus)
Primary genera ​
',3)),t("details",_,[t("summary",null,[s[112]||(s[112]=t("a",{id:"is_primary_with_prime-Tuple{ZZGenus}",href:"#is_primary_with_prime-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),s[113]||(s[113]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[128]||(s[128]=i('
julia
is_primary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
',1)),t("p",null,[s[116]||(s[116]=e("Given a genus of ")),t("mjx-container",ss,[(o(),n("svg",ts,s[114]||(s[114]=[i('',1)]))),s[115]||(s[115]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[117]||(s[117]=e("-lattices ")),s[118]||(s[118]=t("code",null,"G",-1)),s[119]||(s[119]=e(", return whether it is primary, that is whether the bilinear form is integral and the associated discriminant form (see ")),s[120]||(s[120]=t("a",{href:"/Hecke.jl/v0.34.8/manual/quad_forms/discriminant_group#discriminant_group-Tuple{ZZLat}"},[t("code",null,"discriminant_group")],-1)),s[121]||(s[121]=e(") is a ")),s[122]||(s[122]=t("code",null,"p",-1)),s[123]||(s[123]=e("-group for some prime number ")),s[124]||(s[124]=t("code",null,"p",-1)),s[125]||(s[125]=e(". In case it is, ")),s[126]||(s[126]=t("code",null,"p",-1)),s[127]||(s[127]=e(" is also returned as second output."))]),s[129]||(s[129]=t("p",null,[e("Note that for unimodular genera, this function returns "),t("code",null,"(true, 1)"),e(". If the genus is not primary, the second return value is "),t("code",null,"-1"),e(" by default.")],-1)),s[130]||(s[130]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",es,[t("summary",null,[s[131]||(s[131]=t("a",{id:"is_primary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}",href:"#is_primary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"is_primary")],-1)),s[132]||(s[132]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[137]||(s[137]=i('
julia
is_primary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
',1)),t("p",null,[s[135]||(s[135]=e("Given a genus of integral ")),t("mjx-container",is,[(o(),n("svg",as,s[133]||(s[133]=[i('',1)]))),s[134]||(s[134]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[136]||(s[136]=i('-lattices G and a prime number p, return whether G is p-primary, that is whether the associated discriminant form (see discriminant_group) is a p-group.',13))]),s[138]||(s[138]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",ls,[t("summary",null,[s[139]||(s[139]=t("a",{id:"is_elementary_with_prime-Tuple{ZZGenus}",href:"#is_elementary_with_prime-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_elementary_with_prime")],-1)),s[140]||(s[140]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[155]||(s[155]=i('
julia
is_elementary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
',1)),t("p",null,[s[143]||(s[143]=e("Given a genus of ")),t("mjx-container",ns,[(o(),n("svg",os,s[141]||(s[141]=[i('',1)]))),s[142]||(s[142]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[144]||(s[144]=e("-lattices ")),s[145]||(s[145]=t("code",null,"G",-1)),s[146]||(s[146]=e(", return whether it is elementary, that is whether the bilinear form is inegtral and the associated discriminant form (see ")),s[147]||(s[147]=t("a",{href:"/Hecke.jl/v0.34.8/manual/quad_forms/discriminant_group#discriminant_group-Tuple{ZZLat}"},[t("code",null,"discriminant_group")],-1)),s[148]||(s[148]=e(") is an elementary ")),s[149]||(s[149]=t("code",null,"p",-1)),s[150]||(s[150]=e("-group for some prime number ")),s[151]||(s[151]=t("code",null,"p",-1)),s[152]||(s[152]=e(". In case it is, ")),s[153]||(s[153]=t("code",null,"p",-1)),s[154]||(s[154]=e(" is also returned as second output."))]),s[156]||(s[156]=t("p",null,[e("Note that for unimodular genera, this function returns "),t("code",null,"(true, 1)"),e(". If the genus is not elementary, the second return value is "),t("code",null,"-1"),e(" by default.")],-1)),s[157]||(s[157]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",rs,[t("summary",null,[s[158]||(s[158]=t("a",{id:"is_elementary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}",href:"#is_elementary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"is_elementary")],-1)),s[159]||(s[159]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[164]||(s[164]=i('
julia
is_elementary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
',1)),t("p",null,[s[162]||(s[162]=e("Given a genus of integral ")),t("mjx-container",ps,[(o(),n("svg",ds,s[160]||(s[160]=[i('',1)]))),s[161]||(s[161]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[163]||(s[163]=i('-lattices G and a prime number p, return whether G is p-elementary, that is whether its associated discriminant form (see discriminant_group) is an elementary p-group.',13))]),s[165]||(s[165]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[325]||(s[325]=t("h3",{id:"local-Symbol",tabindex:"-1"},[e("local Symbol "),t("a",{class:"header-anchor",href:"#local-Symbol","aria-label":'Permalink to "local Symbol {#local-Symbol}"'},"​")],-1)),t("details",hs,[t("summary",null,[s[166]||(s[166]=t("a",{id:"local_symbol-Tuple{ZZGenus, Any}",href:"#local_symbol-Tuple{ZZGenus, Any}"},[t("span",{class:"jlbinding"},"local_symbol")],-1)),s[167]||(s[167]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=i('
julia
local_symbol(G::ZZGenus, p) -> ZZLocalGenus
Return the local symbol at p.
source
',3))]),s[326]||(s[326]=t("h2",{id:"representative-s",tabindex:"-1"},[e("Representative(s) "),t("a",{class:"header-anchor",href:"#representative-s","aria-label":'Permalink to "Representative(s)"'},"​")],-1)),t("details",Qs,[t("summary",null,[s[169]||(s[169]=t("a",{id:"quadratic_space-Tuple{ZZGenus}",href:"#quadratic_space-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"quadratic_space")],-1)),s[170]||(s[170]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[171]||(s[171]=i('
julia
quadratic_space(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}
Return the quadratic space defined by this genus.
source
',3))]),t("details",ms,[t("summary",null,[s[172]||(s[172]=t("a",{id:"rational_representative-Tuple{ZZGenus}",href:"#rational_representative-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"rational_representative")],-1)),s[173]||(s[173]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[174]||(s[174]=i('
julia
rational_representative(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}
Return the quadratic space defined by this genus.
source
',3))]),t("details",gs,[t("summary",null,[s[175]||(s[175]=t("a",{id:"representative-Tuple{ZZGenus}",href:"#representative-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"representative")],-1)),s[176]||(s[176]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[177]||(s[177]=i('
julia
representative(G::ZZGenus) -> ZZLat
Compute a representative of this genus && cache it.
source
',3))]),t("details",ks,[t("summary",null,[s[178]||(s[178]=t("a",{id:"representatives-Tuple{ZZGenus}",href:"#representatives-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"representatives")],-1)),s[179]||(s[179]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[180]||(s[180]=i('
julia
representatives(G::ZZGenus) -> Vector{ZZLat}
Return a list of representatives of the isometry classes in this genus.
source
',3))]),t("details",Ts,[t("summary",null,[s[181]||(s[181]=t("a",{id:"mass-Tuple{ZZGenus}",href:"#mass-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"mass")],-1)),s[182]||(s[182]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[189]||(s[189]=i('
julia
mass(G::ZZGenus) -> QQFieldElem
Return the mass of this genus.
',2)),t("p",null,[s[185]||(s[185]=e("The genus must be definite. Let ")),s[186]||(s[186]=t("code",null,"L_1, ... L_n",-1)),s[187]||(s[187]=e(" be a complete list of representatives of the isometry classes in this genus. Its mass is defined as ")),t("mjx-container",us,[(o(),n("svg",ys,s[183]||(s[183]=[i('',1)]))),s[184]||(s[184]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("munderover",null,[t("mo",{"data-mjx-texclass":"OP"},"∑"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mo",null,"="),t("mn",null,"1")]),t("mi",null,"n")]),t("mfrac",null,[t("mn",null,"1"),t("mrow",null,[t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mi",null,"O"),t("mo",{stretchy:"false"},"("),t("msub",null,[t("mi",null,"L"),t("mi",null,"i")]),t("mo",{stretchy:"false"},")"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])])])],-1))]),s[188]||(s[188]=e("."))]),s[190]||(s[190]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",cs,[t("summary",null,[s[191]||(s[191]=t("a",{id:"rescale-Tuple{ZZGenus, Union{Integer, QQFieldElem, ZZRingElem, Rational}}",href:"#rescale-Tuple{ZZGenus, Union{Integer, QQFieldElem, ZZRingElem, Rational}}"},[t("span",{class:"jlbinding"},"rescale")],-1)),s[192]||(s[192]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[203]||(s[203]=i('
julia
rescale(G::ZZGenus, a::RationalUnion) -> ZZGenus
',1)),t("p",null,[s[195]||(s[195]=e("Given a genus symbol ")),s[196]||(s[196]=t("code",null,"G",-1)),s[197]||(s[197]=e(" of ")),t("mjx-container",bs,[(o(),n("svg",xs,s[193]||(s[193]=[i('',1)]))),s[194]||(s[194]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[198]||(s[198]=e("-lattices, return the genus symbol of any representative of ")),s[199]||(s[199]=t("code",null,"G",-1)),s[200]||(s[200]=e(" rescaled by ")),s[201]||(s[201]=t("code",null,"a",-1)),s[202]||(s[202]=e("."))]),s[204]||(s[204]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[327]||(s[327]=t("h2",{id:"Embeddings-and-Representations",tabindex:"-1"},[e("Embeddings and Representations "),t("a",{class:"header-anchor",href:"#Embeddings-and-Representations","aria-label":'Permalink to "Embeddings and Representations {#Embeddings-and-Representations}"'},"​")],-1)),t("details",fs,[t("summary",null,[s[205]||(s[205]=t("a",{id:"represents-Tuple{ZZGenus, ZZGenus}",href:"#represents-Tuple{ZZGenus, ZZGenus}"},[t("span",{class:"jlbinding"},"represents")],-1)),s[206]||(s[206]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[207]||(s[207]=i('
julia
represents(G1::ZZGenus, G2::ZZGenus) -> Bool
Return if G1 represents G2. That is if some element in the genus of G1 represents some element in the genus of G2.
source
',3))]),s[328]||(s[328]=t("h2",{id:"Local-genus-Symbols",tabindex:"-1"},[e("Local genus Symbols "),t("a",{class:"header-anchor",href:"#Local-genus-Symbols","aria-label":'Permalink to "Local genus Symbols {#Local-genus-Symbols}"'},"​")],-1)),t("details",Es,[t("summary",null,[s[208]||(s[208]=t("a",{id:"ZZLocalGenus",href:"#ZZLocalGenus"},[t("span",{class:"jlbinding"},"ZZLocalGenus")],-1)),s[209]||(s[209]=e()),l(a,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[210]||(s[210]=i('
julia
ZZLocalGenus
Local genus symbol over a p-adic ring.
The genus symbol of a component p^m A for odd prime = p is of the form (m,n,d), where
  • m = valuation of the component
  • n = rank of A
  • d = det(A) \\in \\{1,u\\} for a normalized quadratic non-residue u.
The genus symbol of a component 2^m A is of the form (m, n, s, d, o), where
  • m = valuation of the component
  • n = rank of A
  • d = det(A) in {1,3,5,7}
  • s = 0 (or 1) if even (or odd)
  • o = oddity of A (= 0 if s = 0) in Z/8Z = the trace of the diagonalization of A
The genus symbol is a list of such symbols (ordered by m) for each of the Jordan blocks A_1,...,A_t.
Reference: [5] Chapter 15, Section 7.
Arguments
  • prime: a prime number
  • symbol: the list of invariants for Jordan blocks A_t,...,A_t given as a list of lists of integers
source
',11))]),s[329]||(s[329]=t("h3",{id:"creation",tabindex:"-1"},[e("Creation "),t("a",{class:"header-anchor",href:"#creation","aria-label":'Permalink to "Creation"'},"​")],-1)),t("details",vs,[t("summary",null,[s[211]||(s[211]=t("a",{id:"genus-Tuple{ZZLat, Union{Integer, ZZRingElem}}",href:"#genus-Tuple{ZZLat, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[212]||(s[212]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[213]||(s[213]=i('
julia
genus(L::ZZLat, p::IntegerUnion) -> ZZLocalGenus
Return the local genus symbol of L at the prime p.
source
',3))]),t("details",ws,[t("summary",null,[s[214]||(s[214]=t("a",{id:"genus-Tuple{QQMatrix, Union{Integer, ZZRingElem}}",href:"#genus-Tuple{QQMatrix, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[215]||(s[215]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[216]||(s[216]=i('
julia
genus(A::QQMatrix, p::IntegerUnion) -> ZZLocalGenus
Return the local genus symbol of a Z-lattice with gram matrix A at the prime p.
source
',3))]),s[330]||(s[330]=t("h3",{id:"attributes",tabindex:"-1"},[e("Attributes "),t("a",{class:"header-anchor",href:"#attributes","aria-label":'Permalink to "Attributes"'},"​")],-1)),t("details",Zs,[t("summary",null,[s[217]||(s[217]=t("a",{id:"prime-Tuple{ZZLocalGenus}",href:"#prime-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"prime")],-1)),s[218]||(s[218]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[219]||(s[219]=i('
julia
prime(S::ZZLocalGenus) -> ZZRingElem
Return the prime p of this p-adic genus.
source
',3))]),t("details",js,[t("summary",null,[s[220]||(s[220]=t("a",{id:"iseven-Tuple{ZZLocalGenus}",href:"#iseven-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"iseven")],-1)),s[221]||(s[221]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[222]||(s[222]=i('
julia
iseven(S::ZZLocalGenus) -> Bool
Return if the underlying p-adic lattice is even.
If p is odd, every lattice is even.
source
',4))]),t("details",Ls,[t("summary",null,[s[223]||(s[223]=t("a",{id:"symbol-Tuple{ZZLocalGenus, Int64}",href:"#symbol-Tuple{ZZLocalGenus, Int64}"},[t("span",{class:"jlbinding"},"symbol")],-1)),s[224]||(s[224]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[225]||(s[225]=i('
julia
symbol(S::ZZLocalGenus, scale::Int) -> Vector{Int}
Return the underlying lists of integers for the Jordan block of the given scale
source
',3))]),t("details",Cs,[t("summary",null,[s[226]||(s[226]=t("a",{id:"hasse_invariant-Tuple{ZZLocalGenus}",href:"#hasse_invariant-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"hasse_invariant")],-1)),s[227]||(s[227]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[230]||(s[230]=i('
julia
hasse_invariant(S::ZZLocalGenus) -> Int
Return the Hasse invariant of a representative. If the representative is diagonal (a_1, ... , a_n) Then the Hasse invariant is
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julia
det(S::ZZLocalGenus) -> QQFieldElem
Return an rational representing the determinant of this genus.
source
',3))]),t("details",Gs,[t("summary",null,[s[236]||(s[236]=t("a",{id:"dim-Tuple{ZZLocalGenus}",href:"#dim-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"dim")],-1)),s[237]||(s[237]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[238]||(s[238]=i('
julia
dim(S::ZZLocalGenus) -> Int
Return the dimension of this genus.
source
',3))]),t("details",Ds,[t("summary",null,[s[239]||(s[239]=t("a",{id:"rank-Tuple{ZZLocalGenus}",href:"#rank-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"rank")],-1)),s[240]||(s[240]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[241]||(s[241]=i('
julia
rank(S::ZZLocalGenus) -> Int
Return the rank of (a representative of) S.
source
',3))]),t("details",Bs,[t("summary",null,[s[242]||(s[242]=t("a",{id:"excess-Tuple{ZZLocalGenus}",href:"#excess-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"excess")],-1)),s[243]||(s[243]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[244]||(s[244]=i('
julia
excess(S::ZZLocalGenus) -> zzModRingElem
Return the p-excess of the quadratic form whose Hessian matrix is the symmetric matrix A.
When p = 2 the p-excess is called the oddity. The p-excess is always even && is divisible by 4 if p is congruent 1 mod 4.
Reference
[5] pp 370-371.
source
',6))]),t("details",As,[t("summary",null,[s[245]||(s[245]=t("a",{id:"signature-Tuple{ZZLocalGenus}",href:"#signature-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"signature")],-1)),s[246]||(s[246]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[254]||(s[254]=i('
julia
signature(S::ZZLocalGenus) -> zzModRingElem
',1)),t("p",null,[s[251]||(s[251]=e("Return the ")),t("mjx-container",Vs,[(o(),n("svg",Rs,s[247]||(s[247]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z",style:{"stroke-width":"3"}})])])],-1)]))),s[248]||(s[248]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"p")])],-1))]),s[252]||(s[252]=e("-signature of this ")),t("mjx-container",Os,[(o(),n("svg",Ss,s[249]||(s[249]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z",style:{"stroke-width":"3"}})])])],-1)]))),s[250]||(s[250]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"p")])],-1))]),s[253]||(s[253]=e("-adic form."))]),s[255]||(s[255]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Is,[t("summary",null,[s[256]||(s[256]=t("a",{id:"oddity-Tuple{ZZLocalGenus}",href:"#oddity-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"oddity")],-1)),s[257]||(s[257]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[262]||(s[262]=i('
julia
oddity(S::ZZLocalGenus) -> zzModRingElem
',1)),t("p",null,[s[260]||(s[260]=e("Return the oddity of this even form. The oddity is also called the ")),t("mjx-container",zs,[(o(),n("svg",Js,s[258]||(s[258]=[t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mn"},[t("path",{"data-c":"32",d:"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z",style:{"stroke-width":"3"}})])])],-1)]))),s[259]||(s[259]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mn",null,"2")])],-1))]),s[261]||(s[261]=e("-signature"))]),s[263]||(s[263]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Ps,[t("summary",null,[s[264]||(s[264]=t("a",{id:"scale-Tuple{ZZLocalGenus}",href:"#scale-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"scale")],-1)),s[265]||(s[265]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[266]||(s[266]=i('
julia
scale(S::ZZLocalGenus) -> QQFieldElem
Return the scale of this local genus.
Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).
source
',4))]),t("details",Us,[t("summary",null,[s[267]||(s[267]=t("a",{id:"norm-Tuple{ZZLocalGenus}",href:"#norm-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"norm")],-1)),s[268]||(s[268]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[279]||(s[279]=i('
julia
norm(S::ZZLocalGenus) -> QQFieldElem
Return the norm of this local genus.
',2)),t("p",null,[s[271]||(s[271]=e("Let ")),s[272]||(s[272]=t("code",null,"L",-1)),s[273]||(s[273]=e(" be a lattice with bilinear form ")),s[274]||(s[274]=t("code",null,"b",-1)),s[275]||(s[275]=e(". The norm of ")),s[276]||(s[276]=t("code",null,"(L,b)",-1)),s[277]||(s[277]=e(" is defined as the ideal generated by ")),t("mjx-container",Xs,[(o(),n("svg",qs,s[269]||(s[269]=[i('',1)]))),s[270]||(s[270]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mo",{fence:"false",stretchy:"false"},"{"),t("mi",null,"b"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",null,","),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mi",null,"x"),t("mo",null,"∈"),t("mi",null,"L"),t("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[278]||(s[278]=e("."))]),s[280]||(s[280]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",Ns,[t("summary",null,[s[281]||(s[281]=t("a",{id:"level-Tuple{ZZLocalGenus}",href:"#level-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"level")],-1)),s[282]||(s[282]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[283]||(s[283]=i('
julia
level(S::ZZLocalGenus) -> QQFieldElem
Return the maximal scale of a jordan component.
source
',3))]),s[331]||(s[331]=t("h3",{id:"representative",tabindex:"-1"},[e("Representative "),t("a",{class:"header-anchor",href:"#representative","aria-label":'Permalink to "Representative"'},"​")],-1)),t("details",$s,[t("summary",null,[s[284]||(s[284]=t("a",{id:"representative-Tuple{ZZLocalGenus}",href:"#representative-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"representative")],-1)),s[285]||(s[285]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[286]||(s[286]=i('
julia
representative(S::ZZLocalGenus) -> ZZLat
Return an integer lattice which represents this local genus.
source
',3))]),t("details",Ws,[t("summary",null,[s[287]||(s[287]=t("a",{id:"gram_matrix-Tuple{ZZLocalGenus}",href:"#gram_matrix-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"gram_matrix")],-1)),s[288]||(s[288]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[289]||(s[289]=i('
julia
gram_matrix(S::ZZLocalGenus) -> MatElem
Return a gram matrix of some representative of this local genus.
source
',3))]),t("details",Ks,[t("summary",null,[s[290]||(s[290]=t("a",{id:"rescale-Tuple{ZZLocalGenus, Union{Integer, QQFieldElem, ZZRingElem, Rational}}",href:"#rescale-Tuple{ZZLocalGenus, Union{Integer, QQFieldElem, ZZRingElem, Rational}}"},[t("span",{class:"jlbinding"},"rescale")],-1)),s[291]||(s[291]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[302]||(s[302]=i('
julia
rescale(G::ZZLocalGenus, a::RationalUnion) -> ZZLocalGenus
',1)),t("p",null,[s[294]||(s[294]=e("Given a local genus symbol ")),s[295]||(s[295]=t("code",null,"G",-1)),s[296]||(s[296]=e(" of ")),t("mjx-container",Ys,[(o(),n("svg",_s,s[292]||(s[292]=[i('',1)]))),s[293]||(s[293]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[297]||(s[297]=e("-lattices, return the local genus symbol of any representative of ")),s[298]||(s[298]=t("code",null,"G",-1)),s[299]||(s[299]=e(" rescaled by ")),s[300]||(s[300]=t("code",null,"a",-1)),s[301]||(s[301]=e("."))]),s[303]||(s[303]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[332]||(s[332]=t("h3",{id:"Direct-sums",tabindex:"-1"},[e("Direct sums "),t("a",{class:"header-anchor",href:"#Direct-sums","aria-label":'Permalink to "Direct sums {#Direct-sums}"'},"​")],-1)),t("details",s1,[t("summary",null,[s[304]||(s[304]=t("a",{id:"direct_sum-Tuple{ZZLocalGenus, ZZLocalGenus}",href:"#direct_sum-Tuple{ZZLocalGenus, ZZLocalGenus}"},[t("span",{class:"jlbinding"},"direct_sum")],-1)),s[305]||(s[305]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[306]||(s[306]=i('
julia
direct_sum(S1::ZZLocalGenus, S2::ZZLocalGenus) -> ZZLocalGenus
Return the local genus of the direct sum of two representatives.
source
',3))]),s[333]||(s[333]=t("h3",{id:"embeddings-representations",tabindex:"-1"},[e("Embeddings/Representations "),t("a",{class:"header-anchor",href:"#embeddings-representations","aria-label":'Permalink to "Embeddings/Representations"'},"​")],-1)),t("details",t1,[t("summary",null,[s[307]||(s[307]=t("a",{id:"represents-Tuple{ZZLocalGenus, ZZLocalGenus}",href:"#represents-Tuple{ZZLocalGenus, ZZLocalGenus}"},[t("span",{class:"jlbinding"},"represents")],-1)),s[308]||(s[308]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[315]||(s[315]=i('
julia
represents(g1::ZZLocalGenus, g2::ZZLocalGenus) -> Bool
Return whether g1 represents g2.
',2)),t("p",null,[s[311]||(s[311]=e("Based on O'Meara Integral Representations of Quadratic Forms Over Local Fields Note that for ")),s[312]||(s[312]=t("code",null,"p == 2",-1)),s[313]||(s[313]=e(" there is a typo in O'Meara Theorem 3 (V). The correct statement is (V) ")),t("mjx-container",e1,[(o(),n("svg",i1,s[309]||(s[309]=[i('',1)]))),s[310]||(s[310]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mn",null,"2"),t("mi",null,"i")]),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",null,"+"),t("mn",null,"4"),t("mi",null,"ω"),t("mo",{stretchy:"false"},")"),t("mo",{accent:"false",stretchy:"false"},"→"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"L")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mo",null,"+"),t("mn",null,"1")])]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"l")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",{stretchy:"false"},"["),t("mi",null,"i"),t("mo",{stretchy:"false"},"]")])])])],-1))]),s[314]||(s[314]=e("."))]),s[316]||(s[316]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))])])}const Q1=r(d,[["render",a1]]);export{h1 as __pageData,Q1 as default};
+       even=false)                                         -> Vector{ZZGenus}
`,1)),t("p",null,[s[54]||(s[54]=e("Return a list of all genera with the given conditions. Genera of non-integral ")),t("mjx-container",B,[(o(),n("svg",A,s[52]||(s[52]=[i('',1)]))),s[53]||(s[53]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[55]||(s[55]=e("-lattices are also supported."))]),s[57]||(s[57]=i('

Arguments

  • sig_pair: a pair of non-negative integers giving the signature

  • determinant: a rational number; the sign is ignored

  • min_scale: a rational number; return only genera whose scale is an integer multiple of min_scale (default: min(one(QQ), QQ(abs(determinant))))

  • max_scale: a rational number; return only genera such that max_scale is an integer multiple of the scale (default: max(one(QQ), QQ(abs(determinant))))

  • even: boolean; if set to true, return only the even genera (default: false)

source

',3))]),s[322]||(s[322]=t("h3",{id:"From-other-genus-symbols",tabindex:"-1"},[e("From other genus symbols "),t("a",{class:"header-anchor",href:"#From-other-genus-symbols","aria-label":'Permalink to "From other genus symbols {#From-other-genus-symbols}"'},"​")],-1)),t("details",V,[t("summary",null,[s[58]||(s[58]=t("a",{id:"direct_sum-Tuple{ZZGenus, ZZGenus}",href:"#direct_sum-Tuple{ZZGenus, ZZGenus}"},[t("span",{class:"jlbinding"},"direct_sum")],-1)),s[59]||(s[59]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[60]||(s[60]=i('
julia
direct_sum(G1::ZZGenus, G2::ZZGenus) -> ZZGenus

Return the genus of the direct sum of G1 and G2.

The direct sum is defined via representatives.

source

',4))]),s[323]||(s[323]=t("h2",{id:"Attributes-of-the-genus",tabindex:"-1"},[e("Attributes of the genus "),t("a",{class:"header-anchor",href:"#Attributes-of-the-genus","aria-label":'Permalink to "Attributes of the genus {#Attributes-of-the-genus}"'},"​")],-1)),t("details",R,[t("summary",null,[s[61]||(s[61]=t("a",{id:"dim-Tuple{ZZGenus}",href:"#dim-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"dim")],-1)),s[62]||(s[62]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[63]||(s[63]=i('
julia
dim(G::ZZGenus) -> Int

Return the dimension of this genus.

source

',3))]),t("details",O,[t("summary",null,[s[64]||(s[64]=t("a",{id:"rank-Tuple{ZZGenus}",href:"#rank-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"rank")],-1)),s[65]||(s[65]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[66]||(s[66]=i('
julia
rank(G::ZZGenus) -> Int

Return the rank of a (representative of) the genus G.

source

',3))]),t("details",S,[t("summary",null,[s[67]||(s[67]=t("a",{id:"signature-Tuple{ZZGenus}",href:"#signature-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"signature")],-1)),s[68]||(s[68]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[69]||(s[69]=i('
julia
signature(G::ZZGenus) -> Int

Return the signature of this genus.

The signature is p - n where p is the number of positive eigenvalues and n the number of negative eigenvalues.

source

',4))]),t("details",I,[t("summary",null,[s[70]||(s[70]=t("a",{id:"det-Tuple{ZZGenus}",href:"#det-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"det")],-1)),s[71]||(s[71]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[72]||(s[72]=i('
julia
det(G::ZZGenus) -> QQFieldElem

Return the determinant of this genus.

source

',3))]),t("details",z,[t("summary",null,[s[73]||(s[73]=t("a",{id:"iseven-Tuple{ZZGenus}",href:"#iseven-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"iseven")],-1)),s[74]||(s[74]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[75]||(s[75]=i('
julia
iseven(G::ZZGenus) -> Bool

Return if this genus is even.

source

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julia
is_definite(G::ZZGenus) -> Bool

Return if this genus is definite.

source

',3))]),t("details",P,[t("summary",null,[s[79]||(s[79]=t("a",{id:"level-Tuple{ZZGenus}",href:"#level-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"level")],-1)),s[80]||(s[80]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[81]||(s[81]=i('
julia
level(G::ZZGenus) -> QQFieldElem

Return the level of this genus.

This is the denominator of the inverse gram matrix of a representative.

source

',4))]),t("details",U,[t("summary",null,[s[82]||(s[82]=t("a",{id:"scale-Tuple{ZZGenus}",href:"#scale-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"scale")],-1)),s[83]||(s[83]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[84]||(s[84]=i('
julia
scale(G::ZZGenus) -> QQFieldElem

Return the scale of this genus.

Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).

source

',4))]),t("details",X,[t("summary",null,[s[85]||(s[85]=t("a",{id:"norm-Tuple{ZZGenus}",href:"#norm-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"norm")],-1)),s[86]||(s[86]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[97]||(s[97]=i('
julia
norm(G::ZZGenus) -> QQFieldElem

Return the norm of this genus.

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julia
primes(G::ZZGenus) -> Vector{ZZRingElem}

Return the list of primes of the local symbols of G.

Note that 2 is always in the output since the 2-adic symbol of a ZZGenus is, by convention, always defined.

source

',4))]),t("details",W,[t("summary",null,[s[102]||(s[102]=t("a",{id:"is_integral-Tuple{ZZGenus}",href:"#is_integral-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_integral")],-1)),s[103]||(s[103]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[110]||(s[110]=i('
julia
is_integral(G::ZZGenus) -> Bool
',1)),t("p",null,[s[106]||(s[106]=e("Return whether ")),s[107]||(s[107]=t("code",null,"G",-1)),s[108]||(s[108]=e(" is a genus of integral ")),t("mjx-container",K,[(o(),n("svg",Y,s[104]||(s[104]=[i('',1)]))),s[105]||(s[105]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[109]||(s[109]=e("-lattices."))]),s[111]||(s[111]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[324]||(s[324]=i('

Discriminant group ​

discriminant_group(::ZZGenus)

Primary genera ​

',3)),t("details",_,[t("summary",null,[s[112]||(s[112]=t("a",{id:"is_primary_with_prime-Tuple{ZZGenus}",href:"#is_primary_with_prime-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),s[113]||(s[113]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[128]||(s[128]=i('
julia
is_primary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
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julia
is_primary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
',1)),t("p",null,[s[135]||(s[135]=e("Given a genus of integral ")),t("mjx-container",is,[(o(),n("svg",as,s[133]||(s[133]=[i('',1)]))),s[134]||(s[134]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[136]||(s[136]=i('-lattices G and a prime number p, return whether G is p-primary, that is whether the associated discriminant form (see discriminant_group) is a p-group.',13))]),s[138]||(s[138]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",ls,[t("summary",null,[s[139]||(s[139]=t("a",{id:"is_elementary_with_prime-Tuple{ZZGenus}",href:"#is_elementary_with_prime-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"is_elementary_with_prime")],-1)),s[140]||(s[140]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[155]||(s[155]=i('
julia
is_elementary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
',1)),t("p",null,[s[143]||(s[143]=e("Given a genus of ")),t("mjx-container",ns,[(o(),n("svg",os,s[141]||(s[141]=[i('',1)]))),s[142]||(s[142]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[144]||(s[144]=e("-lattices ")),s[145]||(s[145]=t("code",null,"G",-1)),s[146]||(s[146]=e(", return whether it is elementary, that is whether the bilinear form is inegtral and the associated discriminant form (see ")),s[147]||(s[147]=t("a",{href:"/v0.34.8/manual/quad_forms/discriminant_group#discriminant_group-Tuple{ZZLat}"},[t("code",null,"discriminant_group")],-1)),s[148]||(s[148]=e(") is an elementary ")),s[149]||(s[149]=t("code",null,"p",-1)),s[150]||(s[150]=e("-group for some prime number ")),s[151]||(s[151]=t("code",null,"p",-1)),s[152]||(s[152]=e(". In case it is, ")),s[153]||(s[153]=t("code",null,"p",-1)),s[154]||(s[154]=e(" is also returned as second output."))]),s[156]||(s[156]=t("p",null,[e("Note that for unimodular genera, this function returns "),t("code",null,"(true, 1)"),e(". If the genus is not elementary, the second return value is "),t("code",null,"-1"),e(" by default.")],-1)),s[157]||(s[157]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),t("details",rs,[t("summary",null,[s[158]||(s[158]=t("a",{id:"is_elementary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}",href:"#is_elementary-Tuple{ZZGenus, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"is_elementary")],-1)),s[159]||(s[159]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[164]||(s[164]=i('
julia
is_elementary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
',1)),t("p",null,[s[162]||(s[162]=e("Given a genus of integral ")),t("mjx-container",ps,[(o(),n("svg",ds,s[160]||(s[160]=[i('',1)]))),s[161]||(s[161]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[163]||(s[163]=i('-lattices G and a prime number p, return whether G is p-elementary, that is whether its associated discriminant form (see discriminant_group) is an elementary p-group.',13))]),s[165]||(s[165]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[325]||(s[325]=t("h3",{id:"local-Symbol",tabindex:"-1"},[e("local Symbol "),t("a",{class:"header-anchor",href:"#local-Symbol","aria-label":'Permalink to "local Symbol {#local-Symbol}"'},"​")],-1)),t("details",hs,[t("summary",null,[s[166]||(s[166]=t("a",{id:"local_symbol-Tuple{ZZGenus, Any}",href:"#local_symbol-Tuple{ZZGenus, Any}"},[t("span",{class:"jlbinding"},"local_symbol")],-1)),s[167]||(s[167]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[168]||(s[168]=i('
julia
local_symbol(G::ZZGenus, p) -> ZZLocalGenus

Return the local symbol at p.

source

',3))]),s[326]||(s[326]=t("h2",{id:"representative-s",tabindex:"-1"},[e("Representative(s) "),t("a",{class:"header-anchor",href:"#representative-s","aria-label":'Permalink to "Representative(s)"'},"​")],-1)),t("details",Qs,[t("summary",null,[s[169]||(s[169]=t("a",{id:"quadratic_space-Tuple{ZZGenus}",href:"#quadratic_space-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"quadratic_space")],-1)),s[170]||(s[170]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[171]||(s[171]=i('
julia
quadratic_space(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}

Return the quadratic space defined by this genus.

source

',3))]),t("details",ms,[t("summary",null,[s[172]||(s[172]=t("a",{id:"rational_representative-Tuple{ZZGenus}",href:"#rational_representative-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"rational_representative")],-1)),s[173]||(s[173]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[174]||(s[174]=i('
julia
rational_representative(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}

Return the quadratic space defined by this genus.

source

',3))]),t("details",gs,[t("summary",null,[s[175]||(s[175]=t("a",{id:"representative-Tuple{ZZGenus}",href:"#representative-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"representative")],-1)),s[176]||(s[176]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[177]||(s[177]=i('
julia
representative(G::ZZGenus) -> ZZLat

Compute a representative of this genus && cache it.

source

',3))]),t("details",ks,[t("summary",null,[s[178]||(s[178]=t("a",{id:"representatives-Tuple{ZZGenus}",href:"#representatives-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"representatives")],-1)),s[179]||(s[179]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[180]||(s[180]=i('
julia
representatives(G::ZZGenus) -> Vector{ZZLat}

Return a list of representatives of the isometry classes in this genus.

source

',3))]),t("details",Ts,[t("summary",null,[s[181]||(s[181]=t("a",{id:"mass-Tuple{ZZGenus}",href:"#mass-Tuple{ZZGenus}"},[t("span",{class:"jlbinding"},"mass")],-1)),s[182]||(s[182]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[189]||(s[189]=i('
julia
mass(G::ZZGenus) -> QQFieldElem

Return the mass of this genus.

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julia
rescale(G::ZZGenus, a::RationalUnion) -> ZZGenus
',1)),t("p",null,[s[195]||(s[195]=e("Given a genus symbol ")),s[196]||(s[196]=t("code",null,"G",-1)),s[197]||(s[197]=e(" of ")),t("mjx-container",bs,[(o(),n("svg",xs,s[193]||(s[193]=[i('',1)]))),s[194]||(s[194]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[198]||(s[198]=e("-lattices, return the genus symbol of any representative of ")),s[199]||(s[199]=t("code",null,"G",-1)),s[200]||(s[200]=e(" rescaled by ")),s[201]||(s[201]=t("code",null,"a",-1)),s[202]||(s[202]=e("."))]),s[204]||(s[204]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[327]||(s[327]=t("h2",{id:"Embeddings-and-Representations",tabindex:"-1"},[e("Embeddings and Representations "),t("a",{class:"header-anchor",href:"#Embeddings-and-Representations","aria-label":'Permalink to "Embeddings and Representations {#Embeddings-and-Representations}"'},"​")],-1)),t("details",fs,[t("summary",null,[s[205]||(s[205]=t("a",{id:"represents-Tuple{ZZGenus, ZZGenus}",href:"#represents-Tuple{ZZGenus, ZZGenus}"},[t("span",{class:"jlbinding"},"represents")],-1)),s[206]||(s[206]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[207]||(s[207]=i('
julia
represents(G1::ZZGenus, G2::ZZGenus) -> Bool

Return if G1 represents G2. That is if some element in the genus of G1 represents some element in the genus of G2.

source

',3))]),s[328]||(s[328]=t("h2",{id:"Local-genus-Symbols",tabindex:"-1"},[e("Local genus Symbols "),t("a",{class:"header-anchor",href:"#Local-genus-Symbols","aria-label":'Permalink to "Local genus Symbols {#Local-genus-Symbols}"'},"​")],-1)),t("details",Es,[t("summary",null,[s[208]||(s[208]=t("a",{id:"ZZLocalGenus",href:"#ZZLocalGenus"},[t("span",{class:"jlbinding"},"ZZLocalGenus")],-1)),s[209]||(s[209]=e()),l(a,{type:"info",class:"jlObjectType jlType",text:"Type"})]),s[210]||(s[210]=i('
julia
ZZLocalGenus

Local genus symbol over a p-adic ring.

The genus symbol of a component p^m A for odd prime = p is of the form (m,n,d), where

  • m = valuation of the component

  • n = rank of A

  • d = det(A) \\in \\{1,u\\} for a normalized quadratic non-residue u.

The genus symbol of a component 2^m A is of the form (m, n, s, d, o), where

  • m = valuation of the component

  • n = rank of A

  • d = det(A) in {1,3,5,7}

  • s = 0 (or 1) if even (or odd)

  • o = oddity of A (= 0 if s = 0) in Z/8Z = the trace of the diagonalization of A

The genus symbol is a list of such symbols (ordered by m) for each of the Jordan blocks A_1,...,A_t.

Reference: [5] Chapter 15, Section 7.

Arguments

  • prime: a prime number

  • symbol: the list of invariants for Jordan blocks A_t,...,A_t given as a list of lists of integers

source

',11))]),s[329]||(s[329]=t("h3",{id:"creation",tabindex:"-1"},[e("Creation "),t("a",{class:"header-anchor",href:"#creation","aria-label":'Permalink to "Creation"'},"​")],-1)),t("details",vs,[t("summary",null,[s[211]||(s[211]=t("a",{id:"genus-Tuple{ZZLat, Union{Integer, ZZRingElem}}",href:"#genus-Tuple{ZZLat, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[212]||(s[212]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[213]||(s[213]=i('
julia
genus(L::ZZLat, p::IntegerUnion) -> ZZLocalGenus

Return the local genus symbol of L at the prime p.

source

',3))]),t("details",ws,[t("summary",null,[s[214]||(s[214]=t("a",{id:"genus-Tuple{QQMatrix, Union{Integer, ZZRingElem}}",href:"#genus-Tuple{QQMatrix, Union{Integer, ZZRingElem}}"},[t("span",{class:"jlbinding"},"genus")],-1)),s[215]||(s[215]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[216]||(s[216]=i('
julia
genus(A::QQMatrix, p::IntegerUnion) -> ZZLocalGenus

Return the local genus symbol of a Z-lattice with gram matrix A at the prime p.

source

',3))]),s[330]||(s[330]=t("h3",{id:"attributes",tabindex:"-1"},[e("Attributes "),t("a",{class:"header-anchor",href:"#attributes","aria-label":'Permalink to "Attributes"'},"​")],-1)),t("details",Zs,[t("summary",null,[s[217]||(s[217]=t("a",{id:"prime-Tuple{ZZLocalGenus}",href:"#prime-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"prime")],-1)),s[218]||(s[218]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[219]||(s[219]=i('
julia
prime(S::ZZLocalGenus) -> ZZRingElem

Return the prime p of this p-adic genus.

source

',3))]),t("details",js,[t("summary",null,[s[220]||(s[220]=t("a",{id:"iseven-Tuple{ZZLocalGenus}",href:"#iseven-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"iseven")],-1)),s[221]||(s[221]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[222]||(s[222]=i('
julia
iseven(S::ZZLocalGenus) -> Bool

Return if the underlying p-adic lattice is even.

If p is odd, every lattice is even.

source

',4))]),t("details",Ls,[t("summary",null,[s[223]||(s[223]=t("a",{id:"symbol-Tuple{ZZLocalGenus, Int64}",href:"#symbol-Tuple{ZZLocalGenus, Int64}"},[t("span",{class:"jlbinding"},"symbol")],-1)),s[224]||(s[224]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[225]||(s[225]=i('
julia
symbol(S::ZZLocalGenus, scale::Int) -> Vector{Int}

Return the underlying lists of integers for the Jordan block of the given scale

source

',3))]),t("details",Cs,[t("summary",null,[s[226]||(s[226]=t("a",{id:"hasse_invariant-Tuple{ZZLocalGenus}",href:"#hasse_invariant-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"hasse_invariant")],-1)),s[227]||(s[227]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[230]||(s[230]=i('
julia
hasse_invariant(S::ZZLocalGenus) -> Int

Return the Hasse invariant of a representative. If the representative is diagonal (a_1, ... , a_n) Then the Hasse invariant is

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julia
det(S::ZZLocalGenus) -> QQFieldElem

Return an rational representing the determinant of this genus.

source

',3))]),t("details",Gs,[t("summary",null,[s[236]||(s[236]=t("a",{id:"dim-Tuple{ZZLocalGenus}",href:"#dim-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"dim")],-1)),s[237]||(s[237]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[238]||(s[238]=i('
julia
dim(S::ZZLocalGenus) -> Int

Return the dimension of this genus.

source

',3))]),t("details",Ds,[t("summary",null,[s[239]||(s[239]=t("a",{id:"rank-Tuple{ZZLocalGenus}",href:"#rank-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"rank")],-1)),s[240]||(s[240]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[241]||(s[241]=i('
julia
rank(S::ZZLocalGenus) -> Int

Return the rank of (a representative of) S.

source

',3))]),t("details",Bs,[t("summary",null,[s[242]||(s[242]=t("a",{id:"excess-Tuple{ZZLocalGenus}",href:"#excess-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"excess")],-1)),s[243]||(s[243]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[244]||(s[244]=i('
julia
excess(S::ZZLocalGenus) -> zzModRingElem

Return the p-excess of the quadratic form whose Hessian matrix is the symmetric matrix A.

When p = 2 the p-excess is called the oddity. The p-excess is always even && is divisible by 4 if p is congruent 1 mod 4.

Reference

[5] pp 370-371.

source

',6))]),t("details",As,[t("summary",null,[s[245]||(s[245]=t("a",{id:"signature-Tuple{ZZLocalGenus}",href:"#signature-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"signature")],-1)),s[246]||(s[246]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[254]||(s[254]=i('
julia
signature(S::ZZLocalGenus) -> zzModRingElem
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julia
oddity(S::ZZLocalGenus) -> zzModRingElem
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julia
scale(S::ZZLocalGenus) -> QQFieldElem

Return the scale of this local genus.

Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).

source

',4))]),t("details",Us,[t("summary",null,[s[267]||(s[267]=t("a",{id:"norm-Tuple{ZZLocalGenus}",href:"#norm-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"norm")],-1)),s[268]||(s[268]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[279]||(s[279]=i('
julia
norm(S::ZZLocalGenus) -> QQFieldElem

Return the norm of this local genus.

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julia
level(S::ZZLocalGenus) -> QQFieldElem

Return the maximal scale of a jordan component.

source

',3))]),s[331]||(s[331]=t("h3",{id:"representative",tabindex:"-1"},[e("Representative "),t("a",{class:"header-anchor",href:"#representative","aria-label":'Permalink to "Representative"'},"​")],-1)),t("details",$s,[t("summary",null,[s[284]||(s[284]=t("a",{id:"representative-Tuple{ZZLocalGenus}",href:"#representative-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"representative")],-1)),s[285]||(s[285]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[286]||(s[286]=i('
julia
representative(S::ZZLocalGenus) -> ZZLat

Return an integer lattice which represents this local genus.

source

',3))]),t("details",Ws,[t("summary",null,[s[287]||(s[287]=t("a",{id:"gram_matrix-Tuple{ZZLocalGenus}",href:"#gram_matrix-Tuple{ZZLocalGenus}"},[t("span",{class:"jlbinding"},"gram_matrix")],-1)),s[288]||(s[288]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[289]||(s[289]=i('
julia
gram_matrix(S::ZZLocalGenus) -> MatElem

Return a gram matrix of some representative of this local genus.

source

',3))]),t("details",Ks,[t("summary",null,[s[290]||(s[290]=t("a",{id:"rescale-Tuple{ZZLocalGenus, Union{Integer, QQFieldElem, ZZRingElem, Rational}}",href:"#rescale-Tuple{ZZLocalGenus, Union{Integer, QQFieldElem, ZZRingElem, Rational}}"},[t("span",{class:"jlbinding"},"rescale")],-1)),s[291]||(s[291]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[302]||(s[302]=i('
julia
rescale(G::ZZLocalGenus, a::RationalUnion) -> ZZLocalGenus
',1)),t("p",null,[s[294]||(s[294]=e("Given a local genus symbol ")),s[295]||(s[295]=t("code",null,"G",-1)),s[296]||(s[296]=e(" of ")),t("mjx-container",Ys,[(o(),n("svg",_s,s[292]||(s[292]=[i('',1)]))),s[293]||(s[293]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[297]||(s[297]=e("-lattices, return the local genus symbol of any representative of ")),s[298]||(s[298]=t("code",null,"G",-1)),s[299]||(s[299]=e(" rescaled by ")),s[300]||(s[300]=t("code",null,"a",-1)),s[301]||(s[301]=e("."))]),s[303]||(s[303]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[332]||(s[332]=t("h3",{id:"Direct-sums",tabindex:"-1"},[e("Direct sums "),t("a",{class:"header-anchor",href:"#Direct-sums","aria-label":'Permalink to "Direct sums {#Direct-sums}"'},"​")],-1)),t("details",s1,[t("summary",null,[s[304]||(s[304]=t("a",{id:"direct_sum-Tuple{ZZLocalGenus, ZZLocalGenus}",href:"#direct_sum-Tuple{ZZLocalGenus, ZZLocalGenus}"},[t("span",{class:"jlbinding"},"direct_sum")],-1)),s[305]||(s[305]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[306]||(s[306]=i('
julia
direct_sum(S1::ZZLocalGenus, S2::ZZLocalGenus) -> ZZLocalGenus

Return the local genus of the direct sum of two representatives.

source

',3))]),s[333]||(s[333]=t("h3",{id:"embeddings-representations",tabindex:"-1"},[e("Embeddings/Representations "),t("a",{class:"header-anchor",href:"#embeddings-representations","aria-label":'Permalink to "Embeddings/Representations"'},"​")],-1)),t("details",t1,[t("summary",null,[s[307]||(s[307]=t("a",{id:"represents-Tuple{ZZLocalGenus, ZZLocalGenus}",href:"#represents-Tuple{ZZLocalGenus, ZZLocalGenus}"},[t("span",{class:"jlbinding"},"represents")],-1)),s[308]||(s[308]=e()),l(a,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[315]||(s[315]=i('
julia
represents(g1::ZZLocalGenus, g2::ZZLocalGenus) -> Bool

Return whether g1 represents g2.

',2)),t("p",null,[s[311]||(s[311]=e("Based on O'Meara Integral Representations of Quadratic Forms Over Local Fields Note that for ")),s[312]||(s[312]=t("code",null,"p == 2",-1)),s[313]||(s[313]=e(" there is a typo in O'Meara Theorem 3 (V). The correct statement is (V) ")),t("mjx-container",e1,[(o(),n("svg",i1,s[309]||(s[309]=[i('',1)]))),s[310]||(s[310]=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mn",null,"2"),t("mi",null,"i")]),t("mo",{stretchy:"false"},"("),t("mn",null,"1"),t("mo",null,"+"),t("mn",null,"4"),t("mi",null,"ω"),t("mo",{stretchy:"false"},")"),t("mo",{accent:"false",stretchy:"false"},"→"),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"L")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mo",null,"+"),t("mn",null,"1")])]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"/")]),t("msub",null,[t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",{mathvariant:"fraktur"},"l")]),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",{stretchy:"false"},"["),t("mi",null,"i"),t("mo",{stretchy:"false"},"]")])])])],-1))]),s[314]||(s[314]=e("."))]),s[316]||(s[316]=t("p",null,[t("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))])])}const Q1=r(d,[["render",a1]]);export{h1 as __pageData,Q1 as default}; diff --git a/v0.34.8/assets/manual_quad_forms_discriminant_group.md.COmgBTDB.js b/v0.34.8/assets/manual_quad_forms_discriminant_group.md.COmgBTDB.js index 80d10ae754..f192c40a4f 100644 --- a/v0.34.8/assets/manual_quad_forms_discriminant_group.md.COmgBTDB.js +++ b/v0.34.8/assets/manual_quad_forms_discriminant_group.md.COmgBTDB.js @@ -35,7 +35,7 @@ import{_ as Q,c as l,j as s,a as t,a5 as a,G as h,B as p,o as e}from"./chunks/fr 1 3//2 1`,3)),s("p",null,[i[87]||(i[87]=t("N.B. Since there are no elements of ")),s("mjx-container",P,[(e(),l("svg",U,i[85]||(i[85]=[a('',1)]))),i[86]||(i[86]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),i[88]||(i[88]=t("-lattices, we think of elements of ")),i[89]||(i[89]=s("code",null,"M",-1)),i[90]||(i[90]=t(" as elements of the ambient vector space. Thus if ")),i[91]||(i[91]=s("code",null,"v::Vector",-1)),i[92]||(i[92]=t(" is such an element then the coordinates with respec to the basis of ")),i[93]||(i[93]=s("code",null,"M",-1)),i[94]||(i[94]=t(" are given by ")),i[95]||(i[95]=s("code",null,"solve(basis_matrix(M), v; side = :left)",-1)),i[96]||(i[96]=t("."))]),i[99]||(i[99]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i[405]||(i[405]=s("p",null,[t("Most of the functionality mirrors that of "),s("code",null,"AbGrp"),t(" its elements and homomorphisms. Here we display the part that is specific to elements of torsion quadratic modules.")],-1)),i[406]||(i[406]=s("h3",{id:"attributes",tabindex:"-1"},[t("Attributes "),s("a",{class:"header-anchor",href:"#attributes","aria-label":'Permalink to "Attributes"'},"​")],-1)),s("details",$,[s("summary",null,[i[100]||(i[100]=s("a",{id:"abelian_group-Tuple{TorQuadModule}",href:"#abelian_group-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"abelian_group")],-1)),i[101]||(i[101]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[102]||(i[102]=a('
julia
abelian_group(T::TorQuadModule) -> FinGenAbGroup

Return the underlying abelian group of T.

source

',3))]),s("details",K,[s("summary",null,[i[103]||(i[103]=s("a",{id:"cover-Tuple{TorQuadModule}",href:"#cover-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"cover")],-1)),i[104]||(i[104]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[112]||(i[112]=a('
julia
cover(T::TorQuadModule) -> ZZLat
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julia
relations(T::TorQuadModule) -> ZZLat
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julia
value_module(T::TorQuadModule) -> QmodnZ

Return the value module Q/nZ of the bilinear form of T.

source

',3))]),s("details",h1,[s("summary",null,[i[128]||(i[128]=s("a",{id:"value_module_quadratic_form-Tuple{TorQuadModule}",href:"#value_module_quadratic_form-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"value_module_quadratic_form")],-1)),i[129]||(i[129]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[130]||(i[130]=a('
julia
value_module_quadratic_form(T::TorQuadModule) -> QmodnZ

Return the value module Q/mZ of the quadratic form of T.

source

',3))]),s("details",Q1,[s("summary",null,[i[131]||(i[131]=s("a",{id:"gram_matrix_bilinear-Tuple{TorQuadModule}",href:"#gram_matrix_bilinear-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"gram_matrix_bilinear")],-1)),i[132]||(i[132]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[133]||(i[133]=a('
julia
gram_matrix_bilinear(T::TorQuadModule) -> QQMatrix

Return the gram matrix of the bilinear form of T.

source

',3))]),s("details",p1,[s("summary",null,[i[134]||(i[134]=s("a",{id:"gram_matrix_quadratic-Tuple{TorQuadModule}",href:"#gram_matrix_quadratic-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"gram_matrix_quadratic")],-1)),i[135]||(i[135]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[136]||(i[136]=a('
julia
gram_matrix_quadratic(T::TorQuadModule) -> QQMatrix

Return the 'gram matrix' of the quadratic form of T.

The off diagonal entries are given by the bilinear form whereas the diagonal entries are given by the quadratic form.

source

',4))]),s("details",r1,[s("summary",null,[i[137]||(i[137]=s("a",{id:"modulus_bilinear_form-Tuple{TorQuadModule}",href:"#modulus_bilinear_form-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"modulus_bilinear_form")],-1)),i[138]||(i[138]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[139]||(i[139]=a('
julia
modulus_bilinear_form(T::TorQuadModule) -> QQFieldElem

Return the modulus of the value module of the bilinear form ofT.

source

',3))]),s("details",d1,[s("summary",null,[i[140]||(i[140]=s("a",{id:"modulus_quadratic_form-Tuple{TorQuadModule}",href:"#modulus_quadratic_form-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"modulus_quadratic_form")],-1)),i[141]||(i[141]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[142]||(i[142]=a('
julia
modulus_quadratic_form(T::TorQuadModule) -> QQFieldElem

Return the modulus of the value module of the quadratic form of T.

source

',3))]),i[407]||(i[407]=s("h3",{id:"elements",tabindex:"-1"},[t("Elements "),s("a",{class:"header-anchor",href:"#elements","aria-label":'Permalink to "Elements"'},"​")],-1)),s("details",o1,[s("summary",null,[i[143]||(i[143]=s("a",{id:"quadratic_product-Tuple{TorQuadModuleElem}",href:"#quadratic_product-Tuple{TorQuadModuleElem}"},[s("span",{class:"jlbinding"},"quadratic_product")],-1)),i[144]||(i[144]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[152]||(i[152]=a('
julia
quadratic_product(a::TorQuadModuleElem) -> QmodnZElem

Return the quadratic product of a.

',2)),s("p",null,[i[149]||(i[149]=t("It is defined in terms of a representative: for ")),s("mjx-container",k1,[(e(),l("svg",T1,i[145]||(i[145]=[a('',1)]))),i[146]||(i[146]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"b"),s("mo",null,"+"),s("mi",null,"M"),s("mo",null,"∈"),s("mi",null,"M"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"N"),s("mo",null,"="),s("mi",null,"T")])],-1))]),i[150]||(i[150]=t(", this returns ")),s("mjx-container",m1,[(e(),l("svg",g1,i[147]||(i[147]=[a('',1)]))),i[148]||(i[148]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",{mathvariant:"normal"},"Φ"),s("mo",{stretchy:"false"},"("),s("mi",null,"b"),s("mo",null,","),s("mi",null,"b"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mi",null,"n"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),i[151]||(i[151]=t("."))]),i[153]||(i[153]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",u1,[s("summary",null,[i[154]||(i[154]=s("a",{id:"inner_product-Tuple{TorQuadModuleElem, TorQuadModuleElem}",href:"#inner_product-Tuple{TorQuadModuleElem, TorQuadModuleElem}"},[s("span",{class:"jlbinding"},"inner_product")],-1)),i[155]||(i[155]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[156]||(i[156]=a('
julia
inner_product(a::TorQuadModuleElem, b::TorQuadModuleElem) -> QmodnZElem

Return the inner product of a and b.

source

',3))]),i[408]||(i[408]=s("h3",{id:"Lift-to-the-cover",tabindex:"-1"},[t("Lift to the cover "),s("a",{class:"header-anchor",href:"#Lift-to-the-cover","aria-label":'Permalink to "Lift to the cover {#Lift-to-the-cover}"'},"​")],-1)),s("details",y1,[s("summary",null,[i[157]||(i[157]=s("a",{id:"lift-Tuple{TorQuadModuleElem}",href:"#lift-Tuple{TorQuadModuleElem}"},[s("span",{class:"jlbinding"},"lift")],-1)),i[158]||(i[158]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[166]||(i[166]=a('
julia
lift(a::TorQuadModuleElem) -> Vector{QQFieldElem}

Lift a to the ambient space of cover(parent(a)).

',2)),s("p",null,[i[163]||(i[163]=t("For ")),s("mjx-container",E1,[(e(),l("svg",c1,i[159]||(i[159]=[a('',1)]))),i[160]||(i[160]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"a"),s("mo",null,"+"),s("mi",null,"N"),s("mo",null,"∈"),s("mi",null,"M"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"N")])],-1))]),i[164]||(i[164]=t(" this returns the representative ")),s("mjx-container",x1,[(e(),l("svg",F1,i[161]||(i[161]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),i[162]||(i[162]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"a")])],-1))]),i[165]||(i[165]=t("."))]),i[167]||(i[167]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",w1,[s("summary",null,[i[168]||(i[168]=s("a",{id:"representative-Tuple{TorQuadModuleElem}",href:"#representative-Tuple{TorQuadModuleElem}"},[s("span",{class:"jlbinding"},"representative")],-1)),i[169]||(i[169]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[179]||(i[179]=a('
julia
representative(a::TorQuadModuleElem) -> Vector{QQFieldElem}
',1)),s("p",null,[i[174]||(i[174]=t("For ")),s("mjx-container",C1,[(e(),l("svg",f1,i[170]||(i[170]=[a('',1)]))),i[171]||(i[171]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"a"),s("mo",null,"+"),s("mi",null,"N"),s("mo",null,"∈"),s("mi",null,"M"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"N")])],-1))]),i[175]||(i[175]=t(" this returns the representative ")),s("mjx-container",H1,[(e(),l("svg",b1,i[172]||(i[172]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),i[173]||(i[173]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"a")])],-1))]),i[176]||(i[176]=t(". An alias for ")),i[177]||(i[177]=s("code",null,"lift(a)",-1)),i[178]||(i[178]=t("."))]),i[180]||(i[180]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i[409]||(i[409]=s("h3",{id:"Orthogonal-submodules",tabindex:"-1"},[t("Orthogonal submodules "),s("a",{class:"header-anchor",href:"#Orthogonal-submodules","aria-label":'Permalink to "Orthogonal submodules {#Orthogonal-submodules}"'},"​")],-1)),s("details",M1,[s("summary",null,[i[181]||(i[181]=s("a",{id:"orthogonal_submodule-Tuple{TorQuadModule, TorQuadModule}",href:"#orthogonal_submodule-Tuple{TorQuadModule, TorQuadModule}"},[s("span",{class:"jlbinding"},"orthogonal_submodule")],-1)),i[182]||(i[182]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[183]||(i[183]=a('
julia
orthogonal_submodule(T::TorQuadModule, S::TorQuadModule)-> TorQuadModule

Return the orthogonal submodule to the submodule S of T.

source

',3))]),i[410]||(i[410]=s("h3",{id:"isometry",tabindex:"-1"},[t("Isometry "),s("a",{class:"header-anchor",href:"#isometry","aria-label":'Permalink to "Isometry"'},"​")],-1)),s("details",L1,[s("summary",null,[i[184]||(i[184]=s("a",{id:"is_isometric_with_isometry-Tuple{TorQuadModule, TorQuadModule}",href:"#is_isometric_with_isometry-Tuple{TorQuadModule, TorQuadModule}"},[s("span",{class:"jlbinding"},"is_isometric_with_isometry")],-1)),i[185]||(i[185]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[194]||(i[194]=a(`
julia
is_isometric_with_isometry(T::TorQuadModule, U::TorQuadModule)
-                                               -> Bool, TorQuadModuleMap
`,1)),s("p",null,[i[188]||(i[188]=t("Return whether the torsion quadratic modules ")),i[189]||(i[189]=s("code",null,"T",-1)),i[190]||(i[190]=t(" and ")),i[191]||(i[191]=s("code",null,"U",-1)),i[192]||(i[192]=t(" are isometric. If yes, it also returns an isometry ")),s("mjx-container",v1,[(e(),l("svg",D1,i[186]||(i[186]=[a('',1)]))),i[187]||(i[187]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"T"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"U")])],-1))]),i[193]||(i[193]=t("."))]),i[195]||(i[195]=a(`

If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).

It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).

Examples

julia
julia> T = torsion_quadratic_module(QQ[2//3 2//3    0    0    0;
+                                               -> Bool, TorQuadModuleMap
`,1)),s("p",null,[i[188]||(i[188]=t("Return whether the torsion quadratic modules ")),i[189]||(i[189]=s("code",null,"T",-1)),i[190]||(i[190]=t(" and ")),i[191]||(i[191]=s("code",null,"U",-1)),i[192]||(i[192]=t(" are isometric. If yes, it also returns an isometry ")),s("mjx-container",v1,[(e(),l("svg",D1,i[186]||(i[186]=[a('',1)]))),i[187]||(i[187]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"T"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"U")])],-1))]),i[193]||(i[193]=t("."))]),i[195]||(i[195]=a(`

If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).

It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).

Examples

julia
julia> T = torsion_quadratic_module(QQ[2//3 2//3    0    0    0;
                                        2//3 2//3 2//3    0 2//3;
                                           0 2//3 2//3 2//3    0;
                                           0    0 2//3 2//3    0;
@@ -86,7 +86,7 @@ import{_ as Q,c as l,j as s,a as t,a5 as a,G as h,B as p,o as e}from"./chunks/fr
 
 julia> is_bijective(phi)
 true

source

`,5))]),s("details",B1,[s("summary",null,[i[196]||(i[196]=s("a",{id:"is_anti_isometric_with_anti_isometry-Tuple{TorQuadModule, TorQuadModule}",href:"#is_anti_isometric_with_anti_isometry-Tuple{TorQuadModule, TorQuadModule}"},[s("span",{class:"jlbinding"},"is_anti_isometric_with_anti_isometry")],-1)),i[197]||(i[197]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[206]||(i[206]=a(`
julia
is_anti_isometric_with_anti_isometry(T::TorQuadModule, U::TorQuadModule)
-                                                 -> Bool, TorQuadModuleMap
`,1)),s("p",null,[i[200]||(i[200]=t("Return whether there exists an anti-isometry between the torsion quadratic modules ")),i[201]||(i[201]=s("code",null,"T",-1)),i[202]||(i[202]=t(" and ")),i[203]||(i[203]=s("code",null,"U",-1)),i[204]||(i[204]=t(". If yes, it returns such an anti-isometry ")),s("mjx-container",j1,[(e(),l("svg",Z1,i[198]||(i[198]=[a('',1)]))),i[199]||(i[199]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"T"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"U")])],-1))]),i[205]||(i[205]=t("."))]),i[207]||(i[207]=a(`

If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).

It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).

Examples

julia
julia> T = torsion_quadratic_module(QQ[4//5;])
+                                                 -> Bool, TorQuadModuleMap
`,1)),s("p",null,[i[200]||(i[200]=t("Return whether there exists an anti-isometry between the torsion quadratic modules ")),i[201]||(i[201]=s("code",null,"T",-1)),i[202]||(i[202]=t(" and ")),i[203]||(i[203]=s("code",null,"U",-1)),i[204]||(i[204]=t(". If yes, it returns such an anti-isometry ")),s("mjx-container",j1,[(e(),l("svg",Z1,i[198]||(i[198]=[a('',1)]))),i[199]||(i[199]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"T"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"U")])],-1))]),i[205]||(i[205]=t("."))]),i[207]||(i[207]=a(`

If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).

It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).

Examples

julia
julia> T = torsion_quadratic_module(QQ[4//5;])
 Finite quadratic module
   over integer ring
 Abelian group: Z/5
@@ -146,7 +146,7 @@ import{_ as Q,c as l,j as s,a as t,a5 as a,G as h,B as p,o as e}from"./chunks/fr
 julia> a = gens(T)[1];
 
 julia> a*a == -phi(a)*phi(a)
-true

source

`,5))]),i[411]||(i[411]=s("h3",{id:"Primary-and-elementary-modules",tabindex:"-1"},[t("Primary and elementary modules "),s("a",{class:"header-anchor",href:"#Primary-and-elementary-modules","aria-label":'Permalink to "Primary and elementary modules {#Primary-and-elementary-modules}"'},"​")],-1)),s("details",A1,[s("summary",null,[i[208]||(i[208]=s("a",{id:"is_primary_with_prime-Tuple{TorQuadModule}",href:"#is_primary_with_prime-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),i[209]||(i[209]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[210]||(i[210]=a('
julia
is_primary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem

Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group for some prime number p. In case it is, p is also returned as second output.

Note that in the case of trivial groups, this function returns (true, 1). If T is not primary, the second return value is -1 by default.

source

',4))]),s("details",V1,[s("summary",null,[i[211]||(i[211]=s("a",{id:"is_primary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}",href:"#is_primary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}"},[s("span",{class:"jlbinding"},"is_primary")],-1)),i[212]||(i[212]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[213]||(i[213]=a('
julia
is_primary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool

Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group.

source

',3))]),s("details",q1,[s("summary",null,[i[214]||(i[214]=s("a",{id:"is_elementary_with_prime-Tuple{TorQuadModule}",href:"#is_elementary_with_prime-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_elementary_with_prime")],-1)),i[215]||(i[215]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[216]||(i[216]=a('
julia
is_elementary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem

Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group, for some prime number p. In case it is, p is also returned as second output.

Note that in the case of trivial groups, this function returns (true, 1). If T is not elementary, the second return value is -1 by default.

source

',4))]),s("details",O1,[s("summary",null,[i[217]||(i[217]=s("a",{id:"is_elementary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}",href:"#is_elementary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}"},[s("span",{class:"jlbinding"},"is_elementary")],-1)),i[218]||(i[218]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[219]||(i[219]=a('
julia
is_elementary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool

Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group.

source

',3))]),i[412]||(i[412]=s("h3",{id:"Smith-normal-form",tabindex:"-1"},[t("Smith normal form "),s("a",{class:"header-anchor",href:"#Smith-normal-form","aria-label":'Permalink to "Smith normal form {#Smith-normal-form}"'},"​")],-1)),s("details",R1,[s("summary",null,[i[220]||(i[220]=s("a",{id:"snf-Tuple{TorQuadModule}",href:"#snf-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"snf")],-1)),i[221]||(i[221]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[234]||(i[234]=a('
julia
snf(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap
',1)),s("p",null,[i[224]||(i[224]=t("Given a torsion quadratic module ")),i[225]||(i[225]=s("code",null,"T",-1)),i[226]||(i[226]=t(", return a torsion quadratic module ")),i[227]||(i[227]=s("code",null,"S",-1)),i[228]||(i[228]=t(", isometric to ")),i[229]||(i[229]=s("code",null,"T",-1)),i[230]||(i[230]=t(", such that the underlying abelian group of ")),i[231]||(i[231]=s("code",null,"S",-1)),i[232]||(i[232]=t(" is in canonical Smith normal form. It comes with an isometry ")),s("mjx-container",G1,[(e(),l("svg",S1,i[222]||(i[222]=[a('',1)]))),i[223]||(i[223]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"f"),s("mo",null,":"),s("mi",null,"S"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"T")])],-1))]),i[233]||(i[233]=t("."))]),i[235]||(i[235]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",z1,[s("summary",null,[i[236]||(i[236]=s("a",{id:"is_snf-Tuple{TorQuadModule}",href:"#is_snf-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_snf")],-1)),i[237]||(i[237]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[238]||(i[238]=a('
julia
is_snf(T::TorQuadModule) -> Bool

Given a torsion quadratic module T, return whether its underlying abelian group is in Smith normal form.

source

',3))]),i[413]||(i[413]=a('

Discriminant Groups ​

See [6] for the general theory of discriminant groups. They are particularly useful to work with primitive embeddings of integral integer quadratic lattices.

From a lattice ​

',3)),s("details",I1,[s("summary",null,[i[239]||(i[239]=s("a",{id:"discriminant_group-Tuple{ZZLat}",href:"#discriminant_group-Tuple{ZZLat}"},[s("span",{class:"jlbinding"},"discriminant_group")],-1)),i[240]||(i[240]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[249]||(i[249]=a('
julia
discriminant_group(L::ZZLat) -> TorQuadModule

Return the discriminant group of L.

The discriminant group of an integral lattice L is the finite abelian group D = dual(L)/L.

It comes equipped with the discriminant bilinear form

',4)),s("mjx-container",J1,[(e(),l("svg",N1,i[241]||(i[241]=[a('',1)]))),i[242]||(i[242]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"D"),s("mo",null,"×"),s("mi",null,"D"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Q")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mstyle",{scriptlevel:"0"},[s("mspace",{width:"2em"})]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"↦"),s("mi",{mathvariant:"normal"},"Φ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mo",null,".")])],-1))]),s("p",null,[i[245]||(i[245]=t("If ")),i[246]||(i[246]=s("code",null,"L",-1)),i[247]||(i[247]=t(" is even, then the discriminant group is equipped with the discriminant quadratic form ")),s("mjx-container",X1,[(e(),l("svg",P1,i[243]||(i[243]=[a('',1)]))),i[244]||(i[244]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"D"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Q")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mn",null,"2"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mo",null,","),s("mi",null,"x"),s("mo",{stretchy:"false"},"↦"),s("mi",{mathvariant:"normal"},"Φ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"2"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),i[248]||(i[248]=t("."))]),i[250]||(i[250]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i[414]||(i[414]=s("h3",{id:"From-a-matrix",tabindex:"-1"},[t("From a matrix "),s("a",{class:"header-anchor",href:"#From-a-matrix","aria-label":'Permalink to "From a matrix {#From-a-matrix}"'},"​")],-1)),s("details",U1,[s("summary",null,[i[251]||(i[251]=s("a",{id:"torsion_quadratic_module-Tuple{QQMatrix}",href:"#torsion_quadratic_module-Tuple{QQMatrix}"},[s("span",{class:"jlbinding"},"torsion_quadratic_module")],-1)),i[252]||(i[252]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[253]||(i[253]=a(`
julia
torsion_quadratic_module(q::QQMatrix) -> TorQuadModule

Return a torsion quadratic module with gram matrix given by q and value module Q/Z. If all the diagonal entries of q have: either even numerator or even denominator, then the value module of the quadratic form is Q/2Z

Example

julia
julia> torsion_quadratic_module(QQ[1//6;])
+true

source

`,5))]),i[411]||(i[411]=s("h3",{id:"Primary-and-elementary-modules",tabindex:"-1"},[t("Primary and elementary modules "),s("a",{class:"header-anchor",href:"#Primary-and-elementary-modules","aria-label":'Permalink to "Primary and elementary modules {#Primary-and-elementary-modules}"'},"​")],-1)),s("details",A1,[s("summary",null,[i[208]||(i[208]=s("a",{id:"is_primary_with_prime-Tuple{TorQuadModule}",href:"#is_primary_with_prime-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),i[209]||(i[209]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[210]||(i[210]=a('
julia
is_primary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem

Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group for some prime number p. In case it is, p is also returned as second output.

Note that in the case of trivial groups, this function returns (true, 1). If T is not primary, the second return value is -1 by default.

source

',4))]),s("details",V1,[s("summary",null,[i[211]||(i[211]=s("a",{id:"is_primary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}",href:"#is_primary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}"},[s("span",{class:"jlbinding"},"is_primary")],-1)),i[212]||(i[212]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[213]||(i[213]=a('
julia
is_primary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool

Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group.

source

',3))]),s("details",q1,[s("summary",null,[i[214]||(i[214]=s("a",{id:"is_elementary_with_prime-Tuple{TorQuadModule}",href:"#is_elementary_with_prime-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_elementary_with_prime")],-1)),i[215]||(i[215]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[216]||(i[216]=a('
julia
is_elementary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem

Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group, for some prime number p. In case it is, p is also returned as second output.

Note that in the case of trivial groups, this function returns (true, 1). If T is not elementary, the second return value is -1 by default.

source

',4))]),s("details",O1,[s("summary",null,[i[217]||(i[217]=s("a",{id:"is_elementary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}",href:"#is_elementary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}"},[s("span",{class:"jlbinding"},"is_elementary")],-1)),i[218]||(i[218]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[219]||(i[219]=a('
julia
is_elementary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool

Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group.

source

',3))]),i[412]||(i[412]=s("h3",{id:"Smith-normal-form",tabindex:"-1"},[t("Smith normal form "),s("a",{class:"header-anchor",href:"#Smith-normal-form","aria-label":'Permalink to "Smith normal form {#Smith-normal-form}"'},"​")],-1)),s("details",R1,[s("summary",null,[i[220]||(i[220]=s("a",{id:"snf-Tuple{TorQuadModule}",href:"#snf-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"snf")],-1)),i[221]||(i[221]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[234]||(i[234]=a('
julia
snf(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap
',1)),s("p",null,[i[224]||(i[224]=t("Given a torsion quadratic module ")),i[225]||(i[225]=s("code",null,"T",-1)),i[226]||(i[226]=t(", return a torsion quadratic module ")),i[227]||(i[227]=s("code",null,"S",-1)),i[228]||(i[228]=t(", isometric to ")),i[229]||(i[229]=s("code",null,"T",-1)),i[230]||(i[230]=t(", such that the underlying abelian group of ")),i[231]||(i[231]=s("code",null,"S",-1)),i[232]||(i[232]=t(" is in canonical Smith normal form. It comes with an isometry ")),s("mjx-container",G1,[(e(),l("svg",S1,i[222]||(i[222]=[a('',1)]))),i[223]||(i[223]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"f"),s("mo",null,":"),s("mi",null,"S"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"T")])],-1))]),i[233]||(i[233]=t("."))]),i[235]||(i[235]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",z1,[s("summary",null,[i[236]||(i[236]=s("a",{id:"is_snf-Tuple{TorQuadModule}",href:"#is_snf-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_snf")],-1)),i[237]||(i[237]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[238]||(i[238]=a('
julia
is_snf(T::TorQuadModule) -> Bool

Given a torsion quadratic module T, return whether its underlying abelian group is in Smith normal form.

source

',3))]),i[413]||(i[413]=a('

Discriminant Groups ​

See [6] for the general theory of discriminant groups. They are particularly useful to work with primitive embeddings of integral integer quadratic lattices.

From a lattice ​

',3)),s("details",I1,[s("summary",null,[i[239]||(i[239]=s("a",{id:"discriminant_group-Tuple{ZZLat}",href:"#discriminant_group-Tuple{ZZLat}"},[s("span",{class:"jlbinding"},"discriminant_group")],-1)),i[240]||(i[240]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[249]||(i[249]=a('
julia
discriminant_group(L::ZZLat) -> TorQuadModule

Return the discriminant group of L.

The discriminant group of an integral lattice L is the finite abelian group D = dual(L)/L.

It comes equipped with the discriminant bilinear form

',4)),s("mjx-container",J1,[(e(),l("svg",N1,i[241]||(i[241]=[a('',1)]))),i[242]||(i[242]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"D"),s("mo",null,"×"),s("mi",null,"D"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Q")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mstyle",{scriptlevel:"0"},[s("mspace",{width:"2em"})]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"↦"),s("mi",{mathvariant:"normal"},"Φ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mo",null,".")])],-1))]),s("p",null,[i[245]||(i[245]=t("If ")),i[246]||(i[246]=s("code",null,"L",-1)),i[247]||(i[247]=t(" is even, then the discriminant group is equipped with the discriminant quadratic form ")),s("mjx-container",X1,[(e(),l("svg",P1,i[243]||(i[243]=[a('',1)]))),i[244]||(i[244]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"D"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Q")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mn",null,"2"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mo",null,","),s("mi",null,"x"),s("mo",{stretchy:"false"},"↦"),s("mi",{mathvariant:"normal"},"Φ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"2"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),i[248]||(i[248]=t("."))]),i[250]||(i[250]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i[414]||(i[414]=s("h3",{id:"From-a-matrix",tabindex:"-1"},[t("From a matrix "),s("a",{class:"header-anchor",href:"#From-a-matrix","aria-label":'Permalink to "From a matrix {#From-a-matrix}"'},"​")],-1)),s("details",U1,[s("summary",null,[i[251]||(i[251]=s("a",{id:"torsion_quadratic_module-Tuple{QQMatrix}",href:"#torsion_quadratic_module-Tuple{QQMatrix}"},[s("span",{class:"jlbinding"},"torsion_quadratic_module")],-1)),i[252]||(i[252]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[253]||(i[253]=a(`
julia
torsion_quadratic_module(q::QQMatrix) -> TorQuadModule

Return a torsion quadratic module with gram matrix given by q and value module Q/Z. If all the diagonal entries of q have: either even numerator or even denominator, then the value module of the quadratic form is Q/2Z

Example

julia
julia> torsion_quadratic_module(QQ[1//6;])
 Finite quadratic module
   over integer ring
 Abelian group: Z/6
@@ -180,9 +180,9 @@ import{_ as Q,c as l,j as s,a as t,a5 as a,G as h,B as p,o as e}from"./chunks/fr
 Bilinear value module: Q/Z
 Quadratic value module: Q/Z
 Gram matrix quadratic form:
-[1//3]

source

`,5))]),i[415]||(i[415]=s("h3",{id:"Rescaling-the-form",tabindex:"-1"},[t("Rescaling the form "),s("a",{class:"header-anchor",href:"#Rescaling-the-form","aria-label":'Permalink to "Rescaling the form {#Rescaling-the-form}"'},"​")],-1)),s("details",$1,[s("summary",null,[i[254]||(i[254]=s("a",{id:"rescale-Tuple{TorQuadModule, RingElement}",href:"#rescale-Tuple{TorQuadModule, RingElement}"},[s("span",{class:"jlbinding"},"rescale")],-1)),i[255]||(i[255]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[264]||(i[264]=a('
julia
rescale(T::TorQuadModule, k::RingElement) -> TorQuadModule
',1)),s("p",null,[i[258]||(i[258]=t("Return the torsion quadratic module with quadratic form scaled by ")),s("mjx-container",K1,[(e(),l("svg",W1,i[256]||(i[256]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z",style:{"stroke-width":"3"}})])])],-1)]))),i[257]||(i[257]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"k")])],-1))]),i[259]||(i[259]=t(", where k is a non-zero rational number. If the old form was defined modulo ")),i[260]||(i[260]=s("code",null,"n",-1)),i[261]||(i[261]=t(", then the new form is defined modulo ")),i[262]||(i[262]=s("code",null,"n k",-1)),i[263]||(i[263]=t("."))]),i[265]||(i[265]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i[416]||(i[416]=s("h3",{id:"invariants",tabindex:"-1"},[t("Invariants "),s("a",{class:"header-anchor",href:"#invariants","aria-label":'Permalink to "Invariants"'},"​")],-1)),s("details",Y1,[s("summary",null,[i[266]||(i[266]=s("a",{id:"is_degenerate-Tuple{TorQuadModule}",href:"#is_degenerate-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_degenerate")],-1)),i[267]||(i[267]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[268]||(i[268]=a('
julia
is_degenerate(T::TorQuadModule) -> Bool

Return true if the underlying bilinear form is degenerate.

source

',3))]),s("details",_1,[s("summary",null,[i[269]||(i[269]=s("a",{id:"is_semi_regular-Tuple{TorQuadModule}",href:"#is_semi_regular-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_semi_regular")],-1)),i[270]||(i[270]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[271]||(i[271]=a('
julia
is_semi_regular(T::TorQuadModule) -> Bool

Return whether T is semi-regular, that is its quadratic radical is trivial (see radical_quadratic).

source

',3))]),s("details",s2,[s("summary",null,[i[272]||(i[272]=s("a",{id:"radical_bilinear-Tuple{TorQuadModule}",href:"#radical_bilinear-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"radical_bilinear")],-1)),i[273]||(i[273]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[274]||(i[274]=a('
julia
radical_bilinear(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap

Return the radical \\{x \\in T | b(x,T) = 0\\} of the bilinear form b on T.

source

',3))]),s("details",i2,[s("summary",null,[i[275]||(i[275]=s("a",{id:"radical_quadratic-Tuple{TorQuadModule}",href:"#radical_quadratic-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"radical_quadratic")],-1)),i[276]||(i[276]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[277]||(i[277]=a('
julia
radical_quadratic(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap

Return the radical \\{x \\in T | b(x,T) = 0 and q(x)=0\\} of the quadratic form q on T.

source

',3))]),s("details",t2,[s("summary",null,[i[278]||(i[278]=s("a",{id:"normal_form-Tuple{TorQuadModule}",href:"#normal_form-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"normal_form")],-1)),i[279]||(i[279]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[280]||(i[280]=a('
julia
normal_form(T::TorQuadModule; partial=false) -> TorQuadModule, TorQuadModuleMap

Return the normal form N of the given torsion quadratic module T along with the projection T -> N.

Let K be the radical of the quadratic form of T. Then N = T/K is half-regular. Two half-regular torsion quadratic modules are isometric if and only if they have equal normal forms.

source

',4))]),i[417]||(i[417]=s("h3",{id:"genus",tabindex:"-1"},[t("Genus "),s("a",{class:"header-anchor",href:"#genus","aria-label":'Permalink to "Genus"'},"​")],-1)),s("details",a2,[s("summary",null,[i[281]||(i[281]=s("a",{id:"genus-Tuple{TorQuadModule, Tuple{Int64, Int64}}",href:"#genus-Tuple{TorQuadModule, Tuple{Int64, Int64}}"},[s("span",{class:"jlbinding"},"genus")],-1)),i[282]||(i[282]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[283]||(i[283]=a(`
julia
genus(T::TorQuadModule, signature_pair::Tuple{Int, Int};
+[1//3]

source

`,5))]),i[415]||(i[415]=s("h3",{id:"Rescaling-the-form",tabindex:"-1"},[t("Rescaling the form "),s("a",{class:"header-anchor",href:"#Rescaling-the-form","aria-label":'Permalink to "Rescaling the form {#Rescaling-the-form}"'},"​")],-1)),s("details",$1,[s("summary",null,[i[254]||(i[254]=s("a",{id:"rescale-Tuple{TorQuadModule, RingElement}",href:"#rescale-Tuple{TorQuadModule, RingElement}"},[s("span",{class:"jlbinding"},"rescale")],-1)),i[255]||(i[255]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[264]||(i[264]=a('
julia
rescale(T::TorQuadModule, k::RingElement) -> TorQuadModule
',1)),s("p",null,[i[258]||(i[258]=t("Return the torsion quadratic module with quadratic form scaled by ")),s("mjx-container",K1,[(e(),l("svg",W1,i[256]||(i[256]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z",style:{"stroke-width":"3"}})])])],-1)]))),i[257]||(i[257]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"k")])],-1))]),i[259]||(i[259]=t(", where k is a non-zero rational number. If the old form was defined modulo ")),i[260]||(i[260]=s("code",null,"n",-1)),i[261]||(i[261]=t(", then the new form is defined modulo ")),i[262]||(i[262]=s("code",null,"n k",-1)),i[263]||(i[263]=t("."))]),i[265]||(i[265]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i[416]||(i[416]=s("h3",{id:"invariants",tabindex:"-1"},[t("Invariants "),s("a",{class:"header-anchor",href:"#invariants","aria-label":'Permalink to "Invariants"'},"​")],-1)),s("details",Y1,[s("summary",null,[i[266]||(i[266]=s("a",{id:"is_degenerate-Tuple{TorQuadModule}",href:"#is_degenerate-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_degenerate")],-1)),i[267]||(i[267]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[268]||(i[268]=a('
julia
is_degenerate(T::TorQuadModule) -> Bool

Return true if the underlying bilinear form is degenerate.

source

',3))]),s("details",_1,[s("summary",null,[i[269]||(i[269]=s("a",{id:"is_semi_regular-Tuple{TorQuadModule}",href:"#is_semi_regular-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_semi_regular")],-1)),i[270]||(i[270]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[271]||(i[271]=a('
julia
is_semi_regular(T::TorQuadModule) -> Bool

Return whether T is semi-regular, that is its quadratic radical is trivial (see radical_quadratic).

source

',3))]),s("details",s2,[s("summary",null,[i[272]||(i[272]=s("a",{id:"radical_bilinear-Tuple{TorQuadModule}",href:"#radical_bilinear-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"radical_bilinear")],-1)),i[273]||(i[273]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[274]||(i[274]=a('
julia
radical_bilinear(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap

Return the radical \\{x \\in T | b(x,T) = 0\\} of the bilinear form b on T.

source

',3))]),s("details",i2,[s("summary",null,[i[275]||(i[275]=s("a",{id:"radical_quadratic-Tuple{TorQuadModule}",href:"#radical_quadratic-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"radical_quadratic")],-1)),i[276]||(i[276]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[277]||(i[277]=a('
julia
radical_quadratic(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap

Return the radical \\{x \\in T | b(x,T) = 0 and q(x)=0\\} of the quadratic form q on T.

source

',3))]),s("details",t2,[s("summary",null,[i[278]||(i[278]=s("a",{id:"normal_form-Tuple{TorQuadModule}",href:"#normal_form-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"normal_form")],-1)),i[279]||(i[279]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[280]||(i[280]=a('
julia
normal_form(T::TorQuadModule; partial=false) -> TorQuadModule, TorQuadModuleMap

Return the normal form N of the given torsion quadratic module T along with the projection T -> N.

Let K be the radical of the quadratic form of T. Then N = T/K is half-regular. Two half-regular torsion quadratic modules are isometric if and only if they have equal normal forms.

source

',4))]),i[417]||(i[417]=s("h3",{id:"genus",tabindex:"-1"},[t("Genus "),s("a",{class:"header-anchor",href:"#genus","aria-label":'Permalink to "Genus"'},"​")],-1)),s("details",a2,[s("summary",null,[i[281]||(i[281]=s("a",{id:"genus-Tuple{TorQuadModule, Tuple{Int64, Int64}}",href:"#genus-Tuple{TorQuadModule, Tuple{Int64, Int64}}"},[s("span",{class:"jlbinding"},"genus")],-1)),i[282]||(i[282]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[283]||(i[283]=a(`
julia
genus(T::TorQuadModule, signature_pair::Tuple{Int, Int};
                         parity::RationalUnion = modulus_quadratic_form(T))
-                                                                -> ZZGenus

Return the genus of an integer lattice whose discriminant group has the bilinear form of T, the given signature_pair and the given parity.

The argument parity is one of the following: either parity == 1 for genera of odd lattices, or parity == 2 for even lattices. By default, parity is set to be as the parity of the quadratic form on T

If no such genus exists, raise an error.

Reference

[6] Corollary 1.9.4 and 1.16.3.

source

`,7))]),s("details",l2,[s("summary",null,[i[284]||(i[284]=s("a",{id:"brown_invariant-Tuple{TorQuadModule}",href:"#brown_invariant-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"brown_invariant")],-1)),i[285]||(i[285]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[288]||(i[288]=a('
julia
brown_invariant(self::TorQuadModule) -> Nemo.zzModRingElem

Return the Brown invariant of this torsion quadratic form.

Let (D,q) be a torsion quadratic module with values in Q / 2Z. The Brown invariant Br(D,q) in Z/8Z is defined by the equation

',3)),s("mjx-container",e2,[(e(),l("svg",n2,i[286]||(i[286]=[a('',1)]))),i[287]||(i[287]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"exp"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("mi",null,"i")]),s("mn",null,"8")]),s("mi",null,"B"),s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mi",null,"q"),s("mo",{stretchy:"false"},")"),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("msqrt",null,[s("mi",null,"D")])]),s("munder",null,[s("mo",{"data-mjx-texclass":"OP"},"∑"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"x"),s("mo",null,"∈"),s("mi",null,"D")])]),s("mi",null,"exp"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"i"),s("mi",null,"π"),s("mi",null,"q"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},")"),s("mo",null,".")])],-1))]),i[289]||(i[289]=a(`

The Brown invariant is additive with respect to direct sums of torsion quadratic modules.

Examples

julia
julia> L = integer_lattice(gram=matrix(ZZ, [[2,-1,0,0],[-1,2,-1,-1],[0,-1,2,0],[0,-1,0,2]]));
+                                                                -> ZZGenus

Return the genus of an integer lattice whose discriminant group has the bilinear form of T, the given signature_pair and the given parity.

The argument parity is one of the following: either parity == 1 for genera of odd lattices, or parity == 2 for even lattices. By default, parity is set to be as the parity of the quadratic form on T

If no such genus exists, raise an error.

Reference

[6] Corollary 1.9.4 and 1.16.3.

source

`,7))]),s("details",l2,[s("summary",null,[i[284]||(i[284]=s("a",{id:"brown_invariant-Tuple{TorQuadModule}",href:"#brown_invariant-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"brown_invariant")],-1)),i[285]||(i[285]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[288]||(i[288]=a('
julia
brown_invariant(self::TorQuadModule) -> Nemo.zzModRingElem

Return the Brown invariant of this torsion quadratic form.

Let (D,q) be a torsion quadratic module with values in Q / 2Z. The Brown invariant Br(D,q) in Z/8Z is defined by the equation

',3)),s("mjx-container",e2,[(e(),l("svg",n2,i[286]||(i[286]=[a('',1)]))),i[287]||(i[287]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"exp"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("mi",null,"i")]),s("mn",null,"8")]),s("mi",null,"B"),s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mi",null,"q"),s("mo",{stretchy:"false"},")"),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("msqrt",null,[s("mi",null,"D")])]),s("munder",null,[s("mo",{"data-mjx-texclass":"OP"},"∑"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"x"),s("mo",null,"∈"),s("mi",null,"D")])]),s("mi",null,"exp"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"i"),s("mi",null,"π"),s("mi",null,"q"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},")"),s("mo",null,".")])],-1))]),i[289]||(i[289]=a(`

The Brown invariant is additive with respect to direct sums of torsion quadratic modules.

Examples

julia
julia> L = integer_lattice(gram=matrix(ZZ, [[2,-1,0,0],[-1,2,-1,-1],[0,-1,2,0],[0,-1,0,2]]));
 
 julia> T = Hecke.discriminant_group(L);
 
diff --git a/v0.34.8/assets/manual_quad_forms_discriminant_group.md.COmgBTDB.lean.js b/v0.34.8/assets/manual_quad_forms_discriminant_group.md.COmgBTDB.lean.js
index 80d10ae754..f192c40a4f 100644
--- a/v0.34.8/assets/manual_quad_forms_discriminant_group.md.COmgBTDB.lean.js
+++ b/v0.34.8/assets/manual_quad_forms_discriminant_group.md.COmgBTDB.lean.js
@@ -35,7 +35,7 @@ import{_ as Q,c as l,j as s,a as t,a5 as a,G as h,B as p,o as e}from"./chunks/fr
  1
  3//2
  1
`,3)),s("p",null,[i[87]||(i[87]=t("N.B. Since there are no elements of ")),s("mjx-container",P,[(e(),l("svg",U,i[85]||(i[85]=[a('',1)]))),i[86]||(i[86]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),i[88]||(i[88]=t("-lattices, we think of elements of ")),i[89]||(i[89]=s("code",null,"M",-1)),i[90]||(i[90]=t(" as elements of the ambient vector space. Thus if ")),i[91]||(i[91]=s("code",null,"v::Vector",-1)),i[92]||(i[92]=t(" is such an element then the coordinates with respec to the basis of ")),i[93]||(i[93]=s("code",null,"M",-1)),i[94]||(i[94]=t(" are given by ")),i[95]||(i[95]=s("code",null,"solve(basis_matrix(M), v; side = :left)",-1)),i[96]||(i[96]=t("."))]),i[99]||(i[99]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i[405]||(i[405]=s("p",null,[t("Most of the functionality mirrors that of "),s("code",null,"AbGrp"),t(" its elements and homomorphisms. Here we display the part that is specific to elements of torsion quadratic modules.")],-1)),i[406]||(i[406]=s("h3",{id:"attributes",tabindex:"-1"},[t("Attributes "),s("a",{class:"header-anchor",href:"#attributes","aria-label":'Permalink to "Attributes"'},"​")],-1)),s("details",$,[s("summary",null,[i[100]||(i[100]=s("a",{id:"abelian_group-Tuple{TorQuadModule}",href:"#abelian_group-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"abelian_group")],-1)),i[101]||(i[101]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[102]||(i[102]=a('
julia
abelian_group(T::TorQuadModule) -> FinGenAbGroup

Return the underlying abelian group of T.

source

',3))]),s("details",K,[s("summary",null,[i[103]||(i[103]=s("a",{id:"cover-Tuple{TorQuadModule}",href:"#cover-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"cover")],-1)),i[104]||(i[104]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[112]||(i[112]=a('
julia
cover(T::TorQuadModule) -> ZZLat
',1)),s("p",null,[i[109]||(i[109]=t("For ")),s("mjx-container",W,[(e(),l("svg",Y,i[105]||(i[105]=[a('',1)]))),i[106]||(i[106]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"T"),s("mo",null,"="),s("mi",null,"M"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"N")])],-1))]),i[110]||(i[110]=t(" this returns ")),s("mjx-container",_,[(e(),l("svg",s1,i[107]||(i[107]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),i[108]||(i[108]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"M")])],-1))]),i[111]||(i[111]=t("."))]),i[113]||(i[113]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",i1,[s("summary",null,[i[114]||(i[114]=s("a",{id:"relations-Tuple{TorQuadModule}",href:"#relations-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"relations")],-1)),i[115]||(i[115]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[123]||(i[123]=a('
julia
relations(T::TorQuadModule) -> ZZLat
',1)),s("p",null,[i[120]||(i[120]=t("For ")),s("mjx-container",t1,[(e(),l("svg",a1,i[116]||(i[116]=[a('',1)]))),i[117]||(i[117]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"T"),s("mo",null,"="),s("mi",null,"M"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"N")])],-1))]),i[121]||(i[121]=t(" this returns ")),s("mjx-container",l1,[(e(),l("svg",e1,i[118]||(i[118]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D441",d:"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z",style:{"stroke-width":"3"}})])])],-1)]))),i[119]||(i[119]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"N")])],-1))]),i[122]||(i[122]=t("."))]),i[124]||(i[124]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",n1,[s("summary",null,[i[125]||(i[125]=s("a",{id:"value_module-Tuple{TorQuadModule}",href:"#value_module-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"value_module")],-1)),i[126]||(i[126]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[127]||(i[127]=a('
julia
value_module(T::TorQuadModule) -> QmodnZ

Return the value module Q/nZ of the bilinear form of T.

source

',3))]),s("details",h1,[s("summary",null,[i[128]||(i[128]=s("a",{id:"value_module_quadratic_form-Tuple{TorQuadModule}",href:"#value_module_quadratic_form-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"value_module_quadratic_form")],-1)),i[129]||(i[129]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[130]||(i[130]=a('
julia
value_module_quadratic_form(T::TorQuadModule) -> QmodnZ

Return the value module Q/mZ of the quadratic form of T.

source

',3))]),s("details",Q1,[s("summary",null,[i[131]||(i[131]=s("a",{id:"gram_matrix_bilinear-Tuple{TorQuadModule}",href:"#gram_matrix_bilinear-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"gram_matrix_bilinear")],-1)),i[132]||(i[132]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[133]||(i[133]=a('
julia
gram_matrix_bilinear(T::TorQuadModule) -> QQMatrix

Return the gram matrix of the bilinear form of T.

source

',3))]),s("details",p1,[s("summary",null,[i[134]||(i[134]=s("a",{id:"gram_matrix_quadratic-Tuple{TorQuadModule}",href:"#gram_matrix_quadratic-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"gram_matrix_quadratic")],-1)),i[135]||(i[135]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[136]||(i[136]=a('
julia
gram_matrix_quadratic(T::TorQuadModule) -> QQMatrix

Return the 'gram matrix' of the quadratic form of T.

The off diagonal entries are given by the bilinear form whereas the diagonal entries are given by the quadratic form.

source

',4))]),s("details",r1,[s("summary",null,[i[137]||(i[137]=s("a",{id:"modulus_bilinear_form-Tuple{TorQuadModule}",href:"#modulus_bilinear_form-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"modulus_bilinear_form")],-1)),i[138]||(i[138]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[139]||(i[139]=a('
julia
modulus_bilinear_form(T::TorQuadModule) -> QQFieldElem

Return the modulus of the value module of the bilinear form ofT.

source

',3))]),s("details",d1,[s("summary",null,[i[140]||(i[140]=s("a",{id:"modulus_quadratic_form-Tuple{TorQuadModule}",href:"#modulus_quadratic_form-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"modulus_quadratic_form")],-1)),i[141]||(i[141]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[142]||(i[142]=a('
julia
modulus_quadratic_form(T::TorQuadModule) -> QQFieldElem

Return the modulus of the value module of the quadratic form of T.

source

',3))]),i[407]||(i[407]=s("h3",{id:"elements",tabindex:"-1"},[t("Elements "),s("a",{class:"header-anchor",href:"#elements","aria-label":'Permalink to "Elements"'},"​")],-1)),s("details",o1,[s("summary",null,[i[143]||(i[143]=s("a",{id:"quadratic_product-Tuple{TorQuadModuleElem}",href:"#quadratic_product-Tuple{TorQuadModuleElem}"},[s("span",{class:"jlbinding"},"quadratic_product")],-1)),i[144]||(i[144]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[152]||(i[152]=a('
julia
quadratic_product(a::TorQuadModuleElem) -> QmodnZElem

Return the quadratic product of a.

',2)),s("p",null,[i[149]||(i[149]=t("It is defined in terms of a representative: for ")),s("mjx-container",k1,[(e(),l("svg",T1,i[145]||(i[145]=[a('',1)]))),i[146]||(i[146]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"b"),s("mo",null,"+"),s("mi",null,"M"),s("mo",null,"∈"),s("mi",null,"M"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"N"),s("mo",null,"="),s("mi",null,"T")])],-1))]),i[150]||(i[150]=t(", this returns ")),s("mjx-container",m1,[(e(),l("svg",g1,i[147]||(i[147]=[a('',1)]))),i[148]||(i[148]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",{mathvariant:"normal"},"Φ"),s("mo",{stretchy:"false"},"("),s("mi",null,"b"),s("mo",null,","),s("mi",null,"b"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mi",null,"n"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),i[151]||(i[151]=t("."))]),i[153]||(i[153]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",u1,[s("summary",null,[i[154]||(i[154]=s("a",{id:"inner_product-Tuple{TorQuadModuleElem, TorQuadModuleElem}",href:"#inner_product-Tuple{TorQuadModuleElem, TorQuadModuleElem}"},[s("span",{class:"jlbinding"},"inner_product")],-1)),i[155]||(i[155]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[156]||(i[156]=a('
julia
inner_product(a::TorQuadModuleElem, b::TorQuadModuleElem) -> QmodnZElem

Return the inner product of a and b.

source

',3))]),i[408]||(i[408]=s("h3",{id:"Lift-to-the-cover",tabindex:"-1"},[t("Lift to the cover "),s("a",{class:"header-anchor",href:"#Lift-to-the-cover","aria-label":'Permalink to "Lift to the cover {#Lift-to-the-cover}"'},"​")],-1)),s("details",y1,[s("summary",null,[i[157]||(i[157]=s("a",{id:"lift-Tuple{TorQuadModuleElem}",href:"#lift-Tuple{TorQuadModuleElem}"},[s("span",{class:"jlbinding"},"lift")],-1)),i[158]||(i[158]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[166]||(i[166]=a('
julia
lift(a::TorQuadModuleElem) -> Vector{QQFieldElem}

Lift a to the ambient space of cover(parent(a)).

',2)),s("p",null,[i[163]||(i[163]=t("For ")),s("mjx-container",E1,[(e(),l("svg",c1,i[159]||(i[159]=[a('',1)]))),i[160]||(i[160]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"a"),s("mo",null,"+"),s("mi",null,"N"),s("mo",null,"∈"),s("mi",null,"M"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"N")])],-1))]),i[164]||(i[164]=t(" this returns the representative ")),s("mjx-container",x1,[(e(),l("svg",F1,i[161]||(i[161]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),i[162]||(i[162]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"a")])],-1))]),i[165]||(i[165]=t("."))]),i[167]||(i[167]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",w1,[s("summary",null,[i[168]||(i[168]=s("a",{id:"representative-Tuple{TorQuadModuleElem}",href:"#representative-Tuple{TorQuadModuleElem}"},[s("span",{class:"jlbinding"},"representative")],-1)),i[169]||(i[169]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[179]||(i[179]=a('
julia
representative(a::TorQuadModuleElem) -> Vector{QQFieldElem}
',1)),s("p",null,[i[174]||(i[174]=t("For ")),s("mjx-container",C1,[(e(),l("svg",f1,i[170]||(i[170]=[a('',1)]))),i[171]||(i[171]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"a"),s("mo",null,"+"),s("mi",null,"N"),s("mo",null,"∈"),s("mi",null,"M"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mi",null,"N")])],-1))]),i[175]||(i[175]=t(" this returns the representative ")),s("mjx-container",H1,[(e(),l("svg",b1,i[172]||(i[172]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D44E",d:"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z",style:{"stroke-width":"3"}})])])],-1)]))),i[173]||(i[173]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"a")])],-1))]),i[176]||(i[176]=t(". An alias for ")),i[177]||(i[177]=s("code",null,"lift(a)",-1)),i[178]||(i[178]=t("."))]),i[180]||(i[180]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i[409]||(i[409]=s("h3",{id:"Orthogonal-submodules",tabindex:"-1"},[t("Orthogonal submodules "),s("a",{class:"header-anchor",href:"#Orthogonal-submodules","aria-label":'Permalink to "Orthogonal submodules {#Orthogonal-submodules}"'},"​")],-1)),s("details",M1,[s("summary",null,[i[181]||(i[181]=s("a",{id:"orthogonal_submodule-Tuple{TorQuadModule, TorQuadModule}",href:"#orthogonal_submodule-Tuple{TorQuadModule, TorQuadModule}"},[s("span",{class:"jlbinding"},"orthogonal_submodule")],-1)),i[182]||(i[182]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[183]||(i[183]=a('
julia
orthogonal_submodule(T::TorQuadModule, S::TorQuadModule)-> TorQuadModule

Return the orthogonal submodule to the submodule S of T.

source

',3))]),i[410]||(i[410]=s("h3",{id:"isometry",tabindex:"-1"},[t("Isometry "),s("a",{class:"header-anchor",href:"#isometry","aria-label":'Permalink to "Isometry"'},"​")],-1)),s("details",L1,[s("summary",null,[i[184]||(i[184]=s("a",{id:"is_isometric_with_isometry-Tuple{TorQuadModule, TorQuadModule}",href:"#is_isometric_with_isometry-Tuple{TorQuadModule, TorQuadModule}"},[s("span",{class:"jlbinding"},"is_isometric_with_isometry")],-1)),i[185]||(i[185]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[194]||(i[194]=a(`
julia
is_isometric_with_isometry(T::TorQuadModule, U::TorQuadModule)
-                                               -> Bool, TorQuadModuleMap
`,1)),s("p",null,[i[188]||(i[188]=t("Return whether the torsion quadratic modules ")),i[189]||(i[189]=s("code",null,"T",-1)),i[190]||(i[190]=t(" and ")),i[191]||(i[191]=s("code",null,"U",-1)),i[192]||(i[192]=t(" are isometric. If yes, it also returns an isometry ")),s("mjx-container",v1,[(e(),l("svg",D1,i[186]||(i[186]=[a('',1)]))),i[187]||(i[187]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"T"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"U")])],-1))]),i[193]||(i[193]=t("."))]),i[195]||(i[195]=a(`

If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).

It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).

Examples

julia
julia> T = torsion_quadratic_module(QQ[2//3 2//3    0    0    0;
+                                               -> Bool, TorQuadModuleMap
`,1)),s("p",null,[i[188]||(i[188]=t("Return whether the torsion quadratic modules ")),i[189]||(i[189]=s("code",null,"T",-1)),i[190]||(i[190]=t(" and ")),i[191]||(i[191]=s("code",null,"U",-1)),i[192]||(i[192]=t(" are isometric. If yes, it also returns an isometry ")),s("mjx-container",v1,[(e(),l("svg",D1,i[186]||(i[186]=[a('',1)]))),i[187]||(i[187]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"T"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"U")])],-1))]),i[193]||(i[193]=t("."))]),i[195]||(i[195]=a(`

If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).

It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).

Examples

julia
julia> T = torsion_quadratic_module(QQ[2//3 2//3    0    0    0;
                                        2//3 2//3 2//3    0 2//3;
                                           0 2//3 2//3 2//3    0;
                                           0    0 2//3 2//3    0;
@@ -86,7 +86,7 @@ import{_ as Q,c as l,j as s,a as t,a5 as a,G as h,B as p,o as e}from"./chunks/fr
 
 julia> is_bijective(phi)
 true

source

`,5))]),s("details",B1,[s("summary",null,[i[196]||(i[196]=s("a",{id:"is_anti_isometric_with_anti_isometry-Tuple{TorQuadModule, TorQuadModule}",href:"#is_anti_isometric_with_anti_isometry-Tuple{TorQuadModule, TorQuadModule}"},[s("span",{class:"jlbinding"},"is_anti_isometric_with_anti_isometry")],-1)),i[197]||(i[197]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[206]||(i[206]=a(`
julia
is_anti_isometric_with_anti_isometry(T::TorQuadModule, U::TorQuadModule)
-                                                 -> Bool, TorQuadModuleMap
`,1)),s("p",null,[i[200]||(i[200]=t("Return whether there exists an anti-isometry between the torsion quadratic modules ")),i[201]||(i[201]=s("code",null,"T",-1)),i[202]||(i[202]=t(" and ")),i[203]||(i[203]=s("code",null,"U",-1)),i[204]||(i[204]=t(". If yes, it returns such an anti-isometry ")),s("mjx-container",j1,[(e(),l("svg",Z1,i[198]||(i[198]=[a('',1)]))),i[199]||(i[199]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"T"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"U")])],-1))]),i[205]||(i[205]=t("."))]),i[207]||(i[207]=a(`

If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).

It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).

Examples

julia
julia> T = torsion_quadratic_module(QQ[4//5;])
+                                                 -> Bool, TorQuadModuleMap
`,1)),s("p",null,[i[200]||(i[200]=t("Return whether there exists an anti-isometry between the torsion quadratic modules ")),i[201]||(i[201]=s("code",null,"T",-1)),i[202]||(i[202]=t(" and ")),i[203]||(i[203]=s("code",null,"U",-1)),i[204]||(i[204]=t(". If yes, it returns such an anti-isometry ")),s("mjx-container",j1,[(e(),l("svg",Z1,i[198]||(i[198]=[a('',1)]))),i[199]||(i[199]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"T"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"U")])],-1))]),i[205]||(i[205]=t("."))]),i[207]||(i[207]=a(`

If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).

It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).

Examples

julia
julia> T = torsion_quadratic_module(QQ[4//5;])
 Finite quadratic module
   over integer ring
 Abelian group: Z/5
@@ -146,7 +146,7 @@ import{_ as Q,c as l,j as s,a as t,a5 as a,G as h,B as p,o as e}from"./chunks/fr
 julia> a = gens(T)[1];
 
 julia> a*a == -phi(a)*phi(a)
-true

source

`,5))]),i[411]||(i[411]=s("h3",{id:"Primary-and-elementary-modules",tabindex:"-1"},[t("Primary and elementary modules "),s("a",{class:"header-anchor",href:"#Primary-and-elementary-modules","aria-label":'Permalink to "Primary and elementary modules {#Primary-and-elementary-modules}"'},"​")],-1)),s("details",A1,[s("summary",null,[i[208]||(i[208]=s("a",{id:"is_primary_with_prime-Tuple{TorQuadModule}",href:"#is_primary_with_prime-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),i[209]||(i[209]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[210]||(i[210]=a('
julia
is_primary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem

Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group for some prime number p. In case it is, p is also returned as second output.

Note that in the case of trivial groups, this function returns (true, 1). If T is not primary, the second return value is -1 by default.

source

',4))]),s("details",V1,[s("summary",null,[i[211]||(i[211]=s("a",{id:"is_primary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}",href:"#is_primary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}"},[s("span",{class:"jlbinding"},"is_primary")],-1)),i[212]||(i[212]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[213]||(i[213]=a('
julia
is_primary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool

Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group.

source

',3))]),s("details",q1,[s("summary",null,[i[214]||(i[214]=s("a",{id:"is_elementary_with_prime-Tuple{TorQuadModule}",href:"#is_elementary_with_prime-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_elementary_with_prime")],-1)),i[215]||(i[215]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[216]||(i[216]=a('
julia
is_elementary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem

Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group, for some prime number p. In case it is, p is also returned as second output.

Note that in the case of trivial groups, this function returns (true, 1). If T is not elementary, the second return value is -1 by default.

source

',4))]),s("details",O1,[s("summary",null,[i[217]||(i[217]=s("a",{id:"is_elementary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}",href:"#is_elementary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}"},[s("span",{class:"jlbinding"},"is_elementary")],-1)),i[218]||(i[218]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[219]||(i[219]=a('
julia
is_elementary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool

Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group.

source

',3))]),i[412]||(i[412]=s("h3",{id:"Smith-normal-form",tabindex:"-1"},[t("Smith normal form "),s("a",{class:"header-anchor",href:"#Smith-normal-form","aria-label":'Permalink to "Smith normal form {#Smith-normal-form}"'},"​")],-1)),s("details",R1,[s("summary",null,[i[220]||(i[220]=s("a",{id:"snf-Tuple{TorQuadModule}",href:"#snf-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"snf")],-1)),i[221]||(i[221]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[234]||(i[234]=a('
julia
snf(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap
',1)),s("p",null,[i[224]||(i[224]=t("Given a torsion quadratic module ")),i[225]||(i[225]=s("code",null,"T",-1)),i[226]||(i[226]=t(", return a torsion quadratic module ")),i[227]||(i[227]=s("code",null,"S",-1)),i[228]||(i[228]=t(", isometric to ")),i[229]||(i[229]=s("code",null,"T",-1)),i[230]||(i[230]=t(", such that the underlying abelian group of ")),i[231]||(i[231]=s("code",null,"S",-1)),i[232]||(i[232]=t(" is in canonical Smith normal form. It comes with an isometry ")),s("mjx-container",G1,[(e(),l("svg",S1,i[222]||(i[222]=[a('',1)]))),i[223]||(i[223]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"f"),s("mo",null,":"),s("mi",null,"S"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"T")])],-1))]),i[233]||(i[233]=t("."))]),i[235]||(i[235]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",z1,[s("summary",null,[i[236]||(i[236]=s("a",{id:"is_snf-Tuple{TorQuadModule}",href:"#is_snf-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_snf")],-1)),i[237]||(i[237]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[238]||(i[238]=a('
julia
is_snf(T::TorQuadModule) -> Bool

Given a torsion quadratic module T, return whether its underlying abelian group is in Smith normal form.

source

',3))]),i[413]||(i[413]=a('

Discriminant Groups ​

See [6] for the general theory of discriminant groups. They are particularly useful to work with primitive embeddings of integral integer quadratic lattices.

From a lattice ​

',3)),s("details",I1,[s("summary",null,[i[239]||(i[239]=s("a",{id:"discriminant_group-Tuple{ZZLat}",href:"#discriminant_group-Tuple{ZZLat}"},[s("span",{class:"jlbinding"},"discriminant_group")],-1)),i[240]||(i[240]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[249]||(i[249]=a('
julia
discriminant_group(L::ZZLat) -> TorQuadModule

Return the discriminant group of L.

The discriminant group of an integral lattice L is the finite abelian group D = dual(L)/L.

It comes equipped with the discriminant bilinear form

',4)),s("mjx-container",J1,[(e(),l("svg",N1,i[241]||(i[241]=[a('',1)]))),i[242]||(i[242]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"D"),s("mo",null,"×"),s("mi",null,"D"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Q")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mstyle",{scriptlevel:"0"},[s("mspace",{width:"2em"})]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"↦"),s("mi",{mathvariant:"normal"},"Φ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mo",null,".")])],-1))]),s("p",null,[i[245]||(i[245]=t("If ")),i[246]||(i[246]=s("code",null,"L",-1)),i[247]||(i[247]=t(" is even, then the discriminant group is equipped with the discriminant quadratic form ")),s("mjx-container",X1,[(e(),l("svg",P1,i[243]||(i[243]=[a('',1)]))),i[244]||(i[244]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"D"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Q")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mn",null,"2"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mo",null,","),s("mi",null,"x"),s("mo",{stretchy:"false"},"↦"),s("mi",{mathvariant:"normal"},"Φ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"2"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),i[248]||(i[248]=t("."))]),i[250]||(i[250]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i[414]||(i[414]=s("h3",{id:"From-a-matrix",tabindex:"-1"},[t("From a matrix "),s("a",{class:"header-anchor",href:"#From-a-matrix","aria-label":'Permalink to "From a matrix {#From-a-matrix}"'},"​")],-1)),s("details",U1,[s("summary",null,[i[251]||(i[251]=s("a",{id:"torsion_quadratic_module-Tuple{QQMatrix}",href:"#torsion_quadratic_module-Tuple{QQMatrix}"},[s("span",{class:"jlbinding"},"torsion_quadratic_module")],-1)),i[252]||(i[252]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[253]||(i[253]=a(`
julia
torsion_quadratic_module(q::QQMatrix) -> TorQuadModule

Return a torsion quadratic module with gram matrix given by q and value module Q/Z. If all the diagonal entries of q have: either even numerator or even denominator, then the value module of the quadratic form is Q/2Z

Example

julia
julia> torsion_quadratic_module(QQ[1//6;])
+true

source

`,5))]),i[411]||(i[411]=s("h3",{id:"Primary-and-elementary-modules",tabindex:"-1"},[t("Primary and elementary modules "),s("a",{class:"header-anchor",href:"#Primary-and-elementary-modules","aria-label":'Permalink to "Primary and elementary modules {#Primary-and-elementary-modules}"'},"​")],-1)),s("details",A1,[s("summary",null,[i[208]||(i[208]=s("a",{id:"is_primary_with_prime-Tuple{TorQuadModule}",href:"#is_primary_with_prime-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),i[209]||(i[209]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[210]||(i[210]=a('
julia
is_primary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem

Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group for some prime number p. In case it is, p is also returned as second output.

Note that in the case of trivial groups, this function returns (true, 1). If T is not primary, the second return value is -1 by default.

source

',4))]),s("details",V1,[s("summary",null,[i[211]||(i[211]=s("a",{id:"is_primary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}",href:"#is_primary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}"},[s("span",{class:"jlbinding"},"is_primary")],-1)),i[212]||(i[212]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[213]||(i[213]=a('
julia
is_primary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool

Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group.

source

',3))]),s("details",q1,[s("summary",null,[i[214]||(i[214]=s("a",{id:"is_elementary_with_prime-Tuple{TorQuadModule}",href:"#is_elementary_with_prime-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_elementary_with_prime")],-1)),i[215]||(i[215]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[216]||(i[216]=a('
julia
is_elementary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem

Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group, for some prime number p. In case it is, p is also returned as second output.

Note that in the case of trivial groups, this function returns (true, 1). If T is not elementary, the second return value is -1 by default.

source

',4))]),s("details",O1,[s("summary",null,[i[217]||(i[217]=s("a",{id:"is_elementary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}",href:"#is_elementary-Tuple{TorQuadModule, Union{Integer, ZZRingElem}}"},[s("span",{class:"jlbinding"},"is_elementary")],-1)),i[218]||(i[218]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[219]||(i[219]=a('
julia
is_elementary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool

Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group.

source

',3))]),i[412]||(i[412]=s("h3",{id:"Smith-normal-form",tabindex:"-1"},[t("Smith normal form "),s("a",{class:"header-anchor",href:"#Smith-normal-form","aria-label":'Permalink to "Smith normal form {#Smith-normal-form}"'},"​")],-1)),s("details",R1,[s("summary",null,[i[220]||(i[220]=s("a",{id:"snf-Tuple{TorQuadModule}",href:"#snf-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"snf")],-1)),i[221]||(i[221]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[234]||(i[234]=a('
julia
snf(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap
',1)),s("p",null,[i[224]||(i[224]=t("Given a torsion quadratic module ")),i[225]||(i[225]=s("code",null,"T",-1)),i[226]||(i[226]=t(", return a torsion quadratic module ")),i[227]||(i[227]=s("code",null,"S",-1)),i[228]||(i[228]=t(", isometric to ")),i[229]||(i[229]=s("code",null,"T",-1)),i[230]||(i[230]=t(", such that the underlying abelian group of ")),i[231]||(i[231]=s("code",null,"S",-1)),i[232]||(i[232]=t(" is in canonical Smith normal form. It comes with an isometry ")),s("mjx-container",G1,[(e(),l("svg",S1,i[222]||(i[222]=[a('',1)]))),i[223]||(i[223]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"f"),s("mo",null,":"),s("mi",null,"S"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mi",null,"T")])],-1))]),i[233]||(i[233]=t("."))]),i[235]||(i[235]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s("details",z1,[s("summary",null,[i[236]||(i[236]=s("a",{id:"is_snf-Tuple{TorQuadModule}",href:"#is_snf-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_snf")],-1)),i[237]||(i[237]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[238]||(i[238]=a('
julia
is_snf(T::TorQuadModule) -> Bool

Given a torsion quadratic module T, return whether its underlying abelian group is in Smith normal form.

source

',3))]),i[413]||(i[413]=a('

Discriminant Groups ​

See [6] for the general theory of discriminant groups. They are particularly useful to work with primitive embeddings of integral integer quadratic lattices.

From a lattice ​

',3)),s("details",I1,[s("summary",null,[i[239]||(i[239]=s("a",{id:"discriminant_group-Tuple{ZZLat}",href:"#discriminant_group-Tuple{ZZLat}"},[s("span",{class:"jlbinding"},"discriminant_group")],-1)),i[240]||(i[240]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[249]||(i[249]=a('
julia
discriminant_group(L::ZZLat) -> TorQuadModule

Return the discriminant group of L.

The discriminant group of an integral lattice L is the finite abelian group D = dual(L)/L.

It comes equipped with the discriminant bilinear form

',4)),s("mjx-container",J1,[(e(),l("svg",N1,i[241]||(i[241]=[a('',1)]))),i[242]||(i[242]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"D"),s("mo",null,"×"),s("mi",null,"D"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Q")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mstyle",{scriptlevel:"0"},[s("mspace",{width:"2em"})]),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},"↦"),s("mi",{mathvariant:"normal"},"Φ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"y"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mo",null,".")])],-1))]),s("p",null,[i[245]||(i[245]=t("If ")),i[246]||(i[246]=s("code",null,"L",-1)),i[247]||(i[247]=t(" is even, then the discriminant group is equipped with the discriminant quadratic form ")),s("mjx-container",X1,[(e(),l("svg",P1,i[243]||(i[243]=[a('',1)]))),i[244]||(i[244]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"D"),s("mo",{accent:"false",stretchy:"false"},"→"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Q")]),s("mrow",{"data-mjx-texclass":"ORD"},[s("mo",null,"/")]),s("mn",null,"2"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")]),s("mo",null,","),s("mi",null,"x"),s("mo",{stretchy:"false"},"↦"),s("mi",{mathvariant:"normal"},"Φ"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",null,","),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",null,"+"),s("mn",null,"2"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),i[248]||(i[248]=t("."))]),i[250]||(i[250]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i[414]||(i[414]=s("h3",{id:"From-a-matrix",tabindex:"-1"},[t("From a matrix "),s("a",{class:"header-anchor",href:"#From-a-matrix","aria-label":'Permalink to "From a matrix {#From-a-matrix}"'},"​")],-1)),s("details",U1,[s("summary",null,[i[251]||(i[251]=s("a",{id:"torsion_quadratic_module-Tuple{QQMatrix}",href:"#torsion_quadratic_module-Tuple{QQMatrix}"},[s("span",{class:"jlbinding"},"torsion_quadratic_module")],-1)),i[252]||(i[252]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[253]||(i[253]=a(`
julia
torsion_quadratic_module(q::QQMatrix) -> TorQuadModule

Return a torsion quadratic module with gram matrix given by q and value module Q/Z. If all the diagonal entries of q have: either even numerator or even denominator, then the value module of the quadratic form is Q/2Z

Example

julia
julia> torsion_quadratic_module(QQ[1//6;])
 Finite quadratic module
   over integer ring
 Abelian group: Z/6
@@ -180,9 +180,9 @@ import{_ as Q,c as l,j as s,a as t,a5 as a,G as h,B as p,o as e}from"./chunks/fr
 Bilinear value module: Q/Z
 Quadratic value module: Q/Z
 Gram matrix quadratic form:
-[1//3]

source

`,5))]),i[415]||(i[415]=s("h3",{id:"Rescaling-the-form",tabindex:"-1"},[t("Rescaling the form "),s("a",{class:"header-anchor",href:"#Rescaling-the-form","aria-label":'Permalink to "Rescaling the form {#Rescaling-the-form}"'},"​")],-1)),s("details",$1,[s("summary",null,[i[254]||(i[254]=s("a",{id:"rescale-Tuple{TorQuadModule, RingElement}",href:"#rescale-Tuple{TorQuadModule, RingElement}"},[s("span",{class:"jlbinding"},"rescale")],-1)),i[255]||(i[255]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[264]||(i[264]=a('
julia
rescale(T::TorQuadModule, k::RingElement) -> TorQuadModule
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julia
is_degenerate(T::TorQuadModule) -> Bool

Return true if the underlying bilinear form is degenerate.

source

',3))]),s("details",_1,[s("summary",null,[i[269]||(i[269]=s("a",{id:"is_semi_regular-Tuple{TorQuadModule}",href:"#is_semi_regular-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_semi_regular")],-1)),i[270]||(i[270]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[271]||(i[271]=a('
julia
is_semi_regular(T::TorQuadModule) -> Bool

Return whether T is semi-regular, that is its quadratic radical is trivial (see radical_quadratic).

source

',3))]),s("details",s2,[s("summary",null,[i[272]||(i[272]=s("a",{id:"radical_bilinear-Tuple{TorQuadModule}",href:"#radical_bilinear-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"radical_bilinear")],-1)),i[273]||(i[273]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[274]||(i[274]=a('
julia
radical_bilinear(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap

Return the radical \\{x \\in T | b(x,T) = 0\\} of the bilinear form b on T.

source

',3))]),s("details",i2,[s("summary",null,[i[275]||(i[275]=s("a",{id:"radical_quadratic-Tuple{TorQuadModule}",href:"#radical_quadratic-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"radical_quadratic")],-1)),i[276]||(i[276]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[277]||(i[277]=a('
julia
radical_quadratic(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap

Return the radical \\{x \\in T | b(x,T) = 0 and q(x)=0\\} of the quadratic form q on T.

source

',3))]),s("details",t2,[s("summary",null,[i[278]||(i[278]=s("a",{id:"normal_form-Tuple{TorQuadModule}",href:"#normal_form-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"normal_form")],-1)),i[279]||(i[279]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[280]||(i[280]=a('
julia
normal_form(T::TorQuadModule; partial=false) -> TorQuadModule, TorQuadModuleMap

Return the normal form N of the given torsion quadratic module T along with the projection T -> N.

Let K be the radical of the quadratic form of T. Then N = T/K is half-regular. Two half-regular torsion quadratic modules are isometric if and only if they have equal normal forms.

source

',4))]),i[417]||(i[417]=s("h3",{id:"genus",tabindex:"-1"},[t("Genus "),s("a",{class:"header-anchor",href:"#genus","aria-label":'Permalink to "Genus"'},"​")],-1)),s("details",a2,[s("summary",null,[i[281]||(i[281]=s("a",{id:"genus-Tuple{TorQuadModule, Tuple{Int64, Int64}}",href:"#genus-Tuple{TorQuadModule, Tuple{Int64, Int64}}"},[s("span",{class:"jlbinding"},"genus")],-1)),i[282]||(i[282]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[283]||(i[283]=a(`
julia
genus(T::TorQuadModule, signature_pair::Tuple{Int, Int};
+[1//3]

source

`,5))]),i[415]||(i[415]=s("h3",{id:"Rescaling-the-form",tabindex:"-1"},[t("Rescaling the form "),s("a",{class:"header-anchor",href:"#Rescaling-the-form","aria-label":'Permalink to "Rescaling the form {#Rescaling-the-form}"'},"​")],-1)),s("details",$1,[s("summary",null,[i[254]||(i[254]=s("a",{id:"rescale-Tuple{TorQuadModule, RingElement}",href:"#rescale-Tuple{TorQuadModule, RingElement}"},[s("span",{class:"jlbinding"},"rescale")],-1)),i[255]||(i[255]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[264]||(i[264]=a('
julia
rescale(T::TorQuadModule, k::RingElement) -> TorQuadModule
',1)),s("p",null,[i[258]||(i[258]=t("Return the torsion quadratic module with quadratic form scaled by ")),s("mjx-container",K1,[(e(),l("svg",W1,i[256]||(i[256]=[s("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[s("g",{"data-mml-node":"math"},[s("g",{"data-mml-node":"mi"},[s("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z",style:{"stroke-width":"3"}})])])],-1)]))),i[257]||(i[257]=s("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[s("mi",null,"k")])],-1))]),i[259]||(i[259]=t(", where k is a non-zero rational number. If the old form was defined modulo ")),i[260]||(i[260]=s("code",null,"n",-1)),i[261]||(i[261]=t(", then the new form is defined modulo ")),i[262]||(i[262]=s("code",null,"n k",-1)),i[263]||(i[263]=t("."))]),i[265]||(i[265]=s("p",null,[s("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i[416]||(i[416]=s("h3",{id:"invariants",tabindex:"-1"},[t("Invariants "),s("a",{class:"header-anchor",href:"#invariants","aria-label":'Permalink to "Invariants"'},"​")],-1)),s("details",Y1,[s("summary",null,[i[266]||(i[266]=s("a",{id:"is_degenerate-Tuple{TorQuadModule}",href:"#is_degenerate-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_degenerate")],-1)),i[267]||(i[267]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[268]||(i[268]=a('
julia
is_degenerate(T::TorQuadModule) -> Bool

Return true if the underlying bilinear form is degenerate.

source

',3))]),s("details",_1,[s("summary",null,[i[269]||(i[269]=s("a",{id:"is_semi_regular-Tuple{TorQuadModule}",href:"#is_semi_regular-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"is_semi_regular")],-1)),i[270]||(i[270]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[271]||(i[271]=a('
julia
is_semi_regular(T::TorQuadModule) -> Bool

Return whether T is semi-regular, that is its quadratic radical is trivial (see radical_quadratic).

source

',3))]),s("details",s2,[s("summary",null,[i[272]||(i[272]=s("a",{id:"radical_bilinear-Tuple{TorQuadModule}",href:"#radical_bilinear-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"radical_bilinear")],-1)),i[273]||(i[273]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[274]||(i[274]=a('
julia
radical_bilinear(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap

Return the radical \\{x \\in T | b(x,T) = 0\\} of the bilinear form b on T.

source

',3))]),s("details",i2,[s("summary",null,[i[275]||(i[275]=s("a",{id:"radical_quadratic-Tuple{TorQuadModule}",href:"#radical_quadratic-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"radical_quadratic")],-1)),i[276]||(i[276]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[277]||(i[277]=a('
julia
radical_quadratic(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap

Return the radical \\{x \\in T | b(x,T) = 0 and q(x)=0\\} of the quadratic form q on T.

source

',3))]),s("details",t2,[s("summary",null,[i[278]||(i[278]=s("a",{id:"normal_form-Tuple{TorQuadModule}",href:"#normal_form-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"normal_form")],-1)),i[279]||(i[279]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[280]||(i[280]=a('
julia
normal_form(T::TorQuadModule; partial=false) -> TorQuadModule, TorQuadModuleMap

Return the normal form N of the given torsion quadratic module T along with the projection T -> N.

Let K be the radical of the quadratic form of T. Then N = T/K is half-regular. Two half-regular torsion quadratic modules are isometric if and only if they have equal normal forms.

source

',4))]),i[417]||(i[417]=s("h3",{id:"genus",tabindex:"-1"},[t("Genus "),s("a",{class:"header-anchor",href:"#genus","aria-label":'Permalink to "Genus"'},"​")],-1)),s("details",a2,[s("summary",null,[i[281]||(i[281]=s("a",{id:"genus-Tuple{TorQuadModule, Tuple{Int64, Int64}}",href:"#genus-Tuple{TorQuadModule, Tuple{Int64, Int64}}"},[s("span",{class:"jlbinding"},"genus")],-1)),i[282]||(i[282]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[283]||(i[283]=a(`
julia
genus(T::TorQuadModule, signature_pair::Tuple{Int, Int};
                         parity::RationalUnion = modulus_quadratic_form(T))
-                                                                -> ZZGenus

Return the genus of an integer lattice whose discriminant group has the bilinear form of T, the given signature_pair and the given parity.

The argument parity is one of the following: either parity == 1 for genera of odd lattices, or parity == 2 for even lattices. By default, parity is set to be as the parity of the quadratic form on T

If no such genus exists, raise an error.

Reference

[6] Corollary 1.9.4 and 1.16.3.

source

`,7))]),s("details",l2,[s("summary",null,[i[284]||(i[284]=s("a",{id:"brown_invariant-Tuple{TorQuadModule}",href:"#brown_invariant-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"brown_invariant")],-1)),i[285]||(i[285]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[288]||(i[288]=a('
julia
brown_invariant(self::TorQuadModule) -> Nemo.zzModRingElem

Return the Brown invariant of this torsion quadratic form.

Let (D,q) be a torsion quadratic module with values in Q / 2Z. The Brown invariant Br(D,q) in Z/8Z is defined by the equation

',3)),s("mjx-container",e2,[(e(),l("svg",n2,i[286]||(i[286]=[a('',1)]))),i[287]||(i[287]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"exp"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("mi",null,"i")]),s("mn",null,"8")]),s("mi",null,"B"),s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mi",null,"q"),s("mo",{stretchy:"false"},")"),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("msqrt",null,[s("mi",null,"D")])]),s("munder",null,[s("mo",{"data-mjx-texclass":"OP"},"∑"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"x"),s("mo",null,"∈"),s("mi",null,"D")])]),s("mi",null,"exp"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"i"),s("mi",null,"π"),s("mi",null,"q"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},")"),s("mo",null,".")])],-1))]),i[289]||(i[289]=a(`

The Brown invariant is additive with respect to direct sums of torsion quadratic modules.

Examples

julia
julia> L = integer_lattice(gram=matrix(ZZ, [[2,-1,0,0],[-1,2,-1,-1],[0,-1,2,0],[0,-1,0,2]]));
+                                                                -> ZZGenus

Return the genus of an integer lattice whose discriminant group has the bilinear form of T, the given signature_pair and the given parity.

The argument parity is one of the following: either parity == 1 for genera of odd lattices, or parity == 2 for even lattices. By default, parity is set to be as the parity of the quadratic form on T

If no such genus exists, raise an error.

Reference

[6] Corollary 1.9.4 and 1.16.3.

source

`,7))]),s("details",l2,[s("summary",null,[i[284]||(i[284]=s("a",{id:"brown_invariant-Tuple{TorQuadModule}",href:"#brown_invariant-Tuple{TorQuadModule}"},[s("span",{class:"jlbinding"},"brown_invariant")],-1)),i[285]||(i[285]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),i[288]||(i[288]=a('
julia
brown_invariant(self::TorQuadModule) -> Nemo.zzModRingElem

Return the Brown invariant of this torsion quadratic form.

Let (D,q) be a torsion quadratic module with values in Q / 2Z. The Brown invariant Br(D,q) in Z/8Z is defined by the equation

',3)),s("mjx-container",e2,[(e(),l("svg",n2,i[286]||(i[286]=[a('',1)]))),i[287]||(i[287]=s("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("mi",null,"exp"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mrow",{"data-mjx-texclass":"INNER"},[s("mo",{"data-mjx-texclass":"OPEN"},"("),s("mfrac",null,[s("mrow",null,[s("mn",null,"2"),s("mi",null,"π"),s("mi",null,"i")]),s("mn",null,"8")]),s("mi",null,"B"),s("mi",null,"r"),s("mo",{stretchy:"false"},"("),s("mi",null,"q"),s("mo",{stretchy:"false"},")"),s("mo",{"data-mjx-texclass":"CLOSE"},")")]),s("mo",null,"="),s("mfrac",null,[s("mn",null,"1"),s("msqrt",null,[s("mi",null,"D")])]),s("munder",null,[s("mo",{"data-mjx-texclass":"OP"},"∑"),s("mrow",{"data-mjx-texclass":"ORD"},[s("mi",null,"x"),s("mo",null,"∈"),s("mi",null,"D")])]),s("mi",null,"exp"),s("mo",{"data-mjx-texclass":"NONE"},"⁡"),s("mo",{stretchy:"false"},"("),s("mi",null,"i"),s("mi",null,"π"),s("mi",null,"q"),s("mo",{stretchy:"false"},"("),s("mi",null,"x"),s("mo",{stretchy:"false"},")"),s("mo",{stretchy:"false"},")"),s("mo",null,".")])],-1))]),i[289]||(i[289]=a(`

The Brown invariant is additive with respect to direct sums of torsion quadratic modules.

Examples

julia
julia> L = integer_lattice(gram=matrix(ZZ, [[2,-1,0,0],[-1,2,-1,-1],[0,-1,2,0],[0,-1,0,2]]));
 
 julia> T = Hecke.discriminant_group(L);
 
diff --git a/v0.34.8/assets/manual_quad_forms_integer_lattices.md.B7hEml57.js b/v0.34.8/assets/manual_quad_forms_integer_lattices.md.B7hEml57.js
index 5d61c04ea0..40222b379c 100644
--- a/v0.34.8/assets/manual_quad_forms_integer_lattices.md.B7hEml57.js
+++ b/v0.34.8/assets/manual_quad_forms_integer_lattices.md.B7hEml57.js
@@ -1,4 +1,4 @@
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")),s[52]||(s[52]=i("a",{href:"/Hecke.jl/v0.34.8/manual/quad_forms/integer_lattices#basis_matrix-Tuple{ZZLat}"},[i("code",null,"basis_matrix(L::ZZLat)")],-1)),s[53]||(s[53]=t("."))]),s[872]||(s[872]=i("h2",{id:"Creation-of-integer-lattices",tabindex:"-1"},[t("Creation of integer lattices "),i("a",{class:"header-anchor",href:"#Creation-of-integer-lattices","aria-label":'Permalink to "Creation of integer lattices {#Creation-of-integer-lattices}"'},"​")],-1)),s[873]||(s[873]=i("h3",{id:"From-a-gram-matrix",tabindex:"-1"},[t("From a gram matrix "),i("a",{class:"header-anchor",href:"#From-a-gram-matrix","aria-label":'Permalink to "From a gram matrix {#From-a-gram-matrix}"'},"​")],-1)),i("details",z,[i("summary",null,[s[54]||(s[54]=i("a",{id:"integer_lattice-Tuple{QQMatrix}",href:"#integer_lattice-Tuple{QQMatrix}"},[i("span",{class:"jlbinding"},"integer_lattice")],-1)),s[55]||(s[55]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[68]||(s[68]=a('
julia
integer_lattice([B::MatElem]; gram) -> ZZLat
',1)),i("p",null,[s[58]||(s[58]=t("Return the Z-lattice with basis matrix ")),i("mjx-container",J,[(e(),l("svg",I,s[56]||(s[56]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),s[57]||(s[57]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"B")])],-1))]),s[59]||(s[59]=t(" inside the quadratic space with Gram matrix ")),s[60]||(s[60]=i("code",null,"gram",-1)),s[61]||(s[61]=t("."))]),i("p",null,[s[64]||(s[64]=t("If the keyword ")),s[65]||(s[65]=i("code",null,"gram",-1)),s[66]||(s[66]=t(" is not specified, the Gram matrix is the identity matrix. If ")),i("mjx-container",N,[(e(),l("svg",q,s[62]||(s[62]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),s[63]||(s[63]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"B")])],-1))]),s[67]||(s[67]=t(" is not specified, the basis matrix is the identity matrix."))]),s[69]||(s[69]=a(`
Examples
julia
julia> L = integer_lattice(matrix(QQ, 2, 2, [1//2, 0, 0, 2]));
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")),i("mjx-container",R,[(e(),l("svg",O,s[45]||(s[45]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 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lattices "),i("a",{class:"header-anchor",href:"#Creation-of-integer-lattices","aria-label":'Permalink to "Creation of integer lattices {#Creation-of-integer-lattices}"'},"​")],-1)),s[873]||(s[873]=i("h3",{id:"From-a-gram-matrix",tabindex:"-1"},[t("From a gram matrix "),i("a",{class:"header-anchor",href:"#From-a-gram-matrix","aria-label":'Permalink to "From a gram matrix {#From-a-gram-matrix}"'},"​")],-1)),i("details",z,[i("summary",null,[s[54]||(s[54]=i("a",{id:"integer_lattice-Tuple{QQMatrix}",href:"#integer_lattice-Tuple{QQMatrix}"},[i("span",{class:"jlbinding"},"integer_lattice")],-1)),s[55]||(s[55]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[68]||(s[68]=a('
julia
integer_lattice([B::MatElem]; gram) -> ZZLat
',1)),i("p",null,[s[58]||(s[58]=t("Return the Z-lattice with basis matrix ")),i("mjx-container",J,[(e(),l("svg",I,s[56]||(s[56]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),s[57]||(s[57]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"B")])],-1))]),s[59]||(s[59]=t(" inside the quadratic space with Gram matrix ")),s[60]||(s[60]=i("code",null,"gram",-1)),s[61]||(s[61]=t("."))]),i("p",null,[s[64]||(s[64]=t("If the keyword ")),s[65]||(s[65]=i("code",null,"gram",-1)),s[66]||(s[66]=t(" is not specified, the Gram matrix is the identity matrix. If ")),i("mjx-container",N,[(e(),l("svg",q,s[62]||(s[62]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),s[63]||(s[63]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"B")])],-1))]),s[67]||(s[67]=t(" is not specified, the basis matrix is the identity matrix."))]),s[69]||(s[69]=a(`
Examples
julia
julia> L = integer_lattice(matrix(QQ, 2, 2, [1//2, 0, 0, 2]));
 
 julia> gram_matrix(L) == matrix(QQ, 2, 2, [1//4, 0, 0, 4])
 true
@@ -16,7 +16,7 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
 
 julia> gram_matrix(L)
 [  0   -13]
-[-13     0]
source
`,3))]),i("details",W,[i("summary",null,[s[88]||(s[88]=i("a",{id:"integer_lattice-Tuple{Symbol, Union{Int64, ZZRingElem}}",href:"#integer_lattice-Tuple{Symbol, Union{Int64, ZZRingElem}}"},[i("span",{class:"jlbinding"},"integer_lattice")],-1)),s[89]||(s[89]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[104]||(s[104]=a('
julia
integer_lattice(S::Symbol, n::RationalUnion = 1) -> ZZlat
',1)),i("p",null,[s[96]||(s[96]=t("Given ")),s[97]||(s[97]=i("code",null,"S = :H",-1)),s[98]||(s[98]=t(" or ")),s[99]||(s[99]=i("code",null,"S = :U",-1)),s[100]||(s[100]=t(", return a ")),i("mjx-container",Y,[(e(),l("svg",_,s[90]||(s[90]=[a('',1)]))),s[91]||(s[91]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[101]||(s[101]=t("-lattice admitting ")),i("mjx-container",ss,[(e(),l("svg",is,s[92]||(s[92]=[a('',1)]))),s[93]||(s[93]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"n"),i("mo",null,"∗"),i("msub",null,[i("mi",null,"J"),i("mn",null,"2")])])],-1))]),s[102]||(s[102]=t(" as Gram matrix in some basis, where ")),i("mjx-container",ts,[(e(),l("svg",as,s[94]||(s[94]=[a('',1)]))),s[95]||(s[95]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"J"),i("mn",null,"2")])])],-1))]),s[103]||(s[103]=t(" is the 2-by-2 matrix with 0's on the main diagonal and 1's elsewhere."))]),s[105]||(s[105]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",ls,[i("summary",null,[s[106]||(s[106]=i("a",{id:"leech_lattice",href:"#leech_lattice"},[i("span",{class:"jlbinding"},"leech_lattice")],-1)),s[107]||(s[107]=t()),h(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[108]||(s[108]=a(`
julia
leech_lattice() -> ZZLat
Return the Leech lattice.
source
julia
leech_lattice(niemeier_lattice::ZZLat) -> ZZLat, QQMatrix, Int
Return a triple L, v, h where L is the Leech lattice.
L is an h-neighbor of the Niemeier lattice N with respect to v. This means that L / L ∩ N ≅ ℤ / h ℤ. Here h is the Coxeter number of the Niemeier lattice.
This implements the 23 holy constructions of the Leech lattice in [5].
Examples
julia
julia> R = integer_lattice(gram=2 * identity_matrix(ZZ, 24));
+[-13     0]
source
`,3))]),i("details",W,[i("summary",null,[s[88]||(s[88]=i("a",{id:"integer_lattice-Tuple{Symbol, Union{Int64, ZZRingElem}}",href:"#integer_lattice-Tuple{Symbol, Union{Int64, ZZRingElem}}"},[i("span",{class:"jlbinding"},"integer_lattice")],-1)),s[89]||(s[89]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[104]||(s[104]=a('
julia
integer_lattice(S::Symbol, n::RationalUnion = 1) -> ZZlat
',1)),i("p",null,[s[96]||(s[96]=t("Given ")),s[97]||(s[97]=i("code",null,"S = :H",-1)),s[98]||(s[98]=t(" or ")),s[99]||(s[99]=i("code",null,"S = :U",-1)),s[100]||(s[100]=t(", return a ")),i("mjx-container",Y,[(e(),l("svg",_,s[90]||(s[90]=[a('',1)]))),s[91]||(s[91]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[101]||(s[101]=t("-lattice admitting ")),i("mjx-container",ss,[(e(),l("svg",is,s[92]||(s[92]=[a('',1)]))),s[93]||(s[93]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"n"),i("mo",null,"∗"),i("msub",null,[i("mi",null,"J"),i("mn",null,"2")])])],-1))]),s[102]||(s[102]=t(" as Gram matrix in some basis, where ")),i("mjx-container",ts,[(e(),l("svg",as,s[94]||(s[94]=[a('',1)]))),s[95]||(s[95]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"J"),i("mn",null,"2")])])],-1))]),s[103]||(s[103]=t(" is the 2-by-2 matrix with 0's on the main diagonal and 1's elsewhere."))]),s[105]||(s[105]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",ls,[i("summary",null,[s[106]||(s[106]=i("a",{id:"leech_lattice",href:"#leech_lattice"},[i("span",{class:"jlbinding"},"leech_lattice")],-1)),s[107]||(s[107]=t()),h(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[108]||(s[108]=a(`
julia
leech_lattice() -> ZZLat
Return the Leech lattice.
source
julia
leech_lattice(niemeier_lattice::ZZLat) -> ZZLat, QQMatrix, Int
Return a triple L, v, h where L is the Leech lattice.
L is an h-neighbor of the Niemeier lattice N with respect to v. This means that L / L ∩ N ≅ ℤ / h ℤ. Here h is the Coxeter number of the Niemeier lattice.
This implements the 23 holy constructions of the Leech lattice in [5].
Examples
julia
julia> R = integer_lattice(gram=2 * identity_matrix(ZZ, 24));
 
 julia> N = maximal_even_lattice(R) # Some Niemeier lattice
 Integer lattice of rank 24 and degree 24
@@ -149,7 +149,7 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
 Genus symbol for integer lattices
 Signatures: (3, 0, 20)
 Local symbol:
-  Local genus symbol at 2: 1^22 4^1_7
source
`,6))]),s[876]||(s[876]=a('
From a genus ​
Integer lattices can be created as representatives of a genus. See (representative(L::ZZGenus))
Rescaling the Quadratic Form ​
',3)),i("details",ps,[i("summary",null,[s[118]||(s[118]=i("a",{id:"rescale-Tuple{ZZLat, Union{Integer, QQFieldElem, ZZRingElem, Rational}}",href:"#rescale-Tuple{ZZLat, Union{Integer, QQFieldElem, ZZRingElem, Rational}}"},[i("span",{class:"jlbinding"},"rescale")],-1)),s[119]||(s[119]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[120]||(s[120]=a(`
julia
rescale(L::ZZLat, r::RationalUnion) -> ZZLat
Return the lattice L in the quadratic space with form r \\Phi.
Examples
This can be useful to apply methods intended for positive definite lattices.
julia
julia> L = integer_lattice(gram=ZZ[-1 0; 0 -1])
+  Local genus symbol at 2: 1^22 4^1_7
source
`,6))]),s[876]||(s[876]=a('
From a genus ​
Integer lattices can be created as representatives of a genus. See (representative(L::ZZGenus))
Rescaling the Quadratic Form ​
',3)),i("details",ps,[i("summary",null,[s[118]||(s[118]=i("a",{id:"rescale-Tuple{ZZLat, Union{Integer, QQFieldElem, ZZRingElem, Rational}}",href:"#rescale-Tuple{ZZLat, Union{Integer, QQFieldElem, ZZRingElem, Rational}}"},[i("span",{class:"jlbinding"},"rescale")],-1)),s[119]||(s[119]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[120]||(s[120]=a(`
julia
rescale(L::ZZLat, r::RationalUnion) -> ZZLat
Return the lattice L in the quadratic space with form r \\Phi.
Examples
This can be useful to apply methods intended for positive definite lattices.
julia
julia> L = integer_lattice(gram=ZZ[-1 0; 0 -1])
 Integer lattice of rank 2 and degree 2
 with gram matrix
 [-1    0]
@@ -169,9 +169,9 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
   over rational field
 with gram matrix
 [ 4   -2]
-[-2    5]
source
`,3))]),s[878]||(s[878]=i("h2",{id:"invariants",tabindex:"-1"},[t("Invariants "),i("a",{class:"header-anchor",href:"#invariants","aria-label":'Permalink to "Invariants"'},"​")],-1)),i("details",bs,[i("summary",null,[s[160]||(s[160]=i("a",{id:"rank-Tuple{ZZLat}",href:"#rank-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"rank")],-1)),s[161]||(s[161]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[162]||(s[162]=a('
julia
rank(L::AbstractLat) -> Int
Return the rank of the underlying module of the lattice L.
source
',3))]),i("details",Bs,[i("summary",null,[s[163]||(s[163]=i("a",{id:"det-Tuple{ZZLat}",href:"#det-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"det")],-1)),s[164]||(s[164]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=a('
julia
det(L::ZZLat) -> QQFieldElem
Return the determinant of the gram matrix of L.
source
',3))]),i("details",fs,[i("summary",null,[s[166]||(s[166]=i("a",{id:"scale-Tuple{ZZLat}",href:"#scale-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"scale")],-1)),s[167]||(s[167]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[177]||(s[177]=a('
julia
scale(L::ZZLat) -> QQFieldElem
Return the scale of L.
',2)),i("p",null,[s[172]||(s[172]=t("The scale of ")),s[173]||(s[173]=i("code",null,"L",-1)),s[174]||(s[174]=t(" is defined as the positive generator of the ")),i("mjx-container",vs,[(e(),l("svg",Ls,s[168]||(s[168]=[a('',1)]))),s[169]||(s[169]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[175]||(s[175]=t("-ideal generated by ")),i("mjx-container",Hs,[(e(),l("svg",Ms,s[170]||(s[170]=[a('',1)]))),s[171]||(s[171]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mo",{fence:"false",stretchy:"false"},"{"),i("mi",{mathvariant:"normal"},"Φ"),i("mo",{stretchy:"false"},"("),i("mi",null,"x"),i("mo",null,","),i("mi",null,"y"),i("mo",{stretchy:"false"},")"),i("mo",null,":"),i("mi",null,"x"),i("mo",null,","),i("mi",null,"y"),i("mo",null,"∈"),i("mi",null,"L"),i("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[176]||(s[176]=t("."))]),s[178]||(s[178]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",js,[i("summary",null,[s[179]||(s[179]=i("a",{id:"norm-Tuple{ZZLat}",href:"#norm-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"norm")],-1)),s[180]||(s[180]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[190]||(s[190]=a('
julia
norm(L::ZZLat) -> QQFieldElem
Return the norm of L.
',2)),i("p",null,[s[185]||(s[185]=t("The norm of ")),s[186]||(s[186]=i("code",null,"L",-1)),s[187]||(s[187]=t(" is defined as the positive generator of the ")),i("mjx-container",Zs,[(e(),l("svg",Ds,s[181]||(s[181]=[a('',1)]))),s[182]||(s[182]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[188]||(s[188]=t("- ideal generated by ")),i("mjx-container",As,[(e(),l("svg",Vs,s[183]||(s[183]=[a('',1)]))),s[184]||(s[184]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mo",{fence:"false",stretchy:"false"},"{"),i("mi",{mathvariant:"normal"},"Φ"),i("mo",{stretchy:"false"},"("),i("mi",null,"x"),i("mo",null,","),i("mi",null,"x"),i("mo",{stretchy:"false"},")"),i("mo",null,":"),i("mi",null,"x"),i("mo",null,"∈"),i("mi",null,"L"),i("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[189]||(s[189]=t("."))]),s[191]||(s[191]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ss,[i("summary",null,[s[192]||(s[192]=i("a",{id:"iseven-Tuple{ZZLat}",href:"#iseven-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"iseven")],-1)),s[193]||(s[193]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[206]||(s[206]=a('
julia
iseven(L::ZZLat) -> Bool
Return whether L is even.
',2)),i("p",null,[s[200]||(s[200]=t("An integer lattice ")),s[201]||(s[201]=i("code",null,"L",-1)),s[202]||(s[202]=t(" in the rational quadratic space ")),i("mjx-container",Gs,[(e(),l("svg",Rs,s[194]||(s[194]=[a('',1)]))),s[195]||(s[195]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mo",{stretchy:"false"},"("),i("mi",null,"V"),i("mo",null,","),i("mi",{mathvariant:"normal"},"Φ"),i("mo",{stretchy:"false"},")")])],-1))]),s[203]||(s[203]=t(" is called even if ")),i("mjx-container",Os,[(e(),l("svg",zs,s[196]||(s[196]=[a('',1)]))),s[197]||(s[197]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",{mathvariant:"normal"},"Φ"),i("mo",{stretchy:"false"},"("),i("mi",null,"x"),i("mo",null,","),i("mi",null,"x"),i("mo",{stretchy:"false"},")"),i("mo",null,"∈"),i("mn",null,"2"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[204]||(s[204]=t(" for all ")),i("mjx-container",Js,[(e(),l("svg",Is,s[198]||(s[198]=[a('',1)]))),s[199]||(s[199]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"x"),i("mi",null,"i"),i("mi",null,"n"),i("mi",null,"L")])],-1))]),s[205]||(s[205]=t("."))]),s[207]||(s[207]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ns,[i("summary",null,[s[208]||(s[208]=i("a",{id:"is_integral-Tuple{ZZLat}",href:"#is_integral-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_integral")],-1)),s[209]||(s[209]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[210]||(s[210]=a('
julia
is_integral(L::AbstractLat) -> Bool
Return whether the lattice L is integral.
source
',3))]),i("details",qs,[i("summary",null,[s[211]||(s[211]=i("a",{id:"is_primary_with_prime-Tuple{ZZLat}",href:"#is_primary_with_prime-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),s[212]||(s[212]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[217]||(s[217]=a('
julia
is_primary_with_prime(L::ZZLat) -> Bool, ZZRingElem
',1)),i("p",null,[s[215]||(s[215]=t("Given a ")),i("mjx-container",Us,[(e(),l("svg",Xs,s[213]||(s[213]=[a('',1)]))),s[214]||(s[214]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[216]||(s[216]=a('-lattice L, return whether L is primary, that is whether L is integral and its discriminant group (see discriminant_group) is a p-group for some prime number p. In case it is, p is also returned as second output.',15))]),s[218]||(s[218]=i("p",null,[t("Note that for unimodular lattices, this function returns "),i("code",null,"(true, 1)"),t(". If the lattice is not primary, the second return value is "),i("code",null,"-1"),t(" by default.")],-1)),s[219]||(s[219]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ps,[i("summary",null,[s[220]||(s[220]=i("a",{id:"is_primary-Tuple{ZZLat, Union{Integer, ZZRingElem}}",href:"#is_primary-Tuple{ZZLat, Union{Integer, ZZRingElem}}"},[i("span",{class:"jlbinding"},"is_primary")],-1)),s[221]||(s[221]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[226]||(s[226]=a('
julia
is_primary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
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julia
is_elementary_with_prime(L::ZZLat) -> Bool, ZZRingElem
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julia
is_elementary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
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julia
mass(L::ZZLat) -> QQFieldElem
Return the mass of the genus of L.
source
',3))]),i("details",li,[i("summary",null,[s[248]||(s[248]=i("a",{id:"genus_representatives-Tuple{ZZLat}",href:"#genus_representatives-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"genus_representatives")],-1)),s[249]||(s[249]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[250]||(s[250]=a('
julia
genus_representatives(L::ZZLat) -> Vector{ZZLat}
Return representatives for the isometry classes in the genus of L.
source
',3))]),s[881]||(s[881]=i("h3",{id:"Real-invariants",tabindex:"-1"},[t("Real invariants "),i("a",{class:"header-anchor",href:"#Real-invariants","aria-label":'Permalink to "Real invariants {#Real-invariants}"'},"​")],-1)),i("details",ei,[i("summary",null,[s[251]||(s[251]=i("a",{id:"signature_tuple-Tuple{ZZLat}",href:"#signature_tuple-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"signature_tuple")],-1)),s[252]||(s[252]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[253]||(s[253]=a('
julia
signature_tuple(L::ZZLat) -> Tuple{Int,Int,Int}
Return the number of (positive, zero, negative) inertia of L.
source
',3))]),i("details",ni,[i("summary",null,[s[254]||(s[254]=i("a",{id:"is_positive_definite-Tuple{ZZLat}",href:"#is_positive_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_positive_definite")],-1)),s[255]||(s[255]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[256]||(s[256]=a('
julia
is_positive_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is positive definite.
source
',3))]),i("details",hi,[i("summary",null,[s[257]||(s[257]=i("a",{id:"is_negative_definite-Tuple{ZZLat}",href:"#is_negative_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_negative_definite")],-1)),s[258]||(s[258]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[259]||(s[259]=a('
julia
is_negative_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is negative definite.
source
',3))]),i("details",pi,[i("summary",null,[s[260]||(s[260]=i("a",{id:"is_definite-Tuple{ZZLat}",href:"#is_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_definite")],-1)),s[261]||(s[261]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[262]||(s[262]=a('
julia
is_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is definite.
source
',3))]),s[882]||(s[882]=i("h2",{id:"isometries",tabindex:"-1"},[t("Isometries "),i("a",{class:"header-anchor",href:"#isometries","aria-label":'Permalink to "Isometries"'},"​")],-1)),i("details",ki,[i("summary",null,[s[263]||(s[263]=i("a",{id:"automorphism_group_generators-Tuple{ZZLat}",href:"#automorphism_group_generators-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"automorphism_group_generators")],-1)),s[264]||(s[264]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[269]||(s[269]=a('
julia
automorphism_group_generators(E::EllipticCurve) -> Vector{EllCrvIso}
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source
julia
automorphism_group_generators(L::AbstractLat; ambient_representation::Bool = true,
+[-2    5]
source
`,3))]),s[878]||(s[878]=i("h2",{id:"invariants",tabindex:"-1"},[t("Invariants "),i("a",{class:"header-anchor",href:"#invariants","aria-label":'Permalink to "Invariants"'},"​")],-1)),i("details",bs,[i("summary",null,[s[160]||(s[160]=i("a",{id:"rank-Tuple{ZZLat}",href:"#rank-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"rank")],-1)),s[161]||(s[161]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[162]||(s[162]=a('
julia
rank(L::AbstractLat) -> Int
Return the rank of the underlying module of the lattice L.
source
',3))]),i("details",Bs,[i("summary",null,[s[163]||(s[163]=i("a",{id:"det-Tuple{ZZLat}",href:"#det-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"det")],-1)),s[164]||(s[164]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=a('
julia
det(L::ZZLat) -> QQFieldElem
Return the determinant of the gram matrix of L.
source
',3))]),i("details",fs,[i("summary",null,[s[166]||(s[166]=i("a",{id:"scale-Tuple{ZZLat}",href:"#scale-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"scale")],-1)),s[167]||(s[167]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[177]||(s[177]=a('
julia
scale(L::ZZLat) -> QQFieldElem
Return the scale of L.
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julia
norm(L::ZZLat) -> QQFieldElem
Return the norm of L.
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julia
iseven(L::ZZLat) -> Bool
Return whether L is even.
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julia
is_integral(L::AbstractLat) -> Bool
Return whether the lattice L is integral.
source
',3))]),i("details",qs,[i("summary",null,[s[211]||(s[211]=i("a",{id:"is_primary_with_prime-Tuple{ZZLat}",href:"#is_primary_with_prime-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),s[212]||(s[212]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[217]||(s[217]=a('
julia
is_primary_with_prime(L::ZZLat) -> Bool, ZZRingElem
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julia
is_primary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
',1)),i("p",null,[s[224]||(s[224]=t("Given an integral ")),i("mjx-container",Ks,[(e(),l("svg",$s,s[222]||(s[222]=[a('',1)]))),s[223]||(s[223]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[225]||(s[225]=a('-lattice L and a prime number p, return whether L is p-primary, that is whether its discriminant group (see discriminant_group) is a p-group.',13))]),s[227]||(s[227]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ws,[i("summary",null,[s[228]||(s[228]=i("a",{id:"is_elementary_with_prime-Tuple{ZZLat}",href:"#is_elementary_with_prime-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_elementary_with_prime")],-1)),s[229]||(s[229]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[234]||(s[234]=a('
julia
is_elementary_with_prime(L::ZZLat) -> Bool, ZZRingElem
',1)),i("p",null,[s[232]||(s[232]=t("Given a ")),i("mjx-container",Ys,[(e(),l("svg",_s,s[230]||(s[230]=[a('',1)]))),s[231]||(s[231]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[233]||(s[233]=a('-lattice L, return whether L is elementary, that is whether L is integral and its discriminant group (see discriminant_group) is an elemenentary p-group for some prime number p. In case it is, p is also returned as second output.',15))]),s[235]||(s[235]=i("p",null,[t("Note that for unimodular lattices, this function returns "),i("code",null,"(true, 1)"),t(". If the lattice is not elementary, the second return value is "),i("code",null,"-1"),t(" by default.")],-1)),s[236]||(s[236]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",si,[i("summary",null,[s[237]||(s[237]=i("a",{id:"is_elementary-Tuple{ZZLat, Union{Integer, ZZRingElem}}",href:"#is_elementary-Tuple{ZZLat, Union{Integer, ZZRingElem}}"},[i("span",{class:"jlbinding"},"is_elementary")],-1)),s[238]||(s[238]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[243]||(s[243]=a('
julia
is_elementary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
',1)),i("p",null,[s[241]||(s[241]=t("Given an integral ")),i("mjx-container",ii,[(e(),l("svg",ti,s[239]||(s[239]=[a('',1)]))),s[240]||(s[240]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[242]||(s[242]=a('-lattice L and a prime number p, return whether L is p-elementary, that is whether its discriminant group (see discriminant_group) is an elementary p-group.',13))]),s[244]||(s[244]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[879]||(s[879]=i("h3",{id:"The-Genus",tabindex:"-1"},[t("The Genus "),i("a",{class:"header-anchor",href:"#The-Genus","aria-label":'Permalink to "The Genus {#The-Genus}"'},"​")],-1)),s[880]||(s[880]=i("p",null,[t("For an integral lattice The genus of an integer lattice collects its local invariants. "),i("a",{href:"/v0.34.8/manual/quad_forms/Zgenera#genus-Tuple{ZZLat}"},[i("code",null,"genus(::ZZLat)")])],-1)),i("details",ai,[i("summary",null,[s[245]||(s[245]=i("a",{id:"mass-Tuple{ZZLat}",href:"#mass-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"mass")],-1)),s[246]||(s[246]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[247]||(s[247]=a('
julia
mass(L::ZZLat) -> QQFieldElem
Return the mass of the genus of L.
source
',3))]),i("details",li,[i("summary",null,[s[248]||(s[248]=i("a",{id:"genus_representatives-Tuple{ZZLat}",href:"#genus_representatives-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"genus_representatives")],-1)),s[249]||(s[249]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[250]||(s[250]=a('
julia
genus_representatives(L::ZZLat) -> Vector{ZZLat}
Return representatives for the isometry classes in the genus of L.
source
',3))]),s[881]||(s[881]=i("h3",{id:"Real-invariants",tabindex:"-1"},[t("Real invariants "),i("a",{class:"header-anchor",href:"#Real-invariants","aria-label":'Permalink to "Real invariants {#Real-invariants}"'},"​")],-1)),i("details",ei,[i("summary",null,[s[251]||(s[251]=i("a",{id:"signature_tuple-Tuple{ZZLat}",href:"#signature_tuple-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"signature_tuple")],-1)),s[252]||(s[252]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[253]||(s[253]=a('
julia
signature_tuple(L::ZZLat) -> Tuple{Int,Int,Int}
Return the number of (positive, zero, negative) inertia of L.
source
',3))]),i("details",ni,[i("summary",null,[s[254]||(s[254]=i("a",{id:"is_positive_definite-Tuple{ZZLat}",href:"#is_positive_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_positive_definite")],-1)),s[255]||(s[255]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[256]||(s[256]=a('
julia
is_positive_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is positive definite.
source
',3))]),i("details",hi,[i("summary",null,[s[257]||(s[257]=i("a",{id:"is_negative_definite-Tuple{ZZLat}",href:"#is_negative_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_negative_definite")],-1)),s[258]||(s[258]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[259]||(s[259]=a('
julia
is_negative_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is negative definite.
source
',3))]),i("details",pi,[i("summary",null,[s[260]||(s[260]=i("a",{id:"is_definite-Tuple{ZZLat}",href:"#is_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_definite")],-1)),s[261]||(s[261]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[262]||(s[262]=a('
julia
is_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is definite.
source
',3))]),s[882]||(s[882]=i("h2",{id:"isometries",tabindex:"-1"},[t("Isometries "),i("a",{class:"header-anchor",href:"#isometries","aria-label":'Permalink to "Isometries"'},"​")],-1)),i("details",ki,[i("summary",null,[s[263]||(s[263]=i("a",{id:"automorphism_group_generators-Tuple{ZZLat}",href:"#automorphism_group_generators-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"automorphism_group_generators")],-1)),s[264]||(s[264]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[269]||(s[269]=a('
julia
automorphism_group_generators(E::EllipticCurve) -> Vector{EllCrvIso}
',1)),i("p",null,[s[267]||(s[267]=t("Return generators of the automorphism group of ")),i("mjx-container",ri,[(e(),l("svg",di,s[265]||(s[265]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D438",d:"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z",style:{"stroke-width":"3"}})])])],-1)]))),s[266]||(s[266]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"E")])],-1))]),s[268]||(s[268]=t("."))]),s[270]||(s[270]=a(`
source
julia
automorphism_group_generators(L::AbstractLat; ambient_representation::Bool = true,
                                               depth::Int = -1, bacher_depth::Int = 0)
-                                                      -> Vector{MatElem}
Given a definite lattice L, return generators for the automorphism group of L. If ambient_representation == true (the default), the transformations are represented with respect to the ambient space of L. Otherwise, the transformations are represented with respect to the (pseudo-)basis of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
`,5))]),i("details",oi,[i("summary",null,[s[271]||(s[271]=i("a",{id:"automorphism_group_order-Tuple{ZZLat}",href:"#automorphism_group_order-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"automorphism_group_order")],-1)),s[272]||(s[272]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[273]||(s[273]=a('
julia
automorphism_group_order(L::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Int
Given a definite lattice L, return the order of the automorphism group of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
',4))]),i("details",gi,[i("summary",null,[s[274]||(s[274]=i("a",{id:"is_isometric-Tuple{ZZLat, ZZLat}",href:"#is_isometric-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"is_isometric")],-1)),s[275]||(s[275]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[276]||(s[276]=a('
julia
is_isometric(L::AbstractLat, M::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Bool
Return whether the lattices L and M are isometric.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
',4))]),i("details",Qi,[i("summary",null,[s[277]||(s[277]=i("a",{id:"is_locally_isometric-Tuple{ZZLat, ZZLat, Int64}",href:"#is_locally_isometric-Tuple{ZZLat, ZZLat, Int64}"},[i("span",{class:"jlbinding"},"is_locally_isometric")],-1)),s[278]||(s[278]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[283]||(s[283]=a('
julia
is_locally_isometric(L::ZZLat, M::ZZLat, p::Int) -> Bool
Return whether L and M are isometric over the p-adic integers.
',2)),i("p",null,[s[281]||(s[281]=t("i.e. whether ")),i("mjx-container",Ti,[(e(),l("svg",mi,s[279]||(s[279]=[a('',1)]))),s[280]||(s[280]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L"),i("mo",null,"⊗"),i("msub",null,[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")]),i("mi",null,"p")]),i("mo",null,"≅"),i("mi",null,"M"),i("mo",null,"⊗"),i("msub",null,[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")]),i("mi",null,"p")])])],-1))]),s[282]||(s[282]=t("."))]),s[284]||(s[284]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[883]||(s[883]=i("h1",{id:"Root-lattices",tabindex:"-1"},[t("Root lattices "),i("a",{class:"header-anchor",href:"#Root-lattices","aria-label":'Permalink to "Root lattices {#Root-lattices}"'},"​")],-1)),i("details",yi,[i("summary",null,[s[285]||(s[285]=i("a",{id:"root_lattice_recognition-Tuple{ZZLat}",href:"#root_lattice_recognition-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"root_lattice_recognition")],-1)),s[286]||(s[286]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[304]||(s[304]=a('
julia
root_lattice_recognition(L::ZZLat)
Return the ADE type of the root sublattice of L.
',2)),i("p",null,[s[293]||(s[293]=t("The root sublattice is the lattice spanned by the vectors of squared length ")),i("mjx-container",Fi,[(e(),l("svg",Ci,s[287]||(s[287]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),s[288]||(s[288]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"1")])],-1))]),s[294]||(s[294]=t(" and ")),i("mjx-container",Ei,[(e(),l("svg",ui,s[289]||(s[289]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"32",d:"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z",style:{"stroke-width":"3"}})])])],-1)]))),s[290]||(s[290]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"2")])],-1))]),s[295]||(s[295]=t(". The odd lattice of rank 1 and determinant ")),i("mjx-container",xi,[(e(),l("svg",ci,s[291]||(s[291]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),s[292]||(s[292]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"1")])],-1))]),s[296]||(s[296]=t(" is denoted by ")),s[297]||(s[297]=i("code",null,"(:I, 1)",-1)),s[298]||(s[298]=t("."))]),s[305]||(s[305]=i("p",null,"Input:",-1)),i("p",null,[s[301]||(s[301]=i("code",null,"L",-1)),s[302]||(s[302]=t(" – a definite and integral ")),i("mjx-container",wi,[(e(),l("svg",bi,s[299]||(s[299]=[a('',1)]))),s[300]||(s[300]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[303]||(s[303]=t("-lattice."))]),s[306]||(s[306]=a(`
Output:
Two lists, the first one containing the ADE types and the second one the irreducible root sublattices.
For more recognizable gram matrices use root_lattice_recognition_fundamental.
Examples
julia
julia> L = integer_lattice(gram=ZZ[4  0 0  0 3  0 3  0;
+                                                      -> Vector{MatElem}
Given a definite lattice L, return generators for the automorphism group of L. If ambient_representation == true (the default), the transformations are represented with respect to the ambient space of L. Otherwise, the transformations are represented with respect to the (pseudo-)basis of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
`,5))]),i("details",oi,[i("summary",null,[s[271]||(s[271]=i("a",{id:"automorphism_group_order-Tuple{ZZLat}",href:"#automorphism_group_order-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"automorphism_group_order")],-1)),s[272]||(s[272]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[273]||(s[273]=a('
julia
automorphism_group_order(L::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Int
Given a definite lattice L, return the order of the automorphism group of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
',4))]),i("details",gi,[i("summary",null,[s[274]||(s[274]=i("a",{id:"is_isometric-Tuple{ZZLat, ZZLat}",href:"#is_isometric-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"is_isometric")],-1)),s[275]||(s[275]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[276]||(s[276]=a('
julia
is_isometric(L::AbstractLat, M::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Bool
Return whether the lattices L and M are isometric.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
',4))]),i("details",Qi,[i("summary",null,[s[277]||(s[277]=i("a",{id:"is_locally_isometric-Tuple{ZZLat, ZZLat, Int64}",href:"#is_locally_isometric-Tuple{ZZLat, ZZLat, Int64}"},[i("span",{class:"jlbinding"},"is_locally_isometric")],-1)),s[278]||(s[278]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[283]||(s[283]=a('
julia
is_locally_isometric(L::ZZLat, M::ZZLat, p::Int) -> Bool
Return whether L and M are isometric over the p-adic integers.
',2)),i("p",null,[s[281]||(s[281]=t("i.e. whether ")),i("mjx-container",Ti,[(e(),l("svg",mi,s[279]||(s[279]=[a('',1)]))),s[280]||(s[280]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L"),i("mo",null,"⊗"),i("msub",null,[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")]),i("mi",null,"p")]),i("mo",null,"≅"),i("mi",null,"M"),i("mo",null,"⊗"),i("msub",null,[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")]),i("mi",null,"p")])])],-1))]),s[282]||(s[282]=t("."))]),s[284]||(s[284]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[883]||(s[883]=i("h1",{id:"Root-lattices",tabindex:"-1"},[t("Root lattices "),i("a",{class:"header-anchor",href:"#Root-lattices","aria-label":'Permalink to "Root lattices {#Root-lattices}"'},"​")],-1)),i("details",yi,[i("summary",null,[s[285]||(s[285]=i("a",{id:"root_lattice_recognition-Tuple{ZZLat}",href:"#root_lattice_recognition-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"root_lattice_recognition")],-1)),s[286]||(s[286]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[304]||(s[304]=a('
julia
root_lattice_recognition(L::ZZLat)
Return the ADE type of the root sublattice of L.
',2)),i("p",null,[s[293]||(s[293]=t("The root sublattice is the lattice spanned by the vectors of squared length ")),i("mjx-container",Fi,[(e(),l("svg",Ci,s[287]||(s[287]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),s[288]||(s[288]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"1")])],-1))]),s[294]||(s[294]=t(" and ")),i("mjx-container",Ei,[(e(),l("svg",ui,s[289]||(s[289]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"32",d:"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z",style:{"stroke-width":"3"}})])])],-1)]))),s[290]||(s[290]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"2")])],-1))]),s[295]||(s[295]=t(". The odd lattice of rank 1 and determinant ")),i("mjx-container",xi,[(e(),l("svg",ci,s[291]||(s[291]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),s[292]||(s[292]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"1")])],-1))]),s[296]||(s[296]=t(" is denoted by ")),s[297]||(s[297]=i("code",null,"(:I, 1)",-1)),s[298]||(s[298]=t("."))]),s[305]||(s[305]=i("p",null,"Input:",-1)),i("p",null,[s[301]||(s[301]=i("code",null,"L",-1)),s[302]||(s[302]=t(" – a definite and integral ")),i("mjx-container",wi,[(e(),l("svg",bi,s[299]||(s[299]=[a('',1)]))),s[300]||(s[300]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[303]||(s[303]=t("-lattice."))]),s[306]||(s[306]=a(`
Output:
Two lists, the first one containing the ADE types and the second one the irreducible root sublattices.
For more recognizable gram matrices use root_lattice_recognition_fundamental.
Examples
julia
julia> L = integer_lattice(gram=ZZ[4  0 0  0 3  0 3  0;
                             0 16 8 12 2 12 6 10;
                             0  8 8  6 2  8 4  5;
                             0 12 6 10 2  9 5  8;
@@ -233,7 +233,7 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
 [0   -1   -1    2   -1    0    0]
 [0    0    0   -1    2   -1    0]
 [0    0    0    0   -1    2   -1]
-[0    0    0    0    0   -1    2]
source
`,5))]),i("details",Li,[i("summary",null,[s[316]||(s[316]=i("a",{id:"ADE_type-Tuple{MatrixElem}",href:"#ADE_type-Tuple{MatrixElem}"},[i("span",{class:"jlbinding"},"ADE_type")],-1)),s[317]||(s[317]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[318]||(s[318]=a(`
julia
ADE_type(G::MatrixElem) -> Tuple{Symbol,Int64}
Return the type of the irreducible root lattice with gram matrix G.
See also root_lattice_recognition.
Examples
julia
julia> Hecke.ADE_type(gram_matrix(root_lattice(:A,3)))
+[0    0    0    0    0   -1    2]
source
`,5))]),i("details",Li,[i("summary",null,[s[316]||(s[316]=i("a",{id:"ADE_type-Tuple{MatrixElem}",href:"#ADE_type-Tuple{MatrixElem}"},[i("span",{class:"jlbinding"},"ADE_type")],-1)),s[317]||(s[317]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[318]||(s[318]=a(`
julia
ADE_type(G::MatrixElem) -> Tuple{Symbol,Int64}
Return the type of the irreducible root lattice with gram matrix G.
See also root_lattice_recognition.
Examples
julia
julia> Hecke.ADE_type(gram_matrix(root_lattice(:A,3)))
 (:A, 3)
source
`,6))]),i("details",Hi,[i("summary",null,[s[319]||(s[319]=i("a",{id:"coxeter_number-Tuple{Symbol, Any}",href:"#coxeter_number-Tuple{Symbol, Any}"},[i("span",{class:"jlbinding"},"coxeter_number")],-1)),s[320]||(s[320]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[339]||(s[339]=a('
julia
coxeter_number(ADE::Symbol, n) -> Int
Return the Coxeter number of the corresponding ADE root lattice.
',2)),i("p",null,[s[331]||(s[331]=t("If ")),i("mjx-container",Mi,[(e(),l("svg",ji,s[321]||(s[321]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[322]||(s[322]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z",style:{"stroke-width":"3"}})])])],-1)]))),s[324]||(s[324]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"R")])],-1))]),s[333]||(s[333]=t(" its set of roots, then the Coxeter number ")),i("mjx-container",Ai,[(e(),l("svg",Vi,s[325]||(s[325]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"210E",d:"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 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0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),i("mi",null,"R"),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mo",null,"/")]),i("mi",null,"n")])],-1))]),s[335]||(s[335]=t(" where ")),s[336]||(s[336]=i("code",null,"n",-1)),s[337]||(s[337]=t(" is the rank of ")),i("mjx-container",Ri,[(e(),l("svg",Oi,s[329]||(s[329]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[330]||(s[330]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[338]||(s[338]=t("."))]),s[340]||(s[340]=a(`
Examples
julia
julia> coxeter_number(:D, 4)
 6
source
`,3))]),i("details",zi,[i("summary",null,[s[341]||(s[341]=i("a",{id:"highest_root-Tuple{Symbol, Any}",href:"#highest_root-Tuple{Symbol, Any}"},[i("span",{class:"jlbinding"},"highest_root")],-1)),s[342]||(s[342]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[343]||(s[343]=a(`
julia
highest_root(ADE::Symbol, n) -> ZZMatrix
Return coordinates of the highest root of root_lattice(ADE, n).
Examples
julia
julia> highest_root(:E, 6)
 [1   2   3   2   1   2]
source
`,5))]),s[884]||(s[884]=i("h2",{id:"Module-operations",tabindex:"-1"},[t("Module operations "),i("a",{class:"header-anchor",href:"#Module-operations","aria-label":'Permalink to "Module operations {#Module-operations}"'},"​")],-1)),s[885]||(s[885]=i("p",null,"Most module operations assume that the lattices live in the same ambient space. For instance only lattices in the same ambient space compare.",-1)),i("details",Ji,[i("summary",null,[s[344]||(s[344]=i("a",{id:"==-Tuple{ZZLat, ZZLat}",href:"#==-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"==")],-1)),s[345]||(s[345]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[346]||(s[346]=i("p",null,[t("Return "),i("code",null,"true"),t(" if both lattices have the same ambient quadratic space and the same underlying module.")],-1)),s[347]||(s[347]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ii,[i("summary",null,[s[348]||(s[348]=i("a",{id:"is_sublattice-Tuple{ZZLat, ZZLat}",href:"#is_sublattice-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"is_sublattice")],-1)),s[349]||(s[349]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[350]||(s[350]=a('
julia
is_sublattice(L::AbstractLat, M::AbstractLat) -> Bool
Return whether M is a sublattice of the lattice L.
source
',3))]),i("details",Ni,[i("summary",null,[s[351]||(s[351]=i("a",{id:"is_sublattice_with_relations-Tuple{ZZLat, ZZLat}",href:"#is_sublattice_with_relations-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"is_sublattice_with_relations")],-1)),s[352]||(s[352]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[378]||(s[378]=a('
julia
is_sublattice_with_relations(M::ZZLat, N::ZZLat) -> Bool, QQMatrix
',1)),i("p",null,[s[369]||(s[369]=t("Returns whether ")),i("mjx-container",qi,[(e(),l("svg",Ui,s[353]||(s[353]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D441",d:"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[354]||(s[354]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"N")])],-1))]),s[370]||(s[370]=t(" is a sublattice of ")),i("mjx-container",Xi,[(e(),l("svg",Pi,s[355]||(s[355]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),s[356]||(s[356]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"M")])],-1))]),s[371]||(s[371]=t(". In this case, the second return value is a matrix ")),i("mjx-container",Ki,[(e(),l("svg",$i,s[357]||(s[357]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),s[358]||(s[358]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"B")])],-1))]),s[372]||(s[372]=t(" such that ")),i("mjx-container",Wi,[(e(),l("svg",Yi,s[359]||(s[359]=[a('',1)]))),s[360]||(s[360]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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")),i("mjx-container",it,[(e(),l("svg",tt,s[363]||(s[363]=[a('',1)]))),s[364]||(s[364]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"B"),i("mi",null,"N")])])],-1))]),s[375]||(s[375]=t(" are the basis matrices of ")),i("mjx-container",at,[(e(),l("svg",lt,s[365]||(s[365]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),s[366]||(s[366]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"M")])],-1))]),s[376]||(s[376]=t(" and ")),i("mjx-container",et,[(e(),l("svg",nt,s[367]||(s[367]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D441",d:"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[368]||(s[368]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"N")])],-1))]),s[377]||(s[377]=t(" respectively."))]),s[379]||(s[379]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",ht,[i("summary",null,[s[380]||(s[380]=i("a",{id:"+-Tuple{ZZLat, ZZLat}",href:"#+-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"+")],-1)),s[381]||(s[381]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[382]||(s[382]=a('
julia
+(L::AbstractLat, M::AbstractLat) -> AbstractLat
Return the sum of the lattices L and M.
The lattices L and M must have the same ambient space.
source
',4))]),i("details",pt,[i("summary",null,[s[383]||(s[383]=i("a",{id:"*-Tuple{Union{Integer, QQFieldElem, ZZRingElem, Rational}, ZZLat}",href:"#*-Tuple{Union{Integer, QQFieldElem, ZZRingElem, Rational}, ZZLat}"},[i("span",{class:"jlbinding"},"*")],-1)),s[384]||(s[384]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[392]||(s[392]=a('
julia
*(a::RationalUnion, L::ZZLat) -> ZZLat
',1)),i("p",null,[s[389]||(s[389]=t("Return the lattice ")),i("mjx-container",kt,[(e(),l("svg",rt,s[385]||(s[385]=[a('',1)]))),s[386]||(s[386]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"a"),i("mi",null,"M")])],-1))]),s[390]||(s[390]=t(" inside the ambient space of ")),i("mjx-container",dt,[(e(),l("svg",ot,s[387]||(s[387]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),s[388]||(s[388]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"M")])],-1))]),s[391]||(s[391]=t("."))]),s[393]||(s[393]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",gt,[i("summary",null,[s[394]||(s[394]=i("a",{id:"intersect-Tuple{ZZLat, ZZLat}",href:"#intersect-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"intersect")],-1)),s[395]||(s[395]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[396]||(s[396]=a('
julia
intersect(L::AbstractLat, M::AbstractLat) -> AbstractLat
Return the intersection of the lattices L and M.
The lattices L and M must have the same ambient space.
source
',4))]),i("details",Qt,[i("summary",null,[s[397]||(s[397]=i("a",{id:"in-Tuple{Vector, ZZLat}",href:"#in-Tuple{Vector, ZZLat}"},[i("span",{class:"jlbinding"},"in")],-1)),s[398]||(s[398]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[399]||(s[399]=a('
julia
Base.in(v::Vector, L::ZZLat) -> Bool
Return whether the vector v lies in the lattice L.
source
',3))]),i("details",Tt,[i("summary",null,[s[400]||(s[400]=i("a",{id:"in-Tuple{QQMatrix, ZZLat}",href:"#in-Tuple{QQMatrix, ZZLat}"},[i("span",{class:"jlbinding"},"in")],-1)),s[401]||(s[401]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[402]||(s[402]=a('
julia
Base.in(v::QQMatrix, L::ZZLat) -> Bool
Return whether the row span of v lies in the lattice L.
source
',3))]),i("details",mt,[i("summary",null,[s[403]||(s[403]=i("a",{id:"primitive_closure-Tuple{ZZLat, ZZLat}",href:"#primitive_closure-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"primitive_closure")],-1)),s[404]||(s[404]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[423]||(s[423]=a('
julia
primitive_closure(M::ZZLat, N::ZZLat) -> ZZLat
',1)),i("p",null,[s[411]||(s[411]=t("Given two ")),i("mjx-container",yt,[(e(),l("svg",Ft,s[405]||(s[405]=[a('',1)]))),s[406]||(s[406]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[412]||(s[412]=t("-lattices ")),s[413]||(s[413]=i("code",null,"M",-1)),s[414]||(s[414]=t(" and ")),s[415]||(s[415]=i("code",null,"N",-1)),s[416]||(s[416]=t(" with ")),i("mjx-container",Ct,[(e(),l("svg",Et,s[407]||(s[407]=[a('',1)]))),s[408]||(s[408]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"N"),i("mo",null,"⊆"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Q")]),i("mi",null,"M")])],-1))]),s[417]||(s[417]=t(", return the primitive closure ")),i("mjx-container",ut,[(e(),l("svg",xt,s[409]||(s[409]=[a('',1)]))),s[410]||(s[410]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"M"),i("mo",null,"∩"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Q")]),i("mi",null,"N")])],-1))]),s[418]||(s[418]=t(" of ")),s[419]||(s[419]=i("code",null,"N",-1)),s[420]||(s[420]=t(" in ")),s[421]||(s[421]=i("code",null,"M",-1)),s[422]||(s[422]=t("."))]),s[424]||(s[424]=a(`
Examples
julia
julia> M = root_lattice(:D, 6);
@@ -282,7 +282,7 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
 true
source
`,3))]),i("details",vt,[i("summary",null,[s[440]||(s[440]=i("a",{id:"is_primitive-Tuple{ZZLat, Union{QQMatrix, Vector}}",href:"#is_primitive-Tuple{ZZLat, Union{QQMatrix, Vector}}"},[i("span",{class:"jlbinding"},"is_primitive")],-1)),s[441]||(s[441]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[462]||(s[462]=a('
julia
is_primitive(L::ZZLat, v::Union{Vector, QQMatrix}) -> Bool
Return whether the vector v is primitive in L.
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julia
divisibility(L::ZZLat, v::Union{Vector, QQMatrix}) -> QQFieldElem
Return the divisibility of v with respect to L.
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julia
direct_sum(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
 direct_sum(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
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julia
direct_product(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
 direct_product(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
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(seen as maps between the corresponding ambient spaces)."))]),i("p",null,[s[544]||(s[544]=t("For objects of type ")),s[545]||(s[545]=i("code",null,"ZZLat",-1)),s[546]||(s[546]=t(", finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain ")),s[547]||(s[547]=i("code",null,"L",-1)),s[548]||(s[548]=t(" as a direct sum with the injections ")),i("mjx-container",oa,[(e(),l("svg",ga,s[538]||(s[538]=[a('',1)]))),s[539]||(s[539]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"L"),i("mi",null,"i")]),i("mo",{accent:"false",stretchy:"false"},"→"),i("mi",null,"L")])],-1))]),s[549]||(s[549]=t(", one should call ")),s[550]||(s[550]=i("code",null,"direct_sum(x)",-1)),s[551]||(s[551]=t(". 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julia
biproduct(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
-biproduct(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
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julia
orthogonal_submodule(L::ZZLat, S::ZZLat) -> ZZLat
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julia
irreducible_components(L::ZZLat) -> Vector{ZZLat}
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julia
dual(L::AbstractLat) -> AbstractLat
Return the dual lattice of the lattice L.
source
',3))]),s[890]||(s[890]=a('
Discriminant group ​
See discriminant_group(L::ZZLat).
Overlattices ​
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julia
glue_map(L::ZZLat, S::ZZLat, R::ZZLat; check=true)
+biproduct(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
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(seen as maps between the corresponding ambient spaces)."))]),i("p",null,[s[582]||(s[582]=t("For objects of type ")),s[583]||(s[583]=i("code",null,"ZZLat",-1)),s[584]||(s[584]=t(", finite direct sums and finite direct products agree and they are therefore called biproducts. 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If one wants to obtain ")),s[590]||(s[590]=i("code",null,"L",-1)),s[591]||(s[591]=t(" as a direct product with the projections ")),i("mjx-container",Ma,[(e(),l("svg",ja,s[580]||(s[580]=[a('',1)]))),s[581]||(s[581]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L"),i("mo",{accent:"false",stretchy:"false"},"→"),i("msub",null,[i("mi",null,"L"),i("mi",null,"i")])])],-1))]),s[592]||(s[592]=t(", one should call ")),s[593]||(s[593]=i("code",null,"direct_product(x)",-1)),s[594]||(s[594]=t("."))]),s[596]||(s[596]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[888]||(s[888]=i("h3",{id:"Orthogonal-sublattices",tabindex:"-1"},[t("Orthogonal sublattices "),i("a",{class:"header-anchor",href:"#Orthogonal-sublattices","aria-label":'Permalink to "Orthogonal sublattices {#Orthogonal-sublattices}"'},"​")],-1)),i("details",Za,[i("summary",null,[s[597]||(s[597]=i("a",{id:"orthogonal_submodule-Tuple{ZZLat, ZZLat}",href:"#orthogonal_submodule-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"orthogonal_submodule")],-1)),s[598]||(s[598]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[606]||(s[606]=a('
julia
orthogonal_submodule(L::ZZLat, S::ZZLat) -> ZZLat
',1)),i("p",null,[s[603]||(s[603]=t("Return the largest submodule of ")),i("mjx-container",Da,[(e(),l("svg",Aa,s[599]||(s[599]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[600]||(s[600]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[604]||(s[604]=t(" orthogonal to ")),i("mjx-container",Va,[(e(),l("svg",Sa,s[601]||(s[601]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D446",d:"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z",style:{"stroke-width":"3"}})])])],-1)]))),s[602]||(s[602]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"S")])],-1))]),s[605]||(s[605]=t("."))]),s[607]||(s[607]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ga,[i("summary",null,[s[608]||(s[608]=i("a",{id:"irreducible_components-Tuple{ZZLat}",href:"#irreducible_components-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"irreducible_components")],-1)),s[609]||(s[609]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[619]||(s[619]=a('
julia
irreducible_components(L::ZZLat) -> Vector{ZZLat}
',1)),i("p",null,[s[614]||(s[614]=t("Return the irreducible components ")),i("mjx-container",Ra,[(e(),l("svg",Oa,s[610]||(s[610]=[a('',1)]))),s[611]||(s[611]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"L"),i("mi",null,"i")])])],-1))]),s[615]||(s[615]=t(" of the positive definite lattice ")),i("mjx-container",za,[(e(),l("svg",Ja,s[612]||(s[612]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[613]||(s[613]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[616]||(s[616]=t("."))]),s[620]||(s[620]=i("p",null,[t("This yields a maximal orthogonal splitting of "),i("code",null,"L"),t(" as")],-1)),i("mjx-container",Ia,[(e(),l("svg",Na,s[617]||(s[617]=[a('',1)]))),s[618]||(s[618]=i("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[i("mi",null,"L"),i("mo",null,"="),i("munder",null,[i("mo",{"data-mjx-texclass":"OP"},"⨁"),i("mi",null,"i")]),i("msub",null,[i("mi",null,"L"),i("mi",null,"i")]),i("mo",null,".")])],-1))]),s[621]||(s[621]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[889]||(s[889]=i("h3",{id:"Dual-lattice",tabindex:"-1"},[t("Dual lattice "),i("a",{class:"header-anchor",href:"#Dual-lattice","aria-label":'Permalink to "Dual lattice {#Dual-lattice}"'},"​")],-1)),i("details",qa,[i("summary",null,[s[622]||(s[622]=i("a",{id:"dual-Tuple{ZZLat}",href:"#dual-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"dual")],-1)),s[623]||(s[623]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[624]||(s[624]=a('
julia
dual(L::AbstractLat) -> AbstractLat
Return the dual lattice of the lattice L.
source
',3))]),s[890]||(s[890]=a('
Discriminant group ​
See discriminant_group(L::ZZLat).
Overlattices ​
',3)),i("details",Ua,[i("summary",null,[s[625]||(s[625]=i("a",{id:"glue_map-Tuple{ZZLat, ZZLat, ZZLat}",href:"#glue_map-Tuple{ZZLat, ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"glue_map")],-1)),s[626]||(s[626]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[644]||(s[644]=a(`
julia
glue_map(L::ZZLat, S::ZZLat, R::ZZLat; check=true)
                        -> Tuple{TorQuadModuleMap, TorQuadModuleMap, TorQuadModuleMap}
`,1)),i("p",null,[s[635]||(s[635]=t("Given three integral ")),i("mjx-container",Xa,[(e(),l("svg",Pa,s[627]||(s[627]=[a('',1)]))),s[628]||(s[628]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[636]||(s[636]=a("-lattices L, S and R, with S and R primitive sublattices of L and such that the sum of the ranks of S and R is equal to the rank of L, return the glue map ",19)),i("mjx-container",Ka,[(e(),l("svg",$a,s[629]||(s[629]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D6FE",d:"M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z",style:{"stroke-width":"3"}})])])],-1)]))),s[630]||(s[630]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"γ")])],-1))]),s[637]||(s[637]=t(" of the primitive extension ")),i("mjx-container",Wa,[(e(),l("svg",Ya,s[631]||(s[631]=[a('',1)]))),s[632]||(s[632]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"S"),i("mo",null,"+"),i("mi",null,"R"),i("mo",null,"⊆"),i("mi",null,"L")])],-1))]),s[638]||(s[638]=t(", as well as the inclusion maps of the domain and codomain of ")),i("mjx-container",_a,[(e(),l("svg",s2,s[633]||(s[633]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D6FE",d:"M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z",style:{"stroke-width":"3"}})])])],-1)]))),s[634]||(s[634]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"γ")])],-1))]),s[639]||(s[639]=t(" into the respective discriminant groups of ")),s[640]||(s[640]=i("code",null,"S",-1)),s[641]||(s[641]=t(" and ")),s[642]||(s[642]=i("code",null,"R",-1)),s[643]||(s[643]=t("."))]),s[645]||(s[645]=a(`
Example
julia
julia> M = root_lattice(:E,8);
 
 julia> f = matrix(QQ, 8, 8, [-1 -1  0  0  0  0  0  0;
@@ -351,7 +351,7 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
                   ambient_representation::Bool = true) -> ZZLat
`,1)),i("p",null,[s[730]||(s[730]=t("Given a ")),i("mjx-container",j2,[(e(),l("svg",Z2,s[716]||(s[716]=[a('',1)]))),s[717]||(s[717]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"bold"},"Z")])])],-1))]),s[731]||(s[731]=t("-lattice ")),i("mjx-container",D2,[(e(),l("svg",A2,s[718]||(s[718]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[719]||(s[719]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[732]||(s[732]=t(" and a list of matrices ")),i("mjx-container",V2,[(e(),l("svg",S2,s[720]||(s[720]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[721]||(s[721]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"G")])],-1))]),s[733]||(s[733]=t(" inducing endomorphisms of ")),i("mjx-container",G2,[(e(),l("svg",R2,s[722]||(s[722]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[723]||(s[723]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[725]||(s[725]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[729]||(s[729]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"G")])],-1))]),s[737]||(s[737]=t("."))]),i("p",null,[s[742]||(s[742]=t("If ")),s[743]||(s[743]=i("code",null,"ambient_representation",-1)),s[744]||(s[744]=t(" is ")),s[745]||(s[745]=i("code",null,"true",-1)),s[746]||(s[746]=t(" (the default), the endomorphism is represented with respect to the ambient space of 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julia
coinvariant_lattice(L::ZZLat, G::Vector{MatElem};
                     ambient_representation::Bool = true) -> ZZLat
 coinvariant_lattice(L::ZZLat, G::MatElem;
-                    ambient_representation::Bool = true) -> ZZLat
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0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[771]||(s[771]=t(" and a list of matrices ")),i("mjx-container",i1,[(e(),l("svg",t1,s[757]||(s[757]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[758]||(s[758]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"G")])],-1))]),s[772]||(s[772]=t(" inducing endomorphisms of ")),i("mjx-container",a1,[(e(),l("svg",l1,s[759]||(s[759]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[760]||(s[760]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[773]||(s[773]=t(" (or just one matrix ")),i("mjx-container",e1,[(e(),l("svg",n1,s[761]||(s[761]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[762]||(s[762]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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Otherwise, the endomorphism is represented with respect to the basis of ")),i("mjx-container",T1,[(e(),l("svg",m1,s[782]||(s[782]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[783]||(s[783]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[790]||(s[790]=t("."))]),s[792]||(s[792]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[892]||(s[892]=i("h3",{id:"Computing-embeddings",tabindex:"-1"},[t("Computing embeddings "),i("a",{class:"header-anchor",href:"#Computing-embeddings","aria-label":'Permalink to "Computing embeddings {#Computing-embeddings}"'},"​")],-1)),i("details",y1,[i("summary",null,[s[793]||(s[793]=i("a",{id:"embed-Tuple{ZZLat, ZZGenus}",href:"#embed-Tuple{ZZLat, ZZGenus}"},[i("span",{class:"jlbinding"},"embed")],-1)),s[794]||(s[794]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[795]||(s[795]=a(`
julia
embed(S::ZZLat, G::Genus, primitive::Bool=true) -> Bool, embedding
Return a (primitive) embedding of the integral lattice S into some lattice in the genus of G.
julia
julia> G = integer_genera((8,0), 1, even=true)[1];
+                    ambient_representation::Bool = true) -> ZZLat
`,1)),i("p",null,[s[769]||(s[769]=t("Given a ")),i("mjx-container",W2,[(e(),l("svg",Y2,s[753]||(s[753]=[a('',1)]))),s[754]||(s[754]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"bold"},"Z")])])],-1))]),s[770]||(s[770]=t("-lattice ")),i("mjx-container",_2,[(e(),l("svg",s1,s[755]||(s[755]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[756]||(s[756]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[771]||(s[771]=t(" and a list of matrices ")),i("mjx-container",i1,[(e(),l("svg",t1,s[757]||(s[757]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[758]||(s[758]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"G")])],-1))]),s[772]||(s[772]=t(" inducing endomorphisms of ")),i("mjx-container",a1,[(e(),l("svg",l1,s[759]||(s[759]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[760]||(s[760]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[773]||(s[773]=t(" (or just one matrix ")),i("mjx-container",e1,[(e(),l("svg",n1,s[761]||(s[761]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[762]||(s[762]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"G")])],-1))]),s[774]||(s[774]=t("), return the orthogonal complement ")),i("mjx-container",h1,[(e(),l("svg",p1,s[763]||(s[763]=[a('',1)]))),s[764]||(s[764]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"L"),i("mi",null,"G")])])],-1))]),s[775]||(s[775]=t(" in ")),i("mjx-container",k1,[(e(),l("svg",r1,s[765]||(s[765]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[766]||(s[766]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[776]||(s[776]=t(" of the fixed lattice ")),i("mjx-container",d1,[(e(),l("svg",o1,s[767]||(s[767]=[a('',1)]))),s[768]||(s[768]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msup",null,[i("mi",null,"L"),i("mi",null,"G")])])],-1))]),s[777]||(s[777]=t(" (see ")),s[778]||(s[778]=i("a",{href:"/v0.34.8/manual/quad_forms/integer_lattices#invariant_lattice-Tuple{ZZLat, Vector{<:MatElem}}"},[i("code",null,"invariant_lattice")],-1)),s[779]||(s[779]=t(")."))]),i("p",null,[s[784]||(s[784]=t("If ")),s[785]||(s[785]=i("code",null,"ambient_representation",-1)),s[786]||(s[786]=t(" is ")),s[787]||(s[787]=i("code",null,"true",-1)),s[788]||(s[788]=t(" (the default), the endomorphism is represented with respect to the ambient space of ")),i("mjx-container",g1,[(e(),l("svg",Q1,s[780]||(s[780]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[781]||(s[781]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[789]||(s[789]=t(". Otherwise, the endomorphism is represented with respect to the basis of ")),i("mjx-container",T1,[(e(),l("svg",m1,s[782]||(s[782]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[783]||(s[783]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[790]||(s[790]=t("."))]),s[792]||(s[792]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[892]||(s[892]=i("h3",{id:"Computing-embeddings",tabindex:"-1"},[t("Computing embeddings "),i("a",{class:"header-anchor",href:"#Computing-embeddings","aria-label":'Permalink to "Computing embeddings {#Computing-embeddings}"'},"​")],-1)),i("details",y1,[i("summary",null,[s[793]||(s[793]=i("a",{id:"embed-Tuple{ZZLat, ZZGenus}",href:"#embed-Tuple{ZZLat, ZZGenus}"},[i("span",{class:"jlbinding"},"embed")],-1)),s[794]||(s[794]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[795]||(s[795]=a(`
julia
embed(S::ZZLat, G::Genus, primitive::Bool=true) -> Bool, embedding
Return a (primitive) embedding of the integral lattice S into some lattice in the genus of G.
julia
julia> G = integer_genera((8,0), 1, even=true)[1];
 
 julia> L, S, i = embed(root_lattice(:A,5), G);
source
`,4))]),i("details",F1,[i("summary",null,[s[796]||(s[796]=i("a",{id:"embed_in_unimodular-Tuple{ZZLat, Union{Integer, ZZRingElem}, Union{Integer, ZZRingElem}}",href:"#embed_in_unimodular-Tuple{ZZLat, Union{Integer, ZZRingElem}, Union{Integer, ZZRingElem}}"},[i("span",{class:"jlbinding"},"embed_in_unimodular")],-1)),s[797]||(s[797]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[798]||(s[798]=a(`
julia
embed_in_unimodular(S::ZZLat, pos::Int, neg::Int, primitive=true, even=true) -> Bool, L, S', iS, iR
Return a (primitive) embedding of the integral lattice S into some (even) unimodular lattice of signature (pos, neg).
For now this works only for even lattices.
julia
julia> NS = direct_sum(integer_lattice(:U), rescale(root_lattice(:A, 16), -1))[1];
 
@@ -387,10 +387,10 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
 
 julia> is_primitive(LK3, iNS)
 true
source
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julia
lll(L::ZZLat, same_ambient::Bool = true) -> ZZLat
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julia
short_vectors(L::ZZLat, [lb = 0], ub, [elem_type = ZZRingElem]; check::Bool = true)
-                                   -> Vector{Tuple{Vector{elem_type}, QQFieldElem}}
`,1)),i("p",null,[s[827]||(s[827]=t("Return all tuples ")),s[828]||(s[828]=i("code",null,"(v, n)",-1)),s[829]||(s[829]=t(" such that ")),i("mjx-container",b1,[(e(),l("svg",B1,s[825]||(s[825]=[a('',1)]))),s[826]||(s[826]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"n"),i("mo",null,"="),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),i("mi",null,"v"),i("mi",null,"G"),i("msup",null,[i("mi",null,"v"),i("mi",null,"t")]),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])],-1))]),s[830]||(s[830]=t(" satisfies ")),s[831]||(s[831]=i("code",null,"lb <= n <= ub",-1)),s[832]||(s[832]=t(", where ")),s[833]||(s[833]=i("code",null,"G",-1)),s[834]||(s[834]=t(" is the Gram matrix of ")),s[835]||(s[835]=i("code",null,"L",-1)),s[836]||(s[836]=t(" and ")),s[837]||(s[837]=i("code",null,"v",-1)),s[838]||(s[838]=t(" is non-zero."))]),s[840]||(s[840]=a('
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors_iterator for an iterator version.
source
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julia
shortest_vectors(L::ZZLat, [elem_type = ZZRingElem]; check::Bool = true)
-                                           -> QQFieldElem, Vector{elem_type}, QQFieldElem}
Return the list of shortest non-zero vectors in absolute value. Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also minimum.
source
`,5))]),i("details",v1,[i("summary",null,[s[844]||(s[844]=i("a",{id:"short_vectors_iterator",href:"#short_vectors_iterator"},[i("span",{class:"jlbinding"},"short_vectors_iterator")],-1)),s[845]||(s[845]=t()),h(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[860]||(s[860]=a(`
julia
short_vectors_iterator(L::ZZLat, [lb = 0], ub,
+                                   -> Vector{Tuple{Vector{elem_type}, QQFieldElem}}
`,1)),i("p",null,[s[827]||(s[827]=t("Return all tuples ")),s[828]||(s[828]=i("code",null,"(v, n)",-1)),s[829]||(s[829]=t(" such that ")),i("mjx-container",b1,[(e(),l("svg",B1,s[825]||(s[825]=[a('',1)]))),s[826]||(s[826]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"n"),i("mo",null,"="),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),i("mi",null,"v"),i("mi",null,"G"),i("msup",null,[i("mi",null,"v"),i("mi",null,"t")]),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])],-1))]),s[830]||(s[830]=t(" satisfies ")),s[831]||(s[831]=i("code",null,"lb <= n <= ub",-1)),s[832]||(s[832]=t(", where ")),s[833]||(s[833]=i("code",null,"G",-1)),s[834]||(s[834]=t(" is the Gram matrix of ")),s[835]||(s[835]=i("code",null,"L",-1)),s[836]||(s[836]=t(" and ")),s[837]||(s[837]=i("code",null,"v",-1)),s[838]||(s[838]=t(" is non-zero."))]),s[840]||(s[840]=a('
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors_iterator for an iterator version.
source
',4))]),i("details",f1,[i("summary",null,[s[841]||(s[841]=i("a",{id:"shortest_vectors",href:"#shortest_vectors"},[i("span",{class:"jlbinding"},"shortest_vectors")],-1)),s[842]||(s[842]=t()),h(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[843]||(s[843]=a(`
julia
shortest_vectors(L::ZZLat, [elem_type = ZZRingElem]; check::Bool = true)
+                                           -> QQFieldElem, Vector{elem_type}, QQFieldElem}
Return the list of shortest non-zero vectors in absolute value. Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also minimum.
source
`,5))]),i("details",v1,[i("summary",null,[s[844]||(s[844]=i("a",{id:"short_vectors_iterator",href:"#short_vectors_iterator"},[i("span",{class:"jlbinding"},"short_vectors_iterator")],-1)),s[845]||(s[845]=t()),h(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[860]||(s[860]=a(`
julia
short_vectors_iterator(L::ZZLat, [lb = 0], ub,
                        [elem_type = ZZRingElem]; check::Bool = true)
-                                -> Tuple{Vector{elem_type}, QQFieldElem} (iterator)
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Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors.
source
',4))]),i("details",M1,[i("summary",null,[s[862]||(s[862]=i("a",{id:"minimum-Tuple{ZZLat}",href:"#minimum-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"minimum")],-1)),s[863]||(s[863]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[864]||(s[864]=a('
julia
minimum(L::ZZLat) -> QQFieldElem
Return the minimum absolute squared length among the non-zero vectors in L.
source
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julia
kissing_number(L::ZZLat) -> Int
Return the Kissing number of the sphere packing defined by L.
This is the number of non-overlapping spheres touching any other given sphere.
source
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julia
close_vectors(L:ZZLat, v:Vector, [lb,], ub; check::Bool = false)
+                                -> Tuple{Vector{elem_type}, QQFieldElem} (iterator)
`,1)),i("p",null,[s[848]||(s[848]=t("Return an iterator for all tuples ")),s[849]||(s[849]=i("code",null,"(v, n)",-1)),s[850]||(s[850]=t(" such that ")),i("mjx-container",L1,[(e(),l("svg",H1,s[846]||(s[846]=[a('',1)]))),s[847]||(s[847]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"n"),i("mo",null,"="),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),i("mi",null,"v"),i("mi",null,"G"),i("msup",null,[i("mi",null,"v"),i("mi",null,"t")]),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])],-1))]),s[851]||(s[851]=t(" satisfies ")),s[852]||(s[852]=i("code",null,"lb <= n <= ub",-1)),s[853]||(s[853]=t(", where ")),s[854]||(s[854]=i("code",null,"G",-1)),s[855]||(s[855]=t(" is the Gram matrix of ")),s[856]||(s[856]=i("code",null,"L",-1)),s[857]||(s[857]=t(" and ")),s[858]||(s[858]=i("code",null,"v",-1)),s[859]||(s[859]=t(" is non-zero."))]),s[861]||(s[861]=a('
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors.
source
',4))]),i("details",M1,[i("summary",null,[s[862]||(s[862]=i("a",{id:"minimum-Tuple{ZZLat}",href:"#minimum-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"minimum")],-1)),s[863]||(s[863]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[864]||(s[864]=a('
julia
minimum(L::ZZLat) -> QQFieldElem
Return the minimum absolute squared length among the non-zero vectors in L.
source
',3))]),i("details",j1,[i("summary",null,[s[865]||(s[865]=i("a",{id:"kissing_number-Tuple{ZZLat}",href:"#kissing_number-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"kissing_number")],-1)),s[866]||(s[866]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[867]||(s[867]=a('
julia
kissing_number(L::ZZLat) -> Int
Return the Kissing number of the sphere packing defined by L.
This is the number of non-overlapping spheres touching any other given sphere.
source
',4))]),s[896]||(s[896]=i("h3",{id:"Close-Vectors",tabindex:"-1"},[t("Close Vectors "),i("a",{class:"header-anchor",href:"#Close-Vectors","aria-label":'Permalink to "Close Vectors {#Close-Vectors}"'},"​")],-1)),i("details",Z1,[i("summary",null,[s[868]||(s[868]=i("a",{id:"close_vectors-Tuple{ZZLat, Vector, Vararg{Any}}",href:"#close_vectors-Tuple{ZZLat, Vector, Vararg{Any}}"},[i("span",{class:"jlbinding"},"close_vectors")],-1)),s[869]||(s[869]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[870]||(s[870]=a(`
julia
close_vectors(L:ZZLat, v:Vector, [lb,], ub; check::Bool = false)
                                         -> Vector{Tuple{Vector{Int}}, QQFieldElem}
Return all tuples (x, d) where x is an element of L such that d = b(v - x, v - x) <= ub. If lb is provided, then also lb <= d.
If filter is not nothing, then only those x with filter(x) evaluating to true are returned.
By default, it will be checked whether L is positive definite. This can be disabled setting check = false.
Both input and output are with respect to the basis matrix of L.
Examples
julia
julia> L = integer_lattice(matrix(QQ, 2, 2, [1, 0, 0, 2]));
 
 julia> close_vectors(L, [1, 1], 1)
diff --git a/v0.34.8/assets/manual_quad_forms_integer_lattices.md.B7hEml57.lean.js b/v0.34.8/assets/manual_quad_forms_integer_lattices.md.B7hEml57.lean.js
index 5d61c04ea0..40222b379c 100644
--- a/v0.34.8/assets/manual_quad_forms_integer_lattices.md.B7hEml57.lean.js
+++ b/v0.34.8/assets/manual_quad_forms_integer_lattices.md.B7hEml57.lean.js
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")),s[52]||(s[52]=i("a",{href:"/Hecke.jl/v0.34.8/manual/quad_forms/integer_lattices#basis_matrix-Tuple{ZZLat}"},[i("code",null,"basis_matrix(L::ZZLat)")],-1)),s[53]||(s[53]=t("."))]),s[872]||(s[872]=i("h2",{id:"Creation-of-integer-lattices",tabindex:"-1"},[t("Creation of integer lattices "),i("a",{class:"header-anchor",href:"#Creation-of-integer-lattices","aria-label":'Permalink to "Creation of integer lattices {#Creation-of-integer-lattices}"'},"​")],-1)),s[873]||(s[873]=i("h3",{id:"From-a-gram-matrix",tabindex:"-1"},[t("From a gram matrix "),i("a",{class:"header-anchor",href:"#From-a-gram-matrix","aria-label":'Permalink to "From a gram matrix {#From-a-gram-matrix}"'},"​")],-1)),i("details",z,[i("summary",null,[s[54]||(s[54]=i("a",{id:"integer_lattice-Tuple{QQMatrix}",href:"#integer_lattice-Tuple{QQMatrix}"},[i("span",{class:"jlbinding"},"integer_lattice")],-1)),s[55]||(s[55]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[68]||(s[68]=a('
julia
integer_lattice([B::MatElem]; gram) -> ZZLat
',1)),i("p",null,[s[58]||(s[58]=t("Return the Z-lattice with basis matrix ")),i("mjx-container",J,[(e(),l("svg",I,s[56]||(s[56]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),s[57]||(s[57]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"B")])],-1))]),s[59]||(s[59]=t(" inside the quadratic space with Gram matrix ")),s[60]||(s[60]=i("code",null,"gram",-1)),s[61]||(s[61]=t("."))]),i("p",null,[s[64]||(s[64]=t("If the keyword ")),s[65]||(s[65]=i("code",null,"gram",-1)),s[66]||(s[66]=t(" is not specified, the Gram matrix is the identity matrix. If ")),i("mjx-container",N,[(e(),l("svg",q,s[62]||(s[62]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),s[63]||(s[63]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"B")])],-1))]),s[67]||(s[67]=t(" is not specified, the basis matrix is the identity matrix."))]),s[69]||(s[69]=a(`
Examples
julia
julia> L = integer_lattice(matrix(QQ, 2, 2, [1//2, 0, 0, 2]));
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")),i("mjx-container",R,[(e(),l("svg",O,s[45]||(s[45]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 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lattices "),i("a",{class:"header-anchor",href:"#Creation-of-integer-lattices","aria-label":'Permalink to "Creation of integer lattices {#Creation-of-integer-lattices}"'},"​")],-1)),s[873]||(s[873]=i("h3",{id:"From-a-gram-matrix",tabindex:"-1"},[t("From a gram matrix "),i("a",{class:"header-anchor",href:"#From-a-gram-matrix","aria-label":'Permalink to "From a gram matrix {#From-a-gram-matrix}"'},"​")],-1)),i("details",z,[i("summary",null,[s[54]||(s[54]=i("a",{id:"integer_lattice-Tuple{QQMatrix}",href:"#integer_lattice-Tuple{QQMatrix}"},[i("span",{class:"jlbinding"},"integer_lattice")],-1)),s[55]||(s[55]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[68]||(s[68]=a('
julia
integer_lattice([B::MatElem]; gram) -> ZZLat
',1)),i("p",null,[s[58]||(s[58]=t("Return the Z-lattice with basis matrix ")),i("mjx-container",J,[(e(),l("svg",I,s[56]||(s[56]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),s[57]||(s[57]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"B")])],-1))]),s[59]||(s[59]=t(" inside the quadratic space with Gram matrix ")),s[60]||(s[60]=i("code",null,"gram",-1)),s[61]||(s[61]=t("."))]),i("p",null,[s[64]||(s[64]=t("If the keyword ")),s[65]||(s[65]=i("code",null,"gram",-1)),s[66]||(s[66]=t(" is not specified, the Gram matrix is the identity matrix. If ")),i("mjx-container",N,[(e(),l("svg",q,s[62]||(s[62]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),s[63]||(s[63]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"B")])],-1))]),s[67]||(s[67]=t(" is not specified, the basis matrix is the identity matrix."))]),s[69]||(s[69]=a(`
Examples
julia
julia> L = integer_lattice(matrix(QQ, 2, 2, [1//2, 0, 0, 2]));
 
 julia> gram_matrix(L) == matrix(QQ, 2, 2, [1//4, 0, 0, 4])
 true
@@ -16,7 +16,7 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
 
 julia> gram_matrix(L)
 [  0   -13]
-[-13     0]
source
`,3))]),i("details",W,[i("summary",null,[s[88]||(s[88]=i("a",{id:"integer_lattice-Tuple{Symbol, Union{Int64, ZZRingElem}}",href:"#integer_lattice-Tuple{Symbol, Union{Int64, ZZRingElem}}"},[i("span",{class:"jlbinding"},"integer_lattice")],-1)),s[89]||(s[89]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[104]||(s[104]=a('
julia
integer_lattice(S::Symbol, n::RationalUnion = 1) -> ZZlat
',1)),i("p",null,[s[96]||(s[96]=t("Given ")),s[97]||(s[97]=i("code",null,"S = :H",-1)),s[98]||(s[98]=t(" or ")),s[99]||(s[99]=i("code",null,"S = :U",-1)),s[100]||(s[100]=t(", return a ")),i("mjx-container",Y,[(e(),l("svg",_,s[90]||(s[90]=[a('',1)]))),s[91]||(s[91]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[101]||(s[101]=t("-lattice admitting ")),i("mjx-container",ss,[(e(),l("svg",is,s[92]||(s[92]=[a('',1)]))),s[93]||(s[93]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"n"),i("mo",null,"∗"),i("msub",null,[i("mi",null,"J"),i("mn",null,"2")])])],-1))]),s[102]||(s[102]=t(" as Gram matrix in some basis, where ")),i("mjx-container",ts,[(e(),l("svg",as,s[94]||(s[94]=[a('',1)]))),s[95]||(s[95]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"J"),i("mn",null,"2")])])],-1))]),s[103]||(s[103]=t(" is the 2-by-2 matrix with 0's on the main diagonal and 1's elsewhere."))]),s[105]||(s[105]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",ls,[i("summary",null,[s[106]||(s[106]=i("a",{id:"leech_lattice",href:"#leech_lattice"},[i("span",{class:"jlbinding"},"leech_lattice")],-1)),s[107]||(s[107]=t()),h(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[108]||(s[108]=a(`
julia
leech_lattice() -> ZZLat
Return the Leech lattice.
source
julia
leech_lattice(niemeier_lattice::ZZLat) -> ZZLat, QQMatrix, Int
Return a triple L, v, h where L is the Leech lattice.
L is an h-neighbor of the Niemeier lattice N with respect to v. This means that L / L ∩ N ≅ ℤ / h ℤ. Here h is the Coxeter number of the Niemeier lattice.
This implements the 23 holy constructions of the Leech lattice in [5].
Examples
julia
julia> R = integer_lattice(gram=2 * identity_matrix(ZZ, 24));
+[-13     0]
source
`,3))]),i("details",W,[i("summary",null,[s[88]||(s[88]=i("a",{id:"integer_lattice-Tuple{Symbol, Union{Int64, ZZRingElem}}",href:"#integer_lattice-Tuple{Symbol, Union{Int64, ZZRingElem}}"},[i("span",{class:"jlbinding"},"integer_lattice")],-1)),s[89]||(s[89]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[104]||(s[104]=a('
julia
integer_lattice(S::Symbol, n::RationalUnion = 1) -> ZZlat
',1)),i("p",null,[s[96]||(s[96]=t("Given ")),s[97]||(s[97]=i("code",null,"S = :H",-1)),s[98]||(s[98]=t(" or ")),s[99]||(s[99]=i("code",null,"S = :U",-1)),s[100]||(s[100]=t(", return a ")),i("mjx-container",Y,[(e(),l("svg",_,s[90]||(s[90]=[a('',1)]))),s[91]||(s[91]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[101]||(s[101]=t("-lattice admitting ")),i("mjx-container",ss,[(e(),l("svg",is,s[92]||(s[92]=[a('',1)]))),s[93]||(s[93]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"n"),i("mo",null,"∗"),i("msub",null,[i("mi",null,"J"),i("mn",null,"2")])])],-1))]),s[102]||(s[102]=t(" as Gram matrix in some basis, where ")),i("mjx-container",ts,[(e(),l("svg",as,s[94]||(s[94]=[a('',1)]))),s[95]||(s[95]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"J"),i("mn",null,"2")])])],-1))]),s[103]||(s[103]=t(" is the 2-by-2 matrix with 0's on the main diagonal and 1's elsewhere."))]),s[105]||(s[105]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",ls,[i("summary",null,[s[106]||(s[106]=i("a",{id:"leech_lattice",href:"#leech_lattice"},[i("span",{class:"jlbinding"},"leech_lattice")],-1)),s[107]||(s[107]=t()),h(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[108]||(s[108]=a(`
julia
leech_lattice() -> ZZLat
Return the Leech lattice.
source
julia
leech_lattice(niemeier_lattice::ZZLat) -> ZZLat, QQMatrix, Int
Return a triple L, v, h where L is the Leech lattice.
L is an h-neighbor of the Niemeier lattice N with respect to v. This means that L / L ∩ N ≅ ℤ / h ℤ. Here h is the Coxeter number of the Niemeier lattice.
This implements the 23 holy constructions of the Leech lattice in [5].
Examples
julia
julia> R = integer_lattice(gram=2 * identity_matrix(ZZ, 24));
 
 julia> N = maximal_even_lattice(R) # Some Niemeier lattice
 Integer lattice of rank 24 and degree 24
@@ -149,7 +149,7 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
 Genus symbol for integer lattices
 Signatures: (3, 0, 20)
 Local symbol:
-  Local genus symbol at 2: 1^22 4^1_7
source
`,6))]),s[876]||(s[876]=a('
From a genus ​
Integer lattices can be created as representatives of a genus. See (representative(L::ZZGenus))
Rescaling the Quadratic Form ​
',3)),i("details",ps,[i("summary",null,[s[118]||(s[118]=i("a",{id:"rescale-Tuple{ZZLat, Union{Integer, QQFieldElem, ZZRingElem, Rational}}",href:"#rescale-Tuple{ZZLat, Union{Integer, QQFieldElem, ZZRingElem, Rational}}"},[i("span",{class:"jlbinding"},"rescale")],-1)),s[119]||(s[119]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[120]||(s[120]=a(`
julia
rescale(L::ZZLat, r::RationalUnion) -> ZZLat
Return the lattice L in the quadratic space with form r \\Phi.
Examples
This can be useful to apply methods intended for positive definite lattices.
julia
julia> L = integer_lattice(gram=ZZ[-1 0; 0 -1])
+  Local genus symbol at 2: 1^22 4^1_7
source
`,6))]),s[876]||(s[876]=a('
From a genus ​
Integer lattices can be created as representatives of a genus. See (representative(L::ZZGenus))
Rescaling the Quadratic Form ​
',3)),i("details",ps,[i("summary",null,[s[118]||(s[118]=i("a",{id:"rescale-Tuple{ZZLat, Union{Integer, QQFieldElem, ZZRingElem, Rational}}",href:"#rescale-Tuple{ZZLat, Union{Integer, QQFieldElem, ZZRingElem, Rational}}"},[i("span",{class:"jlbinding"},"rescale")],-1)),s[119]||(s[119]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[120]||(s[120]=a(`
julia
rescale(L::ZZLat, r::RationalUnion) -> ZZLat
Return the lattice L in the quadratic space with form r \\Phi.
Examples
This can be useful to apply methods intended for positive definite lattices.
julia
julia> L = integer_lattice(gram=ZZ[-1 0; 0 -1])
 Integer lattice of rank 2 and degree 2
 with gram matrix
 [-1    0]
@@ -169,9 +169,9 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
   over rational field
 with gram matrix
 [ 4   -2]
-[-2    5]
source
`,3))]),s[878]||(s[878]=i("h2",{id:"invariants",tabindex:"-1"},[t("Invariants "),i("a",{class:"header-anchor",href:"#invariants","aria-label":'Permalink to "Invariants"'},"​")],-1)),i("details",bs,[i("summary",null,[s[160]||(s[160]=i("a",{id:"rank-Tuple{ZZLat}",href:"#rank-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"rank")],-1)),s[161]||(s[161]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[162]||(s[162]=a('
julia
rank(L::AbstractLat) -> Int
Return the rank of the underlying module of the lattice L.
source
',3))]),i("details",Bs,[i("summary",null,[s[163]||(s[163]=i("a",{id:"det-Tuple{ZZLat}",href:"#det-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"det")],-1)),s[164]||(s[164]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=a('
julia
det(L::ZZLat) -> QQFieldElem
Return the determinant of the gram matrix of L.
source
',3))]),i("details",fs,[i("summary",null,[s[166]||(s[166]=i("a",{id:"scale-Tuple{ZZLat}",href:"#scale-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"scale")],-1)),s[167]||(s[167]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[177]||(s[177]=a('
julia
scale(L::ZZLat) -> QQFieldElem
Return the scale of L.
',2)),i("p",null,[s[172]||(s[172]=t("The scale of ")),s[173]||(s[173]=i("code",null,"L",-1)),s[174]||(s[174]=t(" is defined as the positive generator of the ")),i("mjx-container",vs,[(e(),l("svg",Ls,s[168]||(s[168]=[a('',1)]))),s[169]||(s[169]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[175]||(s[175]=t("-ideal generated by ")),i("mjx-container",Hs,[(e(),l("svg",Ms,s[170]||(s[170]=[a('',1)]))),s[171]||(s[171]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mo",{fence:"false",stretchy:"false"},"{"),i("mi",{mathvariant:"normal"},"Φ"),i("mo",{stretchy:"false"},"("),i("mi",null,"x"),i("mo",null,","),i("mi",null,"y"),i("mo",{stretchy:"false"},")"),i("mo",null,":"),i("mi",null,"x"),i("mo",null,","),i("mi",null,"y"),i("mo",null,"∈"),i("mi",null,"L"),i("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[176]||(s[176]=t("."))]),s[178]||(s[178]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",js,[i("summary",null,[s[179]||(s[179]=i("a",{id:"norm-Tuple{ZZLat}",href:"#norm-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"norm")],-1)),s[180]||(s[180]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[190]||(s[190]=a('
julia
norm(L::ZZLat) -> QQFieldElem
Return the norm of L.
',2)),i("p",null,[s[185]||(s[185]=t("The norm of ")),s[186]||(s[186]=i("code",null,"L",-1)),s[187]||(s[187]=t(" is defined as the positive generator of the ")),i("mjx-container",Zs,[(e(),l("svg",Ds,s[181]||(s[181]=[a('',1)]))),s[182]||(s[182]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[188]||(s[188]=t("- ideal generated by ")),i("mjx-container",As,[(e(),l("svg",Vs,s[183]||(s[183]=[a('',1)]))),s[184]||(s[184]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mo",{fence:"false",stretchy:"false"},"{"),i("mi",{mathvariant:"normal"},"Φ"),i("mo",{stretchy:"false"},"("),i("mi",null,"x"),i("mo",null,","),i("mi",null,"x"),i("mo",{stretchy:"false"},")"),i("mo",null,":"),i("mi",null,"x"),i("mo",null,"∈"),i("mi",null,"L"),i("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),s[189]||(s[189]=t("."))]),s[191]||(s[191]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ss,[i("summary",null,[s[192]||(s[192]=i("a",{id:"iseven-Tuple{ZZLat}",href:"#iseven-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"iseven")],-1)),s[193]||(s[193]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[206]||(s[206]=a('
julia
iseven(L::ZZLat) -> Bool
Return whether L is even.
',2)),i("p",null,[s[200]||(s[200]=t("An integer lattice ")),s[201]||(s[201]=i("code",null,"L",-1)),s[202]||(s[202]=t(" in the rational quadratic space ")),i("mjx-container",Gs,[(e(),l("svg",Rs,s[194]||(s[194]=[a('',1)]))),s[195]||(s[195]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mo",{stretchy:"false"},"("),i("mi",null,"V"),i("mo",null,","),i("mi",{mathvariant:"normal"},"Φ"),i("mo",{stretchy:"false"},")")])],-1))]),s[203]||(s[203]=t(" is called even if ")),i("mjx-container",Os,[(e(),l("svg",zs,s[196]||(s[196]=[a('',1)]))),s[197]||(s[197]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",{mathvariant:"normal"},"Φ"),i("mo",{stretchy:"false"},"("),i("mi",null,"x"),i("mo",null,","),i("mi",null,"x"),i("mo",{stretchy:"false"},")"),i("mo",null,"∈"),i("mn",null,"2"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[204]||(s[204]=t(" for all ")),i("mjx-container",Js,[(e(),l("svg",Is,s[198]||(s[198]=[a('',1)]))),s[199]||(s[199]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"x"),i("mi",null,"i"),i("mi",null,"n"),i("mi",null,"L")])],-1))]),s[205]||(s[205]=t("."))]),s[207]||(s[207]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ns,[i("summary",null,[s[208]||(s[208]=i("a",{id:"is_integral-Tuple{ZZLat}",href:"#is_integral-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_integral")],-1)),s[209]||(s[209]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[210]||(s[210]=a('
julia
is_integral(L::AbstractLat) -> Bool
Return whether the lattice L is integral.
source
',3))]),i("details",qs,[i("summary",null,[s[211]||(s[211]=i("a",{id:"is_primary_with_prime-Tuple{ZZLat}",href:"#is_primary_with_prime-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),s[212]||(s[212]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[217]||(s[217]=a('
julia
is_primary_with_prime(L::ZZLat) -> Bool, ZZRingElem
',1)),i("p",null,[s[215]||(s[215]=t("Given a ")),i("mjx-container",Us,[(e(),l("svg",Xs,s[213]||(s[213]=[a('',1)]))),s[214]||(s[214]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[216]||(s[216]=a('-lattice L, return whether L is primary, that is whether L is integral and its discriminant group (see discriminant_group) is a p-group for some prime number p. In case it is, p is also returned as second output.',15))]),s[218]||(s[218]=i("p",null,[t("Note that for unimodular lattices, this function returns "),i("code",null,"(true, 1)"),t(". If the lattice is not primary, the second return value is "),i("code",null,"-1"),t(" by default.")],-1)),s[219]||(s[219]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ps,[i("summary",null,[s[220]||(s[220]=i("a",{id:"is_primary-Tuple{ZZLat, Union{Integer, ZZRingElem}}",href:"#is_primary-Tuple{ZZLat, Union{Integer, ZZRingElem}}"},[i("span",{class:"jlbinding"},"is_primary")],-1)),s[221]||(s[221]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[226]||(s[226]=a('
julia
is_primary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
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julia
is_elementary_with_prime(L::ZZLat) -> Bool, ZZRingElem
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julia
is_elementary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
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julia
mass(L::ZZLat) -> QQFieldElem
Return the mass of the genus of L.
source
',3))]),i("details",li,[i("summary",null,[s[248]||(s[248]=i("a",{id:"genus_representatives-Tuple{ZZLat}",href:"#genus_representatives-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"genus_representatives")],-1)),s[249]||(s[249]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[250]||(s[250]=a('
julia
genus_representatives(L::ZZLat) -> Vector{ZZLat}
Return representatives for the isometry classes in the genus of L.
source
',3))]),s[881]||(s[881]=i("h3",{id:"Real-invariants",tabindex:"-1"},[t("Real invariants "),i("a",{class:"header-anchor",href:"#Real-invariants","aria-label":'Permalink to "Real invariants {#Real-invariants}"'},"​")],-1)),i("details",ei,[i("summary",null,[s[251]||(s[251]=i("a",{id:"signature_tuple-Tuple{ZZLat}",href:"#signature_tuple-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"signature_tuple")],-1)),s[252]||(s[252]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[253]||(s[253]=a('
julia
signature_tuple(L::ZZLat) -> Tuple{Int,Int,Int}
Return the number of (positive, zero, negative) inertia of L.
source
',3))]),i("details",ni,[i("summary",null,[s[254]||(s[254]=i("a",{id:"is_positive_definite-Tuple{ZZLat}",href:"#is_positive_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_positive_definite")],-1)),s[255]||(s[255]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[256]||(s[256]=a('
julia
is_positive_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is positive definite.
source
',3))]),i("details",hi,[i("summary",null,[s[257]||(s[257]=i("a",{id:"is_negative_definite-Tuple{ZZLat}",href:"#is_negative_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_negative_definite")],-1)),s[258]||(s[258]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[259]||(s[259]=a('
julia
is_negative_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is negative definite.
source
',3))]),i("details",pi,[i("summary",null,[s[260]||(s[260]=i("a",{id:"is_definite-Tuple{ZZLat}",href:"#is_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_definite")],-1)),s[261]||(s[261]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[262]||(s[262]=a('
julia
is_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is definite.
source
',3))]),s[882]||(s[882]=i("h2",{id:"isometries",tabindex:"-1"},[t("Isometries "),i("a",{class:"header-anchor",href:"#isometries","aria-label":'Permalink to "Isometries"'},"​")],-1)),i("details",ki,[i("summary",null,[s[263]||(s[263]=i("a",{id:"automorphism_group_generators-Tuple{ZZLat}",href:"#automorphism_group_generators-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"automorphism_group_generators")],-1)),s[264]||(s[264]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[269]||(s[269]=a('
julia
automorphism_group_generators(E::EllipticCurve) -> Vector{EllCrvIso}
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source
julia
automorphism_group_generators(L::AbstractLat; ambient_representation::Bool = true,
+[-2    5]
source
`,3))]),s[878]||(s[878]=i("h2",{id:"invariants",tabindex:"-1"},[t("Invariants "),i("a",{class:"header-anchor",href:"#invariants","aria-label":'Permalink to "Invariants"'},"​")],-1)),i("details",bs,[i("summary",null,[s[160]||(s[160]=i("a",{id:"rank-Tuple{ZZLat}",href:"#rank-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"rank")],-1)),s[161]||(s[161]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[162]||(s[162]=a('
julia
rank(L::AbstractLat) -> Int
Return the rank of the underlying module of the lattice L.
source
',3))]),i("details",Bs,[i("summary",null,[s[163]||(s[163]=i("a",{id:"det-Tuple{ZZLat}",href:"#det-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"det")],-1)),s[164]||(s[164]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[165]||(s[165]=a('
julia
det(L::ZZLat) -> QQFieldElem
Return the determinant of the gram matrix of L.
source
',3))]),i("details",fs,[i("summary",null,[s[166]||(s[166]=i("a",{id:"scale-Tuple{ZZLat}",href:"#scale-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"scale")],-1)),s[167]||(s[167]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[177]||(s[177]=a('
julia
scale(L::ZZLat) -> QQFieldElem
Return the scale of L.
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julia
norm(L::ZZLat) -> QQFieldElem
Return the norm of L.
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julia
iseven(L::ZZLat) -> Bool
Return whether L is even.
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julia
is_integral(L::AbstractLat) -> Bool
Return whether the lattice L is integral.
source
',3))]),i("details",qs,[i("summary",null,[s[211]||(s[211]=i("a",{id:"is_primary_with_prime-Tuple{ZZLat}",href:"#is_primary_with_prime-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_primary_with_prime")],-1)),s[212]||(s[212]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[217]||(s[217]=a('
julia
is_primary_with_prime(L::ZZLat) -> Bool, ZZRingElem
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julia
is_primary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
',1)),i("p",null,[s[224]||(s[224]=t("Given an integral ")),i("mjx-container",Ks,[(e(),l("svg",$s,s[222]||(s[222]=[a('',1)]))),s[223]||(s[223]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[225]||(s[225]=a('-lattice L and a prime number p, return whether L is p-primary, that is whether its discriminant group (see discriminant_group) is a p-group.',13))]),s[227]||(s[227]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ws,[i("summary",null,[s[228]||(s[228]=i("a",{id:"is_elementary_with_prime-Tuple{ZZLat}",href:"#is_elementary_with_prime-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_elementary_with_prime")],-1)),s[229]||(s[229]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[234]||(s[234]=a('
julia
is_elementary_with_prime(L::ZZLat) -> Bool, ZZRingElem
',1)),i("p",null,[s[232]||(s[232]=t("Given a ")),i("mjx-container",Ys,[(e(),l("svg",_s,s[230]||(s[230]=[a('',1)]))),s[231]||(s[231]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[233]||(s[233]=a('-lattice L, return whether L is elementary, that is whether L is integral and its discriminant group (see discriminant_group) is an elemenentary p-group for some prime number p. In case it is, p is also returned as second output.',15))]),s[235]||(s[235]=i("p",null,[t("Note that for unimodular lattices, this function returns "),i("code",null,"(true, 1)"),t(". If the lattice is not elementary, the second return value is "),i("code",null,"-1"),t(" by default.")],-1)),s[236]||(s[236]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",si,[i("summary",null,[s[237]||(s[237]=i("a",{id:"is_elementary-Tuple{ZZLat, Union{Integer, ZZRingElem}}",href:"#is_elementary-Tuple{ZZLat, Union{Integer, ZZRingElem}}"},[i("span",{class:"jlbinding"},"is_elementary")],-1)),s[238]||(s[238]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[243]||(s[243]=a('
julia
is_elementary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
',1)),i("p",null,[s[241]||(s[241]=t("Given an integral ")),i("mjx-container",ii,[(e(),l("svg",ti,s[239]||(s[239]=[a('',1)]))),s[240]||(s[240]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[242]||(s[242]=a('-lattice L and a prime number p, return whether L is p-elementary, that is whether its discriminant group (see discriminant_group) is an elementary p-group.',13))]),s[244]||(s[244]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[879]||(s[879]=i("h3",{id:"The-Genus",tabindex:"-1"},[t("The Genus "),i("a",{class:"header-anchor",href:"#The-Genus","aria-label":'Permalink to "The Genus {#The-Genus}"'},"​")],-1)),s[880]||(s[880]=i("p",null,[t("For an integral lattice The genus of an integer lattice collects its local invariants. "),i("a",{href:"/v0.34.8/manual/quad_forms/Zgenera#genus-Tuple{ZZLat}"},[i("code",null,"genus(::ZZLat)")])],-1)),i("details",ai,[i("summary",null,[s[245]||(s[245]=i("a",{id:"mass-Tuple{ZZLat}",href:"#mass-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"mass")],-1)),s[246]||(s[246]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[247]||(s[247]=a('
julia
mass(L::ZZLat) -> QQFieldElem
Return the mass of the genus of L.
source
',3))]),i("details",li,[i("summary",null,[s[248]||(s[248]=i("a",{id:"genus_representatives-Tuple{ZZLat}",href:"#genus_representatives-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"genus_representatives")],-1)),s[249]||(s[249]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[250]||(s[250]=a('
julia
genus_representatives(L::ZZLat) -> Vector{ZZLat}
Return representatives for the isometry classes in the genus of L.
source
',3))]),s[881]||(s[881]=i("h3",{id:"Real-invariants",tabindex:"-1"},[t("Real invariants "),i("a",{class:"header-anchor",href:"#Real-invariants","aria-label":'Permalink to "Real invariants {#Real-invariants}"'},"​")],-1)),i("details",ei,[i("summary",null,[s[251]||(s[251]=i("a",{id:"signature_tuple-Tuple{ZZLat}",href:"#signature_tuple-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"signature_tuple")],-1)),s[252]||(s[252]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[253]||(s[253]=a('
julia
signature_tuple(L::ZZLat) -> Tuple{Int,Int,Int}
Return the number of (positive, zero, negative) inertia of L.
source
',3))]),i("details",ni,[i("summary",null,[s[254]||(s[254]=i("a",{id:"is_positive_definite-Tuple{ZZLat}",href:"#is_positive_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_positive_definite")],-1)),s[255]||(s[255]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[256]||(s[256]=a('
julia
is_positive_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is positive definite.
source
',3))]),i("details",hi,[i("summary",null,[s[257]||(s[257]=i("a",{id:"is_negative_definite-Tuple{ZZLat}",href:"#is_negative_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_negative_definite")],-1)),s[258]||(s[258]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[259]||(s[259]=a('
julia
is_negative_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is negative definite.
source
',3))]),i("details",pi,[i("summary",null,[s[260]||(s[260]=i("a",{id:"is_definite-Tuple{ZZLat}",href:"#is_definite-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"is_definite")],-1)),s[261]||(s[261]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[262]||(s[262]=a('
julia
is_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is definite.
source
',3))]),s[882]||(s[882]=i("h2",{id:"isometries",tabindex:"-1"},[t("Isometries "),i("a",{class:"header-anchor",href:"#isometries","aria-label":'Permalink to "Isometries"'},"​")],-1)),i("details",ki,[i("summary",null,[s[263]||(s[263]=i("a",{id:"automorphism_group_generators-Tuple{ZZLat}",href:"#automorphism_group_generators-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"automorphism_group_generators")],-1)),s[264]||(s[264]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[269]||(s[269]=a('
julia
automorphism_group_generators(E::EllipticCurve) -> Vector{EllCrvIso}
',1)),i("p",null,[s[267]||(s[267]=t("Return generators of the automorphism group of ")),i("mjx-container",ri,[(e(),l("svg",di,s[265]||(s[265]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D438",d:"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z",style:{"stroke-width":"3"}})])])],-1)]))),s[266]||(s[266]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"E")])],-1))]),s[268]||(s[268]=t("."))]),s[270]||(s[270]=a(`
source
julia
automorphism_group_generators(L::AbstractLat; ambient_representation::Bool = true,
                                               depth::Int = -1, bacher_depth::Int = 0)
-                                                      -> Vector{MatElem}
Given a definite lattice L, return generators for the automorphism group of L. If ambient_representation == true (the default), the transformations are represented with respect to the ambient space of L. Otherwise, the transformations are represented with respect to the (pseudo-)basis of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
`,5))]),i("details",oi,[i("summary",null,[s[271]||(s[271]=i("a",{id:"automorphism_group_order-Tuple{ZZLat}",href:"#automorphism_group_order-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"automorphism_group_order")],-1)),s[272]||(s[272]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[273]||(s[273]=a('
julia
automorphism_group_order(L::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Int
Given a definite lattice L, return the order of the automorphism group of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
',4))]),i("details",gi,[i("summary",null,[s[274]||(s[274]=i("a",{id:"is_isometric-Tuple{ZZLat, ZZLat}",href:"#is_isometric-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"is_isometric")],-1)),s[275]||(s[275]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[276]||(s[276]=a('
julia
is_isometric(L::AbstractLat, M::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Bool
Return whether the lattices L and M are isometric.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
',4))]),i("details",Qi,[i("summary",null,[s[277]||(s[277]=i("a",{id:"is_locally_isometric-Tuple{ZZLat, ZZLat, Int64}",href:"#is_locally_isometric-Tuple{ZZLat, ZZLat, Int64}"},[i("span",{class:"jlbinding"},"is_locally_isometric")],-1)),s[278]||(s[278]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[283]||(s[283]=a('
julia
is_locally_isometric(L::ZZLat, M::ZZLat, p::Int) -> Bool
Return whether L and M are isometric over the p-adic integers.
',2)),i("p",null,[s[281]||(s[281]=t("i.e. whether ")),i("mjx-container",Ti,[(e(),l("svg",mi,s[279]||(s[279]=[a('',1)]))),s[280]||(s[280]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L"),i("mo",null,"⊗"),i("msub",null,[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")]),i("mi",null,"p")]),i("mo",null,"≅"),i("mi",null,"M"),i("mo",null,"⊗"),i("msub",null,[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")]),i("mi",null,"p")])])],-1))]),s[282]||(s[282]=t("."))]),s[284]||(s[284]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[883]||(s[883]=i("h1",{id:"Root-lattices",tabindex:"-1"},[t("Root lattices "),i("a",{class:"header-anchor",href:"#Root-lattices","aria-label":'Permalink to "Root lattices {#Root-lattices}"'},"​")],-1)),i("details",yi,[i("summary",null,[s[285]||(s[285]=i("a",{id:"root_lattice_recognition-Tuple{ZZLat}",href:"#root_lattice_recognition-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"root_lattice_recognition")],-1)),s[286]||(s[286]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[304]||(s[304]=a('
julia
root_lattice_recognition(L::ZZLat)
Return the ADE type of the root sublattice of L.
',2)),i("p",null,[s[293]||(s[293]=t("The root sublattice is the lattice spanned by the vectors of squared length ")),i("mjx-container",Fi,[(e(),l("svg",Ci,s[287]||(s[287]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),s[288]||(s[288]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"1")])],-1))]),s[294]||(s[294]=t(" and ")),i("mjx-container",Ei,[(e(),l("svg",ui,s[289]||(s[289]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"32",d:"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z",style:{"stroke-width":"3"}})])])],-1)]))),s[290]||(s[290]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"2")])],-1))]),s[295]||(s[295]=t(". The odd lattice of rank 1 and determinant ")),i("mjx-container",xi,[(e(),l("svg",ci,s[291]||(s[291]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),s[292]||(s[292]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"1")])],-1))]),s[296]||(s[296]=t(" is denoted by ")),s[297]||(s[297]=i("code",null,"(:I, 1)",-1)),s[298]||(s[298]=t("."))]),s[305]||(s[305]=i("p",null,"Input:",-1)),i("p",null,[s[301]||(s[301]=i("code",null,"L",-1)),s[302]||(s[302]=t(" – a definite and integral ")),i("mjx-container",wi,[(e(),l("svg",bi,s[299]||(s[299]=[a('',1)]))),s[300]||(s[300]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[303]||(s[303]=t("-lattice."))]),s[306]||(s[306]=a(`
Output:
Two lists, the first one containing the ADE types and the second one the irreducible root sublattices.
For more recognizable gram matrices use root_lattice_recognition_fundamental.
Examples
julia
julia> L = integer_lattice(gram=ZZ[4  0 0  0 3  0 3  0;
+                                                      -> Vector{MatElem}
Given a definite lattice L, return generators for the automorphism group of L. If ambient_representation == true (the default), the transformations are represented with respect to the ambient space of L. Otherwise, the transformations are represented with respect to the (pseudo-)basis of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
`,5))]),i("details",oi,[i("summary",null,[s[271]||(s[271]=i("a",{id:"automorphism_group_order-Tuple{ZZLat}",href:"#automorphism_group_order-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"automorphism_group_order")],-1)),s[272]||(s[272]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[273]||(s[273]=a('
julia
automorphism_group_order(L::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Int
Given a definite lattice L, return the order of the automorphism group of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
',4))]),i("details",gi,[i("summary",null,[s[274]||(s[274]=i("a",{id:"is_isometric-Tuple{ZZLat, ZZLat}",href:"#is_isometric-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"is_isometric")],-1)),s[275]||(s[275]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[276]||(s[276]=a('
julia
is_isometric(L::AbstractLat, M::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Bool
Return whether the lattices L and M are isometric.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
',4))]),i("details",Qi,[i("summary",null,[s[277]||(s[277]=i("a",{id:"is_locally_isometric-Tuple{ZZLat, ZZLat, Int64}",href:"#is_locally_isometric-Tuple{ZZLat, ZZLat, Int64}"},[i("span",{class:"jlbinding"},"is_locally_isometric")],-1)),s[278]||(s[278]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[283]||(s[283]=a('
julia
is_locally_isometric(L::ZZLat, M::ZZLat, p::Int) -> Bool
Return whether L and M are isometric over the p-adic integers.
',2)),i("p",null,[s[281]||(s[281]=t("i.e. whether ")),i("mjx-container",Ti,[(e(),l("svg",mi,s[279]||(s[279]=[a('',1)]))),s[280]||(s[280]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L"),i("mo",null,"⊗"),i("msub",null,[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")]),i("mi",null,"p")]),i("mo",null,"≅"),i("mi",null,"M"),i("mo",null,"⊗"),i("msub",null,[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")]),i("mi",null,"p")])])],-1))]),s[282]||(s[282]=t("."))]),s[284]||(s[284]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[883]||(s[883]=i("h1",{id:"Root-lattices",tabindex:"-1"},[t("Root lattices "),i("a",{class:"header-anchor",href:"#Root-lattices","aria-label":'Permalink to "Root lattices {#Root-lattices}"'},"​")],-1)),i("details",yi,[i("summary",null,[s[285]||(s[285]=i("a",{id:"root_lattice_recognition-Tuple{ZZLat}",href:"#root_lattice_recognition-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"root_lattice_recognition")],-1)),s[286]||(s[286]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[304]||(s[304]=a('
julia
root_lattice_recognition(L::ZZLat)
Return the ADE type of the root sublattice of L.
',2)),i("p",null,[s[293]||(s[293]=t("The root sublattice is the lattice spanned by the vectors of squared length ")),i("mjx-container",Fi,[(e(),l("svg",Ci,s[287]||(s[287]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),s[288]||(s[288]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"1")])],-1))]),s[294]||(s[294]=t(" and ")),i("mjx-container",Ei,[(e(),l("svg",ui,s[289]||(s[289]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"32",d:"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z",style:{"stroke-width":"3"}})])])],-1)]))),s[290]||(s[290]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"2")])],-1))]),s[295]||(s[295]=t(". The odd lattice of rank 1 and determinant ")),i("mjx-container",xi,[(e(),l("svg",ci,s[291]||(s[291]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mn"},[i("path",{"data-c":"31",d:"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z",style:{"stroke-width":"3"}})])])],-1)]))),s[292]||(s[292]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mn",null,"1")])],-1))]),s[296]||(s[296]=t(" is denoted by ")),s[297]||(s[297]=i("code",null,"(:I, 1)",-1)),s[298]||(s[298]=t("."))]),s[305]||(s[305]=i("p",null,"Input:",-1)),i("p",null,[s[301]||(s[301]=i("code",null,"L",-1)),s[302]||(s[302]=t(" – a definite and integral ")),i("mjx-container",wi,[(e(),l("svg",bi,s[299]||(s[299]=[a('',1)]))),s[300]||(s[300]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[303]||(s[303]=t("-lattice."))]),s[306]||(s[306]=a(`
Output:
Two lists, the first one containing the ADE types and the second one the irreducible root sublattices.
For more recognizable gram matrices use root_lattice_recognition_fundamental.
Examples
julia
julia> L = integer_lattice(gram=ZZ[4  0 0  0 3  0 3  0;
                             0 16 8 12 2 12 6 10;
                             0  8 8  6 2  8 4  5;
                             0 12 6 10 2  9 5  8;
@@ -233,7 +233,7 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
 [0   -1   -1    2   -1    0    0]
 [0    0    0   -1    2   -1    0]
 [0    0    0    0   -1    2   -1]
-[0    0    0    0    0   -1    2]
source
`,5))]),i("details",Li,[i("summary",null,[s[316]||(s[316]=i("a",{id:"ADE_type-Tuple{MatrixElem}",href:"#ADE_type-Tuple{MatrixElem}"},[i("span",{class:"jlbinding"},"ADE_type")],-1)),s[317]||(s[317]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[318]||(s[318]=a(`
julia
ADE_type(G::MatrixElem) -> Tuple{Symbol,Int64}
Return the type of the irreducible root lattice with gram matrix G.
See also root_lattice_recognition.
Examples
julia
julia> Hecke.ADE_type(gram_matrix(root_lattice(:A,3)))
+[0    0    0    0    0   -1    2]
source
`,5))]),i("details",Li,[i("summary",null,[s[316]||(s[316]=i("a",{id:"ADE_type-Tuple{MatrixElem}",href:"#ADE_type-Tuple{MatrixElem}"},[i("span",{class:"jlbinding"},"ADE_type")],-1)),s[317]||(s[317]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[318]||(s[318]=a(`
julia
ADE_type(G::MatrixElem) -> Tuple{Symbol,Int64}
Return the type of the irreducible root lattice with gram matrix G.
See also root_lattice_recognition.
Examples
julia
julia> Hecke.ADE_type(gram_matrix(root_lattice(:A,3)))
 (:A, 3)
source
`,6))]),i("details",Hi,[i("summary",null,[s[319]||(s[319]=i("a",{id:"coxeter_number-Tuple{Symbol, Any}",href:"#coxeter_number-Tuple{Symbol, Any}"},[i("span",{class:"jlbinding"},"coxeter_number")],-1)),s[320]||(s[320]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[339]||(s[339]=a('
julia
coxeter_number(ADE::Symbol, n) -> Int
Return the Coxeter number of the corresponding ADE root lattice.
',2)),i("p",null,[s[331]||(s[331]=t("If ")),i("mjx-container",Mi,[(e(),l("svg",ji,s[321]||(s[321]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[322]||(s[322]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z",style:{"stroke-width":"3"}})])])],-1)]))),s[324]||(s[324]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"R")])],-1))]),s[333]||(s[333]=t(" its set of roots, then the Coxeter number ")),i("mjx-container",Ai,[(e(),l("svg",Vi,s[325]||(s[325]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"210E",d:"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 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0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),i("mi",null,"R"),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mo",null,"/")]),i("mi",null,"n")])],-1))]),s[335]||(s[335]=t(" where ")),s[336]||(s[336]=i("code",null,"n",-1)),s[337]||(s[337]=t(" is the rank of ")),i("mjx-container",Ri,[(e(),l("svg",Oi,s[329]||(s[329]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[330]||(s[330]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[338]||(s[338]=t("."))]),s[340]||(s[340]=a(`
Examples
julia
julia> coxeter_number(:D, 4)
 6
source
`,3))]),i("details",zi,[i("summary",null,[s[341]||(s[341]=i("a",{id:"highest_root-Tuple{Symbol, Any}",href:"#highest_root-Tuple{Symbol, Any}"},[i("span",{class:"jlbinding"},"highest_root")],-1)),s[342]||(s[342]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[343]||(s[343]=a(`
julia
highest_root(ADE::Symbol, n) -> ZZMatrix
Return coordinates of the highest root of root_lattice(ADE, n).
Examples
julia
julia> highest_root(:E, 6)
 [1   2   3   2   1   2]
source
`,5))]),s[884]||(s[884]=i("h2",{id:"Module-operations",tabindex:"-1"},[t("Module operations "),i("a",{class:"header-anchor",href:"#Module-operations","aria-label":'Permalink to "Module operations {#Module-operations}"'},"​")],-1)),s[885]||(s[885]=i("p",null,"Most module operations assume that the lattices live in the same ambient space. For instance only lattices in the same ambient space compare.",-1)),i("details",Ji,[i("summary",null,[s[344]||(s[344]=i("a",{id:"==-Tuple{ZZLat, ZZLat}",href:"#==-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"==")],-1)),s[345]||(s[345]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[346]||(s[346]=i("p",null,[t("Return "),i("code",null,"true"),t(" if both lattices have the same ambient quadratic space and the same underlying module.")],-1)),s[347]||(s[347]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ii,[i("summary",null,[s[348]||(s[348]=i("a",{id:"is_sublattice-Tuple{ZZLat, ZZLat}",href:"#is_sublattice-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"is_sublattice")],-1)),s[349]||(s[349]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[350]||(s[350]=a('
julia
is_sublattice(L::AbstractLat, M::AbstractLat) -> Bool
Return whether M is a sublattice of the lattice L.
source
',3))]),i("details",Ni,[i("summary",null,[s[351]||(s[351]=i("a",{id:"is_sublattice_with_relations-Tuple{ZZLat, ZZLat}",href:"#is_sublattice_with_relations-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"is_sublattice_with_relations")],-1)),s[352]||(s[352]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[378]||(s[378]=a('
julia
is_sublattice_with_relations(M::ZZLat, N::ZZLat) -> Bool, QQMatrix
',1)),i("p",null,[s[369]||(s[369]=t("Returns whether ")),i("mjx-container",qi,[(e(),l("svg",Ui,s[353]||(s[353]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D441",d:"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[354]||(s[354]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"N")])],-1))]),s[370]||(s[370]=t(" is a sublattice of ")),i("mjx-container",Xi,[(e(),l("svg",Pi,s[355]||(s[355]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),s[356]||(s[356]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"M")])],-1))]),s[371]||(s[371]=t(". In this case, the second return value is a matrix ")),i("mjx-container",Ki,[(e(),l("svg",$i,s[357]||(s[357]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D435",d:"M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z",style:{"stroke-width":"3"}})])])],-1)]))),s[358]||(s[358]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"B")])],-1))]),s[372]||(s[372]=t(" such that ")),i("mjx-container",Wi,[(e(),l("svg",Yi,s[359]||(s[359]=[a('',1)]))),s[360]||(s[360]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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")),i("mjx-container",it,[(e(),l("svg",tt,s[363]||(s[363]=[a('',1)]))),s[364]||(s[364]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"B"),i("mi",null,"N")])])],-1))]),s[375]||(s[375]=t(" are the basis matrices of ")),i("mjx-container",at,[(e(),l("svg",lt,s[365]||(s[365]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),s[366]||(s[366]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"M")])],-1))]),s[376]||(s[376]=t(" and ")),i("mjx-container",et,[(e(),l("svg",nt,s[367]||(s[367]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D441",d:"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[368]||(s[368]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"N")])],-1))]),s[377]||(s[377]=t(" respectively."))]),s[379]||(s[379]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",ht,[i("summary",null,[s[380]||(s[380]=i("a",{id:"+-Tuple{ZZLat, ZZLat}",href:"#+-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"+")],-1)),s[381]||(s[381]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[382]||(s[382]=a('
julia
+(L::AbstractLat, M::AbstractLat) -> AbstractLat
Return the sum of the lattices L and M.
The lattices L and M must have the same ambient space.
source
',4))]),i("details",pt,[i("summary",null,[s[383]||(s[383]=i("a",{id:"*-Tuple{Union{Integer, QQFieldElem, ZZRingElem, Rational}, ZZLat}",href:"#*-Tuple{Union{Integer, QQFieldElem, ZZRingElem, Rational}, ZZLat}"},[i("span",{class:"jlbinding"},"*")],-1)),s[384]||(s[384]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[392]||(s[392]=a('
julia
*(a::RationalUnion, L::ZZLat) -> ZZLat
',1)),i("p",null,[s[389]||(s[389]=t("Return the lattice ")),i("mjx-container",kt,[(e(),l("svg",rt,s[385]||(s[385]=[a('',1)]))),s[386]||(s[386]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"a"),i("mi",null,"M")])],-1))]),s[390]||(s[390]=t(" inside the ambient space of ")),i("mjx-container",dt,[(e(),l("svg",ot,s[387]||(s[387]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z",style:{"stroke-width":"3"}})])])],-1)]))),s[388]||(s[388]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"M")])],-1))]),s[391]||(s[391]=t("."))]),s[393]||(s[393]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",gt,[i("summary",null,[s[394]||(s[394]=i("a",{id:"intersect-Tuple{ZZLat, ZZLat}",href:"#intersect-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"intersect")],-1)),s[395]||(s[395]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[396]||(s[396]=a('
julia
intersect(L::AbstractLat, M::AbstractLat) -> AbstractLat
Return the intersection of the lattices L and M.
The lattices L and M must have the same ambient space.
source
',4))]),i("details",Qt,[i("summary",null,[s[397]||(s[397]=i("a",{id:"in-Tuple{Vector, ZZLat}",href:"#in-Tuple{Vector, ZZLat}"},[i("span",{class:"jlbinding"},"in")],-1)),s[398]||(s[398]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[399]||(s[399]=a('
julia
Base.in(v::Vector, L::ZZLat) -> Bool
Return whether the vector v lies in the lattice L.
source
',3))]),i("details",Tt,[i("summary",null,[s[400]||(s[400]=i("a",{id:"in-Tuple{QQMatrix, ZZLat}",href:"#in-Tuple{QQMatrix, ZZLat}"},[i("span",{class:"jlbinding"},"in")],-1)),s[401]||(s[401]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[402]||(s[402]=a('
julia
Base.in(v::QQMatrix, L::ZZLat) -> Bool
Return whether the row span of v lies in the lattice L.
source
',3))]),i("details",mt,[i("summary",null,[s[403]||(s[403]=i("a",{id:"primitive_closure-Tuple{ZZLat, ZZLat}",href:"#primitive_closure-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"primitive_closure")],-1)),s[404]||(s[404]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[423]||(s[423]=a('
julia
primitive_closure(M::ZZLat, N::ZZLat) -> ZZLat
',1)),i("p",null,[s[411]||(s[411]=t("Given two ")),i("mjx-container",yt,[(e(),l("svg",Ft,s[405]||(s[405]=[a('',1)]))),s[406]||(s[406]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[412]||(s[412]=t("-lattices ")),s[413]||(s[413]=i("code",null,"M",-1)),s[414]||(s[414]=t(" and ")),s[415]||(s[415]=i("code",null,"N",-1)),s[416]||(s[416]=t(" with ")),i("mjx-container",Ct,[(e(),l("svg",Et,s[407]||(s[407]=[a('',1)]))),s[408]||(s[408]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"N"),i("mo",null,"⊆"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Q")]),i("mi",null,"M")])],-1))]),s[417]||(s[417]=t(", return the primitive closure ")),i("mjx-container",ut,[(e(),l("svg",xt,s[409]||(s[409]=[a('',1)]))),s[410]||(s[410]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"M"),i("mo",null,"∩"),i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Q")]),i("mi",null,"N")])],-1))]),s[418]||(s[418]=t(" of ")),s[419]||(s[419]=i("code",null,"N",-1)),s[420]||(s[420]=t(" in ")),s[421]||(s[421]=i("code",null,"M",-1)),s[422]||(s[422]=t("."))]),s[424]||(s[424]=a(`
Examples
julia
julia> M = root_lattice(:D, 6);
@@ -282,7 +282,7 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
 true
source
`,3))]),i("details",vt,[i("summary",null,[s[440]||(s[440]=i("a",{id:"is_primitive-Tuple{ZZLat, Union{QQMatrix, Vector}}",href:"#is_primitive-Tuple{ZZLat, Union{QQMatrix, Vector}}"},[i("span",{class:"jlbinding"},"is_primitive")],-1)),s[441]||(s[441]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[462]||(s[462]=a('
julia
is_primitive(L::ZZLat, v::Union{Vector, QQMatrix}) -> Bool
Return whether the vector v is primitive in L.
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julia
divisibility(L::ZZLat, v::Union{Vector, QQMatrix}) -> QQFieldElem
Return the divisibility of v with respect to L.
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julia
direct_sum(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
 direct_sum(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
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julia
direct_product(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
 direct_product(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
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(seen as maps between the corresponding ambient spaces)."))]),i("p",null,[s[544]||(s[544]=t("For objects of type ")),s[545]||(s[545]=i("code",null,"ZZLat",-1)),s[546]||(s[546]=t(", finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain ")),s[547]||(s[547]=i("code",null,"L",-1)),s[548]||(s[548]=t(" as a direct sum with the injections ")),i("mjx-container",oa,[(e(),l("svg",ga,s[538]||(s[538]=[a('',1)]))),s[539]||(s[539]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"L"),i("mi",null,"i")]),i("mo",{accent:"false",stretchy:"false"},"→"),i("mi",null,"L")])],-1))]),s[549]||(s[549]=t(", one should call ")),s[550]||(s[550]=i("code",null,"direct_sum(x)",-1)),s[551]||(s[551]=t(". 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julia
biproduct(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
-biproduct(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
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julia
orthogonal_submodule(L::ZZLat, S::ZZLat) -> ZZLat
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julia
irreducible_components(L::ZZLat) -> Vector{ZZLat}
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julia
dual(L::AbstractLat) -> AbstractLat
Return the dual lattice of the lattice L.
source
',3))]),s[890]||(s[890]=a('
Discriminant group ​
See discriminant_group(L::ZZLat).
Overlattices ​
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julia
glue_map(L::ZZLat, S::ZZLat, R::ZZLat; check=true)
+biproduct(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
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(seen as maps between the corresponding ambient spaces)."))]),i("p",null,[s[582]||(s[582]=t("For objects of type ")),s[583]||(s[583]=i("code",null,"ZZLat",-1)),s[584]||(s[584]=t(", finite direct sums and finite direct products agree and they are therefore called biproducts. 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If one wants to obtain ")),s[590]||(s[590]=i("code",null,"L",-1)),s[591]||(s[591]=t(" as a direct product with the projections ")),i("mjx-container",Ma,[(e(),l("svg",ja,s[580]||(s[580]=[a('',1)]))),s[581]||(s[581]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L"),i("mo",{accent:"false",stretchy:"false"},"→"),i("msub",null,[i("mi",null,"L"),i("mi",null,"i")])])],-1))]),s[592]||(s[592]=t(", one should call ")),s[593]||(s[593]=i("code",null,"direct_product(x)",-1)),s[594]||(s[594]=t("."))]),s[596]||(s[596]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[888]||(s[888]=i("h3",{id:"Orthogonal-sublattices",tabindex:"-1"},[t("Orthogonal sublattices "),i("a",{class:"header-anchor",href:"#Orthogonal-sublattices","aria-label":'Permalink to "Orthogonal sublattices {#Orthogonal-sublattices}"'},"​")],-1)),i("details",Za,[i("summary",null,[s[597]||(s[597]=i("a",{id:"orthogonal_submodule-Tuple{ZZLat, ZZLat}",href:"#orthogonal_submodule-Tuple{ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"orthogonal_submodule")],-1)),s[598]||(s[598]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[606]||(s[606]=a('
julia
orthogonal_submodule(L::ZZLat, S::ZZLat) -> ZZLat
',1)),i("p",null,[s[603]||(s[603]=t("Return the largest submodule of ")),i("mjx-container",Da,[(e(),l("svg",Aa,s[599]||(s[599]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[600]||(s[600]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[604]||(s[604]=t(" orthogonal to ")),i("mjx-container",Va,[(e(),l("svg",Sa,s[601]||(s[601]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D446",d:"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z",style:{"stroke-width":"3"}})])])],-1)]))),s[602]||(s[602]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"S")])],-1))]),s[605]||(s[605]=t("."))]),s[607]||(s[607]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),i("details",Ga,[i("summary",null,[s[608]||(s[608]=i("a",{id:"irreducible_components-Tuple{ZZLat}",href:"#irreducible_components-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"irreducible_components")],-1)),s[609]||(s[609]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[619]||(s[619]=a('
julia
irreducible_components(L::ZZLat) -> Vector{ZZLat}
',1)),i("p",null,[s[614]||(s[614]=t("Return the irreducible components ")),i("mjx-container",Ra,[(e(),l("svg",Oa,s[610]||(s[610]=[a('',1)]))),s[611]||(s[611]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"L"),i("mi",null,"i")])])],-1))]),s[615]||(s[615]=t(" of the positive definite lattice ")),i("mjx-container",za,[(e(),l("svg",Ja,s[612]||(s[612]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[613]||(s[613]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[616]||(s[616]=t("."))]),s[620]||(s[620]=i("p",null,[t("This yields a maximal orthogonal splitting of "),i("code",null,"L"),t(" as")],-1)),i("mjx-container",Ia,[(e(),l("svg",Na,s[617]||(s[617]=[a('',1)]))),s[618]||(s[618]=i("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[i("mi",null,"L"),i("mo",null,"="),i("munder",null,[i("mo",{"data-mjx-texclass":"OP"},"⨁"),i("mi",null,"i")]),i("msub",null,[i("mi",null,"L"),i("mi",null,"i")]),i("mo",null,".")])],-1))]),s[621]||(s[621]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[889]||(s[889]=i("h3",{id:"Dual-lattice",tabindex:"-1"},[t("Dual lattice "),i("a",{class:"header-anchor",href:"#Dual-lattice","aria-label":'Permalink to "Dual lattice {#Dual-lattice}"'},"​")],-1)),i("details",qa,[i("summary",null,[s[622]||(s[622]=i("a",{id:"dual-Tuple{ZZLat}",href:"#dual-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"dual")],-1)),s[623]||(s[623]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[624]||(s[624]=a('
julia
dual(L::AbstractLat) -> AbstractLat
Return the dual lattice of the lattice L.
source
',3))]),s[890]||(s[890]=a('
Discriminant group ​
See discriminant_group(L::ZZLat).
Overlattices ​
',3)),i("details",Ua,[i("summary",null,[s[625]||(s[625]=i("a",{id:"glue_map-Tuple{ZZLat, ZZLat, ZZLat}",href:"#glue_map-Tuple{ZZLat, ZZLat, ZZLat}"},[i("span",{class:"jlbinding"},"glue_map")],-1)),s[626]||(s[626]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[644]||(s[644]=a(`
julia
glue_map(L::ZZLat, S::ZZLat, R::ZZLat; check=true)
                        -> Tuple{TorQuadModuleMap, TorQuadModuleMap, TorQuadModuleMap}
`,1)),i("p",null,[s[635]||(s[635]=t("Given three integral ")),i("mjx-container",Xa,[(e(),l("svg",Pa,s[627]||(s[627]=[a('',1)]))),s[628]||(s[628]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"double-struck"},"Z")])])],-1))]),s[636]||(s[636]=a("-lattices L, S and R, with S and R primitive sublattices of L and such that the sum of the ranks of S and R is equal to the rank of L, return the glue map ",19)),i("mjx-container",Ka,[(e(),l("svg",$a,s[629]||(s[629]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D6FE",d:"M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z",style:{"stroke-width":"3"}})])])],-1)]))),s[630]||(s[630]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"γ")])],-1))]),s[637]||(s[637]=t(" of the primitive extension ")),i("mjx-container",Wa,[(e(),l("svg",Ya,s[631]||(s[631]=[a('',1)]))),s[632]||(s[632]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"S"),i("mo",null,"+"),i("mi",null,"R"),i("mo",null,"⊆"),i("mi",null,"L")])],-1))]),s[638]||(s[638]=t(", as well as the inclusion maps of the domain and codomain of ")),i("mjx-container",_a,[(e(),l("svg",s2,s[633]||(s[633]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D6FE",d:"M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z",style:{"stroke-width":"3"}})])])],-1)]))),s[634]||(s[634]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"γ")])],-1))]),s[639]||(s[639]=t(" into the respective discriminant groups of ")),s[640]||(s[640]=i("code",null,"S",-1)),s[641]||(s[641]=t(" and ")),s[642]||(s[642]=i("code",null,"R",-1)),s[643]||(s[643]=t("."))]),s[645]||(s[645]=a(`
Example
julia
julia> M = root_lattice(:E,8);
 
 julia> f = matrix(QQ, 8, 8, [-1 -1  0  0  0  0  0  0;
@@ -351,7 +351,7 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
                   ambient_representation::Bool = true) -> ZZLat
`,1)),i("p",null,[s[730]||(s[730]=t("Given a ")),i("mjx-container",j2,[(e(),l("svg",Z2,s[716]||(s[716]=[a('',1)]))),s[717]||(s[717]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"bold"},"Z")])])],-1))]),s[731]||(s[731]=t("-lattice ")),i("mjx-container",D2,[(e(),l("svg",A2,s[718]||(s[718]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[719]||(s[719]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[732]||(s[732]=t(" and a list of matrices ")),i("mjx-container",V2,[(e(),l("svg",S2,s[720]||(s[720]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[721]||(s[721]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"G")])],-1))]),s[733]||(s[733]=t(" inducing endomorphisms of ")),i("mjx-container",G2,[(e(),l("svg",R2,s[722]||(s[722]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[723]||(s[723]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[725]||(s[725]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[729]||(s[729]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"G")])],-1))]),s[737]||(s[737]=t("."))]),i("p",null,[s[742]||(s[742]=t("If ")),s[743]||(s[743]=i("code",null,"ambient_representation",-1)),s[744]||(s[744]=t(" is ")),s[745]||(s[745]=i("code",null,"true",-1)),s[746]||(s[746]=t(" (the default), the endomorphism is represented with respect to the ambient space of 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julia
coinvariant_lattice(L::ZZLat, G::Vector{MatElem};
                     ambient_representation::Bool = true) -> ZZLat
 coinvariant_lattice(L::ZZLat, G::MatElem;
-                    ambient_representation::Bool = true) -> ZZLat
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0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[771]||(s[771]=t(" and a list of matrices ")),i("mjx-container",i1,[(e(),l("svg",t1,s[757]||(s[757]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[758]||(s[758]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"G")])],-1))]),s[772]||(s[772]=t(" inducing endomorphisms of ")),i("mjx-container",a1,[(e(),l("svg",l1,s[759]||(s[759]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[760]||(s[760]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[773]||(s[773]=t(" (or just one matrix ")),i("mjx-container",e1,[(e(),l("svg",n1,s[761]||(s[761]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[762]||(s[762]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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Otherwise, the endomorphism is represented with respect to the basis of ")),i("mjx-container",T1,[(e(),l("svg",m1,s[782]||(s[782]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[783]||(s[783]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[790]||(s[790]=t("."))]),s[792]||(s[792]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[892]||(s[892]=i("h3",{id:"Computing-embeddings",tabindex:"-1"},[t("Computing embeddings "),i("a",{class:"header-anchor",href:"#Computing-embeddings","aria-label":'Permalink to "Computing embeddings {#Computing-embeddings}"'},"​")],-1)),i("details",y1,[i("summary",null,[s[793]||(s[793]=i("a",{id:"embed-Tuple{ZZLat, ZZGenus}",href:"#embed-Tuple{ZZLat, ZZGenus}"},[i("span",{class:"jlbinding"},"embed")],-1)),s[794]||(s[794]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[795]||(s[795]=a(`
julia
embed(S::ZZLat, G::Genus, primitive::Bool=true) -> Bool, embedding
Return a (primitive) embedding of the integral lattice S into some lattice in the genus of G.
julia
julia> G = integer_genera((8,0), 1, even=true)[1];
+                    ambient_representation::Bool = true) -> ZZLat
`,1)),i("p",null,[s[769]||(s[769]=t("Given a ")),i("mjx-container",W2,[(e(),l("svg",Y2,s[753]||(s[753]=[a('',1)]))),s[754]||(s[754]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mrow",{"data-mjx-texclass":"ORD"},[i("mi",{mathvariant:"bold"},"Z")])])],-1))]),s[770]||(s[770]=t("-lattice ")),i("mjx-container",_2,[(e(),l("svg",s1,s[755]||(s[755]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[756]||(s[756]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[771]||(s[771]=t(" and a list of matrices ")),i("mjx-container",i1,[(e(),l("svg",t1,s[757]||(s[757]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[758]||(s[758]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"G")])],-1))]),s[772]||(s[772]=t(" inducing endomorphisms of ")),i("mjx-container",a1,[(e(),l("svg",l1,s[759]||(s[759]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[760]||(s[760]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[773]||(s[773]=t(" (or just one matrix ")),i("mjx-container",e1,[(e(),l("svg",n1,s[761]||(s[761]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43A",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z",style:{"stroke-width":"3"}})])])],-1)]))),s[762]||(s[762]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"G")])],-1))]),s[774]||(s[774]=t("), return the orthogonal complement ")),i("mjx-container",h1,[(e(),l("svg",p1,s[763]||(s[763]=[a('',1)]))),s[764]||(s[764]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msub",null,[i("mi",null,"L"),i("mi",null,"G")])])],-1))]),s[775]||(s[775]=t(" in ")),i("mjx-container",k1,[(e(),l("svg",r1,s[765]||(s[765]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[766]||(s[766]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[776]||(s[776]=t(" of the fixed lattice ")),i("mjx-container",d1,[(e(),l("svg",o1,s[767]||(s[767]=[a('',1)]))),s[768]||(s[768]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("msup",null,[i("mi",null,"L"),i("mi",null,"G")])])],-1))]),s[777]||(s[777]=t(" (see ")),s[778]||(s[778]=i("a",{href:"/v0.34.8/manual/quad_forms/integer_lattices#invariant_lattice-Tuple{ZZLat, Vector{<:MatElem}}"},[i("code",null,"invariant_lattice")],-1)),s[779]||(s[779]=t(")."))]),i("p",null,[s[784]||(s[784]=t("If ")),s[785]||(s[785]=i("code",null,"ambient_representation",-1)),s[786]||(s[786]=t(" is ")),s[787]||(s[787]=i("code",null,"true",-1)),s[788]||(s[788]=t(" (the default), the endomorphism is represented with respect to the ambient space of ")),i("mjx-container",g1,[(e(),l("svg",Q1,s[780]||(s[780]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[781]||(s[781]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[789]||(s[789]=t(". Otherwise, the endomorphism is represented with respect to the basis of ")),i("mjx-container",T1,[(e(),l("svg",m1,s[782]||(s[782]=[i("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[i("g",{"data-mml-node":"math"},[i("g",{"data-mml-node":"mi"},[i("path",{"data-c":"1D43F",d:"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z",style:{"stroke-width":"3"}})])])],-1)]))),s[783]||(s[783]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"L")])],-1))]),s[790]||(s[790]=t("."))]),s[792]||(s[792]=i("p",null,[i("a",{href:"https://github.com/thofma/Hecke.jl",target:"_blank",rel:"noreferrer"},"source")],-1))]),s[892]||(s[892]=i("h3",{id:"Computing-embeddings",tabindex:"-1"},[t("Computing embeddings "),i("a",{class:"header-anchor",href:"#Computing-embeddings","aria-label":'Permalink to "Computing embeddings {#Computing-embeddings}"'},"​")],-1)),i("details",y1,[i("summary",null,[s[793]||(s[793]=i("a",{id:"embed-Tuple{ZZLat, ZZGenus}",href:"#embed-Tuple{ZZLat, ZZGenus}"},[i("span",{class:"jlbinding"},"embed")],-1)),s[794]||(s[794]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[795]||(s[795]=a(`
julia
embed(S::ZZLat, G::Genus, primitive::Bool=true) -> Bool, embedding
Return a (primitive) embedding of the integral lattice S into some lattice in the genus of G.
julia
julia> G = integer_genera((8,0), 1, even=true)[1];
 
 julia> L, S, i = embed(root_lattice(:A,5), G);
source
`,4))]),i("details",F1,[i("summary",null,[s[796]||(s[796]=i("a",{id:"embed_in_unimodular-Tuple{ZZLat, Union{Integer, ZZRingElem}, Union{Integer, ZZRingElem}}",href:"#embed_in_unimodular-Tuple{ZZLat, Union{Integer, ZZRingElem}, Union{Integer, ZZRingElem}}"},[i("span",{class:"jlbinding"},"embed_in_unimodular")],-1)),s[797]||(s[797]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[798]||(s[798]=a(`
julia
embed_in_unimodular(S::ZZLat, pos::Int, neg::Int, primitive=true, even=true) -> Bool, L, S', iS, iR
Return a (primitive) embedding of the integral lattice S into some (even) unimodular lattice of signature (pos, neg).
For now this works only for even lattices.
julia
julia> NS = direct_sum(integer_lattice(:U), rescale(root_lattice(:A, 16), -1))[1];
 
@@ -387,10 +387,10 @@ import{_ as p,c as l,j as i,a as t,a5 as a,G as h,B as k,o as e}from"./chunks/fr
 
 julia> is_primitive(LK3, iNS)
 true
source
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julia
lll(L::ZZLat, same_ambient::Bool = true) -> ZZLat
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julia
short_vectors(L::ZZLat, [lb = 0], ub, [elem_type = ZZRingElem]; check::Bool = true)
-                                   -> Vector{Tuple{Vector{elem_type}, QQFieldElem}}
`,1)),i("p",null,[s[827]||(s[827]=t("Return all tuples ")),s[828]||(s[828]=i("code",null,"(v, n)",-1)),s[829]||(s[829]=t(" such that ")),i("mjx-container",b1,[(e(),l("svg",B1,s[825]||(s[825]=[a('',1)]))),s[826]||(s[826]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"n"),i("mo",null,"="),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),i("mi",null,"v"),i("mi",null,"G"),i("msup",null,[i("mi",null,"v"),i("mi",null,"t")]),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])],-1))]),s[830]||(s[830]=t(" satisfies ")),s[831]||(s[831]=i("code",null,"lb <= n <= ub",-1)),s[832]||(s[832]=t(", where ")),s[833]||(s[833]=i("code",null,"G",-1)),s[834]||(s[834]=t(" is the Gram matrix of ")),s[835]||(s[835]=i("code",null,"L",-1)),s[836]||(s[836]=t(" and ")),s[837]||(s[837]=i("code",null,"v",-1)),s[838]||(s[838]=t(" is non-zero."))]),s[840]||(s[840]=a('
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors_iterator for an iterator version.
source
',4))]),i("details",f1,[i("summary",null,[s[841]||(s[841]=i("a",{id:"shortest_vectors",href:"#shortest_vectors"},[i("span",{class:"jlbinding"},"shortest_vectors")],-1)),s[842]||(s[842]=t()),h(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[843]||(s[843]=a(`
julia
shortest_vectors(L::ZZLat, [elem_type = ZZRingElem]; check::Bool = true)
-                                           -> QQFieldElem, Vector{elem_type}, QQFieldElem}
Return the list of shortest non-zero vectors in absolute value. Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also minimum.
source
`,5))]),i("details",v1,[i("summary",null,[s[844]||(s[844]=i("a",{id:"short_vectors_iterator",href:"#short_vectors_iterator"},[i("span",{class:"jlbinding"},"short_vectors_iterator")],-1)),s[845]||(s[845]=t()),h(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[860]||(s[860]=a(`
julia
short_vectors_iterator(L::ZZLat, [lb = 0], ub,
+                                   -> Vector{Tuple{Vector{elem_type}, QQFieldElem}}
`,1)),i("p",null,[s[827]||(s[827]=t("Return all tuples ")),s[828]||(s[828]=i("code",null,"(v, n)",-1)),s[829]||(s[829]=t(" such that ")),i("mjx-container",b1,[(e(),l("svg",B1,s[825]||(s[825]=[a('',1)]))),s[826]||(s[826]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"n"),i("mo",null,"="),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),i("mi",null,"v"),i("mi",null,"G"),i("msup",null,[i("mi",null,"v"),i("mi",null,"t")]),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])],-1))]),s[830]||(s[830]=t(" satisfies ")),s[831]||(s[831]=i("code",null,"lb <= n <= ub",-1)),s[832]||(s[832]=t(", where ")),s[833]||(s[833]=i("code",null,"G",-1)),s[834]||(s[834]=t(" is the Gram matrix of ")),s[835]||(s[835]=i("code",null,"L",-1)),s[836]||(s[836]=t(" and ")),s[837]||(s[837]=i("code",null,"v",-1)),s[838]||(s[838]=t(" is non-zero."))]),s[840]||(s[840]=a('
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors_iterator for an iterator version.
source
',4))]),i("details",f1,[i("summary",null,[s[841]||(s[841]=i("a",{id:"shortest_vectors",href:"#shortest_vectors"},[i("span",{class:"jlbinding"},"shortest_vectors")],-1)),s[842]||(s[842]=t()),h(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[843]||(s[843]=a(`
julia
shortest_vectors(L::ZZLat, [elem_type = ZZRingElem]; check::Bool = true)
+                                           -> QQFieldElem, Vector{elem_type}, QQFieldElem}
Return the list of shortest non-zero vectors in absolute value. Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also minimum.
source
`,5))]),i("details",v1,[i("summary",null,[s[844]||(s[844]=i("a",{id:"short_vectors_iterator",href:"#short_vectors_iterator"},[i("span",{class:"jlbinding"},"short_vectors_iterator")],-1)),s[845]||(s[845]=t()),h(n,{type:"info",class:"jlObjectType jlFunction",text:"Function"})]),s[860]||(s[860]=a(`
julia
short_vectors_iterator(L::ZZLat, [lb = 0], ub,
                        [elem_type = ZZRingElem]; check::Bool = true)
-                                -> Tuple{Vector{elem_type}, QQFieldElem} (iterator)
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Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors.
source
',4))]),i("details",M1,[i("summary",null,[s[862]||(s[862]=i("a",{id:"minimum-Tuple{ZZLat}",href:"#minimum-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"minimum")],-1)),s[863]||(s[863]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[864]||(s[864]=a('
julia
minimum(L::ZZLat) -> QQFieldElem
Return the minimum absolute squared length among the non-zero vectors in L.
source
',3))]),i("details",j1,[i("summary",null,[s[865]||(s[865]=i("a",{id:"kissing_number-Tuple{ZZLat}",href:"#kissing_number-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"kissing_number")],-1)),s[866]||(s[866]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[867]||(s[867]=a('
julia
kissing_number(L::ZZLat) -> Int
Return the Kissing number of the sphere packing defined by L.
This is the number of non-overlapping spheres touching any other given sphere.
source
',4))]),s[896]||(s[896]=i("h3",{id:"Close-Vectors",tabindex:"-1"},[t("Close Vectors "),i("a",{class:"header-anchor",href:"#Close-Vectors","aria-label":'Permalink to "Close Vectors {#Close-Vectors}"'},"​")],-1)),i("details",Z1,[i("summary",null,[s[868]||(s[868]=i("a",{id:"close_vectors-Tuple{ZZLat, Vector, Vararg{Any}}",href:"#close_vectors-Tuple{ZZLat, Vector, Vararg{Any}}"},[i("span",{class:"jlbinding"},"close_vectors")],-1)),s[869]||(s[869]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[870]||(s[870]=a(`
julia
close_vectors(L:ZZLat, v:Vector, [lb,], ub; check::Bool = false)
+                                -> Tuple{Vector{elem_type}, QQFieldElem} (iterator)
`,1)),i("p",null,[s[848]||(s[848]=t("Return an iterator for all tuples ")),s[849]||(s[849]=i("code",null,"(v, n)",-1)),s[850]||(s[850]=t(" such that ")),i("mjx-container",L1,[(e(),l("svg",H1,s[846]||(s[846]=[a('',1)]))),s[847]||(s[847]=i("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[i("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[i("mi",null,"n"),i("mo",null,"="),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),i("mi",null,"v"),i("mi",null,"G"),i("msup",null,[i("mi",null,"v"),i("mi",null,"t")]),i("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])],-1))]),s[851]||(s[851]=t(" satisfies ")),s[852]||(s[852]=i("code",null,"lb <= n <= ub",-1)),s[853]||(s[853]=t(", where ")),s[854]||(s[854]=i("code",null,"G",-1)),s[855]||(s[855]=t(" is the Gram matrix of ")),s[856]||(s[856]=i("code",null,"L",-1)),s[857]||(s[857]=t(" and ")),s[858]||(s[858]=i("code",null,"v",-1)),s[859]||(s[859]=t(" is non-zero."))]),s[861]||(s[861]=a('
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors.
source
',4))]),i("details",M1,[i("summary",null,[s[862]||(s[862]=i("a",{id:"minimum-Tuple{ZZLat}",href:"#minimum-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"minimum")],-1)),s[863]||(s[863]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[864]||(s[864]=a('
julia
minimum(L::ZZLat) -> QQFieldElem
Return the minimum absolute squared length among the non-zero vectors in L.
source
',3))]),i("details",j1,[i("summary",null,[s[865]||(s[865]=i("a",{id:"kissing_number-Tuple{ZZLat}",href:"#kissing_number-Tuple{ZZLat}"},[i("span",{class:"jlbinding"},"kissing_number")],-1)),s[866]||(s[866]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[867]||(s[867]=a('
julia
kissing_number(L::ZZLat) -> Int
Return the Kissing number of the sphere packing defined by L.
This is the number of non-overlapping spheres touching any other given sphere.
source
',4))]),s[896]||(s[896]=i("h3",{id:"Close-Vectors",tabindex:"-1"},[t("Close Vectors "),i("a",{class:"header-anchor",href:"#Close-Vectors","aria-label":'Permalink to "Close Vectors {#Close-Vectors}"'},"​")],-1)),i("details",Z1,[i("summary",null,[s[868]||(s[868]=i("a",{id:"close_vectors-Tuple{ZZLat, Vector, Vararg{Any}}",href:"#close_vectors-Tuple{ZZLat, Vector, Vararg{Any}}"},[i("span",{class:"jlbinding"},"close_vectors")],-1)),s[869]||(s[869]=t()),h(n,{type:"info",class:"jlObjectType jlMethod",text:"Method"})]),s[870]||(s[870]=a(`
julia
close_vectors(L:ZZLat, v:Vector, [lb,], ub; check::Bool = false)
                                         -> Vector{Tuple{Vector{Int}}, QQFieldElem}
Return all tuples (x, d) where x is an element of L such that d = b(v - x, v - x) <= ub. If lb is provided, then also lb <= d.
If filter is not nothing, then only those x with filter(x) evaluating to true are returned.
By default, it will be checked whether L is positive definite. This can be disabled setting check = false.
Both input and output are with respect to the basis matrix of L.
Examples
julia
julia> L = integer_lattice(matrix(QQ, 2, 2, [1, 0, 0, 2]));
 
 julia> close_vectors(L, [1, 1], 1)
diff --git a/v0.34.8/examples.html b/v0.34.8/examples.html
index bd6e911255..9a0ba48290 100644
--- a/v0.34.8/examples.html
+++ b/v0.34.8/examples.html
@@ -6,20 +6,20 @@
     Examples and sample code | Hecke
     
     
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Appearance
Examples and sample code ​

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Appearance
Examples and sample code ​

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diff --git a/v0.34.8/howto/index.html b/v0.34.8/howto/index.html
index ba6a0312fa..5a23e6c64a 100644
--- a/v0.34.8/howto/index.html
+++ b/v0.34.8/howto/index.html
@@ -6,20 +6,20 @@
     How-to guides | Hecke
     
     
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Appearance

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Appearance

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diff --git a/v0.34.8/howto/reduction.html b/v0.34.8/howto/reduction.html
index 454ef44538..02b50d2b72 100644
--- a/v0.34.8/howto/reduction.html
+++ b/v0.34.8/howto/reduction.html
@@ -6,19 +6,19 @@
     Reduction of polynomials over number fields modulo a prime ideal | Hecke
     
     
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Appearance
Reduction of polynomials over number fields modulo a prime ideal ​
Given a polynomial f∈K[x] and a prime ideal p of OK, we want to determine the reduction f¯∈F[x], where F=OK/p is the residue field. Concretely, we want to reduce the polynomial f=x3+(1+ζ7+ζ72)x2+(23+55ζ75)x+(ζ7+77)/2 over Q(ζ7). We begin by defining the cyclomotic field and the polynomial.
julia

+    
Hecke
Appearance
Reduction of polynomials over number fields modulo a prime ideal ​
Given a polynomial f∈K[x] and a prime ideal p of OK, we want to determine the reduction f¯∈F[x], where F=OK/p is the residue field. Concretely, we want to reduce the polynomial f=x3+(1+ζ7+ζ72)x2+(23+55ζ75)x+(ζ7+77)/2 over Q(ζ7). We begin by defining the cyclomotic field and the polynomial.
julia

 julia> K, ζ = cyclotomic_field(7);
 
 julia> Kx, x = K['x'];
@@ -55,7 +55,7 @@
 
 julia> base_ring(fbar) === F
 true

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\ No newline at end of file
diff --git a/v0.34.8/index.html b/v0.34.8/index.html
index a507b20e2e..b99a6852b1 100644
--- a/v0.34.8/index.html
+++ b/v0.34.8/index.html
@@ -6,19 +6,19 @@
     Hecke
     
     
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Hecke
Appearance
Hecke
Computational number theory for everyone
Getting Started
Manual
View on Github
What is Hecke?
Hecke is a software package for computational algebraic number theory. It is written in julia and makes use of the computer algebra packages Nemo and AbstractAlgebra.
OSCAR
Hecke is part of the [OSCAR](https://www.oscar-system.org/) system, which covers, in addition to number theory, also commutative algebra, algebraic geometry, group theory and polyhedral geometry.
Features ​
  • Number fields (absolute, relative, simple and non-simple)
  • Orders and ideals in number fields
  • Class and unit group computations of orders
  • Lattice enumeration
  • Sparse linear algebra
  • Class field theory
  • Abelian groups
  • Associative algebras
  • Ideals and orders in (semsimple) associative algebras
  • Locally free class groups of orders in semisimple algebras
  • Quadratic and Hermitian forms and lattices
Citing Hecke ​
If your research depends on computations done with Hecke, please consider giving us a formal citation:
@inproceedings{nemo,
+    
Hecke
Appearance
Hecke
Computational number theory for everyone
Getting Started
Manual
View on Github
What is Hecke?
Hecke is a software package for computational algebraic number theory. It is written in julia and makes use of the computer algebra packages Nemo and AbstractAlgebra.
OSCAR
Hecke is part of the [OSCAR](https://www.oscar-system.org/) system, which covers, in addition to number theory, also commutative algebra, algebraic geometry, group theory and polyhedral geometry.
Features ​
  • Number fields (absolute, relative, simple and non-simple)
  • Orders and ideals in number fields
  • Class and unit group computations of orders
  • Lattice enumeration
  • Sparse linear algebra
  • Class field theory
  • Abelian groups
  • Associative algebras
  • Ideals and orders in (semsimple) associative algebras
  • Locally free class groups of orders in semisimple algebras
  • Quadratic and Hermitian forms and lattices
Citing Hecke ​
If your research depends on computations done with Hecke, please consider giving us a formal citation:
@inproceedings{nemo,
     author = {Fieker, Claus and Hart, William and Hofmann, Tommy and Johansson, Fredrik},
      title = {Nemo/Hecke: Computer Algebra and Number Theory Packages for the Julia Programming Language},
  booktitle = {Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation},
@@ -31,7 +31,7 @@
  publisher = {ACM},
    address = {New York, NY, USA},
 }
Acknowledgement ​
Hecke is part of the OSCAR project and the development is supported by the Deutsche Forschungsgemeinschaft DFG within the Collaborative Research Center TRR 195.

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diff --git a/v0.34.8/manual/abelian/elements.html b/v0.34.8/manual/abelian/elements.html
index 42ac14dc44..ce314eb23e 100644
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     Elements | Hecke
     
     
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Appearance
Elements ​
Elements in a finitely generated abelian group are of type FinGenAbGroupElem and are always given as a linear combination of the generators. Internally this representation is normliased to have a unique representative.
Creation ​
In addition to the standard function id, zero and one that can be used to create the neutral element, we also support more targeted creation:
gens Method
julia
gens(G::FinGenAbGroup) -> Vector{FinGenAbGroupElem}
The sequence of generators of G.
source
FinGenAbGroup Method
julia
(A::FinGenAbGroup)(x::Vector{ZZRingElem}) -> FinGenAbGroupElem
Given an array x of elements of type ZZRingElem of the same length as ngens(A), this function returns the element of A with components x.
source
FinGenAbGroup Method
julia
(A::FinGenAbGroup)(x::ZZMatrix) -> FinGenAbGroupElem
Given a matrix over the integers with either 1 row and ngens(A) columns or ngens(A) rows and 1 column, this function returns the element of A with components x.
source
getindex Method
julia
getindex(A::FinGenAbGroup, i::Int) -> FinGenAbGroupElem
Returns the element of A with components (0,…,0,1,0,…,0), where the 1 is at the i-th position.
source
rand Method
julia
rand(G::FinGenAbGroup) -> FinGenAbGroupElem
Returns an element of G chosen uniformly at random.
source
rand Method
julia
rand(G::FinGenAbGroup, B::ZZRingElem) -> FinGenAbGroupElem
For a (potentially infinite) abelian group G, return an element chosen uniformly at random with coefficients bounded by B.
source
parent Method
julia
parent(x::FinGenAbGroupElem) -> FinGenAbGroup
Returns the parent of x.
source
Access ​
getindex Method
julia
getindex(x::FinGenAbGroupElem, v::AbstractVector{Int}) -> Vector{ZZRingElem}
Returns the i-th components of the element x where i∈v.
Note
This function is inefficient since the elements are internally stored using ZZMatrix but this function outputs a vector.
source
getindex Method
julia
getindex(x::FinGenAbGroupElem, i::Int) -> ZZRingElem
Returns the i-th component of the element x.
source
Predicates ​
We have the standard predicates iszero, isone and is_identity to test an element for being trivial.
Invariants ​
order Method
julia
order(A::FinGenAbGroupElem) -> ZZRingElem
Returns the order of A. It is assumed that the order is finite.
source
Iterator ​
One can iterate over the elements of a finite abelian group.
julia

+    
Hecke
Appearance
Elements ​
Elements in a finitely generated abelian group are of type FinGenAbGroupElem and are always given as a linear combination of the generators. Internally this representation is normliased to have a unique representative.
Creation ​
In addition to the standard function id, zero and one that can be used to create the neutral element, we also support more targeted creation:
gens Method
julia
gens(G::FinGenAbGroup) -> Vector{FinGenAbGroupElem}
The sequence of generators of G.
source
FinGenAbGroup Method
julia
(A::FinGenAbGroup)(x::Vector{ZZRingElem}) -> FinGenAbGroupElem
Given an array x of elements of type ZZRingElem of the same length as ngens(A), this function returns the element of A with components x.
source
FinGenAbGroup Method
julia
(A::FinGenAbGroup)(x::ZZMatrix) -> FinGenAbGroupElem
Given a matrix over the integers with either 1 row and ngens(A) columns or ngens(A) rows and 1 column, this function returns the element of A with components x.
source
getindex Method
julia
getindex(A::FinGenAbGroup, i::Int) -> FinGenAbGroupElem
Returns the element of A with components (0,…,0,1,0,…,0), where the 1 is at the i-th position.
source
rand Method
julia
rand(G::FinGenAbGroup) -> FinGenAbGroupElem
Returns an element of G chosen uniformly at random.
source
rand Method
julia
rand(G::FinGenAbGroup, B::ZZRingElem) -> FinGenAbGroupElem
For a (potentially infinite) abelian group G, return an element chosen uniformly at random with coefficients bounded by B.
source
parent Method
julia
parent(x::FinGenAbGroupElem) -> FinGenAbGroup
Returns the parent of x.
source
Access ​
getindex Method
julia
getindex(x::FinGenAbGroupElem, v::AbstractVector{Int}) -> Vector{ZZRingElem}
Returns the i-th components of the element x where i∈v.
Note
This function is inefficient since the elements are internally stored using ZZMatrix but this function outputs a vector.
source
getindex Method
julia
getindex(x::FinGenAbGroupElem, i::Int) -> ZZRingElem
Returns the i-th component of the element x.
source
Predicates ​
We have the standard predicates iszero, isone and is_identity to test an element for being trivial.
Invariants ​
order Method
julia
order(A::FinGenAbGroupElem) -> ZZRingElem
Returns the order of A. It is assumed that the order is finite.
source
Iterator ​
One can iterate over the elements of a finite abelian group.
julia

 julia> G = abelian_group(ZZRingElem[1 2; 3 4])
 Finitely generated abelian group
   with 2 generators and 2 relations and relation matrix
@@ -29,8 +29,8 @@
          println(g)
        end
 Abelian group element [0, 0]
-Abelian group element [0, 1]

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+Abelian group element [0, 1]

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\ No newline at end of file
diff --git a/v0.34.8/manual/abelian/introduction.html b/v0.34.8/manual/abelian/introduction.html
index feba426734..d3eb446f3b 100644
--- a/v0.34.8/manual/abelian/introduction.html
+++ b/v0.34.8/manual/abelian/introduction.html
@@ -6,19 +6,19 @@
     Introduction | Hecke
     
     
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Appearance
Introduction ​
Within Hecke, abelian groups are of generic abstract type GrpAb which does not have to be finitely generated, Q/Z is an example of a more general abelian group. Having said that, most of the functionality is restricted to abelian groups that are finitely presented as Z-modules.
Basic Creation ​
Finitely presented (as Z-modules) abelian groups are of type FinGenAbGroup with elements of type FinGenAbGroupElem. The creation is mostly via a relation matrix M=(mi,j) for 1≤i≤n and 1≤j≤m. This creates a group with m generators ej and relations
∑i=1nmi,jej=0.
abelian_group Method
julia
abelian_group(::Type{T} = FinGenAbGroup, M::ZZMatrix) -> FinGenAbGroup
Creates the abelian group with relation matrix M. That is, the group will have ncols(M) generators and each row of M describes one relation.
source
abelian_group Method
julia
abelian_group(::Type{T} = FinGenAbGroup, M::AbstractMatrix{<:IntegerUnion})
Creates the abelian group with relation matrix M. That is, the group will have ncols(M) generators and each row of M describes one relation.
source
abelian_group Method
julia
abelian_group(::Type{T} = FinGenAbGroup, M::AbstractMatrix{<:IntegerUnion})
Creates the abelian group with relation matrix M. That is, the group will have ncols(M) generators and each row of M describes one relation.
source
Alternatively, there are shortcuts to create products of cyclic groups:
abelian_group Method
julia
abelian_group(::Type{T} = FinGenAbGroup, M::AbstractVector{<:IntegerUnion}) -> FinGenAbGroup
+    
Hecke
Appearance
Introduction ​
Within Hecke, abelian groups are of generic abstract type GrpAb which does not have to be finitely generated, Q/Z is an example of a more general abelian group. Having said that, most of the functionality is restricted to abelian groups that are finitely presented as Z-modules.
Basic Creation ​
Finitely presented (as Z-modules) abelian groups are of type FinGenAbGroup with elements of type FinGenAbGroupElem. The creation is mostly via a relation matrix M=(mi,j) for 1≤i≤n and 1≤j≤m. This creates a group with m generators ej and relations
∑i=1nmi,jej=0.
abelian_group Method
julia
abelian_group(::Type{T} = FinGenAbGroup, M::ZZMatrix) -> FinGenAbGroup
Creates the abelian group with relation matrix M. That is, the group will have ncols(M) generators and each row of M describes one relation.
source
abelian_group Method
julia
abelian_group(::Type{T} = FinGenAbGroup, M::AbstractMatrix{<:IntegerUnion})
Creates the abelian group with relation matrix M. That is, the group will have ncols(M) generators and each row of M describes one relation.
source
abelian_group Method
julia
abelian_group(::Type{T} = FinGenAbGroup, M::AbstractMatrix{<:IntegerUnion})
Creates the abelian group with relation matrix M. That is, the group will have ncols(M) generators and each row of M describes one relation.
source
Alternatively, there are shortcuts to create products of cyclic groups:
abelian_group Method
julia
abelian_group(::Type{T} = FinGenAbGroup, M::AbstractVector{<:IntegerUnion}) -> FinGenAbGroup
 abelian_group(::Type{T} = FinGenAbGroup, M::IntegerUnion...) -> FinGenAbGroup
Creates the direct product of the cyclic groups Z/mi, where mi is the ith entry of M.
source
julia

 julia> G = abelian_group(2, 2, 6)
 (Z/2)^2 x Z/6
or even
free_abelian_group Method
julia
free_abelian_group(::Type{T} = FinGenAbGroup, n::Int) -> FinGenAbGroup
Creates the free abelian group of rank n.
source
abelian_groups Method
julia
abelian_groups(n::Int) -> Vector{FinGenAbGroup}
Given a positive integer n, return a list of all abelian groups of order n.
source
julia

@@ -26,8 +26,8 @@
 3-element Vector{FinGenAbGroup}:
  (Z/2)^3
  Z/2 x Z/4
- Z/8
Invariants ​
is_snf Method
julia
is_snf(G::FinGenAbGroup) -> Bool
Return whether the current relation matrix of the group G is in Smith normal form.
source
number_of_generators Method
julia
number_of_generators(G::FinGenAbGroup) -> Int
Return the number of generators of G in the current representation.
source
nrels Method
julia
number_of_relations(G::FinGenAbGroup) -> Int
Return the number of relations of G in the current representation.
source
rels Method
julia
rels(A::FinGenAbGroup) -> ZZMatrix
Return the currently used relations of G as a single matrix.
source
is_finite Method
julia
isfinite(A::FinGenAbGroup) -> Bool
Return whether A is finite.
source
torsion_free_rank Method
julia
torsion_free_rank(A::FinGenAbGroup) -> Int
Return the torsion free rank of A, that is, the dimension of the Q-vectorspace A⊗ZQ.
See also rank.
source
order Method
julia
order(A::FinGenAbGroup) -> ZZRingElem
Return the order of A. It is assumed that A is finite.
source
exponent Method
julia
exponent(A::FinGenAbGroup) -> ZZRingElem
Return the exponent of A. It is assumed that A is finite.
source
is_trivial Method
julia
is_trivial(A::FinGenAbGroup) -> Bool
Return whether A is the trivial group.
source
is_torsion Method
julia
is_torsion(G::FinGenAbGroup) -> Bool
Return whether G is a torsion group.
source
is_cyclic Method
julia
is_cyclic(G::FinGenAbGroup) -> Bool
Return whether G is cyclic.
source
elementary_divisors Method
julia
elementary_divisors(G::FinGenAbGroup) -> Vector{ZZRingElem}
Given G, return the elementary divisors of G, that is, the unique non-negative integers e1,…,ek with ei∣ei+1 and ei≠1 such that G≅Z/e1Z×⋯×Z/ekZ.
source

-    
+ Z/8
Invariants ​
is_snf Method
julia
is_snf(G::FinGenAbGroup) -> Bool
Return whether the current relation matrix of the group G is in Smith normal form.
source
number_of_generators Method
julia
number_of_generators(G::FinGenAbGroup) -> Int
Return the number of generators of G in the current representation.
source
nrels Method
julia
number_of_relations(G::FinGenAbGroup) -> Int
Return the number of relations of G in the current representation.
source
rels Method
julia
rels(A::FinGenAbGroup) -> ZZMatrix
Return the currently used relations of G as a single matrix.
source
is_finite Method
julia
isfinite(A::FinGenAbGroup) -> Bool
Return whether A is finite.
source
torsion_free_rank Method
julia
torsion_free_rank(A::FinGenAbGroup) -> Int
Return the torsion free rank of A, that is, the dimension of the Q-vectorspace A⊗ZQ.
See also rank.
source
order Method
julia
order(A::FinGenAbGroup) -> ZZRingElem
Return the order of A. It is assumed that A is finite.
source
exponent Method
julia
exponent(A::FinGenAbGroup) -> ZZRingElem
Return the exponent of A. It is assumed that A is finite.
source
is_trivial Method
julia
is_trivial(A::FinGenAbGroup) -> Bool
Return whether A is the trivial group.
source
is_torsion Method
julia
is_torsion(G::FinGenAbGroup) -> Bool
Return whether G is a torsion group.
source
is_cyclic Method
julia
is_cyclic(G::FinGenAbGroup) -> Bool
Return whether G is cyclic.
source
elementary_divisors Method
julia
elementary_divisors(G::FinGenAbGroup) -> Vector{ZZRingElem}
Given G, return the elementary divisors of G, that is, the unique non-negative integers e1,…,ek with ei∣ei+1 and ei≠1 such that G≅Z/e1Z×⋯×Z/ekZ.
source

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\ No newline at end of file
diff --git a/v0.34.8/manual/abelian/maps.html b/v0.34.8/manual/abelian/maps.html
index fb8c24354e..cb57416e1f 100644
--- a/v0.34.8/manual/abelian/maps.html
+++ b/v0.34.8/manual/abelian/maps.html
@@ -6,19 +6,19 @@
     Maps | Hecke
     
     
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Hecke
Appearance
Maps ​
Maps between abelian groups are mainly of type FinGenAbGroupHom. They allow normal map operations such as image, preimage, domain, codomain and can be created in a variety of situations.
Maps between abelian groups can be constructed via
  • images of the generators
  • pairs of elements
  • via composition
  • and isomorphism/ inclusion testing
hom_direct_sum Method
julia
hom_direct_sum(G::FinGenAbGroup, H::FinGenAbGroup, A::Matrix{ <: Map{FinGenAbGroup, FinGenAbGroup}}) -> Map
Given groups G and H that are created as direct products as well as a matrix A containing maps A[i,j]:Gi→Hj, return the induced homomorphism.
source
is_isomorphic Method
julia
is_isomorphic(G::FinGenAbGroup, H::FinGenAbGroup) -> Bool
Return whether G and H are isomorphic.
source
julia

+    
Hecke
Appearance
Maps ​
Maps between abelian groups are mainly of type FinGenAbGroupHom. They allow normal map operations such as image, preimage, domain, codomain and can be created in a variety of situations.
Maps between abelian groups can be constructed via
  • images of the generators
  • pairs of elements
  • via composition
  • and isomorphism/ inclusion testing
hom_direct_sum Method
julia
hom_direct_sum(G::FinGenAbGroup, H::FinGenAbGroup, A::Matrix{ <: Map{FinGenAbGroup, FinGenAbGroup}}) -> Map
Given groups G and H that are created as direct products as well as a matrix A containing maps A[i,j]:Gi→Hj, return the induced homomorphism.
source
is_isomorphic Method
julia
is_isomorphic(G::FinGenAbGroup, H::FinGenAbGroup) -> Bool
Return whether G and H are isomorphic.
source
julia

 julia> G = free_abelian_group(2)
 Z^2
 
@@ -41,8 +41,8 @@
   to Z^2
 
 julia> (h+h)(gen(G, 1))
-Abelian group element [0, 4]

-    
+Abelian group element [0, 4]

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\ No newline at end of file
diff --git a/v0.34.8/manual/abelian/structural.html b/v0.34.8/manual/abelian/structural.html
index 787666a7fd..abe64401e6 100644
--- a/v0.34.8/manual/abelian/structural.html
+++ b/v0.34.8/manual/abelian/structural.html
@@ -6,19 +6,19 @@
     Structural Computations | Hecke
     
     
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Hecke
Appearance
Structural Computations ​
Abelian groups support a wide range of structural operations such as
  • enumeration of subgroups
  • (outer) direct products
  • tensor and hom constructions
  • free resolutions and general complexes
  • (co)-homology and tensor and hom-functors
snf Method
julia
snf(A::FinGenAbGroup) -> FinGenAbGroup, FinGenAbGroupHom
Return a pair (G,f), where G is an abelian group in canonical Smith normal form isomorphic to A and an isomorphism f:G→A.
source
find_isomorphism Method
julia
find_isomorphism(G, op, A::GrpAb) -> Dict, Dict
Given an abelian group A and a collection G which is an abelian group with the operation op, this functions returns isomorphisms G→A and A→G encoded as dictionaries.
It is assumed that G and A are isomorphic.
source
Subgroups and Quotients ​
torsion_subgroup Method
julia
torsion_subgroup(G::FinGenAbGroup) -> FinGenAbGroup, Map
Return the torsion subgroup of G.
source
sub Method
julia
sub(G::FinGenAbGroup, s::Vector{FinGenAbGroupElem}) -> FinGenAbGroup, FinGenAbGroupHom
Create the subgroup H of G generated by the elements in s together with the injection ι:H→G.
source
sub Method
julia
sub(A::SMat, r::AbstractUnitRange, c::AbstractUnitRange) -> SMat
Return the submatrix of A, where the rows correspond to r and the columns correspond to c.
source
julia
sub(s::Vector{FinGenAbGroupElem}) -> FinGenAbGroup, FinGenAbGroupHom
Assuming that the non-empty array s contains elements of an abelian group G, this functions returns the subgroup H of G generated by the elements in s together with the injection ι:H→G.
source
sub Method
julia
sub(G::FinGenAbGroup, M::ZZMatrix) -> FinGenAbGroup, FinGenAbGroupHom
Create the subgroup H of G generated by the elements corresponding to the rows of M together with the injection ι:H→G.
source
sub Method
julia
sub(G::FinGenAbGroup, n::ZZRingElem) -> FinGenAbGroup, FinGenAbGroupHom
Create the subgroup n⋅G of G together with the injection ι:n⋅G→G.
source
sub Method
julia
sub(G::FinGenAbGroup, n::Integer) -> FinGenAbGroup, Map
Create the subgroup n⋅G of G together with the injection ι:n⋅G→G.
source
sylow_subgroup Method
julia
sylow_subgroup(G::FinGenAbGroup, p::IntegerUnion) -> FinGenAbGroup, FinGenAbGroupHom
Return the Sylow p−subgroup of the finitely generated abelian group G, for a prime p. This is the subgroup of p-power order in G whose index in G is coprime to p.
Examples
julia
julia> A = abelian_group(ZZRingElem[2, 6, 30])
+    
Hecke
Appearance
Structural Computations ​
Abelian groups support a wide range of structural operations such as
  • enumeration of subgroups
  • (outer) direct products
  • tensor and hom constructions
  • free resolutions and general complexes
  • (co)-homology and tensor and hom-functors
snf Method
julia
snf(A::FinGenAbGroup) -> FinGenAbGroup, FinGenAbGroupHom
Return a pair (G,f), where G is an abelian group in canonical Smith normal form isomorphic to A and an isomorphism f:G→A.
source
find_isomorphism Method
julia
find_isomorphism(G, op, A::GrpAb) -> Dict, Dict
Given an abelian group A and a collection G which is an abelian group with the operation op, this functions returns isomorphisms G→A and A→G encoded as dictionaries.
It is assumed that G and A are isomorphic.
source
Subgroups and Quotients ​
torsion_subgroup Method
julia
torsion_subgroup(G::FinGenAbGroup) -> FinGenAbGroup, Map
Return the torsion subgroup of G.
source
sub Method
julia
sub(G::FinGenAbGroup, s::Vector{FinGenAbGroupElem}) -> FinGenAbGroup, FinGenAbGroupHom
Create the subgroup H of G generated by the elements in s together with the injection ι:H→G.
source
sub Method
julia
sub(A::SMat, r::AbstractUnitRange, c::AbstractUnitRange) -> SMat
Return the submatrix of A, where the rows correspond to r and the columns correspond to c.
source
julia
sub(s::Vector{FinGenAbGroupElem}) -> FinGenAbGroup, FinGenAbGroupHom
Assuming that the non-empty array s contains elements of an abelian group G, this functions returns the subgroup H of G generated by the elements in s together with the injection ι:H→G.
source
sub Method
julia
sub(G::FinGenAbGroup, M::ZZMatrix) -> FinGenAbGroup, FinGenAbGroupHom
Create the subgroup H of G generated by the elements corresponding to the rows of M together with the injection ι:H→G.
source
sub Method
julia
sub(G::FinGenAbGroup, n::ZZRingElem) -> FinGenAbGroup, FinGenAbGroupHom
Create the subgroup n⋅G of G together with the injection ι:n⋅G→G.
source
sub Method
julia
sub(G::FinGenAbGroup, n::Integer) -> FinGenAbGroup, Map
Create the subgroup n⋅G of G together with the injection ι:n⋅G→G.
source
sylow_subgroup Method
julia
sylow_subgroup(G::FinGenAbGroup, p::IntegerUnion) -> FinGenAbGroup, FinGenAbGroupHom
Return the Sylow p−subgroup of the finitely generated abelian group G, for a prime p. This is the subgroup of p-power order in G whose index in G is coprime to p.
Examples
julia
julia> A = abelian_group(ZZRingElem[2, 6, 30])
 Z/2 x Z/6 x Z/30
 
 julia> H, j = sylow_subgroup(A, 2);
@@ -28,7 +28,7 @@
 
 julia> divexact(order(A), order(H))
 45
source
has_quotient Method
julia
has_quotient(G::FinGenAbGroup, invariant::Vector{Int}) -> Bool
Given an abelian group G, return true if it has a quotient with given elementary divisors and false otherwise.
source
has_complement Method
julia
has_complement(f::FinGenAbGroupHom) -> Bool, FinGenAbGroupHom
-has_complement(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool, FinGenAbGroupHom
Given a map representing a subgroup of a group G, or a subgroup U of a group G, return either true and an injection of a complement in G, or false.
See also: is_pure
source
is_pure Method
julia
is_pure(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool
A subgroup U of G is called pure if for all n an element in U that is in the image of the multiplication by n map of G is also a multiple of an element in U.
For finite abelian groups this is equivalent to U having a complement in G. They are also know as isolated subgroups and serving subgroups.
See also: is_neat, has_complement
EXAMPLES
julia
julia> G = abelian_group([2, 8]);
+has_complement(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool, FinGenAbGroupHom
Given a map representing a subgroup of a group G, or a subgroup U of a group G, return either true and an injection of a complement in G, or false.
See also: is_pure
source
is_pure Method
julia
is_pure(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool
A subgroup U of G is called pure if for all n an element in U that is in the image of the multiplication by n map of G is also a multiple of an element in U.
For finite abelian groups this is equivalent to U having a complement in G. They are also know as isolated subgroups and serving subgroups.
See also: is_neat, has_complement
EXAMPLES
julia
julia> G = abelian_group([2, 8]);
 
 julia> U, _ = sub(G, [G[1]+2*G[2]]);
 
@@ -41,7 +41,7 @@
 true
 
 julia> has_complement(U, G)[1]
-true
source
is_neat Method
julia
is_neat(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool
A subgroup U of G is called neat if for all primes p an element in U that is in the image of the multiplication by p map of G is also a multiple of an element in U.
See also: is_pure
EXAMPLES
julia
julia> G = abelian_group([2, 8]);
+true
source
is_neat Method
julia
is_neat(U::FinGenAbGroup, G::FinGenAbGroup) -> Bool
A subgroup U of G is called neat if for all primes p an element in U that is in the image of the multiplication by p map of G is also a multiple of an element in U.
See also: is_pure
EXAMPLES
julia
julia> G = abelian_group([2, 8]);
 
 julia> U, _ = sub(G, [G[1] + 2*G[2]]);
 
@@ -49,7 +49,7 @@
 true
 
 julia> is_pure(U, G)
-false
source
saturate Method
julia
saturate(U::FinGenAbGroup, G::FinGenAbGroup) -> FinGenAbGroup
For a subgroup U of G find a minimal overgroup that is pure, and thus has a complement.
See also: is_pure, has_complement
source
A sophisticated algorithm for the enumeration of all (or selected) subgroups of a finite abelian group is available.
psubgroups Method
julia
psubgroups(g::FinGenAbGroup, p::Integer;
+false
source
saturate Method
julia
saturate(U::FinGenAbGroup, G::FinGenAbGroup) -> FinGenAbGroup
For a subgroup U of G find a minimal overgroup that is pure, and thus has a complement.
See also: is_pure, has_complement
source
A sophisticated algorithm for the enumeration of all (or selected) subgroups of a finite abelian group is available.
psubgroups Method
julia
psubgroups(g::FinGenAbGroup, p::Integer;
            subtype = :all,
            quotype = :all,
            index = -1,
@@ -88,8 +88,8 @@
 (U[1], map(U[2], gens(U[1]))) = (Finitely generated abelian group with 3 generators and 3 relations, FinGenAbGroupElem[[0, 3], [0, 4], [2, 0]])
 (U[1], map(U[2], gens(U[1]))) = (Finitely generated abelian group with 4 generators and 4 relations, FinGenAbGroupElem[[3, 6], [0, 6], [0, 4], [2, 0]])
quo Method
julia
quo(G::FinGenAbGroup, s::Vector{FinGenAbGroupElem}) -> FinGenAbGroup, GrpAbfinGemMap
Create the quotient H of G by the subgroup generated by the elements in s, together with the projection p:G→H.
source
quo Method
julia
quo(G::FinGenAbGroup, M::ZZMatrix) -> FinGenAbGroup, FinGenAbGroupHom
Create the quotient H of G by the subgroup generated by the elements corresponding to the rows of M, together with the projection p:G→H.
source
quo Method
julia
quo(G::FinGenAbGroup, n::Integer}) -> FinGenAbGroup, Map
 quo(G::FinGenAbGroup, n::ZZRingElem}) -> FinGenAbGroup, Map
Returns the quotient H=G/nG together with the projection p:G→H.
source
quo Method
julia
quo(G::FinGenAbGroup, n::Integer}) -> FinGenAbGroup, Map
-quo(G::FinGenAbGroup, n::ZZRingElem}) -> FinGenAbGroup, Map
Returns the quotient H=G/nG together with the projection p:G→H.
source
quo Method
julia
quo(G::FinGenAbGroup, U::FinGenAbGroup) -> FinGenAbGroup, Map
Create the quotient H of G by U, together with the projection p:G→H.
source
For 2 subgroups U and V of the same group G, U+V returns the smallest subgroup of G containing both. Similarly, U∩V computes the intersection and U⊂V tests for inclusion. The difference between issubset = ⊂ and is_subgroup is that the inclusion map is also returned in the 2nd call.
intersect Method
julia
intersect(mG::FinGenAbGroupHom, mH::FinGenAbGroupHom) -> FinGenAbGroup, Map
Given two injective maps of abelian groups with the same codomain G, return the intersection of the images as a subgroup of G.
source
Direct Products ​
direct_product Method
julia
direct_product(G::FinGenAbGroup...) -> FinGenAbGroup, Vector{FinGenAbGroupHom}
Return the direct product D of the (finitely many) abelian groups Gi, together with the projections D→Gi.
For finite abelian groups, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain D as a direct sum together with the injections D→Gi, one should call direct_sum(G...). If one wants to obtain D as a biproduct together with the projections and the injections, one should call biproduct(G...).
Otherwise, one could also call canonical_injections(D) or canonical_projections(D) later on.
source
canonical_injection Method
julia
canonical_injection(G::FinGenAbGroup, i::Int) -> FinGenAbGroupHom
Given a group G that was created as a direct product, return the injection from the ith component.
source
canonical_projection Method
julia
canonical_projection(G::FinGenAbGroup, i::Int) -> FinGenAbGroupHom
Given a group G that was created as a direct product, return the projection onto the ith component.
source
flat Method
julia
flat(G::FinGenAbGroup) -> FinGenAbGroupHom
Given a group G that is created using (iterated) direct products, or (iterated) tensor products, return a group isomorphism into a flat product: for G:=(A⊕B)⊕C, it returns the isomorphism G→A⊕B⊕C (resp. ⊗).
source
Tensor Producs ​
tensor_product Method
julia
tensor_product(G::FinGenAbGroup...; task::Symbol = :map) -> FinGenAbGroup, Map
Given groups Gi, compute the tensor product G1⊗⋯⊗Gn. If task is set to ":map", a map ϕ is returned that maps tuples in G1×⋯×Gn to pure tensors g1⊗⋯⊗gn. The map admits a preimage as well.
source
hom_tensor Method
julia
hom_tensor(G::FinGenAbGroup, H::FinGenAbGroup, A::Vector{ <: Map{FinGenAbGroup, FinGenAbGroup}}) -> Map
Given groups G=G1⊗⋯⊗Gn and H=H1⊗⋅⊗Hn as well as maps ϕi:Gi→Hi, compute the tensor product of the maps.
source
Hom-Group ​
hom Method
julia
hom(G::FinGenAbGroup, H::FinGenAbGroup; task::Symbol = :map) -> FinGenAbGroup, Map
Computes the group of all homomorpisms from G to H as an abstract group. If task is ":map", then a map ϕ is computed that can be used to obtain actual homomorphisms. This map also allows preimages. Set task to ":none" to not compute the map.
source

-    
+quo(G::FinGenAbGroup, n::ZZRingElem}) -> FinGenAbGroup, Map
Returns the quotient H=G/nG together with the projection p:G→H.
source
quo Method
julia
quo(G::FinGenAbGroup, U::FinGenAbGroup) -> FinGenAbGroup, Map
Create the quotient H of G by U, together with the projection p:G→H.
source
For 2 subgroups U and V of the same group G, U+V returns the smallest subgroup of G containing both. Similarly, U∩V computes the intersection and U⊂V tests for inclusion. The difference between issubset = ⊂ and is_subgroup is that the inclusion map is also returned in the 2nd call.
intersect Method
julia
intersect(mG::FinGenAbGroupHom, mH::FinGenAbGroupHom) -> FinGenAbGroup, Map
Given two injective maps of abelian groups with the same codomain G, return the intersection of the images as a subgroup of G.
source
Direct Products ​
direct_product Method
julia
direct_product(G::FinGenAbGroup...) -> FinGenAbGroup, Vector{FinGenAbGroupHom}
Return the direct product D of the (finitely many) abelian groups Gi, together with the projections D→Gi.
For finite abelian groups, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain D as a direct sum together with the injections D→Gi, one should call direct_sum(G...). If one wants to obtain D as a biproduct together with the projections and the injections, one should call biproduct(G...).
Otherwise, one could also call canonical_injections(D) or canonical_projections(D) later on.
source
canonical_injection Method
julia
canonical_injection(G::FinGenAbGroup, i::Int) -> FinGenAbGroupHom
Given a group G that was created as a direct product, return the injection from the ith component.
source
canonical_projection Method
julia
canonical_projection(G::FinGenAbGroup, i::Int) -> FinGenAbGroupHom
Given a group G that was created as a direct product, return the projection onto the ith component.
source
flat Method
julia
flat(G::FinGenAbGroup) -> FinGenAbGroupHom
Given a group G that is created using (iterated) direct products, or (iterated) tensor products, return a group isomorphism into a flat product: for G:=(A⊕B)⊕C, it returns the isomorphism G→A⊕B⊕C (resp. ⊗).
source
Tensor Producs ​
tensor_product Method
julia
tensor_product(G::FinGenAbGroup...; task::Symbol = :map) -> FinGenAbGroup, Map
Given groups Gi, compute the tensor product G1⊗⋯⊗Gn. If task is set to ":map", a map ϕ is returned that maps tuples in G1×⋯×Gn to pure tensors g1⊗⋯⊗gn. The map admits a preimage as well.
source
hom_tensor Method
julia
hom_tensor(G::FinGenAbGroup, H::FinGenAbGroup, A::Vector{ <: Map{FinGenAbGroup, FinGenAbGroup}}) -> Map
Given groups G=G1⊗⋯⊗Gn and H=H1⊗⋅⊗Hn as well as maps ϕi:Gi→Hi, compute the tensor product of the maps.
source
Hom-Group ​
hom Method
julia
hom(G::FinGenAbGroup, H::FinGenAbGroup; task::Symbol = :map) -> FinGenAbGroup, Map
Computes the group of all homomorpisms from G to H as an abstract group. If task is ":map", then a map ϕ is computed that can be used to obtain actual homomorphisms. This map also allows preimages. Set task to ":none" to not compute the map.
source

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diff --git a/v0.34.8/manual/algebras/basics.html b/v0.34.8/manual/algebras/basics.html
index e607459cee..9e19ea4fa8 100644
--- a/v0.34.8/manual/algebras/basics.html
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@@ -6,23 +6,23 @@
     Basics | Hecke
     
     
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Basics ​
Creation of algebras ​
See the corresponding sections on structure constant algebras.
zero_algebra Method
julia
zero_algebra([T, ] K::Field) -> AbstractAssociativeAlgebra
Return the zero ring as an algebra over the field K.
The optional first argument determines the type of the algebra, and can be StructureConstantAlgebra (default) or MatrixAlgebra.
Examples
julia
julia> A = zero_algebra(QQ)
-Structure constant algebra of dimension 0 over QQ
source
Basic properties ​
base_ring Method
julia
base_ring(A::AbstractAssociativeAlgebra) -> Field
Given a K-algebra A, return K.
source
basis Method
julia
basis(A::AbstractAssociativeAlgebra) -> Vector
Given a K-algebra A return the K-basis of A. See also coordinates to get the the coordinates of an element with respect to the bases.
source
Predicates ​
is_zero Method
julia
is_zero(A::AbstractAssociativeAlgebra) -> Bool
Return whether A is the zero algebra.
source
is_commutative Method
julia
is_commutative(A::AbstractAssociativeAlgebra) -> Bool
Return whether A is commutative.
Examples
julia
julia> A = matrix_algebra(QQ, 2);
+    
Hecke
Appearance
Basics ​
Creation of algebras ​
See the corresponding sections on structure constant algebras.
zero_algebra Method
julia
zero_algebra([T, ] K::Field) -> AbstractAssociativeAlgebra
Return the zero ring as an algebra over the field K.
The optional first argument determines the type of the algebra, and can be StructureConstantAlgebra (default) or MatrixAlgebra.
Examples
julia
julia> A = zero_algebra(QQ)
+Structure constant algebra of dimension 0 over QQ
source
Basic properties ​
base_ring Method
julia
base_ring(A::AbstractAssociativeAlgebra) -> Field
Given a K-algebra A, return K.
source
basis Method
julia
basis(A::AbstractAssociativeAlgebra) -> Vector
Given a K-algebra A return the K-basis of A. See also coordinates to get the the coordinates of an element with respect to the bases.
source
Predicates ​
is_zero Method
julia
is_zero(A::AbstractAssociativeAlgebra) -> Bool
Return whether A is the zero algebra.
source
is_commutative Method
julia
is_commutative(A::AbstractAssociativeAlgebra) -> Bool
Return whether A is commutative.
Examples
julia
julia> A = matrix_algebra(QQ, 2);
 
 julia> is_commutative(A)
-false
source
is_central Method
julia
is_central(A::AbstractAssociativeAlgebra)
Return whether the K-algebra A is central, that is, whether K is the center of A.
source
Generators ​
gens Method
julia
gens(A::AbstractAssociativeAlgebra; thorough_search::Bool = false) -> Vector
Given a K-algebra A, return a subset of basis(A), which generates A as an algebra over K.
If thorough_search is true, the number of returned generators is possibly smaller. This will in general increase the runtime. It is not guaranteed that the number of generators is minimal in any case.
The gens_with_data function computes additional data for expressing a basis as words in the generators.
Examples
julia
julia> A = matrix_algebra(QQ, 3);
+false
source
is_central Method
julia
is_central(A::AbstractAssociativeAlgebra)
Return whether the K-algebra A is central, that is, whether K is the center of A.
source
Generators ​
gens Method
julia
gens(A::AbstractAssociativeAlgebra; thorough_search::Bool = false) -> Vector
Given a K-algebra A, return a subset of basis(A), which generates A as an algebra over K.
If thorough_search is true, the number of returned generators is possibly smaller. This will in general increase the runtime. It is not guaranteed that the number of generators is minimal in any case.
The gens_with_data function computes additional data for expressing a basis as words in the generators.
Examples
julia
julia> A = matrix_algebra(QQ, 3);
 
 julia> gens(A; thorough_search = true)
 5-element Vector{MatAlgebraElem{QQFieldElem, QQMatrix}}:
@@ -48,8 +48,8 @@
 1
source
dimension_over_center Method
julia
dimension_over_center(A::AbstractAssociativeAlgebra) -> Int
Given a simple K-algebra with center C, return the C-dimension A.
Examples
julia
julia> A = matrix_algebra(QQ, 2);
 
 julia> dimension_of_center(A)
-1
source

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+1
source

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diff --git a/v0.34.8/manual/algebras/groupalgebras.html b/v0.34.8/manual/algebras/groupalgebras.html
index 7096e008a4..fb1e55de21 100644
--- a/v0.34.8/manual/algebras/groupalgebras.html
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     Group algebras | Hecke
     
     
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Group algebras ​
As is natural, the basis of a group algebra K[G] correspond to the elements of G with respect to some arbitrary ordering.
Creation ​
group_algebra Method
julia
group_algebra(K::Ring, G::Group; cached::Bool = true) -> GroupAlgebra
Return the group algebra of the group G over the ring R. Shorthand syntax for this construction is R[G].
Examples
julia
julia> QG = group_algebra(QQ, small_group(8, 5))
+    
Hecke
Appearance
Group algebras ​
As is natural, the basis of a group algebra K[G] correspond to the elements of G with respect to some arbitrary ordering.
Creation ​
group_algebra Method
julia
group_algebra(K::Ring, G::Group; cached::Bool = true) -> GroupAlgebra
Return the group algebra of the group G over the ring R. Shorthand syntax for this construction is R[G].
Examples
julia
julia> QG = group_algebra(QQ, small_group(8, 5))
 Group algebra
   of generic group of order 8 with multiplication table
   over rational field
source
Elements ​
Given a group algebra A and an element of a group g, the corresponding group algebra element can be constructed using the syntax A(g).
julia
julia> G = abelian_group([2, 2]); a = G([0, 1]);
@@ -31,8 +31,8 @@
 true
 
 julia> QG(Dict(a => 2, zero(G) => 1)) == 2 * QG(a) + 1 * QG(zero(G))
-true

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diff --git a/v0.34.8/manual/algebras/intro.html b/v0.34.8/manual/algebras/intro.html
index 89775bb8d0..3b128884ff 100644
--- a/v0.34.8/manual/algebras/intro.html
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@@ -6,20 +6,20 @@
     Introduction | Hecke
     
     
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Introduction ​
Note
The functions described in this section are experimental. While the overall functionality provided will stay the same, names of specific functions or conventions for the return values might change in future versions.
This section describes the functionality for finite-dimensional associative algebras (or just algebras for short). Since different applications have different requirements, the following types of algebras are implemented:
  • structure constant algebras,
  • matrix algebras,
  • group algebras,
  • quaternion algebras.
These share a common interface encompassing a wide range of functions, which is indicated by the use of the type AbstractAssociativeAlgebra in the signature.

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+    
Hecke
Appearance
Introduction ​
Note
The functions described in this section are experimental. While the overall functionality provided will stay the same, names of specific functions or conventions for the return values might change in future versions.
This section describes the functionality for finite-dimensional associative algebras (or just algebras for short). Since different applications have different requirements, the following types of algebras are implemented:
  • structure constant algebras,
  • matrix algebras,
  • group algebras,
  • quaternion algebras.
These share a common interface encompassing a wide range of functions, which is indicated by the use of the type AbstractAssociativeAlgebra in the signature.

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diff --git a/v0.34.8/manual/algebras/structureconstant.html b/v0.34.8/manual/algebras/structureconstant.html
index 9d42a13715..2d5607d013 100644
--- a/v0.34.8/manual/algebras/structureconstant.html
+++ b/v0.34.8/manual/algebras/structureconstant.html
@@ -6,24 +6,24 @@
     Structure constant algebras | Hecke
     
     
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Structure constant algebras ​
Creation ​
structure_constant_algebra Method
julia
structure_constant_algebra(R::Ring, sctable::Array{_, 3}; one::Vector = nothing,
+    
Hecke
Appearance
Structure constant algebras ​
Creation ​
structure_constant_algebra Method
julia
structure_constant_algebra(R::Ring, sctable::Array{_, 3}; one::Vector = nothing,
                                                           check::Bool = true)
Given an array with dimensions (d,d,d) and a ring R, return the d-dimensional structure constant algebra over R. The basis e of R satisfies e[i] * e[j] = sum(sctable[i,j,k] * e[k] for k in 1:d).
Unless check = false, this includes (time consuming) associativity and distributivity checks. If one is given, record the element with the supplied coordinate vector as the one element of the algebra.
Examples
julia
julia> associative_algebra(QQ, reshape([1, 0, 0, 2, 0, 1, 1, 0], (2, 2, 2)))
 Structure constant algebra of dimension 2 over QQ
source
structure_constant_algebra Method
julia
structure_constant_algebra(K::SimpleNumField) -> StructureConstantAlgebra, Map
Given a number field L/K, return L as a K-algebra A together with a K-linear map A→L.
Examples
julia
julia> L, = quadratic_field(2);
 
 julia> structure_constant_algebra(L)
-(Structure constant algebra of dimension 2 over QQ, Map: structure constant algebra -> real quadratic field)
source
Structure constant table ​
structure_constant_table Method
julia
structure_constant_table(A::StructureConstantAlgebra; copy::Bool = true) -> Array{_, 3}
Given an algebra A, return the structure constant table of A. See structure_constant_algebra for the defining property.
Examples
julia
julia> A = associative_algebra(QQ, reshape([1, 0, 0, 2, 0, 1, 1, 0], (2, 2, 2)));
+(Structure constant algebra of dimension 2 over QQ, Map: structure constant algebra -> real quadratic field)
source
Structure constant table ​
structure_constant_table Method
julia
structure_constant_table(A::StructureConstantAlgebra; copy::Bool = true) -> Array{_, 3}
Given an algebra A, return the structure constant table of A. See structure_constant_algebra for the defining property.
Examples
julia
julia> A = associative_algebra(QQ, reshape([1, 0, 0, 2, 0, 1, 1, 0], (2, 2, 2)));
 
 julia> structure_constant_table(A)
 2×2×2 Array{QQFieldElem, 3}:
@@ -33,8 +33,8 @@
 
 [:, :, 2] =
  0  1
- 1  0
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diff --git a/v0.34.8/manual/developer/documentation.html b/v0.34.8/manual/developer/documentation.html
index 8f8a004c1b..386213f090 100644
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     Documentation | Hecke
     
     
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Documentation ​
The files for the documentation are located in the docs/src/manual/ directory.
Adding files to the documentation ​
To add files to the documentation edit directly the file docs/src/.vitepress/config.mts.
Building the documentation ​
  1. Run julia and execute (with Hecke developed in your current environment)
julia
julia> using Hecke
+    
Hecke
Appearance
Documentation ​
The files for the documentation are located in the docs/src/manual/ directory.
Adding files to the documentation ​
To add files to the documentation edit directly the file docs/src/.vitepress/config.mts.
Building the documentation ​
  1. Run julia and execute (with Hecke developed in your current environment)
julia
julia> using Hecke
 
-julia> Hecke.build_doc() # or Hecke.build_doc(;doctest = false) to speed things up
  1. In the terminal, navigate to docs/ and run
bash
Hecke/docs> npm run docs:build
(This step takes place outside of julia.)
Note
To speed up the development process, step 1 can be repeated within the same julia session.

-    
+julia> Hecke.build_doc() # or Hecke.build_doc(;doctest = false) to speed things up
  1. In the terminal, navigate to docs/ and run
bash
Hecke/docs> npm run docs:build
(This step takes place outside of julia.)
Note
To speed up the development process, step 1 can be repeated within the same julia session.

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     Testing | Hecke
     
     
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Testing ​
Structure ​
The Hecke tests can be found in Hecke/test/ and are organized in such a way that the file hierarchy mirrors the source directory Hecke/src/. For example, here is a subset of the src/QuadForm and the test/QuadForm directories:
├── src
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Hecke
Appearance
Testing ​
Structure ​
The Hecke tests can be found in Hecke/test/ and are organized in such a way that the file hierarchy mirrors the source directory Hecke/src/. For example, here is a subset of the src/QuadForm and the test/QuadForm directories:
├── src
 │   ├── QuadForm
 │   │   ├── Enumeration.jl
 │   │   ├── Herm
@@ -51,8 +51,8 @@
 │   │   │   └── ZLattices.jl
 │   │   ├── QuadBin.jl
 │   │   └── Torsion.jl
-│   ├── QuadForm.jl
Adding tests ​
  • If one adds functionality to a file, say src/QuadForm/Quad/Genus.jl, a corresponding a test should be added to the corresponding test file. In this case this would be test/QuadForm/Quad/Genus.jl.
  • Assume one adds a new file, say src/QuadForm/New.jl, which is included in src/QuadForm.jl. Then a corresponding file test/QuadForm/Test.jl containing the tests must be added. This new file must then also be included in test/QuadForm.jl.
  • Similar to the above, if a new directory in src/ is added, the same must apply in test/.
Adding long tests ​
If one knows that running a particular test will take a long time, one can use @long_test instead of @test inside the test suite. When running the test suite, tests annotated with @long_test will not be run, unless specifically asked for (see below). The continuous integration servers will run at least one job including the long tests.
Running the tests ​
Running all tests ​
All tests can be run as usual with Pkg.test("Hecke"). The whole test suite can be run in parallel using the following options:
  • Set the environment variable HECKE_TEST_VARIABLE=n, where n is the number of processes.
  • On julia >= 1.3, run Pkg.test("Hecke", test_args = ["-j$(n)"]), where n is the number of processes.
The tests annotated with @long_test can be invoked by setting HECKE_TESTLONG=1 or adding "long" to the test_args keyword argument on julia >= 1.3.
Running a subset of tests ​
Because the test structure mirrors the source directory, it is easy to run only a subset of tests. For example, to run all the tests in test/QuadForm/Quad/Genus.jl, one can invoke:
julia
julia> Hecke.test_module("QuadForm/Quad/Genus")
This also works on the directory level. If one wants to add run all tests for quadratic forms, one can just run
julia
julia> Hecke.test_module("QuadForm")

-    
+│   ├── QuadForm.jl
Adding tests ​
  • If one adds functionality to a file, say src/QuadForm/Quad/Genus.jl, a corresponding a test should be added to the corresponding test file. In this case this would be test/QuadForm/Quad/Genus.jl.
  • Assume one adds a new file, say src/QuadForm/New.jl, which is included in src/QuadForm.jl. Then a corresponding file test/QuadForm/Test.jl containing the tests must be added. This new file must then also be included in test/QuadForm.jl.
  • Similar to the above, if a new directory in src/ is added, the same must apply in test/.
Adding long tests ​
If one knows that running a particular test will take a long time, one can use @long_test instead of @test inside the test suite. When running the test suite, tests annotated with @long_test will not be run, unless specifically asked for (see below). The continuous integration servers will run at least one job including the long tests.
Running the tests ​
Running all tests ​
All tests can be run as usual with Pkg.test("Hecke"). The whole test suite can be run in parallel using the following options:
  • Set the environment variable HECKE_TEST_VARIABLE=n, where n is the number of processes.
  • On julia >= 1.3, run Pkg.test("Hecke", test_args = ["-j$(n)"]), where n is the number of processes.
The tests annotated with @long_test can be invoked by setting HECKE_TESTLONG=1 or adding "long" to the test_args keyword argument on julia >= 1.3.
Running a subset of tests ​
Because the test structure mirrors the source directory, it is easy to run only a subset of tests. For example, to run all the tests in test/QuadForm/Quad/Genus.jl, one can invoke:
julia
julia> Hecke.test_module("QuadForm/Quad/Genus")
This also works on the directory level. If one wants to add run all tests for quadratic forms, one can just run
julia
julia> Hecke.test_module("QuadForm")

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Basics ​
Creation ​
elliptic_curve Function
julia
elliptic_curve([K::Field], x::Vector; check::Bool = true) -> EllipticCurve
Construct an elliptic curve with Weierstrass equation specified by the coefficients in x, which must have either length 2 or 5.
Per default, it is checked whether the discriminant is non-zero. This can be disabled by setting check = false.
Examples
julia
julia> elliptic_curve(QQ, [1, 2, 3, 4, 5])
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Basics ​
Creation ​
elliptic_curve Function
julia
elliptic_curve([K::Field], x::Vector; check::Bool = true) -> EllipticCurve
Construct an elliptic curve with Weierstrass equation specified by the coefficients in x, which must have either length 2 or 5.
Per default, it is checked whether the discriminant is non-zero. This can be disabled by setting check = false.
Examples
julia
julia> elliptic_curve(QQ, [1, 2, 3, 4, 5])
 Elliptic curve with equation
 y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5
 
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  Point  (0 : 1 : 0)  of Elliptic curve with equation
 y^2 = x^3 + x + 2
  Point  (-1 : 0 : 1)  of Elliptic curve with equation
-y^2 = x^3 + x + 2
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Elliptic curves over finite fields ​
Random points ​
  rand(E::EllipticCurve{<: FinFieldElem})
Return a random point on the elliptic curve E defined over a finite field.
julia
julia> E = elliptic_curve(GF(3), [1, 2]);
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Elliptic curves over finite fields ​
Random points ​
  rand(E::EllipticCurve{<: FinFieldElem})
Return a random point on the elliptic curve E defined over a finite field.
julia
julia> E = elliptic_curve(GF(3), [1, 2]);
 
 julia> rand(E)
 Point  (2 : 0 : 1)  of Elliptic curve with equation
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 y^2 = x^3 + x + 2
 
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     Introduction | Hecke
     
     
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Introduction ​
This chapter deals with functionality for elliptic curves, which is available over arbitrary fields, with specific features available for curvers over the rationals and number fields, and finite fields.
An elliptic curve E is the projective closure of the curve given by the Weierstrass equation
y2+a1xy+a3y=x3+a2x2+a4x+a6
specified by the list of coefficients [a1, a2, a3, a4, a6]. If a1=a2=a3=0, this simplifies to
y2=x3+a4x+a6
which we refer to as a short Weierstrass equation and which is specified by the two element list [a4, a6].

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Introduction ​
This chapter deals with functionality for elliptic curves, which is available over arbitrary fields, with specific features available for curvers over the rationals and number fields, and finite fields.
An elliptic curve E is the projective closure of the curve given by the Weierstrass equation
y2+a1xy+a3y=x3+a2x2+a4x+a6
specified by the list of coefficients [a1, a2, a3, a4, a6]. If a1=a2=a3=0, this simplifies to
y2=x3+a4x+a6
which we refer to as a short Weierstrass equation and which is specified by the two element list [a4, a6].

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     Manual | Hecke
     
     
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     Factored Elements | Hecke
     
     
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Factored Elements ​
In many applications in number theory related to the multiplicative structure of number fields, interesting elements, e.g. units, are extremely large when written wrt. to a fxied basis for the field: for the fundamental unit in Q[d] it is known that the coefficients wrt. the canonical basis 1,d can have O(exp⁡d) many digits. All currently known, fast methods construct those elements as power products of smaller elements, allowing the computer to handle them.
Mathematically, one can think of factored elements to formally live in the ring Z[K] the group ring of the non-zero field elements. Thus elements are of the form $ \prod a_i^{e_i}$ where ai are elements in K, typically small and the ei∈Z are frequently large exponents. We refer to the ai as the base and the ei as the exponents of the factored element.
Since K is, in general, no PID, this presentation is non-unique, elements in this form can easily be multiplied, raised to large powers, but in general not compared and not added.
In Hecke, this is caputured more generally by the type FacElem, parametrized by the type of the elements in the base and the type of their parent.
Important special cases are
  • FacElem{ZZRingElem, ZZRing}, factored integers
  • FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, factored algerbaic numbers
  • FacElem{AbsNumFieldOrderIdeal, AbsNumFieldOrderIdealSet}, factored ideals
It should be noted that an object of type `FacElemZZRingElem,ZZRing will, in general, not represent an integer as the exponents can be negative.
Construction ​
In general one can define factored elements by giving 2 arrays, the base and the exponent, or a dictionary containing the pairs:
FacElem Type
julia
FacElem{B, S}
Type for factored elements, that is elements of the form prod a_i^k_i for elements a_i of type B in a ring of type S.
source
FacElem Method
julia
FacElem{B}(R, base::Vector{B}, exp::Vector{ZZRingElem}) -> FacElem{B}
Returns the element ∏biei, un-expanded.
source
julia
FacElem{B}(base::Vector{B}, exp::Vector{ZZRingElem}) -> FacElem{B}
Returns the element ∏biei, un-expanded.
source
julia
FacElem{B}(R, d::Dict{B, ZZRingElem}) -> FacElem{B}
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Appearance
Factored Elements ​
In many applications in number theory related to the multiplicative structure of number fields, interesting elements, e.g. units, are extremely large when written wrt. to a fxied basis for the field: for the fundamental unit in Q[d] it is known that the coefficients wrt. the canonical basis 1,d can have O(exp⁡d) many digits. All currently known, fast methods construct those elements as power products of smaller elements, allowing the computer to handle them.
Mathematically, one can think of factored elements to formally live in the ring Z[K] the group ring of the non-zero field elements. Thus elements are of the form $ \prod a_i^{e_i}$ where ai are elements in K, typically small and the ei∈Z are frequently large exponents. We refer to the ai as the base and the ei as the exponents of the factored element.
Since K is, in general, no PID, this presentation is non-unique, elements in this form can easily be multiplied, raised to large powers, but in general not compared and not added.
In Hecke, this is caputured more generally by the type FacElem, parametrized by the type of the elements in the base and the type of their parent.
Important special cases are
  • FacElem{ZZRingElem, ZZRing}, factored integers
  • FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, factored algerbaic numbers
  • FacElem{AbsNumFieldOrderIdeal, AbsNumFieldOrderIdealSet}, factored ideals
It should be noted that an object of type `FacElemZZRingElem,ZZRing will, in general, not represent an integer as the exponents can be negative.
Construction ​
In general one can define factored elements by giving 2 arrays, the base and the exponent, or a dictionary containing the pairs:
FacElem Type
julia
FacElem{B, S}
Type for factored elements, that is elements of the form prod a_i^k_i for elements a_i of type B in a ring of type S.
source
FacElem Method
julia
FacElem{B}(R, base::Vector{B}, exp::Vector{ZZRingElem}) -> FacElem{B}
Returns the element ∏biei, un-expanded.
source
julia
FacElem{B}(base::Vector{B}, exp::Vector{ZZRingElem}) -> FacElem{B}
Returns the element ∏biei, un-expanded.
source
julia
FacElem{B}(R, d::Dict{B, ZZRingElem}) -> FacElem{B}
 FacElem{B}(R, d::Dict{B, Integer}) -> FacElem{B}
Returns the element ∏bd[p], un-expanded.
source
julia
FacElem{B}(d::Dict{B, ZZRingElem}) -> FacElem{B}
 FacElem{B}(d::Dict{B, Integer}) -> FacElem{B}
Returns the element ∏bd[p], un-expanded.
source
ideal Method
julia
 ideal(O::AbsSimpleNumFieldOrder, a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField)
The factored fractional ideal a∗O.
source
Conversion ​
The process of computing the value defined by a factored element is available as evaluate. Depending on the types involved this can be very efficient.
evaluate Method
julia
evaluate{T}(x::FacElem{T}) -> T
Expands or evaluates the factored element, i.e. actually computes the value. Does "square-and-multiply" on the exponent vectors.
source
evaluate Method
julia
evaluate(x::FacElem{QQFieldElem}) -> QQFieldElem
 evaluate(x::FacElem{ZZRingElem}) -> ZZRingElem
Expands or evaluates the factored element, i.e. actually computes the the element. Works by first obtaining a simplified version of the power product into coprime base elements.
source
evaluate Method
julia
evaluate{T}(x::FacElem{T}) -> T
Expands or evaluates the factored element, i.e. actually computes the value. Does "square-and-multiply" on the exponent vectors.
source
evaluate_naive Method
julia
evaluate_naive{T}(x::FacElem{T}) -> T
Expands or evaluates the factored element, i.e. actually computes the value. Uses the obvious naive algorithm. Faster for input in finite rings.
source
Special functions ​
In the case where the parent of the base allows for efficient gcd computation, power products can be made unique:
simplify Method
julia
simplify(x::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> FacElem
 simplify(x::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> FacElem
Uses coprime_base to obtain a simplified version of x, ie. in the simplified version all base ideals will be pariwise coprime but not necessarily prime!.
source
simplify Method
julia
simplify(x::FacElem{QQFieldElem}) -> FacElem{QQFieldElem}
 simplify(x::FacElem{ZZRingElem}) -> FacElem{ZZRingElem}
Simplfies the factored element, i.e. arranges for the base to be coprime.
source
The simplified version can then be used further:
isone Method
julia
isone(x::FacElem{QQFieldElem}) -> Bool
 isone(x::FacElem{ZZRingElem}) -> Bool
Tests if x represents 1 without an evaluation.
source
factor_coprime Method
julia
factor_coprime(x::FacElem{ZZRingElem}) -> Fac{ZZRingElem}
Computed a partial factorisation of x, ie. writes x as a product of pariwise coprime integers.
source
factor_coprime Method
julia
factor_coprime(x::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}
Computed a partial factorisation of x, ie. writes x as a product of pariwise coprime integral ideals.
source
factor_coprime Method
julia
factor_coprime(Q::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}
A coprime factorisation of Q: each ideal in Q is split using \code{integral_split} and then a coprime basis is computed. This does {\bf not} use any factorisation.
source
factor_coprime Method
julia
factor_coprime(I::AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}, a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}
Factors the rincipal ideal generated by a into coprimes by computing a coprime basis from the principal ideals in the factorisation of a.
source
factor Method
julia
 factor(Q::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}
The factorisation of Q, by refining a coprime factorisation.
source
factor Method
julia
factor(I::AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}, a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}) -> Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, ZZRingElem}
Factors the principal ideal generated by a by refining a coprime factorisation.
source
For factorised algebraic numbers a unique simplification is not possible, however, this allows still do obtain partial results:
compact_presentation Function
julia
compact_presentation(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, n::Int = 2; decom, arb_prec = 100, short_prec = 1000) -> FacElem
Computes a presentation a=∏aini where all the exponents ni are powers of n and, the elements ai are "small", generically, they have a norm bounded by dn/2 where d is the discriminant of the maximal order. As the algorithm needs the factorisation of the principal ideal generated by a, it can be passed in in \code{decom}.
source
valuation Method
julia
valuation(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
The valuation of a at P.
source
valuation Method
julia
valuation(A::FacElem{AbsSimpleNumFieldOrderFractionalIdeal, AbsNumFieldOrderFractionalIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem})
-valuation(A::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, AbsNumFieldOrderIdealSet{AbsSimpleNumField, AbsSimpleNumFieldElem}}, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem})
The valuation of A at P.
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evaluate_mod Method
julia
evaluate_mod(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, B::AbsSimpleNumFieldOrderFractionalIdeal) -> AbsSimpleNumFieldElem
Evaluates a using CRT and small primes. Assumes that the ideal generated by a is in fact B. Useful in cases where a has huge exponents, but the evaluated element is actually "small".
source
reduce_ideal Method
julia
reduce_ideal(A::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, FacElem{AbsSimpleNumFieldElem}
Computes B and α in factored form, such that αB=A.
source
modular_proj Method
julia
modular_proj(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, me::modular_env) -> Vector{fqPolyRepFieldElem}
Given an algebraic number a in factored form and data \code{me} as computed by \code{modular_init}, project a onto the residue class fields.
source
Positivity & Signs ​
Factored elements can be used instead of number field elements for the functions sign, signs, is_positive, is_negative and is_totally_positive, see Positivity & Signs
Miscellaneous ​
max_exp Method
julia
max_exp(a::FacElem)
Finds the largest exponent in the factored element a.
source
min_exp Method
julia
min_exp(a::FacElem)
Finds the smallest exponent in the factored element a.
source
maxabs_exp Method
julia
maxabs_exp(a::FacElem)
Finds the largest exponent by absolute value in the factored element a.
source

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The valuation of A at P.
source
evaluate_mod Method
julia
evaluate_mod(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, B::AbsSimpleNumFieldOrderFractionalIdeal) -> AbsSimpleNumFieldElem
Evaluates a using CRT and small primes. Assumes that the ideal generated by a is in fact B. Useful in cases where a has huge exponents, but the evaluated element is actually "small".
source
reduce_ideal Method
julia
reduce_ideal(A::FacElem{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, FacElem{AbsSimpleNumFieldElem}
Computes B and α in factored form, such that αB=A.
source
modular_proj Method
julia
modular_proj(a::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, me::modular_env) -> Vector{fqPolyRepFieldElem}
Given an algebraic number a in factored form and data \code{me} as computed by \code{modular_init}, project a onto the residue class fields.
source
Positivity & Signs ​
Factored elements can be used instead of number field elements for the functions sign, signs, is_positive, is_negative and is_totally_positive, see Positivity & Signs
Miscellaneous ​
max_exp Method
julia
max_exp(a::FacElem)
Finds the largest exponent in the factored element a.
source
min_exp Method
julia
min_exp(a::FacElem)
Finds the smallest exponent in the factored element a.
source
maxabs_exp Method
julia
maxabs_exp(a::FacElem)
Finds the largest exponent by absolute value in the factored element a.
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diff --git a/v0.34.8/manual/misc/conjugacy.html b/v0.34.8/manual/misc/conjugacy.html
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Conjugacy of integral matrices ​
is_GLZ_conjugate Method
julia
is_GLZ_conjugate(A::MatElem, B::MatElem) -> Bool, MatElem
Given two integral or rational matrices, determine whether there exists an invertible integral matrix T with TA=BT. If true, the second argument is such a matrix T. Otherwise, the second argument is unspecified.
julia
julia> A = matrix(ZZ, 4, 4, [ 0, 1,  0, 0,
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Conjugacy of integral matrices ​
is_GLZ_conjugate Method
julia
is_GLZ_conjugate(A::MatElem, B::MatElem) -> Bool, MatElem
Given two integral or rational matrices, determine whether there exists an invertible integral matrix T with TA=BT. If true, the second argument is such a matrix T. Otherwise, the second argument is unspecified.
julia
julia> A = matrix(ZZ, 4, 4, [ 0, 1,  0, 0,
                              -4, 0,  0, 0,
                               0, 0,  0, 1,
                               0, 0, -4, 0]);
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 julia> fl, T = is_GLZ_conjugate(A, B);
 
 julia> isone(abs(det(T))) && T * A == B * T
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Multi-sets and sub-set iterators ​
Multi-sets ​
Type and constructors ​
Objects of type MSet consists of a dictionary whose keys are the elements in the set, and the values are their respective multiplicity.
MSet Type
julia
MSet{T} <: AbstractSet{T}
Type for a multi-set, i.e. a set where elements are not unique, they (can) have a multiplicity. MSets can be created from any finite iterator.
Examples
julia
julia> MSet([1,1,2,3,4,4,5])
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Multi-sets and sub-set iterators ​
Multi-sets ​
Type and constructors ​
Objects of type MSet consists of a dictionary whose keys are the elements in the set, and the values are their respective multiplicity.
MSet Type
julia
MSet{T} <: AbstractSet{T}
Type for a multi-set, i.e. a set where elements are not unique, they (can) have a multiplicity. MSets can be created from any finite iterator.
Examples
julia
julia> MSet([1,1,2,3,4,4,5])
 MSet{Int64} with 7 elements:
   5
   4 : 2
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   'y'
   'r'
   ' '
-  'v'
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Sub-set iterators ​
Sub-multi-sets ​
subsets Method
julia
subsets(s::MSet) -> MSubSetIt{T}
Return an iterator on all sub-multi-sets of s.
source
julia
subsets(s::Set) -> SubSetItr{T}
Return an iterator for all sub-sets of s.
source
julia
subsets(s::Set, k::Int) -> SubSetSizeItr{T}
Return an iterator on all sub-sets of size k of s.
source
julia
subsets(v::Vector{T}, k::Int) where T
Return a vector of all ordered k-element sub-vectors of v.
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Sub-sets ​
subsets Method
julia
subsets(s::MSet) -> MSubSetIt{T}
Return an iterator on all sub-multi-sets of s.
source
julia
subsets(s::Set) -> SubSetItr{T}
Return an iterator for all sub-sets of s.
source
julia
subsets(s::Set, k::Int) -> SubSetSizeItr{T}
Return an iterator on all sub-sets of size k of s.
source
julia
subsets(v::Vector{T}, k::Int) where T
Return a vector of all ordered k-element sub-vectors of v.
source
Sub-sets of a given size ​
subsets Method
julia
subsets(s::Set, k::Int) -> SubSetSizeItr{T}
Return an iterator on all sub-sets of size k of s.
source

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Sub-set iterators ​
Sub-multi-sets ​
subsets Method
julia
subsets(s::MSet) -> MSubSetIt{T}
Return an iterator on all sub-multi-sets of s.
source
julia
subsets(s::Set) -> SubSetItr{T}
Return an iterator for all sub-sets of s.
source
julia
subsets(s::Set, k::Int) -> SubSetSizeItr{T}
Return an iterator on all sub-sets of size k of s.
source
julia
subsets(v::Vector{T}, k::Int) where T
Return a vector of all ordered k-element sub-vectors of v.
source
Sub-sets ​
subsets Method
julia
subsets(s::MSet) -> MSubSetIt{T}
Return an iterator on all sub-multi-sets of s.
source
julia
subsets(s::Set) -> SubSetItr{T}
Return an iterator for all sub-sets of s.
source
julia
subsets(s::Set, k::Int) -> SubSetSizeItr{T}
Return an iterator on all sub-sets of size k of s.
source
julia
subsets(v::Vector{T}, k::Int) where T
Return a vector of all ordered k-element sub-vectors of v.
source
Sub-sets of a given size ​
subsets Method
julia
subsets(s::Set, k::Int) -> SubSetSizeItr{T}
Return an iterator on all sub-sets of size k of s.
source

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     Pseudo-matrices | Hecke
     
     
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Pseudo-matrices ​
This chapter deals with pseudo-matrices. We follow the common terminology and conventions introduced in [1], however, we operate on rows, not on columns.
Let R be a Dedekind domain, typically, the maximal order of some number field K, further fix some finite dimensional K-vectorspace V (with some basis), frequently Kn or the K-structure of some extension of K. Since in general R is not a PID, the R-modules in V are usually not free, but still projective.
Any finitely generated R-module M⊂V can be represented as a pseudo-matrix PMat as follows: The structure theory of R-modules gives the existence of (fractional) R-ideals Ai and elements ωi∈V such that M=∑Aiωi and the sum is direct.
Following Cohen we call modules of the form Aω for some ideal A and ω∈V a pseudo element. A system (Ai,ωi) is called a pseudo-generating system for M if ⟨Aiωi|i⟨=M. A pseudo-generating system is called a pseudo-basis if the ωi are K-linear independent.
A pseudo-matrix X is a tuple containing a vector of ideals Ai (1≤i≤r) and a matrix U∈Kr×n. The i-th row together with the i-th ideal defines a pseudo-element, thus an R-module, all of them together generate a module M.
A pseudo-matrix X=((Ai)i,U) is said to be in pseudo-hnf if U is essentially upper triangular. Similar to the classical hnf, there is an algorithm that transforms any pseudo-matrix into one in pseudo-hnf while maintaining the module.
Creation ​
In general to create a PMat one has to specify a matrix and a vector of ideals:
pseudo_matrix Method
julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
Returns the (row) pseudo matrix representing the Zk-module ∑cimi where ci are the ideals in c and mi the rows of M.
source
pseudo_matrix Method
julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
Returns the (row) pseudo matrix representing the Zk-module ∑cimi where ci are the ideals in c and mi the rows of M.
source
pseudo_matrix Method
julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
Returns the free (row) pseudo matrix representing the Zk-module ∑Zkmi where mi are the rows of M.
source
(Those functions are also available as pseudo_matrix)
Operations ​
coefficient_ideals Method
julia
coefficient_ideals(M::PMat)
Returns the vector of coefficient ideals.
source
matrix Method
julia
matrix(M::PMat)
Returns the matrix part of the PMat.
source
base_ring Method
julia
base_ring(M::PMat)
The PMat M defines an R-module for some maximal order R. This function returns the R that was used to defined M.
source
pseudo_hnf Method
julia
pseudo_hnf(P::PMat)
Transforms P into pseudo-Hermite form as defined by Cohen. Essentially the matrix part of P will be upper triangular with some technical normalisation for the off-diagonal elements. This operation preserves the module.
A optional second argument can be specified as a symbols, indicating the desired shape of the echelon form. Possible are :upperright (the default) and :lowerleft
source
pseudo_hnf_with_transform Method
julia
pseudo_hnf_with_transform(P::PMat)
Transforms P into pseudo-Hermite form as defined by Cohen. Essentially the matrix part of P will be upper triangular with some technical normalisation for the off-diagonal elements. This operation preserves the module. The used transformation is returned as a second return value.
A optional second argument can be specified as a symbol, indicating the desired shape of the echelon form. Possible are :upperright (the default) and :lowerleft
source
Examples ​

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Hecke
Appearance
Pseudo-matrices ​
This chapter deals with pseudo-matrices. We follow the common terminology and conventions introduced in [1], however, we operate on rows, not on columns.
Let R be a Dedekind domain, typically, the maximal order of some number field K, further fix some finite dimensional K-vectorspace V (with some basis), frequently Kn or the K-structure of some extension of K. Since in general R is not a PID, the R-modules in V are usually not free, but still projective.
Any finitely generated R-module M⊂V can be represented as a pseudo-matrix PMat as follows: The structure theory of R-modules gives the existence of (fractional) R-ideals Ai and elements ωi∈V such that M=∑Aiωi and the sum is direct.
Following Cohen we call modules of the form Aω for some ideal A and ω∈V a pseudo element. A system (Ai,ωi) is called a pseudo-generating system for M if ⟨Aiωi|i⟨=M. A pseudo-generating system is called a pseudo-basis if the ωi are K-linear independent.
A pseudo-matrix X is a tuple containing a vector of ideals Ai (1≤i≤r) and a matrix U∈Kr×n. The i-th row together with the i-th ideal defines a pseudo-element, thus an R-module, all of them together generate a module M.
A pseudo-matrix X=((Ai)i,U) is said to be in pseudo-hnf if U is essentially upper triangular. Similar to the classical hnf, there is an algorithm that transforms any pseudo-matrix into one in pseudo-hnf while maintaining the module.
Creation ​
In general to create a PMat one has to specify a matrix and a vector of ideals:
pseudo_matrix Method
julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
Returns the (row) pseudo matrix representing the Zk-module ∑cimi where ci are the ideals in c and mi the rows of M.
source
pseudo_matrix Method
julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}, c::Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
Returns the (row) pseudo matrix representing the Zk-module ∑cimi where ci are the ideals in c and mi the rows of M.
source
pseudo_matrix Method
julia
pseudo_matrix(m::Generic.Mat{AbsSimpleNumFieldOrderElem}) -> PMat{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal}
Returns the free (row) pseudo matrix representing the Zk-module ∑Zkmi where mi are the rows of M.
source
(Those functions are also available as pseudo_matrix)
Operations ​
coefficient_ideals Method
julia
coefficient_ideals(M::PMat)
Returns the vector of coefficient ideals.
source
matrix Method
julia
matrix(M::PMat)
Returns the matrix part of the PMat.
source
base_ring Method
julia
base_ring(M::PMat)
The PMat M defines an R-module for some maximal order R. This function returns the R that was used to defined M.
source
pseudo_hnf Method
julia
pseudo_hnf(P::PMat)
Transforms P into pseudo-Hermite form as defined by Cohen. Essentially the matrix part of P will be upper triangular with some technical normalisation for the off-diagonal elements. This operation preserves the module.
A optional second argument can be specified as a symbols, indicating the desired shape of the echelon form. Possible are :upperright (the default) and :lowerleft
source
pseudo_hnf_with_transform Method
julia
pseudo_hnf_with_transform(P::PMat)
Transforms P into pseudo-Hermite form as defined by Cohen. Essentially the matrix part of P will be upper triangular with some technical normalisation for the off-diagonal elements. This operation preserves the module. The used transformation is returned as a second return value.
A optional second argument can be specified as a symbol, indicating the desired shape of the echelon form. Possible are :upperright (the default) and :lowerleft
source
Examples ​

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     Sparse linear algebra | Hecke
     
     
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Sparse linear algebra ​
Introduction ​
This chapter deals with sparse linear algebra over commutative rings and fields.
Sparse linear algebra, that is, linear algebra with sparse matrices, plays an important role in various algorithms in algebraic number theory. For example, it is one of the key ingredients in the computation of class groups and discrete logarithms using index calculus methods.
Sparse rows ​
Building blocks for sparse matrices are sparse rows, which are modelled by objects of type SRow. More precisely, the type is of parametrized form SRow{T}, where T is the element type of the base ring R. For example, SRow{ZZRingElem} is the type for sparse rows over the integers.
It is important to note that sparse rows do not have a fixed number of columns, that is, they represent elements of {(xi)i∈RN∣xi=0 for almost all i}. In particular any two sparse rows over the same base ring can be added.
Creation ​
sparse_row Method
julia
sparse_row(R::Ring, J::Vector{Tuple{Int, T}}) -> SRow{T}
Constructs the sparse row (ai)i with aij=xj, where J=(ij,xj)j. The elements xi must belong to the ring R.
source
sparse_row Method
julia
sparse_row(R::Ring, J::Vector{Tuple{Int, Int}}) -> SRow
Constructs the sparse row (ai)i over R with aij=xj, where J=(ij,xj)j.
source
sparse_row Method
julia
sparse_row(R::NCRing, J::Vector{Int}, V::Vector{T}) -> SRow{T}
Constructs the sparse row (ai)i over R with aij=xj, where J=(ij)j and V=(xj)j.
source
Basic operations ​
Rows support the usual operations:
  • +, -, ==
  • multiplication by scalars
  • div, divexact
getindex Method
julia
getindex(A::SRow, j::Int) -> RingElem
Given a sparse row (ai)i and an index j return aj.
source
add_scaled_row Method
julia
add_scaled_row(A::SRow{T}, B::SRow{T}, c::T) -> SRow{T}
Returns the row cA+B.
source
add_scaled_row Method
julia
add_scaled_row(A::SRow{T}, B::SRow{T}, c::T) -> SRow{T}
Returns the row cA+B.
source
transform_row Method
julia
transform_row(A::SRow{T}, B::SRow{T}, i::Int, j::Int, a::T, b::T, c::T, d::T)
Returns the tuple (aA+bB,cA+dB).
source
length Method
julia
length(A::SRow)
Returns the number of nonzero entries of A.
source
Change of base ring ​
change_base_ring Method
julia
change_base_ring(R::Ring, A::SRow) -> SRow
Create a new sparse row by coercing all elements into the ring R.
source
Maximum, minimum and 2-norm ​
maximum Method
julia
maximum(A::SRow{T}) -> T
Returns the largest entry of A.
source
maximum Method
julia
maximum(A::SRow{T}) -> T
Returns the largest entry of A.
source
minimum Method
julia
minimum(A::SRow{T}) -> T
Returns the smallest entry of A.
source
julia
  minimum(A::RelNumFieldOrderIdeal) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
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Hecke
Appearance
Sparse linear algebra ​
Introduction ​
This chapter deals with sparse linear algebra over commutative rings and fields.
Sparse linear algebra, that is, linear algebra with sparse matrices, plays an important role in various algorithms in algebraic number theory. For example, it is one of the key ingredients in the computation of class groups and discrete logarithms using index calculus methods.
Sparse rows ​
Building blocks for sparse matrices are sparse rows, which are modelled by objects of type SRow. More precisely, the type is of parametrized form SRow{T}, where T is the element type of the base ring R. For example, SRow{ZZRingElem} is the type for sparse rows over the integers.
It is important to note that sparse rows do not have a fixed number of columns, that is, they represent elements of {(xi)i∈RN∣xi=0 for almost all i}. In particular any two sparse rows over the same base ring can be added.
Creation ​
sparse_row Method
julia
sparse_row(R::Ring, J::Vector{Tuple{Int, T}}) -> SRow{T}
Constructs the sparse row (ai)i with aij=xj, where J=(ij,xj)j. The elements xi must belong to the ring R.
source
sparse_row Method
julia
sparse_row(R::Ring, J::Vector{Tuple{Int, Int}}) -> SRow
Constructs the sparse row (ai)i over R with aij=xj, where J=(ij,xj)j.
source
sparse_row Method
julia
sparse_row(R::NCRing, J::Vector{Int}, V::Vector{T}) -> SRow{T}
Constructs the sparse row (ai)i over R with aij=xj, where J=(ij)j and V=(xj)j.
source
Basic operations ​
Rows support the usual operations:
  • +, -, ==
  • multiplication by scalars
  • div, divexact
getindex Method
julia
getindex(A::SRow, j::Int) -> RingElem
Given a sparse row (ai)i and an index j return aj.
source
add_scaled_row Method
julia
add_scaled_row(A::SRow{T}, B::SRow{T}, c::T) -> SRow{T}
Returns the row cA+B.
source
add_scaled_row Method
julia
add_scaled_row(A::SRow{T}, B::SRow{T}, c::T) -> SRow{T}
Returns the row cA+B.
source
transform_row Method
julia
transform_row(A::SRow{T}, B::SRow{T}, i::Int, j::Int, a::T, b::T, c::T, d::T)
Returns the tuple (aA+bB,cA+dB).
source
length Method
julia
length(A::SRow)
Returns the number of nonzero entries of A.
source
Change of base ring ​
change_base_ring Method
julia
change_base_ring(R::Ring, A::SRow) -> SRow
Create a new sparse row by coercing all elements into the ring R.
source
Maximum, minimum and 2-norm ​
maximum Method
julia
maximum(A::SRow{T}) -> T
Returns the largest entry of A.
source
maximum Method
julia
maximum(A::SRow{T}) -> T
Returns the largest entry of A.
source
minimum Method
julia
minimum(A::SRow{T}) -> T
Returns the smallest entry of A.
source
julia
  minimum(A::RelNumFieldOrderIdeal) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
   minimum(A::RelNumFieldOrderIdeal) -> RelNumFieldOrderIdeal
Returns the ideal A∩O where O is the maximal order of the coefficient ideals of A.
source
minimum Method
julia
minimum(A::SRow{T}) -> T
Returns the smallest entry of A.
source
norm2 Method
julia
norm2(A::SRow{T} -> T
Returns A⋅At.
source
Functionality for integral sparse rows ​
lift Method
julia
lift(A::SRow{zzModRingElem}) -> SRow{ZZRingElem}
Return the sparse row obtained by lifting all entries in A.
source
mod! Method
julia
mod!(A::SRow{ZZRingElem}, n::ZZRingElem) -> SRow{ZZRingElem}
Inplace reduction of all entries of A modulo n to the positive residue system.
source
mod_sym! Method
julia
mod_sym!(A::SRow{ZZRingElem}, n::ZZRingElem) -> SRow{ZZRingElem}
Inplace reduction of all entries of A modulo n to the symmetric residue system.
source
mod_sym! Method
julia
mod_sym!(A::SRow{ZZRingElem}, n::Integer) -> SRow{ZZRingElem}
Inplace reduction of all entries of A modulo n to the symmetric residue system.
source
maximum Method
julia
maximum(abs, A::SRow{ZZRingElem}) -> ZZRingElem
Returns the largest, in absolute value, entry of A.
source
Conversion to/from julia and AbstractAlgebra types ​
Vector Method
julia
Vector(a::SMat{T}, n::Int) -> Vector{T}
The first n entries of a, as a julia vector.
source
sparse_row Method
julia
sparse_row(A::MatElem)
Convert A to a sparse row. nrows(A) == 1 must hold.
source
dense_row Method
julia
dense_row(r::SRow, n::Int)
Convert r[1:n] to a dense row, that is an AbstractAlgebra matrix.
source
Sparse matrices ​
Let R be a commutative ring. Sparse matrices with base ring R are modelled by objects of type SMat. More precisely, the type is of parametrized form SRow{T}, where T is the element type of the base ring. For example, SMat{ZZRingElem} is the type for sparse matrices over the integers.
In contrast to sparse rows, sparse matrices have a fixed number of rows and columns, that is, they represent elements of the matrices space Matn×m(R). Internally, sparse matrices are implemented as an array of sparse rows. As a consequence, unlike their dense counterparts, sparse matrices have a mutable number of rows and it is very performant to add additional rows.
Construction ​
sparse_matrix Method
julia
sparse_matrix(R::Ring) -> SMat
Return an empty sparse matrix with base ring R.
source
sparse_matrix Method
julia
sparse_matrix(R::Ring, n::Int, m::Int) -> SMat
Return a sparse n times m zero matrix over R.
source
Sparse matrices can also be created from dense matrices as well as from julia arrays:
sparse_matrix Method
julia
sparse_matrix(A::MatElem; keepzrows::Bool = true)
Constructs the sparse matrix corresponding to the dense matrix A. If keepzrows is false, then the constructor will drop any zero row of A.
source
sparse_matrix Method
julia
sparse_matrix(R::Ring, A::Matrix{T}) -> SMat
Constructs the sparse matrix over R corresponding to A.
source
sparse_matrix Method
julia
sparse_matrix(R::Ring, A::Matrix{T}) -> SMat
Constructs the sparse matrix over R corresponding to A.
source
The normal way however, is to add rows:
push! Method
julia
push!(A::SMat{T}, B::SRow{T}) where T
Appends the sparse row B to A.
source
Sparse matrices can also be concatenated to form larger ones:
vcat! Method
julia
vcat!(A::SMat, B::SMat) -> SMat
Vertically joins A and B inplace, that is, the rows of B are appended to A.
source
vcat Method
julia
vcat(A::SMat, B::SMat) -> SMat
Vertically joins A and B.
source
hcat! Method
julia
hcat!(A::SMat, B::SMat) -> SMat
Horizontally concatenates A and B, inplace, changing A.
source
hcat Method
julia
hcat(A::SMat, B::SMat) -> SMat
Horizontally concatenates A and B.
source
(Normal julia cat is also supported)
There are special constructors:
identity_matrix Method
julia
identity_matrix(::Type{SMat}, R::Ring, n::Int)
Return a sparse n times n identity matrix over R.
source
zero_matrix Method
julia
zero_matrix(::Type{SMat}, R::Ring, n::Int)
Return a sparse n times n zero matrix over R.
source
zero_matrix Method
julia
zero_matrix(::Type{SMat}, R::Ring, n::Int, m::Int)
Return a sparse n times m zero matrix over R.
source
block_diagonal_matrix Method
julia
block_diagonal_matrix(xs::Vector{SMat})
Return the block diagonal matrix with the matrices in xs on the diagonal. Requires all blocks to have the same base ring.
source
Slices:
sub Method
julia
sub(A::SMat, r::AbstractUnitRange, c::AbstractUnitRange) -> SMat
Return the submatrix of A, where the rows correspond to r and the columns correspond to c.
source
Transpose:
transpose Method
julia
transpose(A::SMat) -> SMat
Returns the transpose of A.
source
Elementary Properties ​
sparsity Method
julia
sparsity(A::SMat) -> Float64
Return the sparsity of A, that is, the number of zero-valued elements divided by the number of all elements.
source
density Method
julia
density(A::SMat) -> Float64
Return the density of A, that is, the number of nonzero-valued elements divided by the number of all elements.
source
nnz Method
julia
nnz(A::SMat) -> Int
Return the number of non-zero entries of A.
source
number_of_rows Method
julia
number_of_rows(A::SMat) -> Int
Return the number of rows of A.
source
number_of_columns Method
julia
number_of_columns(A::SMat) -> Int
Return the number of columns of A.
source
isone Method
julia
isone(A::SMat)
Tests if A is an identity matrix.
source
iszero Method
julia
iszero(A::SMat)
Tests if A is a zero matrix.
source
is_upper_triangular Method
julia
is_upper_triangular(A::SMat)
Returns true if and only if A is upper (right) triangular.
source
maximum Method
julia
maximum(A::SMat{T}) -> T
Finds the largest entry of A.
source
minimum Method
julia
minimum(A::SMat{T}) -> T
Finds the smallest entry of A.
source
maximum Method
julia
maximum(abs, A::SMat{ZZRingElem}) -> ZZRingElem
Finds the largest, in absolute value, entry of A.
source
elementary_divisors Method
julia
elementary_divisors(A::SMat{ZZRingElem}) -> Vector{ZZRingElem}
The elementary divisors of A, i.e. the diagonal elements of the Smith normal form of A.
source
solve_dixon_sf Method
julia
solve_dixon_sf(A::SMat{ZZRingElem}, b::SRow{ZZRingElem}, is_int::Bool = false) -> SRow{ZZRingElem}, ZZRingElem
 solve_dixon_sf(A::SMat{ZZRingElem}, B::SMat{ZZRingElem}, is_int::Bool = false) -> SMat{ZZRingElem}, ZZRingElem
For a sparse square matrix A of full rank and a sparse matrix (row), find a sparse matrix (row) x and an integer d s.th. xA=bd holds. The algorithm is a Dixon-based linear p-adic lifting method. If \code{is_int} is given, then d is assumed to be 1. In this case rational reconstruction is avoided.
source
hadamard_bound2 Method
julia
hadamard_bound2(A::SMat{T}) -> T
The square of the product of the norms of the rows of A.
source
echelon_with_transform Method
julia
echelon_with_transform(A::SMat{zzModRingElem}) -> SMat, SMat
Find a unimodular matrix T and an upper-triangular E s.th. TA=E holds.
source
reduce_full Method
julia
reduce_full(A::SMat{ZZRingElem}, g::SRow{ZZRingElem},
-                      with_transform = Val(false)) -> SRow{ZZRingElem}, Vector{Int}
Reduces g modulo A and assumes that A is upper triangular.
The second return value is the array of pivot elements of A that changed.
If with_transform is set to Val(true), then additionally an array of transformations is returned.
source
hnf! Method
julia
hnf!(A::SMat{ZZRingElem})
Inplace transform of A into upper right Hermite normal form.
source
hnf Method
julia
hnf(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}
Return the upper right Hermite normal form of A.
source
snf Method
julia
snf(A::SMat{ZZRingElem})
The Smith normal form (snf) of A.
source
hnf_extend! Method
julia
hnf_extend!(A::SMat{ZZRingElem}, b::SMat{ZZRingElem}, offset::Int = 0) -> SMat{ZZRingElem}
Given a matrix A in HNF, extend this to get the HNF of the concatenation with b.
source
is_diagonal Method
julia
is_diagonal(A::SMat) -> Bool
True iff only the i-th entry in the i-th row is non-zero.
source
det Method
julia
det(A::SMat{ZZRingElem})
The determinant of A using a modular algorithm. Uses the dense (zzModMatrix) determinant on A for various primes p.
source
det_mc Method
julia
det_mc(A::SMat{ZZRingElem})
Computes the determinant of A using a LasVegas style algorithm, i.e. the result is not proven to be correct. Uses the dense (zzModMatrix) determinant on A for various primes p.
source
valence_mc Method
julia
valence_mc{T}(A::SMat{T}; extra_prime = 2, trans = Vector{SMatSLP_add_row{T}}()) -> T
Uses a Monte-Carlo algorithm to compute the valence of A. The valence is the valence of the minimal polynomial f of transpose(A)∗A, thus the last non-zero coefficient, typically f(0).
The valence is computed modulo various primes until the computation stabilises for extra_prime many.
trans, if given, is a SLP (straight-line-program) in GL(n, Z). Then the valence of trans * A is computed instead.
source
saturate Method
julia
saturate(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}
Computes the saturation of A, that is, a basis for Q⊗M∩Zn, where M is the row span of A and n the number of rows of A.
Equivalently, return TA for an invertible rational matrix T, such that TA is integral and the elementary divisors of TA are all trivial.
source
hnf_kannan_bachem Method
julia
hnf_kannan_bachem(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}
Compute the Hermite normal form of A using the Kannan-Bachem algorithm.
source
diagonal_form Method
julia
diagonal_form(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}
A matrix D that is diagonal and obtained via unimodular row and column operations. Like a snf without the divisibility condition.
source
Manipulation/ Access ​
getindex Method
julia
getindex(A::SMat, i::Int, j::Int)
Given a sparse matrix A=(aij)i,j, return the entry aij.
source
getindex Method
julia
getindex(A::SMat, i::Int) -> SRow
Given a sparse matrix A and an index i, return the i-th row of A.
source
setindex! Method
julia
setindex!(A::SMat, b::SRow, i::Int)
Given a sparse matrix A, a sparse row b and an index i, set the i-th row of A equal to b.
source
swap_rows! Method
julia
swap_rows!(A::SMat{T}, i::Int, j::Int)
Swap the i-th and j-th row of A inplace.
source
swap_cols! Method
julia
swap_cols!(A::SMat, i::Int, j::Int)
Swap the i-th and j-th column of A inplace.
source
scale_row! Method
julia
scale_row!(A::SMat{T}, i::Int, c::T)
Multiply the i-th row of A by c inplace.
source
add_scaled_col! Method
julia
add_scaled_col!(A::SMat{T}, i::Int, j::Int, c::T)
Add c times the i-th column to the j-th column of A inplace, that is, Aj→Aj+c⋅Ai, where (Ai)i denote the columns of A.
source
add_scaled_row! Method
julia
add_scaled_row!(A::SMat{T}, i::Int, j::Int, c::T)
Add c times the i-th row to the j-th row of A inplace, that is, Aj→Aj+c⋅Ai, where (Ai)i denote the rows of A.
source
transform_row! Method
julia
transform_row!(A::SMat{T}, i::Int, j::Int, a::T, b::T, c::T, d::T)
Applies the transformation (Ai,Aj)→(aAi+bAj,cAi+dAj) to A, where (Ai)i are the rows of A.
source
diagonal Method
julia
diagonal(A::SMat) -> ZZRingElem[]
The diagonal elements of A in an array.
source
reverse_rows! Method
julia
reverse_rows!(A::SMat)
Inplace inversion of the rows of A.
source
mod_sym! Method
julia
mod_sym!(A::SMat{ZZRingElem}, n::ZZRingElem)
Inplace reduction of all entries of A modulo n to the symmetric residue system.
source
find_row_starting_with Method
julia
find_row_starting_with(A::SMat, p::Int) -> Int
Tries to find the index i such that Ai,p≠0 and Ai,p−j=0 for all j>1. It is assumed that A is upper triangular. If such an index does not exist, find the smallest index larger.
source
reduce Method
julia
reduce(A::SMat{ZZRingElem}, g::SRow{ZZRingElem}, m::ZZRingElem) -> SRow{ZZRingElem}
Given an upper triangular matrix A over the integers, a sparse row g and an integer m, this function reduces g modulo A and returns g modulo m with respect to the symmetric residue system.
source
reduce Method
julia
reduce(A::SMat{ZZRingElem}, g::SRow{ZZRingElem}) -> SRow{ZZRingElem}
Given an upper triangular matrix A over a field and a sparse row g, this function reduces g modulo A.
source
reduce Method
julia
reduce(A::SMat{T}, g::SRow{T}) -> SRow{T}
Given an upper triangular matrix A over a field and a sparse row g, this function reduces g modulo A.
source
rand_row Method
julia
rand_row(A::SMat) -> SRow
Return a random row of the sparse matrix A.
source
Changing of the ring:
map_entries Method
julia
map_entries(f, A::SMat) -> SMat
Given a sparse matrix A and a callable object f, this function will construct a new sparse matrix by applying f to all elements of A.
source
change_base_ring Method
julia
change_base_ring(R::Ring, A::SMat)
Create a new sparse matrix by coercing all elements into the ring R.
source
Arithmetic ​
Matrices support the usual operations as well
  • +, -, ==
  • div, divexact by scalars
  • multiplication by scalars
Various products:
* Method
julia
*(A::SMat{T}, b::AbstractVector{T}) -> Vector{T}
Return the product A⋅b as a dense vector.
source
* Method
julia
*(A::SMat{T}, b::AbstractMatrix{T}) -> Matrix{T}
Return the product A⋅b as a dense array.
source
* Method
julia
*(A::SMat{T}, b::MatElem{T}) -> MatElem
Return the product A⋅b as a dense matrix.
source
* Method
julia
*(A::SRow, B::SMat) -> SRow
Return the product A⋅B as a sparse row.
source
dot Method
julia
dot(x::SRow{T}, A::SMat{T}, y::SRow{T}) where T -> T
Return the generalized dot product dot(x, A*y).
source
dot Method
julia
dot(x::MatrixElem{T}, A::SMat{T}, y::MatrixElem{T}) where T -> T
Return the generalized dot product dot(x, A*y).
source
dot Method
julia
dot(x::AbstractVector{T}, A::SMat{T}, y::AbstractVector{T}) where T -> T
Return the generalized dot product dot(x, A*y).
source
Other:
sparse Method
julia
sparse(A::SMat) -> SparseMatrixCSC
The same matrix, but as a sparse matrix of julia type SparseMatrixCSC.
source
ZZMatrix Method
julia
ZZMatrix(A::SMat{ZZRingElem})
The same matrix A, but as an ZZMatrix.
source
ZZMatrix Method
julia
ZZMatrix(A::SMat{T}) where {T <: Integer}
The same matrix A, but as an ZZMatrix. Requires a conversion from the base ring of A to ZZ.
source
Matrix Method
julia
Matrix(A::SMat{T}) -> Matrix{T}
The same matrix, but as a julia matrix.
source
Array Method
julia
Array(A::SMat{T}) -> Matrix{T}
The same matrix, but as a two-dimensional julia array.
source

-    
+                      with_transform = Val(false)) -> SRow{ZZRingElem}, Vector{Int}
Reduces g modulo A and assumes that A is upper triangular.
The second return value is the array of pivot elements of A that changed.
If with_transform is set to Val(true), then additionally an array of transformations is returned.
source
hnf! Method
julia
hnf!(A::SMat{ZZRingElem})
Inplace transform of A into upper right Hermite normal form.
source
hnf Method
julia
hnf(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}
Return the upper right Hermite normal form of A.
source
snf Method
julia
snf(A::SMat{ZZRingElem})
The Smith normal form (snf) of A.
source
hnf_extend! Method
julia
hnf_extend!(A::SMat{ZZRingElem}, b::SMat{ZZRingElem}, offset::Int = 0) -> SMat{ZZRingElem}
Given a matrix A in HNF, extend this to get the HNF of the concatenation with b.
source
is_diagonal Method
julia
is_diagonal(A::SMat) -> Bool
True iff only the i-th entry in the i-th row is non-zero.
source
det Method
julia
det(A::SMat{ZZRingElem})
The determinant of A using a modular algorithm. Uses the dense (zzModMatrix) determinant on A for various primes p.
source
det_mc Method
julia
det_mc(A::SMat{ZZRingElem})
Computes the determinant of A using a LasVegas style algorithm, i.e. the result is not proven to be correct. Uses the dense (zzModMatrix) determinant on A for various primes p.
source
valence_mc Method
julia
valence_mc{T}(A::SMat{T}; extra_prime = 2, trans = Vector{SMatSLP_add_row{T}}()) -> T
Uses a Monte-Carlo algorithm to compute the valence of A. The valence is the valence of the minimal polynomial f of transpose(A)∗A, thus the last non-zero coefficient, typically f(0).
The valence is computed modulo various primes until the computation stabilises for extra_prime many.
trans, if given, is a SLP (straight-line-program) in GL(n, Z). Then the valence of trans * A is computed instead.
source
saturate Method
julia
saturate(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}
Computes the saturation of A, that is, a basis for Q⊗M∩Zn, where M is the row span of A and n the number of rows of A.
Equivalently, return TA for an invertible rational matrix T, such that TA is integral and the elementary divisors of TA are all trivial.
source
hnf_kannan_bachem Method
julia
hnf_kannan_bachem(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}
Compute the Hermite normal form of A using the Kannan-Bachem algorithm.
source
diagonal_form Method
julia
diagonal_form(A::SMat{ZZRingElem}) -> SMat{ZZRingElem}
A matrix D that is diagonal and obtained via unimodular row and column operations. Like a snf without the divisibility condition.
source
Manipulation/ Access ​
getindex Method
julia
getindex(A::SMat, i::Int, j::Int)
Given a sparse matrix A=(aij)i,j, return the entry aij.
source
getindex Method
julia
getindex(A::SMat, i::Int) -> SRow
Given a sparse matrix A and an index i, return the i-th row of A.
source
setindex! Method
julia
setindex!(A::SMat, b::SRow, i::Int)
Given a sparse matrix A, a sparse row b and an index i, set the i-th row of A equal to b.
source
swap_rows! Method
julia
swap_rows!(A::SMat{T}, i::Int, j::Int)
Swap the i-th and j-th row of A inplace.
source
swap_cols! Method
julia
swap_cols!(A::SMat, i::Int, j::Int)
Swap the i-th and j-th column of A inplace.
source
scale_row! Method
julia
scale_row!(A::SMat{T}, i::Int, c::T)
Multiply the i-th row of A by c inplace.
source
add_scaled_col! Method
julia
add_scaled_col!(A::SMat{T}, i::Int, j::Int, c::T)
Add c times the i-th column to the j-th column of A inplace, that is, Aj→Aj+c⋅Ai, where (Ai)i denote the columns of A.
source
add_scaled_row! Method
julia
add_scaled_row!(A::SMat{T}, i::Int, j::Int, c::T)
Add c times the i-th row to the j-th row of A inplace, that is, Aj→Aj+c⋅Ai, where (Ai)i denote the rows of A.
source
transform_row! Method
julia
transform_row!(A::SMat{T}, i::Int, j::Int, a::T, b::T, c::T, d::T)
Applies the transformation (Ai,Aj)→(aAi+bAj,cAi+dAj) to A, where (Ai)i are the rows of A.
source
diagonal Method
julia
diagonal(A::SMat) -> ZZRingElem[]
The diagonal elements of A in an array.
source
reverse_rows! Method
julia
reverse_rows!(A::SMat)
Inplace inversion of the rows of A.
source
mod_sym! Method
julia
mod_sym!(A::SMat{ZZRingElem}, n::ZZRingElem)
Inplace reduction of all entries of A modulo n to the symmetric residue system.
source
find_row_starting_with Method
julia
find_row_starting_with(A::SMat, p::Int) -> Int
Tries to find the index i such that Ai,p≠0 and Ai,p−j=0 for all j>1. It is assumed that A is upper triangular. If such an index does not exist, find the smallest index larger.
source
reduce Method
julia
reduce(A::SMat{ZZRingElem}, g::SRow{ZZRingElem}, m::ZZRingElem) -> SRow{ZZRingElem}
Given an upper triangular matrix A over the integers, a sparse row g and an integer m, this function reduces g modulo A and returns g modulo m with respect to the symmetric residue system.
source
reduce Method
julia
reduce(A::SMat{ZZRingElem}, g::SRow{ZZRingElem}) -> SRow{ZZRingElem}
Given an upper triangular matrix A over a field and a sparse row g, this function reduces g modulo A.
source
reduce Method
julia
reduce(A::SMat{T}, g::SRow{T}) -> SRow{T}
Given an upper triangular matrix A over a field and a sparse row g, this function reduces g modulo A.
source
rand_row Method
julia
rand_row(A::SMat) -> SRow
Return a random row of the sparse matrix A.
source
Changing of the ring:
map_entries Method
julia
map_entries(f, A::SMat) -> SMat
Given a sparse matrix A and a callable object f, this function will construct a new sparse matrix by applying f to all elements of A.
source
change_base_ring Method
julia
change_base_ring(R::Ring, A::SMat)
Create a new sparse matrix by coercing all elements into the ring R.
source
Arithmetic ​
Matrices support the usual operations as well
  • +, -, ==
  • div, divexact by scalars
  • multiplication by scalars
Various products:
* Method
julia
*(A::SMat{T}, b::AbstractVector{T}) -> Vector{T}
Return the product A⋅b as a dense vector.
source
* Method
julia
*(A::SMat{T}, b::AbstractMatrix{T}) -> Matrix{T}
Return the product A⋅b as a dense array.
source
* Method
julia
*(A::SMat{T}, b::MatElem{T}) -> MatElem
Return the product A⋅b as a dense matrix.
source
* Method
julia
*(A::SRow, B::SMat) -> SRow
Return the product A⋅B as a sparse row.
source
dot Method
julia
dot(x::SRow{T}, A::SMat{T}, y::SRow{T}) where T -> T
Return the generalized dot product dot(x, A*y).
source
dot Method
julia
dot(x::MatrixElem{T}, A::SMat{T}, y::MatrixElem{T}) where T -> T
Return the generalized dot product dot(x, A*y).
source
dot Method
julia
dot(x::AbstractVector{T}, A::SMat{T}, y::AbstractVector{T}) where T -> T
Return the generalized dot product dot(x, A*y).
source
Other:
sparse Method
julia
sparse(A::SMat) -> SparseMatrixCSC
The same matrix, but as a sparse matrix of julia type SparseMatrixCSC.
source
ZZMatrix Method
julia
ZZMatrix(A::SMat{ZZRingElem})
The same matrix A, but as an ZZMatrix.
source
ZZMatrix Method
julia
ZZMatrix(A::SMat{T}) where {T <: Integer}
The same matrix A, but as an ZZMatrix. Requires a conversion from the base ring of A to ZZ.
source
Matrix Method
julia
Matrix(A::SMat{T}) -> Matrix{T}
The same matrix, but as a julia matrix.
source
Array Method
julia
Array(A::SMat{T}) -> Matrix{T}
The same matrix, but as a two-dimensional julia array.
source

+    
     
   
 
\ No newline at end of file
diff --git a/v0.34.8/manual/number_fields/class_fields.html b/v0.34.8/manual/number_fields/class_fields.html
index 042a598e39..e68642330a 100644
--- a/v0.34.8/manual/number_fields/class_fields.html
+++ b/v0.34.8/manual/number_fields/class_fields.html
@@ -6,19 +6,19 @@
     Class Field Theory | Hecke
     
     
-    
-    
+    
+    
     
-    
-    
-    
-    
-    
+    
+    
+    
+    
+    
     
     
   
   
-    
Hecke
Appearance
Class Field Theory ​
Introduction ​
This chapter deals with abelian extensions of number fields and the rational numbers.
Class Field Theory, here specifically, class field theory of global number fields, deals with abelian extension, ie. fields where the group of automorphisms is abelian. For extensions of Q, the famous Kronnecker-Weber theorem classifies all such fields: a field is abelian if and only if it is contained in some cyclotomic field. For general number fields this is more involved and even for extensions of Q is is not practical.
In Hecke, abelian extensions are parametrized by quotients of so called ray class groups. The language of ray class groups while dated is more applicable to algorithms than the modern language of idel class groups and quotients.
Ray Class Groups ​
Given an integral ideal m0≤ZK and a list of real places m∞, the ray class group modulo (m0,m∞), C(m) is defined as the group of ideals coprime to m0 modulo the elements a∈K∗ s.th. vp(a−1)≥vp(m0) and for all v∈m∞, a(v)>0. This is a finite abelian group. For m0=ZK and m∞={} we get C() is the class group, if m∞ contains all real places, we obtain the narrow class group, or strict class group.
ray_class_group Method
julia
ray_class_group(m::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, inf_plc::Vector{InfPlc}; n_quo::Int, lp::Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}) -> FinGenAbGroup, MapRayClassGrp
Given an ideal m and a set of infinite places of K, this function returns the corresponding ray class group as an abstract group Clm and a map going from the group into the group of ideals of K that are coprime to m. If n_quo is set, it will return the group modulo n_quo. The factorization of m can be given with the keyword argument lp.
source
class_group Method
julia
class_group(K::AbsSimpleNumField) -> FinGenAbGroup, Map
Shortcut for class_group(maximal_order(K)): returns the class group as an abelian group and a map from this group to the set of ideals of the maximal order.
source
norm_group Method
julia
norm_group(f::Nemo.PolyRingElem, mR::Hecke.MapRayClassGrp, is_abelian::Bool = true; of_closure::Bool = false) -> Hecke.FinGenGrpAb, Hecke.FinGenGrpAbMap
+    
Hecke
Appearance
Class Field Theory ​
Introduction ​
This chapter deals with abelian extensions of number fields and the rational numbers.
Class Field Theory, here specifically, class field theory of global number fields, deals with abelian extension, ie. fields where the group of automorphisms is abelian. For extensions of Q, the famous Kronnecker-Weber theorem classifies all such fields: a field is abelian if and only if it is contained in some cyclotomic field. For general number fields this is more involved and even for extensions of Q is is not practical.
In Hecke, abelian extensions are parametrized by quotients of so called ray class groups. The language of ray class groups while dated is more applicable to algorithms than the modern language of idel class groups and quotients.
Ray Class Groups ​
Given an integral ideal m0≤ZK and a list of real places m∞, the ray class group modulo (m0,m∞), C(m) is defined as the group of ideals coprime to m0 modulo the elements a∈K∗ s.th. vp(a−1)≥vp(m0) and for all v∈m∞, a(v)>0. This is a finite abelian group. For m0=ZK and m∞={} we get C() is the class group, if m∞ contains all real places, we obtain the narrow class group, or strict class group.
ray_class_group Method
julia
ray_class_group(m::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, inf_plc::Vector{InfPlc}; n_quo::Int, lp::Dict{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Int}) -> FinGenAbGroup, MapRayClassGrp
Given an ideal m and a set of infinite places of K, this function returns the corresponding ray class group as an abstract group Clm and a map going from the group into the group of ideals of K that are coprime to m. If n_quo is set, it will return the group modulo n_quo. The factorization of m can be given with the keyword argument lp.
source
class_group Method
julia
class_group(K::AbsSimpleNumField) -> FinGenAbGroup, Map
Shortcut for class_group(maximal_order(K)): returns the class group as an abelian group and a map from this group to the set of ideals of the maximal order.
source
norm_group Method
julia
norm_group(f::Nemo.PolyRingElem, mR::Hecke.MapRayClassGrp, is_abelian::Bool = true; of_closure::Bool = false) -> Hecke.FinGenGrpAb, Hecke.FinGenGrpAbMap
 
 norm_group(f::Array{PolyRingElem{AbsSimpleNumFieldElem}}, mR::Hecke.MapRayClassGrp, is_abelian::Bool = true; of_closure::Bool = false) -> Hecke.FinGenGrpAb, Hecke.FinGenGrpAbMap
Computes the subgroup of the Ray Class Group R given by the norm of the extension generated by a/the roots of f. If is_abelian is set to true, then the code assumes the field to be abelian, hence the algorithm stops when the quotient by the norm group has the correct order. Even though the algorithm is probabilistic by nature, in this case the result is guaranteed. If of_closure is given, then the norm group of the splitting field of the polynomial(s) is computed. It is the callers responsibility to ensure that the ray class group passed in is large enough.
source
norm_group Method
julia
norm_group(K::RelSimpleNumField{AbsSimpleNumFieldElem}, mR::Hecke.MapRayClassGrp) -> Hecke.FinGenGrpAb, Hecke.FinGenGrpAbMap
 
@@ -57,8 +57,8 @@
 
 julia> isone(discriminant(ZK))
 true
ray_class_field Method
julia
ray_class_field(K::RelSimpleNumField{AbsSimpleNumFieldElem}) -> ClassField
-ray_class_field(K::AbsSimpleNumField) -> ClassField
For a (relative) abelian extension, compute an abstract representation as a class field.
source
genus_field Method
julia
genus_field(A::ClassField, k::AbsSimpleNumField) -> ClassField
The maximal extension contained in A that is the compositum of K with an abelian extension of k.
source
maximal_abelian_subfield Method
julia
maximal_abelian_subfield(A::ClassField, k::AbsSimpleNumField) -> ClassField
The maximal abelian extension of k contained in A. k must be a subfield of the base field of A.
source
maximal_abelian_subfield Method
julia
maximal_abelian_subfield(K::RelSimpleNumField{AbsSimpleNumFieldElem}; of_closure::Bool = false) -> ClassField
Using a probabilistic algorithm for the norm group computation, determine the maximal abelian subfield in K over its base field. If of_closure is set to true, then the algorithm is applied to the normal closure of K (without computing it).
source
Invariants ​
degree Method
julia
degree(A::ClassField)
The degree of A over its base field, i.e. the size of the defining ideal group.
source
base_ring Method
julia
base_ring(A::ClassField)
The maximal order of the field that A is defined over.
source
base_field Method
julia
base_field(A::ClassField)
The number field that A is defined over.
source
discriminant Method
julia
discriminant(C::ClassField) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
Using the conductor-discriminant formula, compute the (relative) discriminant of C. This does not use the defining equations.
source
conductor Method
julia
conductor(C::ClassField) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Vector{InfPlc}
Return the conductor of the abelian extension corresponding to C.
source
defining_modulus Method
julia
defining_modulus(CF::ClassField)
The modulus, i.e. an ideal of the set of real places, used to create the class field.
source
is_cyclic Method
julia
is_cyclic(C::ClassField)
Tests if the (relative) automorphism group of C is cyclic (by checking the defining ideal group).
source
is_conductor Method
julia
is_conductor(C::Hecke.ClassField, m::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, inf_plc::Vector{InfPlc}=InfPlc[]; check) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Vector{InfPlc}
Checks if (m, inf_plc) is the conductor of the abelian extension corresponding to C. If check is false, it assumes that the given modulus is a multiple of the conductor. This is usually faster than computing the conductor.
source
is_normal Method
julia
is_normal(C::ClassField) -> Bool
For a class field C defined over a normal base field k, decide if C is normal over Q.
source
is_central Method
julia
is_central(C::ClassField) -> Bool
For a class field C defined over a normal base field k, decide if C is central over Q.
source
Operations ​
* Method
julia
*(A::ClassField, B::ClassField) -> ClassField
The compositum of a and b as a (formal) class field.
source
compositum Method
julia
compositum(a::ClassField, b::ClassField) -> ClassField
The compositum of a and b as a (formal) class field.
source
== Method
julia
==(a::ClassField, b::ClassField)
Tests if a and b are equal.
source
intersect Method
julia
intersect(a::ClassField, b::ClassField) -> ClassField
The intersection of a and b as a class field.
source
prime_decomposition_type Method
julia
prime_decomposition_type(C::ClassField, p::AbsNumFieldOrderIdeal) -> (Int, Int, Int)
For a prime p in the base ring of r, determine the splitting type of p in r. ie. the tuple (e,f,g) giving the ramification degree, the inertia and the number of primes above p.
source
is_subfield Method
julia
is_subfield(a::ClassField, b::ClassField) -> Bool
Determines if a is a subfield of b.
source
is_local_norm Method
julia
is_local_norm(r::ClassField, a::AbsNumFieldOrderElem) -> Bool
Tests if a is a local norm at all finite places in the extension implicitly given by r.
source
is_local_norm Method
julia
is_local_norm(r::ClassField, a::AbsNumFieldOrderElem, p::AbsNumFieldOrderIdeal) -> Bool
Tests if a is a local norm at p in the extension implicitly given by r. Currently the conductor cannot have infinite places.
source
normal_closure Method
julia
normal_closure(C::ClassField) -> ClassField
For a ray class field C extending a normal base field k, compute the normal closure over Q.
source
subfields Method
julia
subfields(C::ClassField; degree::Int, is_normal, type) -> Vector{ClassField}
Find all subfields of C over the base field.
If the optional keyword argument degree is positive, then only those with prescribed degree will be returned.
If the optional keyword is_normal is given, then only those that are normal over the field fixed by the automorphisms is returned. For normal base fields, this amounts to extensions that are normal over Q.
If the optional keyword is_normal is set to a list of automorphisms, then only those wil be considered.
type can be set to the desired relative Galois group, given as a vector of integers descibing the structure.
Note
This will not find all subfields over Q, but only the ones sharing the same base field.
source

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For a (relative) abelian extension, compute an abstract representation as a class field.
source
genus_field Method
julia
genus_field(A::ClassField, k::AbsSimpleNumField) -> ClassField
The maximal extension contained in A that is the compositum of K with an abelian extension of k.
source
maximal_abelian_subfield Method
julia
maximal_abelian_subfield(A::ClassField, k::AbsSimpleNumField) -> ClassField
The maximal abelian extension of k contained in A. k must be a subfield of the base field of A.
source
maximal_abelian_subfield Method
julia
maximal_abelian_subfield(K::RelSimpleNumField{AbsSimpleNumFieldElem}; of_closure::Bool = false) -> ClassField
Using a probabilistic algorithm for the norm group computation, determine the maximal abelian subfield in K over its base field. If of_closure is set to true, then the algorithm is applied to the normal closure of K (without computing it).
source
Invariants ​
degree Method
julia
degree(A::ClassField)
The degree of A over its base field, i.e. the size of the defining ideal group.
source
base_ring Method
julia
base_ring(A::ClassField)
The maximal order of the field that A is defined over.
source
base_field Method
julia
base_field(A::ClassField)
The number field that A is defined over.
source
discriminant Method
julia
discriminant(C::ClassField) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
Using the conductor-discriminant formula, compute the (relative) discriminant of C. This does not use the defining equations.
source
conductor Method
julia
conductor(C::ClassField) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Vector{InfPlc}
Return the conductor of the abelian extension corresponding to C.
source
defining_modulus Method
julia
defining_modulus(CF::ClassField)
The modulus, i.e. an ideal of the set of real places, used to create the class field.
source
is_cyclic Method
julia
is_cyclic(C::ClassField)
Tests if the (relative) automorphism group of C is cyclic (by checking the defining ideal group).
source
is_conductor Method
julia
is_conductor(C::Hecke.ClassField, m::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, inf_plc::Vector{InfPlc}=InfPlc[]; check) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, Vector{InfPlc}
Checks if (m, inf_plc) is the conductor of the abelian extension corresponding to C. If check is false, it assumes that the given modulus is a multiple of the conductor. This is usually faster than computing the conductor.
source
is_normal Method
julia
is_normal(C::ClassField) -> Bool
For a class field C defined over a normal base field k, decide if C is normal over Q.
source
is_central Method
julia
is_central(C::ClassField) -> Bool
For a class field C defined over a normal base field k, decide if C is central over Q.
source
Operations ​
* Method
julia
*(A::ClassField, B::ClassField) -> ClassField
The compositum of a and b as a (formal) class field.
source
compositum Method
julia
compositum(a::ClassField, b::ClassField) -> ClassField
The compositum of a and b as a (formal) class field.
source
== Method
julia
==(a::ClassField, b::ClassField)
Tests if a and b are equal.
source
intersect Method
julia
intersect(a::ClassField, b::ClassField) -> ClassField
The intersection of a and b as a class field.
source
prime_decomposition_type Method
julia
prime_decomposition_type(C::ClassField, p::AbsNumFieldOrderIdeal) -> (Int, Int, Int)
For a prime p in the base ring of r, determine the splitting type of p in r. ie. the tuple (e,f,g) giving the ramification degree, the inertia and the number of primes above p.
source
is_subfield Method
julia
is_subfield(a::ClassField, b::ClassField) -> Bool
Determines if a is a subfield of b.
source
is_local_norm Method
julia
is_local_norm(r::ClassField, a::AbsNumFieldOrderElem) -> Bool
Tests if a is a local norm at all finite places in the extension implicitly given by r.
source
is_local_norm Method
julia
is_local_norm(r::ClassField, a::AbsNumFieldOrderElem, p::AbsNumFieldOrderIdeal) -> Bool
Tests if a is a local norm at p in the extension implicitly given by r. Currently the conductor cannot have infinite places.
source
normal_closure Method
julia
normal_closure(C::ClassField) -> ClassField
For a ray class field C extending a normal base field k, compute the normal closure over Q.
source
subfields Method
julia
subfields(C::ClassField; degree::Int, is_normal, type) -> Vector{ClassField}
Find all subfields of C over the base field.
If the optional keyword argument degree is positive, then only those with prescribed degree will be returned.
If the optional keyword is_normal is given, then only those that are normal over the field fixed by the automorphisms is returned. For normal base fields, this amounts to extensions that are normal over Q.
If the optional keyword is_normal is set to a list of automorphisms, then only those wil be considered.
type can be set to the desired relative Galois group, given as a vector of integers descibing the structure.
Note
This will not find all subfields over Q, but only the ones sharing the same base field.
source

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diff --git a/v0.34.8/manual/number_fields/complex_embeddings.html b/v0.34.8/manual/number_fields/complex_embeddings.html
index 977900dc49..ea6cf10ae3 100644
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     Complex embedding | Hecke
     
     
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Appearance
Complex embedding ​
We describe functionality for complex embeddings of arbitrary number fields. Note that a complex embeddding of a number field L is a morphism ι:L→C. Such an embedding is called real if im(ι)⊆R and imaginary otherwise.
Construction of complex embeddings ​
complex_embeddings Method
julia
complex_embeddings(K::NumField; conjugates::Bool = true) -> Vector{NumFieldEmb}
Return the complex embeddings of K. If conjugates is false, only one imaginary embedding per conjugated pairs is returned.
Examples
julia
julia> K, a = quadratic_field(-3);
+    
Hecke
Appearance
Complex embedding ​
We describe functionality for complex embeddings of arbitrary number fields. Note that a complex embeddding of a number field L is a morphism ι:L→C. Such an embedding is called real if im(ι)⊆R and imaginary otherwise.
Construction of complex embeddings ​
complex_embeddings Method
julia
complex_embeddings(K::NumField; conjugates::Bool = true) -> Vector{NumFieldEmb}
Return the complex embeddings of K. If conjugates is false, only one imaginary embedding per conjugated pairs is returned.
Examples
julia
julia> K, a = quadratic_field(-3);
 
 julia> complex_embeddings(K)
 2-element Vector{AbsSimpleNumFieldEmbedding}:
@@ -147,8 +147,8 @@
 
 julia> restrict(emb[3], i)
 Complex embedding corresponding to -1.00 * i
-  of imaginary quadratic field defined by x^2 + 1

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+  of imaginary quadratic field defined by x^2 + 1

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diff --git a/v0.34.8/manual/number_fields/conventions.html b/v0.34.8/manual/number_fields/conventions.html
index 94be8479b7..93460c6186 100644
--- a/v0.34.8/manual/number_fields/conventions.html
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     Conventions | Hecke
     
     
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Appearance
Conventions ​
By an absolute number field we mean finite extensions of Q, which is of type AbsSimpleNumField and whose elements are of type AbsSimpleNumFieldElem. Such an absolute number field K is always given in the form K=Q(α)=Q[X]/(f), where f∈Q[X] is an irreducible polynomial. See here for more information on the different types of fields supported.
We call (1,α,α2,…,αd−1), where d is the degree [K:Q] the power basis of K. If β is any element of K, then the representation matrix of β is the matrix representing K→K,γ↦βγ with respect to the power basis, that is,
β⋅(1,α,…,αd−1)=Mα(1,α,…,αd−1).
Let (r,s) be the signature of K, that is, K has r real embeddings σi:K→R, 1≤i≤r, and 2s complex embeddings σi:K→C, 1≤i≤2s. In Hecke the complex embeddings are always ordered such that σi=σi+s― for r+1≤i≤r+s. The Q-linear function
K⟶Rdα⟼(σ1(α),…,σr(α),2Re(σr+1(α)),2Im(σr+1(α)),…,2Re(σr+s(α)),2Im(σr+s(α)))
is called the Minkowski map (or Minkowski embedding).
If K=Q(α) is an absolute number field, then an order O of K is a subring of the ring of integers OK, which is free of rank [K:Q] as a Z-module. The natural order Z[α] is called the equation order of K. In Hecke orders of absolute number fields are constructed (implicitly) by specifying a Z-basis, which is referred to as the basis of O. If (ω1,…,ωd) is the basis of O, then the matrix B∈Matd×d(Q) with
is called the basis matrix of O. We call det(B) the generalized index of O. In case Z[α]⊆O, the determinant det(B)−1 is in fact equal to [O:Z[α]] and is called the index of O. The matrix
(σ1(ω1)…σr(ω1)2Re(σr+1(ω1))2Im(σr+1(ω1))…2Im(σr+s(ω1))σ1(ω2)…σr(ω2)2Re(σr+1(ω2))2Im(σr+1(ω2))…2Im(σr+s(ω2))⋮⋱⋮⋮⋮⋱⋮σ1(ωd)…σr(ωd)2Re(σr+1(ωd))2Im(σr+2(ωd))…2Im(σr+s(ωd)))∈Matd×d(R).
is called the Minkowski matrix of O.

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Hecke
Appearance
Conventions ​
By an absolute number field we mean finite extensions of Q, which is of type AbsSimpleNumField and whose elements are of type AbsSimpleNumFieldElem. Such an absolute number field K is always given in the form K=Q(α)=Q[X]/(f), where f∈Q[X] is an irreducible polynomial. See here for more information on the different types of fields supported.
We call (1,α,α2,…,αd−1), where d is the degree [K:Q] the power basis of K. If β is any element of K, then the representation matrix of β is the matrix representing K→K,γ↦βγ with respect to the power basis, that is,
β⋅(1,α,…,αd−1)=Mα(1,α,…,αd−1).
Let (r,s) be the signature of K, that is, K has r real embeddings σi:K→R, 1≤i≤r, and 2s complex embeddings σi:K→C, 1≤i≤2s. In Hecke the complex embeddings are always ordered such that σi=σi+s― for r+1≤i≤r+s. The Q-linear function
K⟶Rdα⟼(σ1(α),…,σr(α),2Re(σr+1(α)),2Im(σr+1(α)),…,2Re(σr+s(α)),2Im(σr+s(α)))
is called the Minkowski map (or Minkowski embedding).
If K=Q(α) is an absolute number field, then an order O of K is a subring of the ring of integers OK, which is free of rank [K:Q] as a Z-module. The natural order Z[α] is called the equation order of K. In Hecke orders of absolute number fields are constructed (implicitly) by specifying a Z-basis, which is referred to as the basis of O. If (ω1,…,ωd) is the basis of O, then the matrix B∈Matd×d(Q) with
is called the basis matrix of O. We call det(B) the generalized index of O. In case Z[α]⊆O, the determinant det(B)−1 is in fact equal to [O:Z[α]] and is called the index of O. The matrix
(σ1(ω1)…σr(ω1)2Re(σr+1(ω1))2Im(σr+1(ω1))…2Im(σr+s(ω1))σ1(ω2)…σr(ω2)2Re(σr+1(ω2))2Im(σr+1(ω2))…2Im(σr+s(ω2))⋮⋱⋮⋮⋮⋱⋮σ1(ωd)…σr(ωd)2Re(σr+1(ωd))2Im(σr+2(ωd))…2Im(σr+s(ωd)))∈Matd×d(R).
is called the Minkowski matrix of O.

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diff --git a/v0.34.8/manual/number_fields/elements.html b/v0.34.8/manual/number_fields/elements.html
index 3ddcc88a7d..45df06a403 100644
--- a/v0.34.8/manual/number_fields/elements.html
+++ b/v0.34.8/manual/number_fields/elements.html
@@ -6,19 +6,19 @@
     Element operations | Hecke
     
     
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Element operations ​
Creation ​
gen Method
julia
gen(L::SimpleNumField) -> NumFieldElem
Given a simple number field L=K[x]/(f) over K, this functions returns the class of x, which is the canonical primitive element of L over K.
source
gens Method
julia
gens(L::NonSimpleNumField) -> Vector{NumFieldElem}
Given a non-simple number field L=K[x1,…,xn]/(f1,…,fn) over K, this functions returns the list x¯1,…,x¯n.
source
Elements can also be created by specifying the coordinates with respect to the basis of the number field:
julia
    (L::number_field)(c::Vector{NumFieldElem}) -> NumFieldElem
Given a number field L/K of degree d and a vector c length d, this constructs the element a with coordinates(a) == c.
julia
julia> Qx, x = QQ["x"];
+    
Hecke
Appearance
Element operations ​
Creation ​
gen Method
julia
gen(L::SimpleNumField) -> NumFieldElem
Given a simple number field L=K[x]/(f) over K, this functions returns the class of x, which is the canonical primitive element of L over K.
source
gens Method
julia
gens(L::NonSimpleNumField) -> Vector{NumFieldElem}
Given a non-simple number field L=K[x1,…,xn]/(f1,…,fn) over K, this functions returns the list x¯1,…,x¯n.
source
Elements can also be created by specifying the coordinates with respect to the basis of the number field:
julia
    (L::number_field)(c::Vector{NumFieldElem}) -> NumFieldElem
Given a number field L/K of degree d and a vector c length d, this constructs the element a with coordinates(a) == c.
julia
julia> Qx, x = QQ["x"];
 
 julia> K, a = number_field(x^2 - 2, "a");
 
@@ -29,9 +29,9 @@
 (Relative number field of degree 3 over number field, b)
 
 julia> L([a, 1, 1//2])
-1//2*b^2 + b + a
quadratic_defect Method
julia
quadratic_defect(a::Union{NumFieldElem,Rational,QQFieldElem}, p) -> Union{Inf, PosInf}
Returns the valuation of the quadratic defect of the element a at p, which can either be prime object or an infinite place of the parent of a.
source
hilbert_symbol Method
julia
hilbert_symbol(a::NumFieldElem, b::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Int
Returns the local Hilbert symbol (a,b)p.
source
representation_matrix Method
julia
representation_matrix(a::NumFieldElem) -> MatElem
Returns the representation matrix of a, that is, the matrix representing multiplication with a with respect to the canonical basis of the parent of a.
source
basis_matrix Method
julia
basis_matrix(v::Vector{NumFieldElem}) -> Mat
Given a vector v of n elements of a number field K of degree d, this function returns an n×d matrix with entries in the base field of K, where row i contains the coefficients of v[i] with respect of the canonical basis of K.
source
coefficients Method
julia
coefficients(a::SimpleNumFieldElem, i::Int) -> Vector{FieldElem}
Given a number field element a of a simple number field extension L/K, this function returns the coefficients of a, when expanded in the canonical power basis of L.
source
coordinates Method
julia
coordinates(x::NumFieldElem{T}) -> Vector{T}
Given an element x in a number field K, this function returns the coordinates of x with respect to the basis of K (the output of the 'basis' function).
source
absolute_coordinates Method
julia
absolute_coordinates(x::NumFieldElem{T}) -> Vector{T}
Given an element x in a number field K, this function returns the coordinates of x with respect to the basis of K over the rationals (the output of the absolute_basis function).
source
coeff Method
julia
coeff(a::SimpleNumFieldElem, i::Int) -> FieldElem
Given a number field element a of a simple number field extension L/K, this function returns the i-th coefficient of a, when expanded in the canonical power basis of L. The result is an element of K.
source
valuation Method
julia
valuation(a::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
Computes the p-adic valuation of a, that is, the largest i such that a is contained in pi.
source
torsion_unit_order Method
julia
torsion_unit_order(x::AbsSimpleNumFieldElem, n::Int)
Given a torsion unit x together with a multiple n of its order, compute the order of x, that is, the smallest k∈Z≥1 such that xk=1.
It is not checked whether x is a torsion unit.
source
tr Method
julia
tr(a::NumFieldElem) -> NumFieldElem
Returns the trace of an element a of a number field extension L/K. This will be an element of K.
source
absolute_tr Method
julia
absolute_tr(a::NumFieldElem) -> QQFieldElem
Given a number field element a, returns the absolute trace of a.
source
algebraic_split Method
julia
algebraic_split(a::AbsSimpleNumFieldElem) -> AbsSimpleNumFieldElem, AbsSimpleNumFieldElem
Writes the input as a quotient of two "small" algebraic integers.
source
Conjugates ​
conjugates Method
julia
conjugates(x::AbsSimpleNumFieldElem, C::AcbField) -> Vector{AcbFieldElem}
Compute the conjugates of x as elements of type AcbFieldElem. Recall that we order the complex conjugates σr+1(x),...,σr+2s(x) such that σi(x)=σi+s(x)― for r+1≤i≤r+s.
Let p be the precision of C, then every entry y of the vector returned satisfies radius(real(y)) < 2^-p and radius(imag(y)) < 2^-p respectively.
source
conjugates Method
julia
conjugates(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
Compute the conjugates of x as elements of type AcbFieldElem. Recall that we order the complex conjugates σr+1(x),...,σr+2s(x) such that σi(x)=σi+s(x)― for r+1≤i≤r+s.
Every entry y of the vector returned satisfies radius(real(y)) < 2^-abs_tol and radius(imag(y)) < 2^-abs_tol respectively.
source
conjugates_log Method
julia
conjugates_arb_log(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
Returns the elements (log⁡(|σ1(x)|),…,log⁡(|σr(x)|),…,2log⁡(|σr+1(x)|),…,2log⁡(|σr+s(x)|)) as elements of type ArbFieldElem with radius less then 2^-abs_tol.
source
conjugates_real Method
julia
conjugates_arb_real(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
Compute the real conjugates of x as elements of type ArbFieldElem.
Every entry y of the array returned satisfies radius(y) < 2^-abs_tol.
source
conjugates_complex Method
julia
conjugates_complex(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
Compute the complex conjugates of x as elements of type AcbFieldElem. Recall that we order the complex conjugates σr+1(x),...,σr+2s(x) such that σi(x)=σi+s(x)― for r+1≤i≤r+s.
Every entry y of the array returned satisfies radius(real(y)) < 2^-abs_tol and radius(imag(y)) < 2^-abs_tol.
source
conjugates_arb_log_normalise Method
julia
conjugates_arb_log_normalise(x::AbsSimpleNumFieldElem, p::Int = 10)
-conjugates_arb_log_normalise(x::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, p::Int = 10)
The "normalised" logarithms, i.e. the array cilog⁡|x(i)|−1/nlog⁡|N(x)|, so the (weighted) sum adds up to zero.
source
minkowski_map Method
julia
minkowski_map(a::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
Returns the image of a under the Minkowski embedding. Every entry of the array returned is of type ArbFieldElem with radius less then 2^(-abs_tol).
source
Predicates ​
is_integral Method
julia
is_integral(a::NumFieldElem) -> Bool
Returns whether a is integral, that is, whether the minimal polynomial of a has integral coefficients.
source
is_torsion_unit Method
julia
is_torsion_unit(x::AbsSimpleNumFieldElem, checkisunit::Bool = false) -> Bool
Returns whether x is a torsion unit, that is, whether there exists n such that xn=1.
If checkisunit is true, it is first checked whether x is a unit of the maximal order of the number field x is lying in.
source
is_local_norm Method
julia
is_local_norm(L::NumField, a::NumFieldElem, P)
Given a number field L/K, an element a∈K and a prime ideal P of K, returns whether a is a local norm at P.
The number field L/K must be a simple extension of degree 2.
source
is_norm_divisible Method
julia
is_norm_divisible(a::AbsSimpleNumFieldElem, n::ZZRingElem) -> Bool
Checks if the norm of a is divisible by n, assuming that the norm of a is an integer.
source
is_norm Method
julia
is_norm(K::AbsSimpleNumField, a::ZZRingElem; extra::Vector{ZZRingElem}) -> Bool, AbsSimpleNumFieldElem
For a ZZRingElem a, try to find T∈K s.th. N(T)=a holds. If successful, return true and T, otherwise false and some element. In \testtt{extra} one can pass in additional prime numbers that are allowed to occur in the solution. This will then be supplemented. The element will be returned in factored form.
source
Invariants ​
norm Method
julia
norm(a::NumFieldElem) -> NumFieldElem
Returns the norm of an element a of a number field extension L/K. This will be an element of K.
source
absolute_norm Method
julia
absolute_norm(a::NumFieldElem) -> QQFieldElem
Given a number field element a, returns the absolute norm of a.
source
minpoly Method
julia
minpoly(a::NumFieldElem) -> PolyRingElem
Given a number field element a of a number field K, this function returns the minimal polynomial of a over the base field of K.
source
absolute_minpoly Method
julia
absolute_minpoly(a::NumFieldElem) -> PolyRingElem
Given a number field element a of a number field K, this function returns the minimal polynomial of a over the rationals Q.
source
charpoly Method
julia
charpoly(a::NumFieldElem) -> PolyRingElem
Given a number field element a of a number field K, this function returns the characteristic polynomial of a over the base field of K.
source
absolute_charpoly Method
julia
absolute_charpoly(a::NumFieldElem) -> PolyRingElem
Given a number field element a of a number field K, this function returns the characteristic polynomial of a over the rationals Q.
source
norm Method
julia
norm(a::NumFieldElem, k::NumField) -> NumFieldElem
Returns the norm of an element a of a number field L with respect to a subfield k of L. This will be an element of k.
source

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+1//2*b^2 + b + a
quadratic_defect Method
julia
quadratic_defect(a::Union{NumFieldElem,Rational,QQFieldElem}, p) -> Union{Inf, PosInf}
Returns the valuation of the quadratic defect of the element a at p, which can either be prime object or an infinite place of the parent of a.
source
hilbert_symbol Method
julia
hilbert_symbol(a::NumFieldElem, b::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Int
Returns the local Hilbert symbol (a,b)p.
source
representation_matrix Method
julia
representation_matrix(a::NumFieldElem) -> MatElem
Returns the representation matrix of a, that is, the matrix representing multiplication with a with respect to the canonical basis of the parent of a.
source
basis_matrix Method
julia
basis_matrix(v::Vector{NumFieldElem}) -> Mat
Given a vector v of n elements of a number field K of degree d, this function returns an n×d matrix with entries in the base field of K, where row i contains the coefficients of v[i] with respect of the canonical basis of K.
source
coefficients Method
julia
coefficients(a::SimpleNumFieldElem, i::Int) -> Vector{FieldElem}
Given a number field element a of a simple number field extension L/K, this function returns the coefficients of a, when expanded in the canonical power basis of L.
source
coordinates Method
julia
coordinates(x::NumFieldElem{T}) -> Vector{T}
Given an element x in a number field K, this function returns the coordinates of x with respect to the basis of K (the output of the 'basis' function).
source
absolute_coordinates Method
julia
absolute_coordinates(x::NumFieldElem{T}) -> Vector{T}
Given an element x in a number field K, this function returns the coordinates of x with respect to the basis of K over the rationals (the output of the absolute_basis function).
source
coeff Method
julia
coeff(a::SimpleNumFieldElem, i::Int) -> FieldElem
Given a number field element a of a simple number field extension L/K, this function returns the i-th coefficient of a, when expanded in the canonical power basis of L. The result is an element of K.
source
valuation Method
julia
valuation(a::NumFieldElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
Computes the p-adic valuation of a, that is, the largest i such that a is contained in pi.
source
torsion_unit_order Method
julia
torsion_unit_order(x::AbsSimpleNumFieldElem, n::Int)
Given a torsion unit x together with a multiple n of its order, compute the order of x, that is, the smallest k∈Z≥1 such that xk=1.
It is not checked whether x is a torsion unit.
source
tr Method
julia
tr(a::NumFieldElem) -> NumFieldElem
Returns the trace of an element a of a number field extension L/K. This will be an element of K.
source
absolute_tr Method
julia
absolute_tr(a::NumFieldElem) -> QQFieldElem
Given a number field element a, returns the absolute trace of a.
source
algebraic_split Method
julia
algebraic_split(a::AbsSimpleNumFieldElem) -> AbsSimpleNumFieldElem, AbsSimpleNumFieldElem
Writes the input as a quotient of two "small" algebraic integers.
source
Conjugates ​
conjugates Method
julia
conjugates(x::AbsSimpleNumFieldElem, C::AcbField) -> Vector{AcbFieldElem}
Compute the conjugates of x as elements of type AcbFieldElem. Recall that we order the complex conjugates σr+1(x),...,σr+2s(x) such that σi(x)=σi+s(x)― for r+1≤i≤r+s.
Let p be the precision of C, then every entry y of the vector returned satisfies radius(real(y)) < 2^-p and radius(imag(y)) < 2^-p respectively.
source
conjugates Method
julia
conjugates(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
Compute the conjugates of x as elements of type AcbFieldElem. Recall that we order the complex conjugates σr+1(x),...,σr+2s(x) such that σi(x)=σi+s(x)― for r+1≤i≤r+s.
Every entry y of the vector returned satisfies radius(real(y)) < 2^-abs_tol and radius(imag(y)) < 2^-abs_tol respectively.
source
conjugates_log Method
julia
conjugates_arb_log(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
Returns the elements (log⁡(|σ1(x)|),…,log⁡(|σr(x)|),…,2log⁡(|σr+1(x)|),…,2log⁡(|σr+s(x)|)) as elements of type ArbFieldElem with radius less then 2^-abs_tol.
source
conjugates_real Method
julia
conjugates_arb_real(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
Compute the real conjugates of x as elements of type ArbFieldElem.
Every entry y of the array returned satisfies radius(y) < 2^-abs_tol.
source
conjugates_complex Method
julia
conjugates_complex(x::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{AcbFieldElem}
Compute the complex conjugates of x as elements of type AcbFieldElem. Recall that we order the complex conjugates σr+1(x),...,σr+2s(x) such that σi(x)=σi+s(x)― for r+1≤i≤r+s.
Every entry y of the array returned satisfies radius(real(y)) < 2^-abs_tol and radius(imag(y)) < 2^-abs_tol.
source
conjugates_arb_log_normalise Method
julia
conjugates_arb_log_normalise(x::AbsSimpleNumFieldElem, p::Int = 10)
+conjugates_arb_log_normalise(x::FacElem{AbsSimpleNumFieldElem, AbsSimpleNumField}, p::Int = 10)
The "normalised" logarithms, i.e. the array cilog⁡|x(i)|−1/nlog⁡|N(x)|, so the (weighted) sum adds up to zero.
source
minkowski_map Method
julia
minkowski_map(a::AbsSimpleNumFieldElem, abs_tol::Int) -> Vector{ArbFieldElem}
Returns the image of a under the Minkowski embedding. Every entry of the array returned is of type ArbFieldElem with radius less then 2^(-abs_tol).
source
Predicates ​
is_integral Method
julia
is_integral(a::NumFieldElem) -> Bool
Returns whether a is integral, that is, whether the minimal polynomial of a has integral coefficients.
source
is_torsion_unit Method
julia
is_torsion_unit(x::AbsSimpleNumFieldElem, checkisunit::Bool = false) -> Bool
Returns whether x is a torsion unit, that is, whether there exists n such that xn=1.
If checkisunit is true, it is first checked whether x is a unit of the maximal order of the number field x is lying in.
source
is_local_norm Method
julia
is_local_norm(L::NumField, a::NumFieldElem, P)
Given a number field L/K, an element a∈K and a prime ideal P of K, returns whether a is a local norm at P.
The number field L/K must be a simple extension of degree 2.
source
is_norm_divisible Method
julia
is_norm_divisible(a::AbsSimpleNumFieldElem, n::ZZRingElem) -> Bool
Checks if the norm of a is divisible by n, assuming that the norm of a is an integer.
source
is_norm Method
julia
is_norm(K::AbsSimpleNumField, a::ZZRingElem; extra::Vector{ZZRingElem}) -> Bool, AbsSimpleNumFieldElem
For a ZZRingElem a, try to find T∈K s.th. N(T)=a holds. If successful, return true and T, otherwise false and some element. In \testtt{extra} one can pass in additional prime numbers that are allowed to occur in the solution. This will then be supplemented. The element will be returned in factored form.
source
Invariants ​
norm Method
julia
norm(a::NumFieldElem) -> NumFieldElem
Returns the norm of an element a of a number field extension L/K. This will be an element of K.
source
absolute_norm Method
julia
absolute_norm(a::NumFieldElem) -> QQFieldElem
Given a number field element a, returns the absolute norm of a.
source
minpoly Method
julia
minpoly(a::NumFieldElem) -> PolyRingElem
Given a number field element a of a number field K, this function returns the minimal polynomial of a over the base field of K.
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absolute_minpoly Method
julia
absolute_minpoly(a::NumFieldElem) -> PolyRingElem
Given a number field element a of a number field K, this function returns the minimal polynomial of a over the rationals Q.
source
charpoly Method
julia
charpoly(a::NumFieldElem) -> PolyRingElem
Given a number field element a of a number field K, this function returns the characteristic polynomial of a over the base field of K.
source
absolute_charpoly Method
julia
absolute_charpoly(a::NumFieldElem) -> PolyRingElem
Given a number field element a of a number field K, this function returns the characteristic polynomial of a over the rationals Q.
source
norm Method
julia
norm(a::NumFieldElem, k::NumField) -> NumFieldElem
Returns the norm of an element a of a number field L with respect to a subfield k of L. This will be an element of k.
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     Number field operations | Hecke
     
     
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Number field operations ​
Creation of number fields ​
General number fields can be created using the function number_field. To create a simple number field given by a defining polynomial or a non-simple number field given by defining polynomials, the following functions can be used.
number_field Method
julia
number_field(f::Poly{NumFieldElem}, s::VarName;
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Number field operations ​
Creation of number fields ​
General number fields can be created using the function number_field. To create a simple number field given by a defining polynomial or a non-simple number field given by defining polynomials, the following functions can be used.
number_field Method
julia
number_field(f::Poly{NumFieldElem}, s::VarName;
             cached::Bool = false, check::Bool = false) -> NumField, NumFieldElem
Given an irreducible polynomial f∈K[x] over some number field K, this function creates the simple number field L=K[x]/(f) and returns (L,b), where b is the class of x in L. The string s is used only for printing the primitive element b.
  • check: Controls whether irreducibility of f is checked.
  • cached: Controls whether the result is cached.
Examples
julia
julia> K, a = quadratic_field(5);
 
 julia> Kt, t = K["t"];
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 julia> complex_places(K)
 1-element Vector{InfPlc{AbsSimpleNumField, AbsSimpleNumFieldEmbedding}}:
- Infinite place corresponding to (Complex embedding corresponding to 0.00 + 2.24 * i of imaginary quadratic field)
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isreal Method
julia
isreal(P::Plc)
Return whether the embedding into C defined by P is real or not.
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is_complex Method
julia
is_complex(P::Plc) -> Bool
Return whether the embedding into C defined by P is complex or not.
source
Miscellaneous ​
norm_equation Method
julia
norm_equation(K::AnticNumerField, a) -> AbsSimpleNumFieldElem
For a an integer or rational, try to find T∈K s.th. N(T)=a. Raises an error if unsuccessful.
source
lorenz_module Method
julia
lorenz_module(k::AbsSimpleNumField, n::Int) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
Finds an ideal A s.th. for all positive units e=1modA we have that e is an n-th power. Uses Lorenz, number theory, 9.3.1. If containing is set, it has to be an integral ideal. The resulting ideal will be a multiple of this.
source
kummer_failure Method
julia
kummer_failure(x::AbsSimpleNumFieldElem, M::Int, N::Int) -> Int
Computes the quotient of N and [K(ζM,(Nx)):K(ζM)], where K is the field containing x and N divides M.
source
is_defining_polynomial_nice Method
julia
is_defining_polynomial_nice(K::AbsSimpleNumField)
Tests if the defining polynomial of K is integral and monic.
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isreal Method
julia
isreal(P::Plc)
Return whether the embedding into C defined by P is real or not.
source
is_complex Method
julia
is_complex(P::Plc) -> Bool
Return whether the embedding into C defined by P is complex or not.
source
Miscellaneous ​
norm_equation Method
julia
norm_equation(K::AnticNumerField, a) -> AbsSimpleNumFieldElem
For a an integer or rational, try to find T∈K s.th. N(T)=a. Raises an error if unsuccessful.
source
lorenz_module Method
julia
lorenz_module(k::AbsSimpleNumField, n::Int) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
Finds an ideal A s.th. for all positive units e=1modA we have that e is an n-th power. Uses Lorenz, number theory, 9.3.1. If containing is set, it has to be an integral ideal. The resulting ideal will be a multiple of this.
source
kummer_failure Method
julia
kummer_failure(x::AbsSimpleNumFieldElem, M::Int, N::Int) -> Int
Computes the quotient of N and [K(ζM,(Nx)):K(ζM)], where K is the field containing x and N divides M.
source
is_defining_polynomial_nice Method
julia
is_defining_polynomial_nice(K::AbsSimpleNumField)
Tests if the defining polynomial of K is integral and monic.
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     Internals | Hecke
     
     
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Internals ​
Types of number fields ​
Number fields, in Hecke, come in several different types:
  • AbsSimpleNumField: a finite simple extension of the rational numbers Q
  • AbsNonSimpleNumField: a finite extension of Q given by several polynomials. We will refer to this as a non-simple field - even though mathematically we can find a primitive elements.
  • RelSimpleNumField: a finite simple extension of a number field. This is actually parametried by the (element) type of the coefficient field. The complete type of an extension of an absolute field (AbsSimpleNumField) is RelSimpleNumField{AbsSimpleNumFieldElem}. The next extension thus will be RelSimpleNumField{RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}.
  • RelNonSimpleNumField: extensions of number fields given by several polynomials. This too will be referred to as a non-simple field.
The simple types AbsSimpleNumField and RelSimpleNumField are also called simple fields in the rest of this document, RelSimpleNumField and RelNonSimpleNumField are referred to as relative extensions while AbsSimpleNumField and AbsNonSimpleNumField are called absolute.
Internally, simple fields are essentially just (univariate) polynomial quotients in a dense representation, while non-simple fields are multivariate quotient rings, thus have a sparse presentation. In general, simple fields allow much faster arithmetic, while the non-simple fields give easy access to large degree fields.
Absolute simple fields ​
The most basic number field type is that of AbsSimpleNumField. Internally this is essentially represented as a unvariate quotient with the arithmetic provided by the C-library antic with the binding provided by Nemo.

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Internals ​
Types of number fields ​
Number fields, in Hecke, come in several different types:
  • AbsSimpleNumField: a finite simple extension of the rational numbers Q
  • AbsNonSimpleNumField: a finite extension of Q given by several polynomials. We will refer to this as a non-simple field - even though mathematically we can find a primitive elements.
  • RelSimpleNumField: a finite simple extension of a number field. This is actually parametried by the (element) type of the coefficient field. The complete type of an extension of an absolute field (AbsSimpleNumField) is RelSimpleNumField{AbsSimpleNumFieldElem}. The next extension thus will be RelSimpleNumField{RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}.
  • RelNonSimpleNumField: extensions of number fields given by several polynomials. This too will be referred to as a non-simple field.
The simple types AbsSimpleNumField and RelSimpleNumField are also called simple fields in the rest of this document, RelSimpleNumField and RelNonSimpleNumField are referred to as relative extensions while AbsSimpleNumField and AbsNonSimpleNumField are called absolute.
Internally, simple fields are essentially just (univariate) polynomial quotients in a dense representation, while non-simple fields are multivariate quotient rings, thus have a sparse presentation. In general, simple fields allow much faster arithmetic, while the non-simple fields give easy access to large degree fields.
Absolute simple fields ​
The most basic number field type is that of AbsSimpleNumField. Internally this is essentially represented as a unvariate quotient with the arithmetic provided by the C-library antic with the binding provided by Nemo.

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diff --git a/v0.34.8/manual/number_fields/intro.html b/v0.34.8/manual/number_fields/intro.html
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     Introduction | Hecke
     
     
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Introduction ​
By definition, mathematically a number field is just a finite extension of the rational Q. In Hecke, a number field L is recursively defined as being the field of rational numbers Q or a finite extension of a number field K. In the second case, the extension can be defined in the one of the following two ways:
  • We have L=K[x]/(f), where f∈K[x] is an irreducible polynomial (simple extension), or
  • We have L=K[x1,…,xn]/(f1(x1),…,fn(xn)), where f1,…,fn∈K[x] are univariate polynomials (non-simple extension).
In both cases we refer to K as the base field of the number field L. Another useful dichotomy comes from the type of the base field. We call L an absolute number field, if the base field is equal to the rational numbers Q.

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Introduction ​
By definition, mathematically a number field is just a finite extension of the rational Q. In Hecke, a number field L is recursively defined as being the field of rational numbers Q or a finite extension of a number field K. In the second case, the extension can be defined in the one of the following two ways:
  • We have L=K[x]/(f), where f∈K[x] is an irreducible polynomial (simple extension), or
  • We have L=K[x1,…,xn]/(f1(x1),…,fn(xn)), where f1,…,fn∈K[x] are univariate polynomials (non-simple extension).
In both cases we refer to K as the base field of the number field L. Another useful dichotomy comes from the type of the base field. We call L an absolute number field, if the base field is equal to the rational numbers Q.

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     Elements | Hecke
     
     
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Elements ​
Elements in orders have two representations: they can be viewed as elements in the Zn giving the coefficients wrt to the order basis where they are elements in. On the other hand, as every order is in a field, they also have a representation as number field elements. Since, asymptotically, operations are more efficient in the field (due to fast polynomial arithmetic) than in the order, the primary representation is that as a field element.
Creation ​
Elements are constructed either as linear combinations of basis elements or via explicit coercion. Elements will be of type AbsNumFieldOrderElem, the type if actually parametrized by the type of the surrounding field and the type of the field elements. E.g. the type of any element in any order of an absolute simple field will be AbsSimpleNumFieldOrderElem
AbsNumFieldOrder Type
julia
  (O::NumFieldOrder)(a::NumFieldElem, check::Bool = true) -> NumFieldOrderElem
Given an element a of the ambient number field of O, this function coerces the element into O. It will be checked that a is contained in O if and only if check is true.
source
julia
  (O::NumFieldOrder)(a::NumFieldOrderElem, check::Bool = true) -> NumFieldOrderElem
Given an element a of some order in the ambient number field of O, this function coerces the element into O. It will be checked that a is contained in O if and only if check is true.
source
julia
  (O::NumFieldOrder)(a::IntegerUnion) -> NumFieldOrderElem
Given an element a of type ZZRingElem or Integer, this function coerces the element into O.
source
julia
  (O::AbsNumFieldOrder)(arr::Vector{ZZRingElem})
Returns the element of O with coefficient vector arr.
source
julia
  (O::AbsNumFieldOrder)(arr::Vector{Integer})
Returns the element of O with coefficient vector arr.
source
Basic properties ​
parent Method
julia
parent(a::NumFieldOrderElem) -> NumFieldOrder
Returns the order of which a is an element.
source
elem_in_nf Method
julia
elem_in_nf(a::NumFieldOrderElem) -> NumFieldElem
Returns the element a considered as an element of the ambient number field.
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coordinates Method
julia
coordinates(a::AbsNumFieldOrderElem) -> Vector{ZZRingElem}
Returns the coefficient vector of a with respect to the basis of the order.
source
discriminant Method
julia
discriminant(B::Vector{NumFieldOrderElem})
Returns the discriminant of the family B of algebraic numbers, i.e. det((tr(B[i]∗B[j]))i,j)2.
source
julia
discriminant(E::EllipticCurve) -> FieldElem
Return the discriminant of E.
source
julia
discriminant(C::HypellCrv{T}) -> T
Compute the discriminant of C.
source
julia
discriminant(O::AlgssRelOrd)
Returns the discriminant of O.
source
== Method
julia
==(x::NumFieldOrderElem, y::NumFieldOrderElem) -> Bool
Returns whether x and y are equal.
source
Arithmetic ​
All the usual arithmetic operatinos are defined:
  • -(::NUmFieldOrdElem)
  • +(::NumFieldOrderElem, ::NumFieldOrderElem)
  • -(::NumFieldOrderElem, ::NumFieldOrderElem)
  • *(::NumFieldOrderElem, ::NumFieldOrderElem)
  • ^(::NumFieldOrderElem, ::Int)
  • mod(::AbsNumFieldOrderElem, ::Int)
  • mod_sym(::NumFieldOrderElem, ::ZZRingElem)
  • powermod(::AbsNumFieldOrderElem, ::ZZRingElem, ::Int)
Miscellaneous ​
representation_matrix Method
julia
representation_matrix(a::AbsNumFieldOrderElem) -> ZZMatrix
Returns the representation matrix of the element a.
source
representation_matrix Method
julia
representation_matrix(a::AbsNumFieldOrderElem, K::AbsSimpleNumField) -> FakeFmpqMat
Returns the representation matrix of the element a considered as an element of the ambient number field K. It is assumed that K is the ambient number field of the order of a.
source
tr Method
julia
tr(a::NumFieldOrderElem)
Returns the trace of a as an element of the base ring.
source
norm Method
julia
norm(a::NumFieldOrderElem)
Returns the norm of a as an element in the base ring.
source
absolute_norm Method
julia
absolute_norm(a::NumFieldOrderElem) -> ZZRingElem
Return the absolute norm as an integer.
source
absolute_tr Method
julia
absolute_tr(a::NumFieldOrderElem) -> ZZRingElem
Return the absolute trace as an integer.
source
rand Method
julia
rand(O::AbsSimpleNumFieldOrder, n::IntegerUnion) -> AbsNumFieldOrderElem
Computes a coefficient vector with entries uniformly distributed in {−n,…,−1,0,1,…,n} and returns the corresponding element of the order O.
source
minkowski_map Method
julia
minkowski_map(a::NumFieldOrderElem, abs_tol::Int) -> Vector{ArbFieldElem}
Returns the image of a under the Minkowski embedding. Every entry of the array returned is of type ArbFieldElem with radius less then 2^-abs_tol.
source
conjugates_arb Method
julia
conjugates_arb(x::NumFieldOrderElem, abs_tol::Int) -> Vector{AcbFieldElem}
Compute the conjugates of x as elements of type AcbFieldElem. Recall that we order the complex conjugates σr+1(x),...,σr+2s(x) such that σi(x)=σi+s(x)― for r+2≤i≤r+s.
Every entry y of the array returned satisfies radius(real(y)) < 2^-abs_tol, radius(imag(y)) < 2^-abs_tol respectively.
source
conjugates_arb_log Method
julia
conjugates_arb_log(x::NumFieldOrderElem, abs_tol::Int) -> Vector{ArbFieldElem}
Returns the elements (log⁡(|σ1(x)|),…,log⁡(|σr(x)|),…,2log⁡(|σr+1(x)|),…,2log⁡(|σr+s(x)|)) as elements of type ArbFieldElem radius less then 2^-abs_tol.
source
t2 Method
julia
t2(x::NumFieldOrderElem, abs_tol::Int = 32) -> ArbFieldElem
Return the T2-norm of x. The radius of the result will be less than 2^-abs_tol.
source
minpoly Method
julia
minpoly(a::AbsNumFieldOrderElem) -> ZZPolyRingElem
The minimal polynomial of a.
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charpoly Method
julia
charpoly(a::AbsNumFieldOrderElem) -> ZZPolyRingElem
-charpoly(a::AbsNumFieldOrderElem, ZZ) -> ZZPolyRingElem
The characteristic polynomial of a.
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factor Method
julia
factor(a::AbsSimpleNumFieldOrderElem) -> Fac{AbsSimpleNumFieldOrderElem}
Computes a factorization of a into irreducible elements. The return value is a factorization fac, which satisfies a = unit(fac) * prod(p^e for (p, e) in fac).
The function requires that a is non-zero and that all prime ideals containing a are principal, which is for example satisfied if class group of the order of a is trivial.
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denominator Method
julia
denominator(a::NumFieldElem, O::AbsSimpleNumFieldOrder) -> ZZRingElem
Returns the smallest positive integer k such that k⋅a is contained in O.
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discriminant Method
julia
discriminant(B::Vector{NumFieldOrderElem})
Returns the discriminant of the family B of algebraic numbers, i.e. det((tr(B[i]∗B[j]))i,j)2.
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julia
discriminant(E::EllipticCurve) -> FieldElem
Return the discriminant of E.
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julia
discriminant(C::HypellCrv{T}) -> T
Compute the discriminant of C.
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julia
discriminant(O::AlgssRelOrd)
Returns the discriminant of O.
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Elements ​
Elements in orders have two representations: they can be viewed as elements in the Zn giving the coefficients wrt to the order basis where they are elements in. On the other hand, as every order is in a field, they also have a representation as number field elements. Since, asymptotically, operations are more efficient in the field (due to fast polynomial arithmetic) than in the order, the primary representation is that as a field element.
Creation ​
Elements are constructed either as linear combinations of basis elements or via explicit coercion. Elements will be of type AbsNumFieldOrderElem, the type if actually parametrized by the type of the surrounding field and the type of the field elements. E.g. the type of any element in any order of an absolute simple field will be AbsSimpleNumFieldOrderElem
AbsNumFieldOrder Type
julia
  (O::NumFieldOrder)(a::NumFieldElem, check::Bool = true) -> NumFieldOrderElem
Given an element a of the ambient number field of O, this function coerces the element into O. It will be checked that a is contained in O if and only if check is true.
source
julia
  (O::NumFieldOrder)(a::NumFieldOrderElem, check::Bool = true) -> NumFieldOrderElem
Given an element a of some order in the ambient number field of O, this function coerces the element into O. It will be checked that a is contained in O if and only if check is true.
source
julia
  (O::NumFieldOrder)(a::IntegerUnion) -> NumFieldOrderElem
Given an element a of type ZZRingElem or Integer, this function coerces the element into O.
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julia
  (O::AbsNumFieldOrder)(arr::Vector{ZZRingElem})
Returns the element of O with coefficient vector arr.
source
julia
  (O::AbsNumFieldOrder)(arr::Vector{Integer})
Returns the element of O with coefficient vector arr.
source
Basic properties ​
parent Method
julia
parent(a::NumFieldOrderElem) -> NumFieldOrder
Returns the order of which a is an element.
source
elem_in_nf Method
julia
elem_in_nf(a::NumFieldOrderElem) -> NumFieldElem
Returns the element a considered as an element of the ambient number field.
source
coordinates Method
julia
coordinates(a::AbsNumFieldOrderElem) -> Vector{ZZRingElem}
Returns the coefficient vector of a with respect to the basis of the order.
source
discriminant Method
julia
discriminant(B::Vector{NumFieldOrderElem})
Returns the discriminant of the family B of algebraic numbers, i.e. det((tr(B[i]∗B[j]))i,j)2.
source
julia
discriminant(E::EllipticCurve) -> FieldElem
Return the discriminant of E.
source
julia
discriminant(C::HypellCrv{T}) -> T
Compute the discriminant of C.
source
julia
discriminant(O::AlgssRelOrd)
Returns the discriminant of O.
source
== Method
julia
==(x::NumFieldOrderElem, y::NumFieldOrderElem) -> Bool
Returns whether x and y are equal.
source
Arithmetic ​
All the usual arithmetic operatinos are defined:
  • -(::NUmFieldOrdElem)
  • +(::NumFieldOrderElem, ::NumFieldOrderElem)
  • -(::NumFieldOrderElem, ::NumFieldOrderElem)
  • *(::NumFieldOrderElem, ::NumFieldOrderElem)
  • ^(::NumFieldOrderElem, ::Int)
  • mod(::AbsNumFieldOrderElem, ::Int)
  • mod_sym(::NumFieldOrderElem, ::ZZRingElem)
  • powermod(::AbsNumFieldOrderElem, ::ZZRingElem, ::Int)
Miscellaneous ​
representation_matrix Method
julia
representation_matrix(a::AbsNumFieldOrderElem) -> ZZMatrix
Returns the representation matrix of the element a.
source
representation_matrix Method
julia
representation_matrix(a::AbsNumFieldOrderElem, K::AbsSimpleNumField) -> FakeFmpqMat
Returns the representation matrix of the element a considered as an element of the ambient number field K. It is assumed that K is the ambient number field of the order of a.
source
tr Method
julia
tr(a::NumFieldOrderElem)
Returns the trace of a as an element of the base ring.
source
norm Method
julia
norm(a::NumFieldOrderElem)
Returns the norm of a as an element in the base ring.
source
absolute_norm Method
julia
absolute_norm(a::NumFieldOrderElem) -> ZZRingElem
Return the absolute norm as an integer.
source
absolute_tr Method
julia
absolute_tr(a::NumFieldOrderElem) -> ZZRingElem
Return the absolute trace as an integer.
source
rand Method
julia
rand(O::AbsSimpleNumFieldOrder, n::IntegerUnion) -> AbsNumFieldOrderElem
Computes a coefficient vector with entries uniformly distributed in {−n,…,−1,0,1,…,n} and returns the corresponding element of the order O.
source
minkowski_map Method
julia
minkowski_map(a::NumFieldOrderElem, abs_tol::Int) -> Vector{ArbFieldElem}
Returns the image of a under the Minkowski embedding. Every entry of the array returned is of type ArbFieldElem with radius less then 2^-abs_tol.
source
conjugates_arb Method
julia
conjugates_arb(x::NumFieldOrderElem, abs_tol::Int) -> Vector{AcbFieldElem}
Compute the conjugates of x as elements of type AcbFieldElem. Recall that we order the complex conjugates σr+1(x),...,σr+2s(x) such that σi(x)=σi+s(x)― for r+2≤i≤r+s.
Every entry y of the array returned satisfies radius(real(y)) < 2^-abs_tol, radius(imag(y)) < 2^-abs_tol respectively.
source
conjugates_arb_log Method
julia
conjugates_arb_log(x::NumFieldOrderElem, abs_tol::Int) -> Vector{ArbFieldElem}
Returns the elements (log⁡(|σ1(x)|),…,log⁡(|σr(x)|),…,2log⁡(|σr+1(x)|),…,2log⁡(|σr+s(x)|)) as elements of type ArbFieldElem radius less then 2^-abs_tol.
source
t2 Method
julia
t2(x::NumFieldOrderElem, abs_tol::Int = 32) -> ArbFieldElem
Return the T2-norm of x. The radius of the result will be less than 2^-abs_tol.
source
minpoly Method
julia
minpoly(a::AbsNumFieldOrderElem) -> ZZPolyRingElem
The minimal polynomial of a.
source
charpoly Method
julia
charpoly(a::AbsNumFieldOrderElem) -> ZZPolyRingElem
+charpoly(a::AbsNumFieldOrderElem, ZZ) -> ZZPolyRingElem
The characteristic polynomial of a.
source
factor Method
julia
factor(a::AbsSimpleNumFieldOrderElem) -> Fac{AbsSimpleNumFieldOrderElem}
Computes a factorization of a into irreducible elements. The return value is a factorization fac, which satisfies a = unit(fac) * prod(p^e for (p, e) in fac).
The function requires that a is non-zero and that all prime ideals containing a are principal, which is for example satisfied if class group of the order of a is trivial.
source
denominator Method
julia
denominator(a::NumFieldElem, O::AbsSimpleNumFieldOrder) -> ZZRingElem
Returns the smallest positive integer k such that k⋅a is contained in O.
source
discriminant Method
julia
discriminant(B::Vector{NumFieldOrderElem})
Returns the discriminant of the family B of algebraic numbers, i.e. det((tr(B[i]∗B[j]))i,j)2.
source
julia
discriminant(E::EllipticCurve) -> FieldElem
Return the discriminant of E.
source
julia
discriminant(C::HypellCrv{T}) -> T
Compute the discriminant of C.
source
julia
discriminant(O::AlgssRelOrd)
Returns the discriminant of O.
source

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diff --git a/v0.34.8/manual/orders/frac_ideals.html b/v0.34.8/manual/orders/frac_ideals.html
index ad36421903..97e9ae7235 100644
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     Fractional ideals | Hecke
     
     
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Hecke
Appearance
Fractional ideals ​
A fractional ideal in the number field K is a ZK-module A such that there exists an integer d>0 which dA is an (integral) ideal in ZK. Due to the Dedekind property of ZK, the ideals for a multiplicative group.
Fractional ideals are represented as an integral ideal and an additional denominator. They are of type AbsSimpleNumFieldOrderFractionalIdeal.
Creation ​
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, M::ZZMatrix, b::ZZRingElem; M_in_hnf::Bool = false) -> AbsNumFieldOrderFractionalIdeal
Creates the fractional ideal of O with basis matrix M/b. If M_in_hnf is set, then it is assumed that A is already in lower left HNF.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, M::ZZMatrix, b::ZZRingElem; M_in_hnf::Bool = false) -> AbsNumFieldOrderFractionalIdeal
Creates the fractional ideal of O with basis matrix M/b. If M_in_hnf is set, then it is assumed that A is already in lower left HNF.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, M::QQMatrix) -> AbsNumFieldOrderFractionalIdeal
Creates the fractional ideal of O generated by the elements corresponding to the rows of M.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsSimpleNumFieldOrder, I::AbsNumFieldOrderIdeal) -> AbsSimpleNumFieldOrderFractionalIdeal
The fractional ideal of O generated by a Z-basis of I.
source
julia
fractional_ideal(O::AbsNumFieldOrder, I::AbsNumFieldOrderIdeal) -> AbsNumFieldOrderFractionalIdeal
Turns the ideal I into a fractional ideal of O.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, I::AbsNumFieldOrderIdeal, b::ZZRingElem) -> AbsNumFieldOrderFractionalIdeal
Creates the fractional ideal I/b of O.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, a::AbsSimpleNumFieldElem) -> AbsNumFieldOrderFractionalIdeal
Creates the principal fractional ideal (a) of O.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, a::AbsNumFieldOrderElem) -> AbsNumFieldOrderFractionalIdeal
Creates the principal fractional ideal (a) of O.
source
inv Method
julia
inv(A::AbsNumFieldOrderIdeal) -> AbsSimpleNumFieldOrderFractionalIdeal
Computes the inverse of A, that is, the fractional ideal B such that AB=OK.
source
Arithmetic ​
All the normal operations are provided as well.
inv Method
julia
inv(A::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrderFractionalIdeal
Returns the fractional ideal B such that AB=O.
source
integral_split Method
julia
integral_split(A::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrderIdeal, AbsNumFieldOrderIdeal
Computes the unique coprime integral ideals N and D s.th. A=ND−1
source
numerator Method
julia
numerator(a::RelNumFieldOrderFractionalIdeal) -> RelNumFieldOrderIdeal
Returns the ideal d∗a where d is the denominator of a.
source
denominator Method
julia
denominator(a::RelNumFieldOrderFractionalIdeal) -> ZZRingElem
Returns the smallest positive integer d such that da is contained in the order of a.
source
Miscaellenous ​
order Method
julia
order(a::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrder
The order that was used to define the ideal a.
source
basis_matrix Method
julia
basis_matrix(I::AbsNumFieldOrderFractionalIdeal) -> FakeFmpqMat
Returns the basis matrix of I with respect to the basis of the order.
source
basis_mat_inv Method
julia
basis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat
Return the inverse of the basis matrix of A.
source
basis Method
julia
basis(I::AbsNumFieldOrderFractionalIdeal) -> Vector{AbsSimpleNumFieldElem}
Returns the Z-basis of I.
source
norm Method
julia
norm(I::AbsNumFieldOrderFractionalIdeal) -> QQFieldElem
Returns the norm of I.
source
julia
norm(a::RelNumFieldOrderIdeal) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
Returns the norm of a.
source
julia
norm(a::RelNumFieldOrderFractionalIdeal{T, S}) -> S
Returns the norm of a.
source
julia
norm(a::AlgAssAbsOrdIdl, O::AlgAssAbsOrd; copy::Bool = true) -> QQFieldElem
Returns the norm of a considered as an (possibly fractional) ideal of O.
source
julia
norm(a::AlgAssRelOrdIdl{S, T, U}, O::AlgAssRelOrd{S, T, U}; copy::Bool = true)
-  where { S, T, U } -> T
Returns the norm of a considered as an (possibly fractional) ideal of O.
source

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Hecke
Appearance
Fractional ideals ​
A fractional ideal in the number field K is a ZK-module A such that there exists an integer d>0 which dA is an (integral) ideal in ZK. Due to the Dedekind property of ZK, the ideals for a multiplicative group.
Fractional ideals are represented as an integral ideal and an additional denominator. They are of type AbsSimpleNumFieldOrderFractionalIdeal.
Creation ​
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, M::ZZMatrix, b::ZZRingElem; M_in_hnf::Bool = false) -> AbsNumFieldOrderFractionalIdeal
Creates the fractional ideal of O with basis matrix M/b. If M_in_hnf is set, then it is assumed that A is already in lower left HNF.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, M::ZZMatrix, b::ZZRingElem; M_in_hnf::Bool = false) -> AbsNumFieldOrderFractionalIdeal
Creates the fractional ideal of O with basis matrix M/b. If M_in_hnf is set, then it is assumed that A is already in lower left HNF.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, M::QQMatrix) -> AbsNumFieldOrderFractionalIdeal
Creates the fractional ideal of O generated by the elements corresponding to the rows of M.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsSimpleNumFieldOrder, I::AbsNumFieldOrderIdeal) -> AbsSimpleNumFieldOrderFractionalIdeal
The fractional ideal of O generated by a Z-basis of I.
source
julia
fractional_ideal(O::AbsNumFieldOrder, I::AbsNumFieldOrderIdeal) -> AbsNumFieldOrderFractionalIdeal
Turns the ideal I into a fractional ideal of O.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, I::AbsNumFieldOrderIdeal, b::ZZRingElem) -> AbsNumFieldOrderFractionalIdeal
Creates the fractional ideal I/b of O.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, a::AbsSimpleNumFieldElem) -> AbsNumFieldOrderFractionalIdeal
Creates the principal fractional ideal (a) of O.
source
fractional_ideal Method
julia
fractional_ideal(O::AbsNumFieldOrder, a::AbsNumFieldOrderElem) -> AbsNumFieldOrderFractionalIdeal
Creates the principal fractional ideal (a) of O.
source
inv Method
julia
inv(A::AbsNumFieldOrderIdeal) -> AbsSimpleNumFieldOrderFractionalIdeal
Computes the inverse of A, that is, the fractional ideal B such that AB=OK.
source
Arithmetic ​
All the normal operations are provided as well.
inv Method
julia
inv(A::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrderFractionalIdeal
Returns the fractional ideal B such that AB=O.
source
integral_split Method
julia
integral_split(A::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrderIdeal, AbsNumFieldOrderIdeal
Computes the unique coprime integral ideals N and D s.th. A=ND−1
source
numerator Method
julia
numerator(a::RelNumFieldOrderFractionalIdeal) -> RelNumFieldOrderIdeal
Returns the ideal d∗a where d is the denominator of a.
source
denominator Method
julia
denominator(a::RelNumFieldOrderFractionalIdeal) -> ZZRingElem
Returns the smallest positive integer d such that da is contained in the order of a.
source
Miscaellenous ​
order Method
julia
order(a::AbsNumFieldOrderFractionalIdeal) -> AbsNumFieldOrder
The order that was used to define the ideal a.
source
basis_matrix Method
julia
basis_matrix(I::AbsNumFieldOrderFractionalIdeal) -> FakeFmpqMat
Returns the basis matrix of I with respect to the basis of the order.
source
basis_mat_inv Method
julia
basis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat
Return the inverse of the basis matrix of A.
source
basis Method
julia
basis(I::AbsNumFieldOrderFractionalIdeal) -> Vector{AbsSimpleNumFieldElem}
Returns the Z-basis of I.
source
norm Method
julia
norm(I::AbsNumFieldOrderFractionalIdeal) -> QQFieldElem
Returns the norm of I.
source
julia
norm(a::RelNumFieldOrderIdeal) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
Returns the norm of a.
source
julia
norm(a::RelNumFieldOrderFractionalIdeal{T, S}) -> S
Returns the norm of a.
source
julia
norm(a::AlgAssAbsOrdIdl, O::AlgAssAbsOrd; copy::Bool = true) -> QQFieldElem
Returns the norm of a considered as an (possibly fractional) ideal of O.
source
julia
norm(a::AlgAssRelOrdIdl{S, T, U}, O::AlgAssRelOrd{S, T, U}; copy::Bool = true)
+  where { S, T, U } -> T
Returns the norm of a considered as an (possibly fractional) ideal of O.
source

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diff --git a/v0.34.8/manual/orders/ideals.html b/v0.34.8/manual/orders/ideals.html
index 9729b9e681..fc46ab7823 100644
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     Ideals | Hecke
     
     
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Hecke
Appearance
Ideals ​
(Integral) ideals in orders are always free Z-module of the same rank as the order, hence have a representation via a Z-basis. This can be made unique by normalising the corresponding matrix to be in reduced row echelon form (HNF).
For ideals in maximal orders ZK, we also have a second presentation coming from the ZK module structure and the fact that ZK is a Dedekind ring: ideals can be generated by 2 elements, one of which can be any non-zero element in the ideal.
For efficiency, we will choose the 1st generator to be an integer.
Ideals here are of type AbsNumFieldOrderIdeal, which is, similar to the elements above, also indexed by the type of the field and their elements: AbsNumFieldOrderIdeal{AbsSimpleNumField,AbsSimpleNumFieldElem} for ideals in simple absolute fields.
Different to elements, the parentof an ideal is the set of all ideals in the ring, of type AbsNumFieldOrderIdealSet.
Creation ​
ideal Method
julia
ideal(O::AbsSimpleNumFieldOrder, a::ZZRingElem) -> AbsNumFieldOrderIdeal
+    
Hecke
Appearance
Ideals ​
(Integral) ideals in orders are always free Z-module of the same rank as the order, hence have a representation via a Z-basis. This can be made unique by normalising the corresponding matrix to be in reduced row echelon form (HNF).
For ideals in maximal orders ZK, we also have a second presentation coming from the ZK module structure and the fact that ZK is a Dedekind ring: ideals can be generated by 2 elements, one of which can be any non-zero element in the ideal.
For efficiency, we will choose the 1st generator to be an integer.
Ideals here are of type AbsNumFieldOrderIdeal, which is, similar to the elements above, also indexed by the type of the field and their elements: AbsNumFieldOrderIdeal{AbsSimpleNumField,AbsSimpleNumFieldElem} for ideals in simple absolute fields.
Different to elements, the parentof an ideal is the set of all ideals in the ring, of type AbsNumFieldOrderIdealSet.
Creation ​
ideal Method
julia
ideal(O::AbsSimpleNumFieldOrder, a::ZZRingElem) -> AbsNumFieldOrderIdeal
 ideal(O::AbsSimpleNumFieldOrder, a::Integer) -> AbsNumFieldOrderIdeal
Returns the ideal of O which is generated by a.
source
ideal Method
julia
ideal(O::AbsSimpleNumFieldOrder, M::ZZMatrix; check::Bool = false, M_in_hnf::Bool = false) -> AbsNumFieldOrderIdeal
Creates the ideal of O with basis matrix M. If check is set, then it is checked whether M defines an ideal (expensive). If M_in_hnf is set, then it is assumed that M is already in lower left HNF.
source
ideal Method
julia
ideal(O::AbsSimpleNumFieldOrder, x::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
Creates the principal ideal (x) of O.
source
ideal Method
julia
ideal(O::AbsSimpleNumFieldOrder, x::ZZRingElem, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
 ideal(O::AbsSimpleNumFieldOrder, x::Integer, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
Creates the ideal (x,y) of O.
source
ideal Method
julia
ideal(O::AbsSimpleNumFieldOrder, x::ZZRingElem, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
 ideal(O::AbsSimpleNumFieldOrder, x::Integer, y::AbsSimpleNumFieldOrderElem) -> AbsNumFieldOrderIdeal
Creates the ideal (x,y) of O.
source
ideal Method
julia
ideal(O::AbsSimpleNumFieldOrder, a::ZZRingElem) -> AbsNumFieldOrderIdeal
@@ -166,8 +166,8 @@
 valuation(a::ZZRingElem, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
Computes the p-adic valuation of a, that is, the largest i such that a is contained in pi.
source
valuation Method
julia
valuation(A::AbsNumFieldOrderFractionalIdeal, p::AbsNumFieldOrderIdeal)
The valuation of A at p.
source
idempotents Method
julia
idempotents(x::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, y::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderElem, AbsSimpleNumFieldOrderElem
Returns a tuple (e, f) consisting of elements e in x, f in y such that 1 = e + f.
If the ideals are not coprime, an error is raised.
source
Quotient Rings ​
quo Method
julia
quo(O::AbsSimpleNumFieldOrder, I::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderQuoRing, Map
 quo(O::AlgAssAbsOrd, I::AlgAssAbsOrdIdl) -> AbsOrdQuoRing, Map
The quotient ring O/I as a ring together with the section M:O/I→O. The pointwise inverse of M is the canonical projection O→O/I.
source
residue_ring Method
julia
residue_ring(O::AbsSimpleNumFieldOrder, I::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderQuoRing
 residue_ring(O::AlgAssAbsOrd, I::AlgAssAbsOrdIdl) -> AbsOrdQuoRing
The quotient ring O modulo I as a new ring.
source
residue_field Method
julia
residue_field(O::AbsSimpleNumFieldOrder, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, check::Bool = true) -> Field, Map
Returns the residue field of the prime ideal P together with the projection map. If check is true, the ideal is checked for being prime.
source
mod Method
julia
mod(x::AbsSimpleNumFieldOrderElem, I::AbsNumFieldOrderIdeal)
Returns the unique element y of the ambient order of x with x≡ymodI and the following property: If a1,…,ad∈Z≥1 are the diagonal entries of the unique HNF basis matrix of I and (b1,…,bd) is the coefficient vector of y, then 0≤bi<ai for 1≤i≤d.
source
crt Method
julia
crt(r1::AbsSimpleNumFieldOrderElem, i1::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, r2::AbsSimpleNumFieldOrderElem, i2::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbsSimpleNumFieldOrderElem
Find x such that x≡r1modi1 and x≡r2modi2 using idempotents.
source
euler_phi Method
julia
euler_phi(A::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> ZZRingElem
The ideal version of the totient function returns the size of the unit group of the residue ring modulo the ideal.
source
multiplicative_group Method
julia
multiplicative_group(Q::AbsSimpleNumFieldOrderQuoRing) -> FinGenAbGroup, Map{FinGenAbGroup, AbsSimpleNumFieldOrderQuoRing}
-unit_group(Q::AbsSimpleNumFieldOrderQuoRing) -> FinGenAbGroup, Map{FinGenAbGroup, AbsSimpleNumFieldOrderQuoRing}
Returns the unit group of Q as an abstract group A and an isomorphism map f:A→Q×.
source
multiplicative_group_generators Method
julia
multiplicative_group_generators(Q::AbsSimpleNumFieldOrderQuoRing) -> Vector{AbsSimpleNumFieldOrderQuoRingElem}
Return a set of generators for Q×.
source

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+unit_group(Q::AbsSimpleNumFieldOrderQuoRing) -> FinGenAbGroup, Map{FinGenAbGroup, AbsSimpleNumFieldOrderQuoRing}
Returns the unit group of Q as an abstract group A and an isomorphism map f:A→Q×.
source
multiplicative_group_generators Method
julia
multiplicative_group_generators(Q::AbsSimpleNumFieldOrderQuoRing) -> Vector{AbsSimpleNumFieldOrderQuoRingElem}
Return a set of generators for Q×.
source

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diff --git a/v0.34.8/manual/orders/introduction.html b/v0.34.8/manual/orders/introduction.html
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@@ -6,19 +6,19 @@
     Introduction | Hecke
     
     
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Appearance
Introduction ​
This chapter deals with number fields and orders there of. We follow the common terminology and conventions as e.g. used in [2], [1], [3] or [4].
If K is a number field, then an order O of K is a subring of the ring of integers OK of K, which is free of rank [K:Q] as a Z-module. Depending on whether K is an absolute field or relative field, orders are treated differently. As far as possible, the interaction and the interface for orders of absolute number fields and of relative number fields is the same.
Orders of absolute number fields ​
Assume that K is defined as an absolute field. An order O of such a field are constructed (implicitly) by specifying a Z-basis, which is referred to as the basis of O. If (ω1,…,ωd) is the basis of O and (α1,…,αd) the basis of K, then the matrix B∈Matd×d(Q) with
(ω1⋮ωd)=B(α1⋮αd)
is the basis matrix of K. If K=Q(α)=Q[x]/(f) is simple with f∈Z[x], then natural order Z[α]=Z[x]/(f) is called the equation order of K.
Orders of relative number fields ​
Orders in non-absolute number fields, that is, relative extensions, are represented differently. Let L/K be a finite extension of number fields, then currently we require any order in L to contain OK, the ring of integers of K. In this case, an order O in L is a finitly generated torsion-free module over the Dedekind domain OK. As a ring, the order O is unitary and has L as a fraction field. Due to OK in general not being a principal ideal domain, the module structure is more complicated and requires so called pseudo-matrices. See here for details on pseudo-matrices, or [1], Chapter 1 for an introduction.
In short, O is represented as ∑aiωi with fractional OK ideals ai⊂K and K-linear independent elements ωi∈L. In general it is impossible to have both ai integral and ωi∈O, thus coefficients will not be integral and/or generators not in the structure.
Examples ​
Usually, to create an order, one starts with a field (or a polynomial):
julia

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Hecke
Appearance
Introduction ​
This chapter deals with number fields and orders there of. We follow the common terminology and conventions as e.g. used in [2], [1], [3] or [4].
If K is a number field, then an order O of K is a subring of the ring of integers OK of K, which is free of rank [K:Q] as a Z-module. Depending on whether K is an absolute field or relative field, orders are treated differently. As far as possible, the interaction and the interface for orders of absolute number fields and of relative number fields is the same.
Orders of absolute number fields ​
Assume that K is defined as an absolute field. An order O of such a field are constructed (implicitly) by specifying a Z-basis, which is referred to as the basis of O. If (ω1,…,ωd) is the basis of O and (α1,…,αd) the basis of K, then the matrix B∈Matd×d(Q) with
(ω1⋮ωd)=B(α1⋮αd)
is the basis matrix of K. If K=Q(α)=Q[x]/(f) is simple with f∈Z[x], then natural order Z[α]=Z[x]/(f) is called the equation order of K.
Orders of relative number fields ​
Orders in non-absolute number fields, that is, relative extensions, are represented differently. Let L/K be a finite extension of number fields, then currently we require any order in L to contain OK, the ring of integers of K. In this case, an order O in L is a finitly generated torsion-free module over the Dedekind domain OK. As a ring, the order O is unitary and has L as a fraction field. Due to OK in general not being a principal ideal domain, the module structure is more complicated and requires so called pseudo-matrices. See here for details on pseudo-matrices, or [1], Chapter 1 for an introduction.
In short, O is represented as ∑aiωi with fractional OK ideals ai⊂K and K-linear independent elements ωi∈L. In general it is impossible to have both ai integral and ωi∈O, thus coefficients will not be integral and/or generators not in the structure.
Examples ​
Usually, to create an order, one starts with a field (or a polynomial):
julia

 julia> Qx, x = polynomial_ring(QQ, "x");
 
 julia> K, a = number_field(x^2 - 10, "a");
@@ -64,8 +64,8 @@
 julia> [ mFp(x) for x = basis(Z_K)]
 2-element Vector{FqFieldElem}:
  1
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diff --git a/v0.34.8/manual/orders/orders.html b/v0.34.8/manual/orders/orders.html
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     Orders | Hecke
     
     
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Hecke
Appearance
Orders ​
Orders, that is, unitary subrings that are free Z-modules of rank equal to the degree of the number field, are at the core of the arithmetic of number fields. In Hecke, orders are always represented using the module structure, be it the Z-module structure for orders of absolute numbers fields, or the structure as a module over the maximal order of the base field in the case of relative number fields. In this chapter we mainly deal with orders of absolute fields. However, many functions apply in same way to relative extensions. There are more general definitions of orders in number fields available, but those are (currently) not implemented in Hecke.
Among all orders in a fixed field, there is a unique maximal order, called the maximal order, or ring of integers of the number field. It is well known that this is the only order that is a Dedekind domain, hence has a rich ideal structure as well. The maximal order is also the integral closure of Z in the number field and can also be interpreted as a normalization of any other order.
Creation and basic properties ​
Order Method
julia
Order(a::Vector{AbsSimpleNumFieldElem}; check::Bool = true, cached::Bool = true, isbasis::Bool = false) -> AbsSimpleNumFieldOrder
+    
Hecke
Appearance
Orders ​
Orders, that is, unitary subrings that are free Z-modules of rank equal to the degree of the number field, are at the core of the arithmetic of number fields. In Hecke, orders are always represented using the module structure, be it the Z-module structure for orders of absolute numbers fields, or the structure as a module over the maximal order of the base field in the case of relative number fields. In this chapter we mainly deal with orders of absolute fields. However, many functions apply in same way to relative extensions. There are more general definitions of orders in number fields available, but those are (currently) not implemented in Hecke.
Among all orders in a fixed field, there is a unique maximal order, called the maximal order, or ring of integers of the number field. It is well known that this is the only order that is a Dedekind domain, hence has a rich ideal structure as well. The maximal order is also the integral closure of Z in the number field and can also be interpreted as a normalization of any other order.
Creation and basic properties ​
Order Method
julia
Order(a::Vector{AbsSimpleNumFieldElem}; check::Bool = true, cached::Bool = true, isbasis::Bool = false) -> AbsSimpleNumFieldOrder
 Order(K::AbsSimpleNumField, a::Vector{AbsSimpleNumFieldElem}; check::Bool = true, cached::Bool = true, isbasis::Bool = false) -> AbsSimpleNumFieldOrder
Returns the order generated by a. If check is set, it is checked whether a defines an order, in particular the integrality of the elements is checked by computing minimal polynomials. If isbasis is set, then elements are assumed to form a Z-basis. If cached is set, then the constructed order is cached for future use.
source
Order Method
julia
Order(K::AbsSimpleNumField, A::QQMatrix; check::Bool = true) -> AbsSimpleNumFieldOrder
Returns the order which has basis matrix A with respect to the power basis of K. If check is set, it is checked whether A defines an order.
source
julia
Order(K::AbsSimpleNumField, A::QQMatrix; check::Bool = true) -> AbsSimpleNumFieldOrder
Returns the order which has basis matrix A with respect to the power basis of K. If check is set, it is checked whether A defines an order.
source
Order Method
julia
Order(K::AbsSimpleNumField, A::ZZMatrix, check::Bool = true) -> AbsSimpleNumFieldOrder
Returns the order which has basis matrix A with respect to the power basis of K. If check is set, it is checked whether A defines an order.
source
julia
Order(A::AbstractAssociativeAlgebra{<: NumFieldElem}, M::PMat{<: NumFieldElem, T})
   -> AlgAssRelOrd
Returns the order of A with basis pseudo-matrix M.
source
EquationOrder Method
julia
EquationOrder(K::number_field) -> NumFieldOrder
 equation_order(K::number_field) -> NumFieldOrder
Returns the equation order of the number field K.
source
MaximalOrder Method
julia
MaximalOrder(K::NumField{QQFieldElem}; discriminant::ZZRingElem, ramified_primes::Vector{ZZRingElem}) -> AbsNumFieldOrder
Returns the maximal order of K. Additional information can be supplied if they are already known, as the ramified primes or the discriminant of the maximal order.
Example
julia
julia> Qx, x = QQ["x"];
@@ -33,8 +33,8 @@
 with Z-basis AbsSimpleNumFieldOrderElem[1, a]
parent Method
julia
parent(O::AbsNumFieldOrder) -> AbsNumFieldOrderSet
Returns the parent of O, that is, the set of orders of the ambient number field.
source
signature Method
julia
signature(O::NumFieldOrder) -> Tuple{Int, Int}
Returns the signature of the ambient number field of O.
source
nf Method
julia
nf(O::NumFieldOrder) -> NumField
Returns the ambient number field of O.
source
basis Method
julia
basis(O::AbsNumFieldOrder) -> Vector{AbsNumFieldOrderElem}
Returns the Z-basis of O.
source
julia
basis(I::AbsNumFieldOrderFractionalIdeal) -> Vector{AbsSimpleNumFieldElem}
Returns the Z-basis of I.
source
lll_basis Method
julia
lll_basis(M::NumFieldOrder) -> Vector{NumFieldElem}
A basis for M that is reduced using the LLL algorithm for the Minkowski metric.
source
basis Method
julia
basis(O::AbsSimpleNumFieldOrder, K::AbsSimpleNumField) -> Vector{AbsSimpleNumFieldElem}
Returns the Z-basis elements of O as elements of the ambient number field.
source
pseudo_basis Method
julia
  pseudo_basis(O::RelNumFieldOrder{T, S}) -> Vector{Tuple{NumFieldElem{T}, S}}
Returns the pseudo-basis of O.
source
basis_pmatrix Method
julia
  basis_pmatrix(O::RelNumFieldOrder) -> PMat
Returns the basis pseudo-matrix of O with respect to the power basis of the ambient number field.
source
basis_nf Method
julia
  basis_nf(O::RelNumFieldOrder) -> Vector{NumFieldElem}
Returns the elements of the pseudo-basis of O as elements of the ambient number field.
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inv_coeff_ideals Method
julia
  inv_coeff_ideals(O::RelNumFieldOrder{T, S}) -> Vector{S}
Returns the inverses of the coefficient ideals of the pseudo basis of O.
source
basis_matrix Method
julia
basis_matrix(O::AbsNumFieldOrder) -> QQMatrix
Returns the basis matrix of O with respect to the basis of the ambient number field.
source
basis_mat_inv Method
julia
basis_mat_inv(A::GenOrdIdl) -> FakeFracFldMat
Return the inverse of the basis matrix of A.
source
gen_index Method
julia
gen_index(O::AbsSimpleNumFieldOrder) -> QQFieldElem
Returns the generalized index of O with respect to the equation order of the ambient number field.
source
is_index_divisor Method
julia
is_index_divisor(O::AbsSimpleNumFieldOrder, d::ZZRingElem) -> Bool
 is_index_divisor(O::AbsSimpleNumFieldOrder, d::Int) -> Bool
Returns whether d is a divisor of the index of O. It is assumed that O contains the equation order of the ambient number field.
source
minkowski_matrix Method
julia
minkowski_matrix(O::AbsNumFieldOrder, abs_tol::Int = 64) -> ArbMatrix
Returns the Minkowski matrix of O. Thus if O has degree d, then the result is a matrix in Matd×d(R). The entries of the matrix are real balls of type ArbFieldElem with radius less then 2^-abs_tol.
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in Method
julia
in(a::NumFieldElem, O::NumFieldOrder) -> Bool
Checks whether a lies in O.
source
norm_change_const Method
julia
norm_change_const(O::AbsSimpleNumFieldOrder) -> (Float64, Float64)
Returns (c1,c2)∈R>02 such that for all x=∑i=1dxiωi∈O we have T2(x)≤c1⋅∑idxi2 and ∑idxi2≤c2⋅T2(x), where (ωi)i is the Z-basis of O.
source
trace_matrix Method
julia
trace_matrix(O::AbsNumFieldOrder) -> ZZMatrix
Returns the trace matrix of O, that is, the matrix (trK/Q(bi⋅bj))1≤i,j≤d.
source
+ Method
julia
+(R::AbsSimpleNumFieldOrder, S::AbsSimpleNumFieldOrder) -> AbsSimpleNumFieldOrder
Given two orders R, S of K, this function returns the smallest order containing both R and S. It is assumed that R, S contain the ambient equation order and have coprime index.
source
poverorder Method
julia
poverorder(O::AbsSimpleNumFieldOrder, p::ZZRingElem) -> AbsSimpleNumFieldOrder
 poverorder(O::AbsSimpleNumFieldOrder, p::Integer) -> AbsSimpleNumFieldOrder
This function tries to find an order that is locally larger than O at the prime p: If p divides the index [OK:O], this function will return an order R such that vp([OK:R])<vp([OK:O]). Otherwise O is returned.
source
poverorders Method
julia
poverorders(O, p) -> Vector{Ord}
Returns all p-overorders of O, that is all overorders M, such that the index of O in M is a p-power.
source
pmaximal_overorder Method
julia
pmaximal_overorder(O::AbsSimpleNumFieldOrder, p::ZZRingElem) -> AbsSimpleNumFieldOrder
-pmaximal_overorder(O::AbsSimpleNumFieldOrder, p::Integer) -> AbsSimpleNumFieldOrder
This function finds a p-maximal order R containing O. That is, the index [OK:R] is not divisible by p.
source
pradical Method
julia
pradical(O::AbsSimpleNumFieldOrder, p::{ZZRingElem|Integer}) -> AbsNumFieldOrderIdeal
Given a prime number p, this function returns the p-radical pO of O, which is just {x∈O∣∃k∈Z≥0:xk∈pO}. It is not checked that p is prime.
source
pradical Method
julia
  pradical(O::RelNumFieldOrder, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> RelNumFieldOrderIdeal
Given a prime ideal P, this function returns the P-radical PO of O, which is just {x∈O∣∃k∈Z≥0:xk∈PO}. It is not checked that P is prime.
source
ring_of_multipliers Method
julia
ring_of_multipliers(I::AbsNumFieldOrderIdeal) -> AbsNumFieldOrder
Computes the order (I:I), which is the set of all x∈K with xI⊆I.
source
Invariants ​
discriminant Method
julia
discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
Returns the discriminant of O.
source
reduced_discriminant Method
julia
reduced_discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
Returns the reduced discriminant, that is, the largest elementary divisor of the trace matrix of O.
source
degree Method
julia
degree(O::NumFieldOrder) -> Int
Returns the degree of O.
source
index Method
julia
index(O::AbsSimpleNumFieldOrder) -> ZZRingElem
Assuming that the order O contains the equation order Z[α] of the ambient number field, this function returns the index [O:Z].
source
different Method
julia
different(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal
The different ideal of R, that is, the ideal generated by all differents of elements in R. For Gorenstein orders, this is also the inverse ideal of the co-different.
source
codifferent Method
julia
codifferent(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
The codifferent ideal of R, i.e. the trace-dual of R.
source
is_gorenstein Method
julia
is_gorenstein(O::AbsSimpleNumFieldOrder) -> Bool
Return whether the order \mathcal{O} is Gorenstein.
source
is_bass Method
julia
is_bass(O::AbsSimpleNumFieldOrder) -> Bool
Return whether the order \mathcal{O} is Bass.
source
is_equation_order Method
julia
is_equation_order(O::NumFieldOrder) -> Bool
Returns whether O is the equation order of the ambient number field K.
source
zeta_log_residue Method
julia
zeta_log_residue(O::AbsSimpleNumFieldOrder, error::Float64) -> ArbFieldElem
Computes the residue of the zeta function of O at 1. The output will be an element of type ArbFieldElem with radius less then error.
source
ramified_primes Method
julia
ramified_primes(O::AbsNumFieldOrder) -> Vector{ZZRingElem}
Returns the list of prime numbers that divide disc(O).
source
Arithmetic ​
Progress and intermediate results of the functions mentioned here can be obtained via verbose_level, supported are
  • ClassGroup
  • UnitGroup
All of the functions have a very similar interface: they return an abelian group and a map converting elements of the group into the objects required. The maps also allow a point-wise inverse to server as the discrete logarithm map. For more information on abelian groups, see here, for ideals, here.
For the processing of units, there are a couple of helper functions also available:
is_independent Function
julia
is_independent{T}(x::Vector{T})
Given an array of non-zero units in a number field, returns whether they are multiplicatively independent.
source
Predicates ​
is_contained Method
julia
is_contained(R::AbsNumFieldOrder, S::AbsNumFieldOrder) -> Bool
Checks if R is contained in S.
source
is_maximal Method
julia
is_maximal(R::AbsNumFieldOrder) -> Bool
Tests if the order R is maximal. This might trigger the computation of the maximal order.
source

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+pmaximal_overorder(O::AbsSimpleNumFieldOrder, p::Integer) -> AbsSimpleNumFieldOrder
This function finds a p-maximal order R containing O. That is, the index [OK:R] is not divisible by p.
source
pradical Method
julia
pradical(O::AbsSimpleNumFieldOrder, p::{ZZRingElem|Integer}) -> AbsNumFieldOrderIdeal
Given a prime number p, this function returns the p-radical pO of O, which is just {x∈O∣∃k∈Z≥0:xk∈pO}. It is not checked that p is prime.
source
pradical Method
julia
  pradical(O::RelNumFieldOrder, P::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> RelNumFieldOrderIdeal
Given a prime ideal P, this function returns the P-radical PO of O, which is just {x∈O∣∃k∈Z≥0:xk∈PO}. It is not checked that P is prime.
source
ring_of_multipliers Method
julia
ring_of_multipliers(I::AbsNumFieldOrderIdeal) -> AbsNumFieldOrder
Computes the order (I:I), which is the set of all x∈K with xI⊆I.
source
Invariants ​
discriminant Method
julia
discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
Returns the discriminant of O.
source
reduced_discriminant Method
julia
reduced_discriminant(O::AbsSimpleNumFieldOrder) -> ZZRingElem
Returns the reduced discriminant, that is, the largest elementary divisor of the trace matrix of O.
source
degree Method
julia
degree(O::NumFieldOrder) -> Int
Returns the degree of O.
source
index Method
julia
index(O::AbsSimpleNumFieldOrder) -> ZZRingElem
Assuming that the order O contains the equation order Z[α] of the ambient number field, this function returns the index [O:Z].
source
different Method
julia
different(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal
The different ideal of R, that is, the ideal generated by all differents of elements in R. For Gorenstein orders, this is also the inverse ideal of the co-different.
source
codifferent Method
julia
codifferent(R::AbsNumFieldOrder) -> AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}
The codifferent ideal of R, i.e. the trace-dual of R.
source
is_gorenstein Method
julia
is_gorenstein(O::AbsSimpleNumFieldOrder) -> Bool
Return whether the order \mathcal{O} is Gorenstein.
source
is_bass Method
julia
is_bass(O::AbsSimpleNumFieldOrder) -> Bool
Return whether the order \mathcal{O} is Bass.
source
is_equation_order Method
julia
is_equation_order(O::NumFieldOrder) -> Bool
Returns whether O is the equation order of the ambient number field K.
source
zeta_log_residue Method
julia
zeta_log_residue(O::AbsSimpleNumFieldOrder, error::Float64) -> ArbFieldElem
Computes the residue of the zeta function of O at 1. The output will be an element of type ArbFieldElem with radius less then error.
source
ramified_primes Method
julia
ramified_primes(O::AbsNumFieldOrder) -> Vector{ZZRingElem}
Returns the list of prime numbers that divide disc(O).
source
Arithmetic ​
Progress and intermediate results of the functions mentioned here can be obtained via verbose_level, supported are
  • ClassGroup
  • UnitGroup
All of the functions have a very similar interface: they return an abelian group and a map converting elements of the group into the objects required. The maps also allow a point-wise inverse to server as the discrete logarithm map. For more information on abelian groups, see here, for ideals, here.
For the processing of units, there are a couple of helper functions also available:
is_independent Function
julia
is_independent{T}(x::Vector{T})
Given an array of non-zero units in a number field, returns whether they are multiplicatively independent.
source
Predicates ​
is_contained Method
julia
is_contained(R::AbsNumFieldOrder, S::AbsNumFieldOrder) -> Bool
Checks if R is contained in S.
source
is_maximal Method
julia
is_maximal(R::AbsNumFieldOrder) -> Bool
Tests if the order R is maximal. This might trigger the computation of the maximal order.
source

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diff --git a/v0.34.8/manual/quad_forms/Zgenera.html b/v0.34.8/manual/quad_forms/Zgenera.html
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     Genera of Integer Lattices | Hecke
     
     
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Appearance
Genera of Integer Lattices ​
Two Z-lattices M and N are said to be in the same genus if their completions M⊗Zp and N⊗Zp are isometric for all prime numbers p as well as M⊗R≅N⊗R.
The genus of a Z-lattice is encoded in its Conway-Sloane genus symbol. The genus symbol itself is a collection of its local genus symbols. See [5] Chapter 15 for the definitions. Note that genera for non-integral lattices are supported.
The class ZZGenus supports genera of Z-lattices.
ZZGenus Type
julia
ZZGenus
A collection of local genus symbols (at primes) and a signature pair. Together they represent the genus of a non-degenerate integer_lattice.
source
Creation of Genera ​
From an integral Lattice ​
genus Method
julia
genus(L::ZZLat) -> ZZGenus
Return the genus of the lattice L.
source
From a gram matrix ​
genus Method
julia
genus(A::MatElem) -> ZZGenus
Return the genus of a Z-lattice with gram matrix A.
source
Enumeration of genus symbols ​
integer_genera Method
julia
integer_genera(sig_pair::Vector{Int}, determinant::RationalUnion;
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Hecke
Appearance
Genera of Integer Lattices ​
Two Z-lattices M and N are said to be in the same genus if their completions M⊗Zp and N⊗Zp are isometric for all prime numbers p as well as M⊗R≅N⊗R.
The genus of a Z-lattice is encoded in its Conway-Sloane genus symbol. The genus symbol itself is a collection of its local genus symbols. See [5] Chapter 15 for the definitions. Note that genera for non-integral lattices are supported.
The class ZZGenus supports genera of Z-lattices.
ZZGenus Type
julia
ZZGenus
A collection of local genus symbols (at primes) and a signature pair. Together they represent the genus of a non-degenerate integer_lattice.
source
Creation of Genera ​
From an integral Lattice ​
genus Method
julia
genus(L::ZZLat) -> ZZGenus
Return the genus of the lattice L.
source
From a gram matrix ​
genus Method
julia
genus(A::MatElem) -> ZZGenus
Return the genus of a Z-lattice with gram matrix A.
source
Enumeration of genus symbols ​
integer_genera Method
julia
integer_genera(sig_pair::Vector{Int}, determinant::RationalUnion;
        min_scale::RationalUnion = min(one(QQ), QQ(abs(determinant))),
        max_scale::RationalUnion = max(one(QQ), QQ(abs(determinant))),
-       even=false)                                         -> Vector{ZZGenus}
Return a list of all genera with the given conditions. Genera of non-integral Z-lattices are also supported.
Arguments
  • sig_pair: a pair of non-negative integers giving the signature
  • determinant: a rational number; the sign is ignored
  • min_scale: a rational number; return only genera whose scale is an integer multiple of min_scale (default: min(one(QQ), QQ(abs(determinant))))
  • max_scale: a rational number; return only genera such that max_scale is an integer multiple of the scale (default: max(one(QQ), QQ(abs(determinant))))
  • even: boolean; if set to true, return only the even genera (default: false)
source
From other genus symbols ​
direct_sum Method
julia
direct_sum(G1::ZZGenus, G2::ZZGenus) -> ZZGenus
Return the genus of the direct sum of G1 and G2.
The direct sum is defined via representatives.
source
Attributes of the genus ​
dim Method
julia
dim(G::ZZGenus) -> Int
Return the dimension of this genus.
source
rank Method
julia
rank(G::ZZGenus) -> Int
Return the rank of a (representative of) the genus G.
source
signature Method
julia
signature(G::ZZGenus) -> Int
Return the signature of this genus.
The signature is p - n where p is the number of positive eigenvalues and n the number of negative eigenvalues.
source
det Method
julia
det(G::ZZGenus) -> QQFieldElem
Return the determinant of this genus.
source
iseven Method
julia
iseven(G::ZZGenus) -> Bool
Return if this genus is even.
source
is_definite Method
julia
is_definite(G::ZZGenus) -> Bool
Return if this genus is definite.
source
level Method
julia
level(G::ZZGenus) -> QQFieldElem
Return the level of this genus.
This is the denominator of the inverse gram matrix of a representative.
source
scale Method
julia
scale(G::ZZGenus) -> QQFieldElem
Return the scale of this genus.
Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).
source
norm Method
julia
norm(G::ZZGenus) -> QQFieldElem
Return the norm of this genus.
Let L be a lattice with bilinear form b. The norm of (L,b) is defined as the ideal generated by {b(x,x)|x∈L}.
source
primes Method
julia
primes(G::ZZGenus) -> Vector{ZZRingElem}
Return the list of primes of the local symbols of G.
Note that 2 is always in the output since the 2-adic symbol of a ZZGenus is, by convention, always defined.
source
is_integral Method
julia
is_integral(G::ZZGenus) -> Bool
Return whether G is a genus of integral Z-lattices.
source
Discriminant group ​
discriminant_group(::ZZGenus)
Primary genera ​
is_primary_with_prime Method
julia
is_primary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
Given a genus of Z-lattices G, return whether it is primary, that is whether the bilinear form is integral and the associated discriminant form (see discriminant_group) is a p-group for some prime number p. In case it is, p is also returned as second output.
Note that for unimodular genera, this function returns (true, 1). If the genus is not primary, the second return value is -1 by default.
source
is_primary Method
julia
is_primary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
Given a genus of integral Z-lattices G and a prime number p, return whether G is p-primary, that is whether the associated discriminant form (see discriminant_group) is a p-group.
source
is_elementary_with_prime Method
julia
is_elementary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
Given a genus of Z-lattices G, return whether it is elementary, that is whether the bilinear form is inegtral and the associated discriminant form (see discriminant_group) is an elementary p-group for some prime number p. In case it is, p is also returned as second output.
Note that for unimodular genera, this function returns (true, 1). If the genus is not elementary, the second return value is -1 by default.
source
is_elementary Method
julia
is_elementary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
Given a genus of integral Z-lattices G and a prime number p, return whether G is p-elementary, that is whether its associated discriminant form (see discriminant_group) is an elementary p-group.
source
local Symbol ​
local_symbol Method
julia
local_symbol(G::ZZGenus, p) -> ZZLocalGenus
Return the local symbol at p.
source
Representative(s) ​
quadratic_space Method
julia
quadratic_space(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}
Return the quadratic space defined by this genus.
source
rational_representative Method
julia
rational_representative(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}
Return the quadratic space defined by this genus.
source
representative Method
julia
representative(G::ZZGenus) -> ZZLat
Compute a representative of this genus && cache it.
source
representatives Method
julia
representatives(G::ZZGenus) -> Vector{ZZLat}
Return a list of representatives of the isometry classes in this genus.
source
mass Method
julia
mass(G::ZZGenus) -> QQFieldElem
Return the mass of this genus.
The genus must be definite. Let L_1, ... L_n be a complete list of representatives of the isometry classes in this genus. Its mass is defined as ∑i=1n1|O(Li)|.
source
rescale Method
julia
rescale(G::ZZGenus, a::RationalUnion) -> ZZGenus
Given a genus symbol G of Z-lattices, return the genus symbol of any representative of G rescaled by a.
source
Embeddings and Representations ​
represents Method
julia
represents(G1::ZZGenus, G2::ZZGenus) -> Bool
Return if G1 represents G2. That is if some element in the genus of G1 represents some element in the genus of G2.
source
Local genus Symbols ​
ZZLocalGenus Type
julia
ZZLocalGenus
Local genus symbol over a p-adic ring.
The genus symbol of a component p^m A for odd prime = p is of the form (m,n,d), where
  • m = valuation of the component
  • n = rank of A
  • d = det(A) \in \{1,u\} for a normalized quadratic non-residue u.
The genus symbol of a component 2^m A is of the form (m, n, s, d, o), where
  • m = valuation of the component
  • n = rank of A
  • d = det(A) in {1,3,5,7}
  • s = 0 (or 1) if even (or odd)
  • o = oddity of A (= 0 if s = 0) in Z/8Z = the trace of the diagonalization of A
The genus symbol is a list of such symbols (ordered by m) for each of the Jordan blocks A_1,...,A_t.
Reference: [5] Chapter 15, Section 7.
Arguments
  • prime: a prime number
  • symbol: the list of invariants for Jordan blocks A_t,...,A_t given as a list of lists of integers
source
Creation ​
genus Method
julia
genus(L::ZZLat, p::IntegerUnion) -> ZZLocalGenus
Return the local genus symbol of L at the prime p.
source
genus Method
julia
genus(A::QQMatrix, p::IntegerUnion) -> ZZLocalGenus
Return the local genus symbol of a Z-lattice with gram matrix A at the prime p.
source
Attributes ​
prime Method
julia
prime(S::ZZLocalGenus) -> ZZRingElem
Return the prime p of this p-adic genus.
source
iseven Method
julia
iseven(S::ZZLocalGenus) -> Bool
Return if the underlying p-adic lattice is even.
If p is odd, every lattice is even.
source
symbol Method
julia
symbol(S::ZZLocalGenus, scale::Int) -> Vector{Int}
Return the underlying lists of integers for the Jordan block of the given scale
source
hasse_invariant Method
julia
hasse_invariant(S::ZZLocalGenus) -> Int
Return the Hasse invariant of a representative. If the representative is diagonal (a_1, ... , a_n) Then the Hasse invariant is
∏i<j(ai,aj)p
.
source
det Method
julia
det(S::ZZLocalGenus) -> QQFieldElem
Return an rational representing the determinant of this genus.
source
dim Method
julia
dim(S::ZZLocalGenus) -> Int
Return the dimension of this genus.
source
rank Method
julia
rank(S::ZZLocalGenus) -> Int
Return the rank of (a representative of) S.
source
excess Method
julia
excess(S::ZZLocalGenus) -> zzModRingElem
Return the p-excess of the quadratic form whose Hessian matrix is the symmetric matrix A.
When p = 2 the p-excess is called the oddity. The p-excess is always even && is divisible by 4 if p is congruent 1 mod 4.
Reference
[5] pp 370-371.
source
signature Method
julia
signature(S::ZZLocalGenus) -> zzModRingElem
Return the p-signature of this p-adic form.
source
oddity Method
julia
oddity(S::ZZLocalGenus) -> zzModRingElem
Return the oddity of this even form. The oddity is also called the 2-signature
source
scale Method
julia
scale(S::ZZLocalGenus) -> QQFieldElem
Return the scale of this local genus.
Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).
source
norm Method
julia
norm(S::ZZLocalGenus) -> QQFieldElem
Return the norm of this local genus.
Let L be a lattice with bilinear form b. The norm of (L,b) is defined as the ideal generated by {b(x,x)|x∈L}.
source
level Method
julia
level(S::ZZLocalGenus) -> QQFieldElem
Return the maximal scale of a jordan component.
source
Representative ​
representative Method
julia
representative(S::ZZLocalGenus) -> ZZLat
Return an integer lattice which represents this local genus.
source
gram_matrix Method
julia
gram_matrix(S::ZZLocalGenus) -> MatElem
Return a gram matrix of some representative of this local genus.
source
rescale Method
julia
rescale(G::ZZLocalGenus, a::RationalUnion) -> ZZLocalGenus
Given a local genus symbol G of Z-lattices, return the local genus symbol of any representative of G rescaled by a.
source
Direct sums ​
direct_sum Method
julia
direct_sum(S1::ZZLocalGenus, S2::ZZLocalGenus) -> ZZLocalGenus
Return the local genus of the direct sum of two representatives.
source
Embeddings/Representations ​
represents Method
julia
represents(g1::ZZLocalGenus, g2::ZZLocalGenus) -> Bool
Return whether g1 represents g2.
Based on O'Meara Integral Representations of Quadratic Forms Over Local Fields Note that for p == 2 there is a typo in O'Meara Theorem 3 (V). The correct statement is (V) 2i(1+4ω)→Li+1/l[i].
source

-    
+       even=false)                                         -> Vector{ZZGenus}
Return a list of all genera with the given conditions. Genera of non-integral Z-lattices are also supported.
Arguments
  • sig_pair: a pair of non-negative integers giving the signature
  • determinant: a rational number; the sign is ignored
  • min_scale: a rational number; return only genera whose scale is an integer multiple of min_scale (default: min(one(QQ), QQ(abs(determinant))))
  • max_scale: a rational number; return only genera such that max_scale is an integer multiple of the scale (default: max(one(QQ), QQ(abs(determinant))))
  • even: boolean; if set to true, return only the even genera (default: false)
source
From other genus symbols ​
direct_sum Method
julia
direct_sum(G1::ZZGenus, G2::ZZGenus) -> ZZGenus
Return the genus of the direct sum of G1 and G2.
The direct sum is defined via representatives.
source
Attributes of the genus ​
dim Method
julia
dim(G::ZZGenus) -> Int
Return the dimension of this genus.
source
rank Method
julia
rank(G::ZZGenus) -> Int
Return the rank of a (representative of) the genus G.
source
signature Method
julia
signature(G::ZZGenus) -> Int
Return the signature of this genus.
The signature is p - n where p is the number of positive eigenvalues and n the number of negative eigenvalues.
source
det Method
julia
det(G::ZZGenus) -> QQFieldElem
Return the determinant of this genus.
source
iseven Method
julia
iseven(G::ZZGenus) -> Bool
Return if this genus is even.
source
is_definite Method
julia
is_definite(G::ZZGenus) -> Bool
Return if this genus is definite.
source
level Method
julia
level(G::ZZGenus) -> QQFieldElem
Return the level of this genus.
This is the denominator of the inverse gram matrix of a representative.
source
scale Method
julia
scale(G::ZZGenus) -> QQFieldElem
Return the scale of this genus.
Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).
source
norm Method
julia
norm(G::ZZGenus) -> QQFieldElem
Return the norm of this genus.
Let L be a lattice with bilinear form b. The norm of (L,b) is defined as the ideal generated by {b(x,x)|x∈L}.
source
primes Method
julia
primes(G::ZZGenus) -> Vector{ZZRingElem}
Return the list of primes of the local symbols of G.
Note that 2 is always in the output since the 2-adic symbol of a ZZGenus is, by convention, always defined.
source
is_integral Method
julia
is_integral(G::ZZGenus) -> Bool
Return whether G is a genus of integral Z-lattices.
source
Discriminant group ​
discriminant_group(::ZZGenus)
Primary genera ​
is_primary_with_prime Method
julia
is_primary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
Given a genus of Z-lattices G, return whether it is primary, that is whether the bilinear form is integral and the associated discriminant form (see discriminant_group) is a p-group for some prime number p. In case it is, p is also returned as second output.
Note that for unimodular genera, this function returns (true, 1). If the genus is not primary, the second return value is -1 by default.
source
is_primary Method
julia
is_primary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
Given a genus of integral Z-lattices G and a prime number p, return whether G is p-primary, that is whether the associated discriminant form (see discriminant_group) is a p-group.
source
is_elementary_with_prime Method
julia
is_elementary_with_prime(G::ZZGenus) -> Bool, ZZRingElem
Given a genus of Z-lattices G, return whether it is elementary, that is whether the bilinear form is inegtral and the associated discriminant form (see discriminant_group) is an elementary p-group for some prime number p. In case it is, p is also returned as second output.
Note that for unimodular genera, this function returns (true, 1). If the genus is not elementary, the second return value is -1 by default.
source
is_elementary Method
julia
is_elementary(G::ZZGenus, p::Union{Integer, ZZRingElem}) -> Bool
Given a genus of integral Z-lattices G and a prime number p, return whether G is p-elementary, that is whether its associated discriminant form (see discriminant_group) is an elementary p-group.
source
local Symbol ​
local_symbol Method
julia
local_symbol(G::ZZGenus, p) -> ZZLocalGenus
Return the local symbol at p.
source
Representative(s) ​
quadratic_space Method
julia
quadratic_space(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}
Return the quadratic space defined by this genus.
source
rational_representative Method
julia
rational_representative(G::ZZGenus) -> QuadSpace{QQField, QQMatrix}
Return the quadratic space defined by this genus.
source
representative Method
julia
representative(G::ZZGenus) -> ZZLat
Compute a representative of this genus && cache it.
source
representatives Method
julia
representatives(G::ZZGenus) -> Vector{ZZLat}
Return a list of representatives of the isometry classes in this genus.
source
mass Method
julia
mass(G::ZZGenus) -> QQFieldElem
Return the mass of this genus.
The genus must be definite. Let L_1, ... L_n be a complete list of representatives of the isometry classes in this genus. Its mass is defined as ∑i=1n1|O(Li)|.
source
rescale Method
julia
rescale(G::ZZGenus, a::RationalUnion) -> ZZGenus
Given a genus symbol G of Z-lattices, return the genus symbol of any representative of G rescaled by a.
source
Embeddings and Representations ​
represents Method
julia
represents(G1::ZZGenus, G2::ZZGenus) -> Bool
Return if G1 represents G2. That is if some element in the genus of G1 represents some element in the genus of G2.
source
Local genus Symbols ​
ZZLocalGenus Type
julia
ZZLocalGenus
Local genus symbol over a p-adic ring.
The genus symbol of a component p^m A for odd prime = p is of the form (m,n,d), where
  • m = valuation of the component
  • n = rank of A
  • d = det(A) \in \{1,u\} for a normalized quadratic non-residue u.
The genus symbol of a component 2^m A is of the form (m, n, s, d, o), where
  • m = valuation of the component
  • n = rank of A
  • d = det(A) in {1,3,5,7}
  • s = 0 (or 1) if even (or odd)
  • o = oddity of A (= 0 if s = 0) in Z/8Z = the trace of the diagonalization of A
The genus symbol is a list of such symbols (ordered by m) for each of the Jordan blocks A_1,...,A_t.
Reference: [5] Chapter 15, Section 7.
Arguments
  • prime: a prime number
  • symbol: the list of invariants for Jordan blocks A_t,...,A_t given as a list of lists of integers
source
Creation ​
genus Method
julia
genus(L::ZZLat, p::IntegerUnion) -> ZZLocalGenus
Return the local genus symbol of L at the prime p.
source
genus Method
julia
genus(A::QQMatrix, p::IntegerUnion) -> ZZLocalGenus
Return the local genus symbol of a Z-lattice with gram matrix A at the prime p.
source
Attributes ​
prime Method
julia
prime(S::ZZLocalGenus) -> ZZRingElem
Return the prime p of this p-adic genus.
source
iseven Method
julia
iseven(S::ZZLocalGenus) -> Bool
Return if the underlying p-adic lattice is even.
If p is odd, every lattice is even.
source
symbol Method
julia
symbol(S::ZZLocalGenus, scale::Int) -> Vector{Int}
Return the underlying lists of integers for the Jordan block of the given scale
source
hasse_invariant Method
julia
hasse_invariant(S::ZZLocalGenus) -> Int
Return the Hasse invariant of a representative. If the representative is diagonal (a_1, ... , a_n) Then the Hasse invariant is
∏i<j(ai,aj)p
.
source
det Method
julia
det(S::ZZLocalGenus) -> QQFieldElem
Return an rational representing the determinant of this genus.
source
dim Method
julia
dim(S::ZZLocalGenus) -> Int
Return the dimension of this genus.
source
rank Method
julia
rank(S::ZZLocalGenus) -> Int
Return the rank of (a representative of) S.
source
excess Method
julia
excess(S::ZZLocalGenus) -> zzModRingElem
Return the p-excess of the quadratic form whose Hessian matrix is the symmetric matrix A.
When p = 2 the p-excess is called the oddity. The p-excess is always even && is divisible by 4 if p is congruent 1 mod 4.
Reference
[5] pp 370-371.
source
signature Method
julia
signature(S::ZZLocalGenus) -> zzModRingElem
Return the p-signature of this p-adic form.
source
oddity Method
julia
oddity(S::ZZLocalGenus) -> zzModRingElem
Return the oddity of this even form. The oddity is also called the 2-signature
source
scale Method
julia
scale(S::ZZLocalGenus) -> QQFieldElem
Return the scale of this local genus.
Let L be a lattice with bilinear form b. The scale of (L,b) is defined as the ideal b(L,L).
source
norm Method
julia
norm(S::ZZLocalGenus) -> QQFieldElem
Return the norm of this local genus.
Let L be a lattice with bilinear form b. The norm of (L,b) is defined as the ideal generated by {b(x,x)|x∈L}.
source
level Method
julia
level(S::ZZLocalGenus) -> QQFieldElem
Return the maximal scale of a jordan component.
source
Representative ​
representative Method
julia
representative(S::ZZLocalGenus) -> ZZLat
Return an integer lattice which represents this local genus.
source
gram_matrix Method
julia
gram_matrix(S::ZZLocalGenus) -> MatElem
Return a gram matrix of some representative of this local genus.
source
rescale Method
julia
rescale(G::ZZLocalGenus, a::RationalUnion) -> ZZLocalGenus
Given a local genus symbol G of Z-lattices, return the local genus symbol of any representative of G rescaled by a.
source
Direct sums ​
direct_sum Method
julia
direct_sum(S1::ZZLocalGenus, S2::ZZLocalGenus) -> ZZLocalGenus
Return the local genus of the direct sum of two representatives.
source
Embeddings/Representations ​
represents Method
julia
represents(g1::ZZLocalGenus, g2::ZZLocalGenus) -> Bool
Return whether g1 represents g2.
Based on O'Meara Integral Representations of Quadratic Forms Over Local Fields Note that for p == 2 there is a typo in O'Meara Theorem 3 (V). The correct statement is (V) 2i(1+4ω)→Li+1/l[i].
source

+    
     
   
 
\ No newline at end of file
diff --git a/v0.34.8/manual/quad_forms/basics.html b/v0.34.8/manual/quad_forms/basics.html
index 1642970750..d5da069919 100644
--- a/v0.34.8/manual/quad_forms/basics.html
+++ b/v0.34.8/manual/quad_forms/basics.html
@@ -6,19 +6,19 @@
     Spaces | Hecke
     
     
-    
-    
+    
+    
     
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-    
-    
-    
-    
+    
+    
+    
+    
+    
     
     
   
   
-    
Hecke
Appearance
Spaces ​
Creation of spaces ​
quadratic_space Method
julia
quadratic_space(K::NumField, n::Int; cached::Bool = true) -> QuadSpace
Create the quadratic space over K with dimension n and Gram matrix equals to the identity matrix.
source
hermitian_space Method
julia
hermitian_space(E::NumField, n::Int; cached::Bool = true) -> HermSpace
Create the hermitian space over E with dimension n and Gram matrix equals to the identity matrix. The number field E must be a quadratic extension, that is, degree(E)==2 must hold.
source
quadratic_space Method
julia
quadratic_space(K::NumField, G::MatElem; cached::Bool = true) -> QuadSpace
Create the quadratic space over K with Gram matrix G. The matrix G must be square and symmetric.
source
hermitian_space Method
julia
hermitian_space(E::NumField, gram::MatElem; cached::Bool = true) -> HermSpace
Create the hermitian space over E with Gram matrix equals to gram. The matrix gram must be square and hermitian with respect to the non-trivial automorphism of E. The number field E must be a quadratic extension, that is, degree(E)==2 must hold.
source
Examples ​
Here are easy examples to see how these constructors work. We will keep the two following spaces for the rest of this section:
julia

+    
Hecke
Appearance
Spaces ​
Creation of spaces ​
quadratic_space Method
julia
quadratic_space(K::NumField, n::Int; cached::Bool = true) -> QuadSpace
Create the quadratic space over K with dimension n and Gram matrix equals to the identity matrix.
source
hermitian_space Method
julia
hermitian_space(E::NumField, n::Int; cached::Bool = true) -> HermSpace
Create the hermitian space over E with dimension n and Gram matrix equals to the identity matrix. The number field E must be a quadratic extension, that is, degree(E)==2 must hold.
source
quadratic_space Method
julia
quadratic_space(K::NumField, G::MatElem; cached::Bool = true) -> QuadSpace
Create the quadratic space over K with Gram matrix G. The matrix G must be square and symmetric.
source
hermitian_space Method
julia
hermitian_space(E::NumField, gram::MatElem; cached::Bool = true) -> HermSpace
Create the hermitian space over E with Gram matrix equals to gram. The matrix gram must be square and hermitian with respect to the non-trivial automorphism of E. The number field E must be a quadratic extension, that is, degree(E)==2 must hold.
source
Examples ​
Here are easy examples to see how these constructors work. We will keep the two following spaces for the rest of this section:
julia

 julia> K, a = cyclotomic_real_subfield(7);
 
 julia> Kt, t = K["t"];
@@ -230,8 +230,8 @@
 julia> p = prime_decomposition(OK, 7)[1][1];
 
 julia> is_locally_hyperbolic(H, p)
-false

-    
+false

+    
     
   
 
\ No newline at end of file
diff --git a/v0.34.8/manual/quad_forms/discriminant_group.html b/v0.34.8/manual/quad_forms/discriminant_group.html
index 55f32b07ef..7e2c5a190d 100644
--- a/v0.34.8/manual/quad_forms/discriminant_group.html
+++ b/v0.34.8/manual/quad_forms/discriminant_group.html
@@ -6,19 +6,19 @@
     Discriminant Groups | Hecke
     
     
-    
-    
+    
+    
     
-    
-    
-    
-    
-    
+    
+    
+    
+    
+    
     
     
   
   
-    
Hecke
Appearance
Discriminant Groups ​
Torsion Quadratic Modules ​
A torsion quadratic module is the quotient M/N of two quadratic integer lattices N⊆M in the quadratic space (V,Φ). It inherits a bilinear form
b:M/N×M/N→Q/nZ
as well as a quadratic form
q:M/N→Q/mZ.
where nZ=Φ(M,N) and mZ=2nZ+∑x∈NZΦ(x,x).
torsion_quadratic_module Method
julia
torsion_quadratic_module(M::ZZLat, N::ZZLat; gens::Union{Nothing, Vector{<:Vector}} = nothing,
+    
Hecke
Appearance
Discriminant Groups ​
Torsion Quadratic Modules ​
A torsion quadratic module is the quotient M/N of two quadratic integer lattices N⊆M in the quadratic space (V,Φ). It inherits a bilinear form
b:M/N×M/N→Q/nZ
as well as a quadratic form
q:M/N→Q/mZ.
where nZ=Φ(M,N) and mZ=2nZ+∑x∈NZΦ(x,x).
torsion_quadratic_module Method
julia
torsion_quadratic_module(M::ZZLat, N::ZZLat; gens::Union{Nothing, Vector{<:Vector}} = nothing,
                                              snf::Bool = true,
                                              modulus::RationalUnion = QQFieldElem(0),
                                              modulus_qf::RationalUnion = QQFieldElem(0),
@@ -55,7 +55,7 @@
  1
  3//2
  1
N.B. Since there are no elements of Z-lattices, we think of elements of M as elements of the ambient vector space. Thus if v::Vector is such an element then the coordinates with respec to the basis of M are given by solve(basis_matrix(M), v; side = :left).
source
Most of the functionality mirrors that of AbGrp its elements and homomorphisms. Here we display the part that is specific to elements of torsion quadratic modules.
Attributes ​
abelian_group Method
julia
abelian_group(T::TorQuadModule) -> FinGenAbGroup
Return the underlying abelian group of T.
source
cover Method
julia
cover(T::TorQuadModule) -> ZZLat
For T=M/N this returns M.
source
relations Method
julia
relations(T::TorQuadModule) -> ZZLat
For T=M/N this returns N.
source
value_module Method
julia
value_module(T::TorQuadModule) -> QmodnZ
Return the value module Q/nZ of the bilinear form of T.
source
value_module_quadratic_form Method
julia
value_module_quadratic_form(T::TorQuadModule) -> QmodnZ
Return the value module Q/mZ of the quadratic form of T.
source
gram_matrix_bilinear Method
julia
gram_matrix_bilinear(T::TorQuadModule) -> QQMatrix
Return the gram matrix of the bilinear form of T.
source
gram_matrix_quadratic Method
julia
gram_matrix_quadratic(T::TorQuadModule) -> QQMatrix
Return the 'gram matrix' of the quadratic form of T.
The off diagonal entries are given by the bilinear form whereas the diagonal entries are given by the quadratic form.
source
modulus_bilinear_form Method
julia
modulus_bilinear_form(T::TorQuadModule) -> QQFieldElem
Return the modulus of the value module of the bilinear form ofT.
source
modulus_quadratic_form Method
julia
modulus_quadratic_form(T::TorQuadModule) -> QQFieldElem
Return the modulus of the value module of the quadratic form of T.
source
Elements ​
quadratic_product Method
julia
quadratic_product(a::TorQuadModuleElem) -> QmodnZElem
Return the quadratic product of a.
It is defined in terms of a representative: for b+M∈M/N=T, this returns Φ(b,b)+nZ.
source
inner_product Method
julia
inner_product(a::TorQuadModuleElem, b::TorQuadModuleElem) -> QmodnZElem
Return the inner product of a and b.
source
Lift to the cover ​
lift Method
julia
lift(a::TorQuadModuleElem) -> Vector{QQFieldElem}
Lift a to the ambient space of cover(parent(a)).
For a+N∈M/N this returns the representative a.
source
representative Method
julia
representative(a::TorQuadModuleElem) -> Vector{QQFieldElem}
For a+N∈M/N this returns the representative a. An alias for lift(a).
source
Orthogonal submodules ​
orthogonal_submodule Method
julia
orthogonal_submodule(T::TorQuadModule, S::TorQuadModule)-> TorQuadModule
Return the orthogonal submodule to the submodule S of T.
source
Isometry ​
is_isometric_with_isometry Method
julia
is_isometric_with_isometry(T::TorQuadModule, U::TorQuadModule)
-                                               -> Bool, TorQuadModuleMap
Return whether the torsion quadratic modules T and U are isometric. If yes, it also returns an isometry T→U.
If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).
It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).
Examples
julia
julia> T = torsion_quadratic_module(QQ[2//3 2//3    0    0    0;
+                                               -> Bool, TorQuadModuleMap
Return whether the torsion quadratic modules T and U are isometric. If yes, it also returns an isometry T→U.
If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).
It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).
Examples
julia
julia> T = torsion_quadratic_module(QQ[2//3 2//3    0    0    0;
                                        2//3 2//3 2//3    0 2//3;
                                           0 2//3 2//3 2//3    0;
                                           0    0 2//3 2//3    0;
@@ -106,7 +106,7 @@
 
 julia> is_bijective(phi)
 true
source
is_anti_isometric_with_anti_isometry Method
julia
is_anti_isometric_with_anti_isometry(T::TorQuadModule, U::TorQuadModule)
-                                                 -> Bool, TorQuadModuleMap
Return whether there exists an anti-isometry between the torsion quadratic modules T and U. If yes, it returns such an anti-isometry T→U.
If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).
It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).
Examples
julia
julia> T = torsion_quadratic_module(QQ[4//5;])
+                                                 -> Bool, TorQuadModuleMap
Return whether there exists an anti-isometry between the torsion quadratic modules T and U. If yes, it returns such an anti-isometry T→U.
If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).
It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).
Examples
julia
julia> T = torsion_quadratic_module(QQ[4//5;])
 Finite quadratic module
   over integer ring
 Abelian group: Z/5
@@ -166,7 +166,7 @@
 julia> a = gens(T)[1];
 
 julia> a*a == -phi(a)*phi(a)
-true
source
Primary and elementary modules ​
is_primary_with_prime Method
julia
is_primary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem
Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group for some prime number p. In case it is, p is also returned as second output.
Note that in the case of trivial groups, this function returns (true, 1). If T is not primary, the second return value is -1 by default.
source
is_primary Method
julia
is_primary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool
Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group.
source
is_elementary_with_prime Method
julia
is_elementary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem
Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group, for some prime number p. In case it is, p is also returned as second output.
Note that in the case of trivial groups, this function returns (true, 1). If T is not elementary, the second return value is -1 by default.
source
is_elementary Method
julia
is_elementary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool
Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group.
source
Smith normal form ​
snf Method
julia
snf(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap
Given a torsion quadratic module T, return a torsion quadratic module S, isometric to T, such that the underlying abelian group of S is in canonical Smith normal form. It comes with an isometry f:S→T.
source
is_snf Method
julia
is_snf(T::TorQuadModule) -> Bool
Given a torsion quadratic module T, return whether its underlying abelian group is in Smith normal form.
source
Discriminant Groups ​
See [6] for the general theory of discriminant groups. They are particularly useful to work with primitive embeddings of integral integer quadratic lattices.
From a lattice ​
discriminant_group Method
julia
discriminant_group(L::ZZLat) -> TorQuadModule
Return the discriminant group of L.
The discriminant group of an integral lattice L is the finite abelian group D = dual(L)/L.
It comes equipped with the discriminant bilinear form
D×D→Q/Z(x,y)↦Φ(x,y)+Z.
If L is even, then the discriminant group is equipped with the discriminant quadratic form D→Q/2Z,x↦Φ(x,x)+2Z.
source
From a matrix ​
torsion_quadratic_module Method
julia
torsion_quadratic_module(q::QQMatrix) -> TorQuadModule
Return a torsion quadratic module with gram matrix given by q and value module Q/Z. If all the diagonal entries of q have: either even numerator or even denominator, then the value module of the quadratic form is Q/2Z
Example
julia
julia> torsion_quadratic_module(QQ[1//6;])
+true
source
Primary and elementary modules ​
is_primary_with_prime Method
julia
is_primary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem
Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group for some prime number p. In case it is, p is also returned as second output.
Note that in the case of trivial groups, this function returns (true, 1). If T is not primary, the second return value is -1 by default.
source
is_primary Method
julia
is_primary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool
Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group.
source
is_elementary_with_prime Method
julia
is_elementary_with_prime(T::TorQuadModule) -> Bool, ZZRingElem
Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group, for some prime number p. In case it is, p is also returned as second output.
Note that in the case of trivial groups, this function returns (true, 1). If T is not elementary, the second return value is -1 by default.
source
is_elementary Method
julia
is_elementary(T::TorQuadModule, p::Union{Integer, ZZRingElem}) -> Bool
Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group.
source
Smith normal form ​
snf Method
julia
snf(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap
Given a torsion quadratic module T, return a torsion quadratic module S, isometric to T, such that the underlying abelian group of S is in canonical Smith normal form. It comes with an isometry f:S→T.
source
is_snf Method
julia
is_snf(T::TorQuadModule) -> Bool
Given a torsion quadratic module T, return whether its underlying abelian group is in Smith normal form.
source
Discriminant Groups ​
See [6] for the general theory of discriminant groups. They are particularly useful to work with primitive embeddings of integral integer quadratic lattices.
From a lattice ​
discriminant_group Method
julia
discriminant_group(L::ZZLat) -> TorQuadModule
Return the discriminant group of L.
The discriminant group of an integral lattice L is the finite abelian group D = dual(L)/L.
It comes equipped with the discriminant bilinear form
D×D→Q/Z(x,y)↦Φ(x,y)+Z.
If L is even, then the discriminant group is equipped with the discriminant quadratic form D→Q/2Z,x↦Φ(x,x)+2Z.
source
From a matrix ​
torsion_quadratic_module Method
julia
torsion_quadratic_module(q::QQMatrix) -> TorQuadModule
Return a torsion quadratic module with gram matrix given by q and value module Q/Z. If all the diagonal entries of q have: either even numerator or even denominator, then the value module of the quadratic form is Q/2Z
Example
julia
julia> torsion_quadratic_module(QQ[1//6;])
 Finite quadratic module
   over integer ring
 Abelian group: Z/6
@@ -200,9 +200,9 @@
 Bilinear value module: Q/Z
 Quadratic value module: Q/Z
 Gram matrix quadratic form:
-[1//3]
source
Rescaling the form ​
rescale Method
julia
rescale(T::TorQuadModule, k::RingElement) -> TorQuadModule
Return the torsion quadratic module with quadratic form scaled by k, where k is a non-zero rational number. If the old form was defined modulo n, then the new form is defined modulo n k.
source
Invariants ​
is_degenerate Method
julia
is_degenerate(T::TorQuadModule) -> Bool
Return true if the underlying bilinear form is degenerate.
source
is_semi_regular Method
julia
is_semi_regular(T::TorQuadModule) -> Bool
Return whether T is semi-regular, that is its quadratic radical is trivial (see radical_quadratic).
source
radical_bilinear Method
julia
radical_bilinear(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap
Return the radical \{x \in T | b(x,T) = 0\} of the bilinear form b on T.
source
radical_quadratic Method
julia
radical_quadratic(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap
Return the radical \{x \in T | b(x,T) = 0 and q(x)=0\} of the quadratic form q on T.
source
normal_form Method
julia
normal_form(T::TorQuadModule; partial=false) -> TorQuadModule, TorQuadModuleMap
Return the normal form N of the given torsion quadratic module T along with the projection T -> N.
Let K be the radical of the quadratic form of T. Then N = T/K is half-regular. Two half-regular torsion quadratic modules are isometric if and only if they have equal normal forms.
source
Genus ​
genus Method
julia
genus(T::TorQuadModule, signature_pair::Tuple{Int, Int};
+[1//3]
source
Rescaling the form ​
rescale Method
julia
rescale(T::TorQuadModule, k::RingElement) -> TorQuadModule
Return the torsion quadratic module with quadratic form scaled by k, where k is a non-zero rational number. If the old form was defined modulo n, then the new form is defined modulo n k.
source
Invariants ​
is_degenerate Method
julia
is_degenerate(T::TorQuadModule) -> Bool
Return true if the underlying bilinear form is degenerate.
source
is_semi_regular Method
julia
is_semi_regular(T::TorQuadModule) -> Bool
Return whether T is semi-regular, that is its quadratic radical is trivial (see radical_quadratic).
source
radical_bilinear Method
julia
radical_bilinear(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap
Return the radical \{x \in T | b(x,T) = 0\} of the bilinear form b on T.
source
radical_quadratic Method
julia
radical_quadratic(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMap
Return the radical \{x \in T | b(x,T) = 0 and q(x)=0\} of the quadratic form q on T.
source
normal_form Method
julia
normal_form(T::TorQuadModule; partial=false) -> TorQuadModule, TorQuadModuleMap
Return the normal form N of the given torsion quadratic module T along with the projection T -> N.
Let K be the radical of the quadratic form of T. Then N = T/K is half-regular. Two half-regular torsion quadratic modules are isometric if and only if they have equal normal forms.
source
Genus ​
genus Method
julia
genus(T::TorQuadModule, signature_pair::Tuple{Int, Int};
                         parity::RationalUnion = modulus_quadratic_form(T))
-                                                                -> ZZGenus
Return the genus of an integer lattice whose discriminant group has the bilinear form of T, the given signature_pair and the given parity.
The argument parity is one of the following: either parity == 1 for genera of odd lattices, or parity == 2 for even lattices. By default, parity is set to be as the parity of the quadratic form on T
If no such genus exists, raise an error.
Reference
[6] Corollary 1.9.4 and 1.16.3.
source
brown_invariant Method
julia
brown_invariant(self::TorQuadModule) -> Nemo.zzModRingElem
Return the Brown invariant of this torsion quadratic form.
Let (D,q) be a torsion quadratic module with values in Q / 2Z. The Brown invariant Br(D,q) in Z/8Z is defined by the equation
exp⁡(2πi8Br(q))=1D∑x∈Dexp⁡(iπq(x)).
The Brown invariant is additive with respect to direct sums of torsion quadratic modules.
Examples
julia
julia> L = integer_lattice(gram=matrix(ZZ, [[2,-1,0,0],[-1,2,-1,-1],[0,-1,2,0],[0,-1,0,2]]));
+                                                                -> ZZGenus
Return the genus of an integer lattice whose discriminant group has the bilinear form of T, the given signature_pair and the given parity.
The argument parity is one of the following: either parity == 1 for genera of odd lattices, or parity == 2 for even lattices. By default, parity is set to be as the parity of the quadratic form on T
If no such genus exists, raise an error.
Reference
[6] Corollary 1.9.4 and 1.16.3.
source
brown_invariant Method
julia
brown_invariant(self::TorQuadModule) -> Nemo.zzModRingElem
Return the Brown invariant of this torsion quadratic form.
Let (D,q) be a torsion quadratic module with values in Q / 2Z. The Brown invariant Br(D,q) in Z/8Z is defined by the equation
exp⁡(2πi8Br(q))=1D∑x∈Dexp⁡(iπq(x)).
The Brown invariant is additive with respect to direct sums of torsion quadratic modules.
Examples
julia
julia> L = integer_lattice(gram=matrix(ZZ, [[2,-1,0,0],[-1,2,-1,-1],[0,-1,2,0],[0,-1,0,2]]));
 
 julia> T = Hecke.discriminant_group(L);
 
@@ -211,8 +211,8 @@
                            parity::RationalUnion = modulus_quadratic_form(T)) -> Bool
Return if there is an integral lattice whose discriminant form has the bilinear form of T, whose signatures match signature_pair and which is of parity parity.
The argument parity is one of the following: either parity == 1 for genera of odd lattices, or parity == 2 for even lattices. By default, parity is set to be as the parity of the quadratic form on T
source
Categorical constructions ​
direct_sum Method
julia
direct_sum(x::Vararg{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMap}
 direct_sum(x::Vector{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMap}
Given a collection of torsion quadratic modules T1,…,Tn, return their direct sum T:=T1⊕…⊕Tn, together with the injections Ti→T.
For objects of type TorQuadModule, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain T as a direct product with the projections T→Ti, one should call direct_product(x). If one wants to obtain T as a biproduct with the injections Ti→T and the projections T→Ti, one should call biproduct(x).
source
direct_product Method
julia
direct_product(x::Vararg{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMap}
 direct_product(x::Vector{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMap}
Given a collection of torsion quadratic modules T1,…,Tn, return their direct product T:=T1×…×Tn, together with the projections T→Ti.
For objects of type TorQuadModule, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain T as a direct sum with the inctions Ti→T, one should call direct_sum(x). If one wants to obtain T as a biproduct with the injections Ti→T and the projections T→Ti, one should call biproduct(x).
source
biproduct Method
julia
biproduct(x::Vararg{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMap}, Vector{TorQuadModuleMap}
-biproduct(x::Vector{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMap}, Vector{TorQuadModuleMap}
Given a collection of torsion quadratic modules T1,…,Tn, return their biproduct T:=T1⊕…⊕Tn, together with the injections Ti→T and the projections T→Ti.
For objects of type TorQuadModule, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain T as a direct sum with the inctions Ti→T, one should call direct_sum(x). If one wants to obtain T as a direct product with the projections T→Ti, one should call direct_product(x).
source
Submodules ​
submodules Method
julia
submodules(T::TorQuadModule; kw...)
Return the submodules of T as an iterator. Possible keyword arguments to restrict the submodules:
  • order::Int: only submodules of order order,
  • index::Int: only submodules of index index,
  • subtype::Vector{Int}: only submodules which are isomorphic as an abelian group to abelian_group(subtype),
  • quotype::Vector{Int}: only submodules whose quotient are isomorphic as an abelian to abelian_group(quotype).
source
stable_submodules Method
julia
stable_submodules(T::TorQuadModule, act::Vector{TorQuadModuleMap}; kw...)
Return the submodules of T stable under the endomorphisms in act as an iterator. Possible keyword arguments to restrict the submodules:
  • quotype::Vector{Int}: only submodules whose quotient are isomorphic as an abelian group to abelian_group(quotype).
source

-    
+biproduct(x::Vector{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMap}, Vector{TorQuadModuleMap}
Given a collection of torsion quadratic modules T1,…,Tn, return their biproduct T:=T1⊕…⊕Tn, together with the injections Ti→T and the projections T→Ti.
For objects of type TorQuadModule, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain T as a direct sum with the inctions Ti→T, one should call direct_sum(x). If one wants to obtain T as a direct product with the projections T→Ti, one should call direct_product(x).
source
Submodules ​
submodules Method
julia
submodules(T::TorQuadModule; kw...)
Return the submodules of T as an iterator. Possible keyword arguments to restrict the submodules:
  • order::Int: only submodules of order order,
  • index::Int: only submodules of index index,
  • subtype::Vector{Int}: only submodules which are isomorphic as an abelian group to abelian_group(subtype),
  • quotype::Vector{Int}: only submodules whose quotient are isomorphic as an abelian to abelian_group(quotype).
source
stable_submodules Method
julia
stable_submodules(T::TorQuadModule, act::Vector{TorQuadModuleMap}; kw...)
Return the submodules of T stable under the endomorphisms in act as an iterator. Possible keyword arguments to restrict the submodules:
  • quotype::Vector{Int}: only submodules whose quotient are isomorphic as an abelian group to abelian_group(quotype).
source

+    
     
   
 
\ No newline at end of file
diff --git a/v0.34.8/manual/quad_forms/genusherm.html b/v0.34.8/manual/quad_forms/genusherm.html
index a358dce4e9..001fbb31e7 100644
--- a/v0.34.8/manual/quad_forms/genusherm.html
+++ b/v0.34.8/manual/quad_forms/genusherm.html
@@ -6,19 +6,19 @@
     Genera for hermitian lattices | Hecke
     
     
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Appearance
Genera for hermitian lattices ​
Local genus symbols ​
Definition 8.3.1 ([Kir16]) Let L be a hermitian lattice over E/K and let p be a prime ideal of OK. Let P be the largest ideal of OE over p being invariant under the involution of E. We suppose that we are given a Jordan decomposition
Lp=⊥i=1tLi
where the Jordan block Li is Psi-modular for 1≤i≤t, for a strictly increasing sequence of integers s1<…<st. In particular, s(Li)=Psi. Then, the local genus symbol g(L,p) of Lp is defined to be:
  • if p is good, i.e. non ramified and non dyadic,
g(L,p):=[(s1,r1,d1),…,(st,rt,dt)]
where di=1 if the determinant (resp. discriminant) of Li is a norm in Kp×, and di=−1 otherwise, and ri:=rank(Li) for all i;
  • if p is bad,
g(L,p):=[(s1,r1,d1,n1),…,(st,rt,dt,nt)]
where for all i, ni:=ordp(n(Li))
Note that we define the scale and the norm of the lattice Li (1≤i≤n) defined over the extension of local fields EP/Kp similarly to the ones of L, by extending by continuity the sesquilinear form of the ambient space of L to the completion. Regarding the determinant (resp. discriminant), it is defined as the determinant of the Gram matrix associated to a basis of Li relatively to the extension of the sesquilinear form (resp. (−1)(m(m−1)/2 times the determinant, where m is the rank of Li).
We call any tuple in g:=g(L,p)=[g1,…,gt] a Jordan block of g since it corresponds to invariants of a Jordan block of the completion of the lattice L at p. For any such block gi, we call respectively si,ri,di,ni the scale, the rank, the determinant class (resp. discriminant class) and the norm of gi. Note that the norm is necessary only when the prime ideal is bad.
We say that two hermitian lattices L and L′ over E/K are in the same local genus at p if g(L,p)=g(L′,p).
Creation of local genus symbols ​
There are two ways of creating a local genus symbol for hermitian lattices:
  • either abstractly, by choosing the extension E/K, the prime ideal p of OK, the Jordan blocks data and the type of the di's (either determinant class :det or discriminant class :disc);
julia
   genus(HermLat, E::NumField, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, data::Vector; type::Symbol = :det,
+    
Hecke
Appearance
Genera for hermitian lattices ​
Local genus symbols ​
Definition 8.3.1 ([Kir16]) Let L be a hermitian lattice over E/K and let p be a prime ideal of OK. Let P be the largest ideal of OE over p being invariant under the involution of E. We suppose that we are given a Jordan decomposition
Lp=⊥i=1tLi
where the Jordan block Li is Psi-modular for 1≤i≤t, for a strictly increasing sequence of integers s1<…<st. In particular, s(Li)=Psi. Then, the local genus symbol g(L,p) of Lp is defined to be:
  • if p is good, i.e. non ramified and non dyadic,
g(L,p):=[(s1,r1,d1),…,(st,rt,dt)]
where di=1 if the determinant (resp. discriminant) of Li is a norm in Kp×, and di=−1 otherwise, and ri:=rank(Li) for all i;
  • if p is bad,
g(L,p):=[(s1,r1,d1,n1),…,(st,rt,dt,nt)]
where for all i, ni:=ordp(n(Li))
Note that we define the scale and the norm of the lattice Li (1≤i≤n) defined over the extension of local fields EP/Kp similarly to the ones of L, by extending by continuity the sesquilinear form of the ambient space of L to the completion. Regarding the determinant (resp. discriminant), it is defined as the determinant of the Gram matrix associated to a basis of Li relatively to the extension of the sesquilinear form (resp. (−1)(m(m−1)/2 times the determinant, where m is the rank of Li).
We call any tuple in g:=g(L,p)=[g1,…,gt] a Jordan block of g since it corresponds to invariants of a Jordan block of the completion of the lattice L at p. For any such block gi, we call respectively si,ri,di,ni the scale, the rank, the determinant class (resp. discriminant class) and the norm of gi. Note that the norm is necessary only when the prime ideal is bad.
We say that two hermitian lattices L and L′ over E/K are in the same local genus at p if g(L,p)=g(L′,p).
Creation of local genus symbols ​
There are two ways of creating a local genus symbol for hermitian lattices:
  • either abstractly, by choosing the extension E/K, the prime ideal p of OK, the Jordan blocks data and the type of the di's (either determinant class :det or discriminant class :disc);
julia
   genus(HermLat, E::NumField, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, data::Vector; type::Symbol = :det,
                                                           check::Bool = false)
                                                              -> HermLocalGenus
  • or by constructing the local genus symbol of the completion of a hermitian lattice L over E/K at a prime ideal p of OK.
julia
   genus(L::HermLat, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> HermLocalGenus
Examples ​
We will construct two examples for the rest of this section. Note that the prime chosen here is bad.
julia

 julia> Qx, x = QQ["x"];
@@ -478,8 +478,8 @@
 with pseudo-basis
 (1, 1//1 * <1, 1>)
 (_$, 1//1 * <1, 1>)
Rescaling ​
rescale Method
julia
rescale(g::HermLocalGenus, a::Union{FieldElem, RationalUnion})
-                                                          -> HermLocalGenus
Given a local genus symbol G of hermitian lattices and an element a lying in the base field E of g, return the local genus symbol at the prime ideal p associated to g of any representative of g rescaled by a.
source
rescale Method
julia
rescale(G::HermGenus, a::Union{FieldElem, RationalUnion}) -> HermGenus
Given a global genus symbol G of hermitian lattices and an element a lying in the base field E of G, return the global genus symbol of any representative of G rescaled by a.
source

-    
+                                                          -> HermLocalGenus
Given a local genus symbol G of hermitian lattices and an element a lying in the base field E of g, return the local genus symbol at the prime ideal p associated to g of any representative of g rescaled by a.
source
rescale Method
julia
rescale(G::HermGenus, a::Union{FieldElem, RationalUnion}) -> HermGenus
Given a global genus symbol G of hermitian lattices and an element a lying in the base field E of G, return the global genus symbol of any representative of G rescaled by a.
source

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\ No newline at end of file
diff --git a/v0.34.8/manual/quad_forms/integer_lattices.html b/v0.34.8/manual/quad_forms/integer_lattices.html
index 60dfd3edf9..f8db338319 100644
--- a/v0.34.8/manual/quad_forms/integer_lattices.html
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     Integer Lattices | Hecke
     
     
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Hecke
Appearance
Integer Lattices ​
An integer lattice L is a finitely generated Z-submodule of a quadratic vector space V=Qn over the rational numbers. Integer lattices are also known as quadratic forms over the integers. We will refer to them as Z-lattices.
A Z-lattice L has the type ZZLat. It is given in terms of its ambient quadratic space V together with a basis matrix B whose rows span L, i.e. L=ZrB where r is the (Z-module) rank of L.
To access V and B see ambient_space(L::ZZLat) and basis_matrix(L::ZZLat).
Creation of integer lattices ​
From a gram matrix ​
integer_lattice Method
julia
integer_lattice([B::MatElem]; gram) -> ZZLat
Return the Z-lattice with basis matrix B inside the quadratic space with Gram matrix gram.
If the keyword gram is not specified, the Gram matrix is the identity matrix. If B is not specified, the basis matrix is the identity matrix.
Examples
julia
julia> L = integer_lattice(matrix(QQ, 2, 2, [1//2, 0, 0, 2]));
+    
Hecke
Appearance
Integer Lattices ​
An integer lattice L is a finitely generated Z-submodule of a quadratic vector space V=Qn over the rational numbers. Integer lattices are also known as quadratic forms over the integers. We will refer to them as Z-lattices.
A Z-lattice L has the type ZZLat. It is given in terms of its ambient quadratic space V together with a basis matrix B whose rows span L, i.e. L=ZrB where r is the (Z-module) rank of L.
To access V and B see ambient_space(L::ZZLat) and basis_matrix(L::ZZLat).
Creation of integer lattices ​
From a gram matrix ​
integer_lattice Method
julia
integer_lattice([B::MatElem]; gram) -> ZZLat
Return the Z-lattice with basis matrix B inside the quadratic space with Gram matrix gram.
If the keyword gram is not specified, the Gram matrix is the identity matrix. If B is not specified, the basis matrix is the identity matrix.
Examples
julia
julia> L = integer_lattice(matrix(QQ, 2, 2, [1//2, 0, 0, 2]));
 
 julia> gram_matrix(L) == matrix(QQ, 2, 2, [1//4, 0, 0, 4])
 true
@@ -36,7 +36,7 @@
 
 julia> gram_matrix(L)
 [  0   -13]
-[-13     0]
source
integer_lattice Method
julia
integer_lattice(S::Symbol, n::RationalUnion = 1) -> ZZlat
Given S = :H or S = :U, return a Z-lattice admitting n∗J2 as Gram matrix in some basis, where J2 is the 2-by-2 matrix with 0's on the main diagonal and 1's elsewhere.
source
leech_lattice Function
julia
leech_lattice() -> ZZLat
Return the Leech lattice.
source
julia
leech_lattice(niemeier_lattice::ZZLat) -> ZZLat, QQMatrix, Int
Return a triple L, v, h where L is the Leech lattice.
L is an h-neighbor of the Niemeier lattice N with respect to v. This means that L / L ∩ N ≅ ℤ / h ℤ. Here h is the Coxeter number of the Niemeier lattice.
This implements the 23 holy constructions of the Leech lattice in [5].
Examples
julia
julia> R = integer_lattice(gram=2 * identity_matrix(ZZ, 24));
+[-13     0]
source
integer_lattice Method
julia
integer_lattice(S::Symbol, n::RationalUnion = 1) -> ZZlat
Given S = :H or S = :U, return a Z-lattice admitting n∗J2 as Gram matrix in some basis, where J2 is the 2-by-2 matrix with 0's on the main diagonal and 1's elsewhere.
source
leech_lattice Function
julia
leech_lattice() -> ZZLat
Return the Leech lattice.
source
julia
leech_lattice(niemeier_lattice::ZZLat) -> ZZLat, QQMatrix, Int
Return a triple L, v, h where L is the Leech lattice.
L is an h-neighbor of the Niemeier lattice N with respect to v. This means that L / L ∩ N ≅ ℤ / h ℤ. Here h is the Coxeter number of the Niemeier lattice.
This implements the 23 holy constructions of the Leech lattice in [5].
Examples
julia
julia> R = integer_lattice(gram=2 * identity_matrix(ZZ, 24));
 
 julia> N = maximal_even_lattice(R) # Some Niemeier lattice
 Integer lattice of rank 24 and degree 24
@@ -169,7 +169,7 @@
 Genus symbol for integer lattices
 Signatures: (3, 0, 20)
 Local symbol:
-  Local genus symbol at 2: 1^22 4^1_7
source
From a genus ​
Integer lattices can be created as representatives of a genus. See (representative(L::ZZGenus))
Rescaling the Quadratic Form ​
rescale Method
julia
rescale(L::ZZLat, r::RationalUnion) -> ZZLat
Return the lattice L in the quadratic space with form r \Phi.
Examples
This can be useful to apply methods intended for positive definite lattices.
julia
julia> L = integer_lattice(gram=ZZ[-1 0; 0 -1])
+  Local genus symbol at 2: 1^22 4^1_7
source
From a genus ​
Integer lattices can be created as representatives of a genus. See (representative(L::ZZGenus))
Rescaling the Quadratic Form ​
rescale Method
julia
rescale(L::ZZLat, r::RationalUnion) -> ZZLat
Return the lattice L in the quadratic space with form r \Phi.
Examples
This can be useful to apply methods intended for positive definite lattices.
julia
julia> L = integer_lattice(gram=ZZ[-1 0; 0 -1])
 Integer lattice of rank 2 and degree 2
 with gram matrix
 [-1    0]
@@ -189,9 +189,9 @@
   over rational field
 with gram matrix
 [ 4   -2]
-[-2    5]
source
Invariants ​
rank Method
julia
rank(L::AbstractLat) -> Int
Return the rank of the underlying module of the lattice L.
source
det Method
julia
det(L::ZZLat) -> QQFieldElem
Return the determinant of the gram matrix of L.
source
scale Method
julia
scale(L::ZZLat) -> QQFieldElem
Return the scale of L.
The scale of L is defined as the positive generator of the Z-ideal generated by {Φ(x,y):x,y∈L}.
source
norm Method
julia
norm(L::ZZLat) -> QQFieldElem
Return the norm of L.
The norm of L is defined as the positive generator of the Z- ideal generated by {Φ(x,x):x∈L}.
source
iseven Method
julia
iseven(L::ZZLat) -> Bool
Return whether L is even.
An integer lattice L in the rational quadratic space (V,Φ) is called even if Φ(x,x)∈2Z for all xinL.
source
is_integral Method
julia
is_integral(L::AbstractLat) -> Bool
Return whether the lattice L is integral.
source
is_primary_with_prime Method
julia
is_primary_with_prime(L::ZZLat) -> Bool, ZZRingElem
Given a Z-lattice L, return whether L is primary, that is whether L is integral and its discriminant group (see discriminant_group) is a p-group for some prime number p. In case it is, p is also returned as second output.
Note that for unimodular lattices, this function returns (true, 1). If the lattice is not primary, the second return value is -1 by default.
source
is_primary Method
julia
is_primary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
Given an integral Z-lattice L and a prime number p, return whether L is p-primary, that is whether its discriminant group (see discriminant_group) is a p-group.
source
is_elementary_with_prime Method
julia
is_elementary_with_prime(L::ZZLat) -> Bool, ZZRingElem
Given a Z-lattice L, return whether L is elementary, that is whether L is integral and its discriminant group (see discriminant_group) is an elemenentary p-group for some prime number p. In case it is, p is also returned as second output.
Note that for unimodular lattices, this function returns (true, 1). If the lattice is not elementary, the second return value is -1 by default.
source
is_elementary Method
julia
is_elementary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
Given an integral Z-lattice L and a prime number p, return whether L is p-elementary, that is whether its discriminant group (see discriminant_group) is an elementary p-group.
source
The Genus ​
For an integral lattice The genus of an integer lattice collects its local invariants. genus(::ZZLat)
mass Method
julia
mass(L::ZZLat) -> QQFieldElem
Return the mass of the genus of L.
source
genus_representatives Method
julia
genus_representatives(L::ZZLat) -> Vector{ZZLat}
Return representatives for the isometry classes in the genus of L.
source
Real invariants ​
signature_tuple Method
julia
signature_tuple(L::ZZLat) -> Tuple{Int,Int,Int}
Return the number of (positive, zero, negative) inertia of L.
source
is_positive_definite Method
julia
is_positive_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is positive definite.
source
is_negative_definite Method
julia
is_negative_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is negative definite.
source
is_definite Method
julia
is_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is definite.
source
Isometries ​
automorphism_group_generators Method
julia
automorphism_group_generators(E::EllipticCurve) -> Vector{EllCrvIso}
Return generators of the automorphism group of E.
source
julia
automorphism_group_generators(L::AbstractLat; ambient_representation::Bool = true,
+[-2    5]
source
Invariants ​
rank Method
julia
rank(L::AbstractLat) -> Int
Return the rank of the underlying module of the lattice L.
source
det Method
julia
det(L::ZZLat) -> QQFieldElem
Return the determinant of the gram matrix of L.
source
scale Method
julia
scale(L::ZZLat) -> QQFieldElem
Return the scale of L.
The scale of L is defined as the positive generator of the Z-ideal generated by {Φ(x,y):x,y∈L}.
source
norm Method
julia
norm(L::ZZLat) -> QQFieldElem
Return the norm of L.
The norm of L is defined as the positive generator of the Z- ideal generated by {Φ(x,x):x∈L}.
source
iseven Method
julia
iseven(L::ZZLat) -> Bool
Return whether L is even.
An integer lattice L in the rational quadratic space (V,Φ) is called even if Φ(x,x)∈2Z for all xinL.
source
is_integral Method
julia
is_integral(L::AbstractLat) -> Bool
Return whether the lattice L is integral.
source
is_primary_with_prime Method
julia
is_primary_with_prime(L::ZZLat) -> Bool, ZZRingElem
Given a Z-lattice L, return whether L is primary, that is whether L is integral and its discriminant group (see discriminant_group) is a p-group for some prime number p. In case it is, p is also returned as second output.
Note that for unimodular lattices, this function returns (true, 1). If the lattice is not primary, the second return value is -1 by default.
source
is_primary Method
julia
is_primary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
Given an integral Z-lattice L and a prime number p, return whether L is p-primary, that is whether its discriminant group (see discriminant_group) is a p-group.
source
is_elementary_with_prime Method
julia
is_elementary_with_prime(L::ZZLat) -> Bool, ZZRingElem
Given a Z-lattice L, return whether L is elementary, that is whether L is integral and its discriminant group (see discriminant_group) is an elemenentary p-group for some prime number p. In case it is, p is also returned as second output.
Note that for unimodular lattices, this function returns (true, 1). If the lattice is not elementary, the second return value is -1 by default.
source
is_elementary Method
julia
is_elementary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
Given an integral Z-lattice L and a prime number p, return whether L is p-elementary, that is whether its discriminant group (see discriminant_group) is an elementary p-group.
source
The Genus ​
For an integral lattice The genus of an integer lattice collects its local invariants. genus(::ZZLat)
mass Method
julia
mass(L::ZZLat) -> QQFieldElem
Return the mass of the genus of L.
source
genus_representatives Method
julia
genus_representatives(L::ZZLat) -> Vector{ZZLat}
Return representatives for the isometry classes in the genus of L.
source
Real invariants ​
signature_tuple Method
julia
signature_tuple(L::ZZLat) -> Tuple{Int,Int,Int}
Return the number of (positive, zero, negative) inertia of L.
source
is_positive_definite Method
julia
is_positive_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is positive definite.
source
is_negative_definite Method
julia
is_negative_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is negative definite.
source
is_definite Method
julia
is_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L is definite.
source
Isometries ​
automorphism_group_generators Method
julia
automorphism_group_generators(E::EllipticCurve) -> Vector{EllCrvIso}
Return generators of the automorphism group of E.
source
julia
automorphism_group_generators(L::AbstractLat; ambient_representation::Bool = true,
                                               depth::Int = -1, bacher_depth::Int = 0)
-                                                      -> Vector{MatElem}
Given a definite lattice L, return generators for the automorphism group of L. If ambient_representation == true (the default), the transformations are represented with respect to the ambient space of L. Otherwise, the transformations are represented with respect to the (pseudo-)basis of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
automorphism_group_order Method
julia
automorphism_group_order(L::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Int
Given a definite lattice L, return the order of the automorphism group of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
is_isometric Method
julia
is_isometric(L::AbstractLat, M::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Bool
Return whether the lattices L and M are isometric.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
is_locally_isometric Method
julia
is_locally_isometric(L::ZZLat, M::ZZLat, p::Int) -> Bool
Return whether L and M are isometric over the p-adic integers.
i.e. whether L⊗Zp≅M⊗Zp.
source
Root lattices ​
root_lattice_recognition Method
julia
root_lattice_recognition(L::ZZLat)
Return the ADE type of the root sublattice of L.
The root sublattice is the lattice spanned by the vectors of squared length 1 and 2. The odd lattice of rank 1 and determinant 1 is denoted by (:I, 1).
Input:
L – a definite and integral Z-lattice.
Output:
Two lists, the first one containing the ADE types and the second one the irreducible root sublattices.
For more recognizable gram matrices use root_lattice_recognition_fundamental.
Examples
julia
julia> L = integer_lattice(gram=ZZ[4  0 0  0 3  0 3  0;
+                                                      -> Vector{MatElem}
Given a definite lattice L, return generators for the automorphism group of L. If ambient_representation == true (the default), the transformations are represented with respect to the ambient space of L. Otherwise, the transformations are represented with respect to the (pseudo-)basis of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
automorphism_group_order Method
julia
automorphism_group_order(L::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Int
Given a definite lattice L, return the order of the automorphism group of L.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
is_isometric Method
julia
is_isometric(L::AbstractLat, M::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Bool
Return whether the lattices L and M are isometric.
Setting the parameters depth and bacher_depth to a positive value may improve performance. If set to -1 (default), the used value of depth is chosen heuristically depending on the rank of L. By default, bacher_depth is set to 0.
source
is_locally_isometric Method
julia
is_locally_isometric(L::ZZLat, M::ZZLat, p::Int) -> Bool
Return whether L and M are isometric over the p-adic integers.
i.e. whether L⊗Zp≅M⊗Zp.
source
Root lattices ​
root_lattice_recognition Method
julia
root_lattice_recognition(L::ZZLat)
Return the ADE type of the root sublattice of L.
The root sublattice is the lattice spanned by the vectors of squared length 1 and 2. The odd lattice of rank 1 and determinant 1 is denoted by (:I, 1).
Input:
L – a definite and integral Z-lattice.
Output:
Two lists, the first one containing the ADE types and the second one the irreducible root sublattices.
For more recognizable gram matrices use root_lattice_recognition_fundamental.
Examples
julia
julia> L = integer_lattice(gram=ZZ[4  0 0  0 3  0 3  0;
                             0 16 8 12 2 12 6 10;
                             0  8 8  6 2  8 4  5;
                             0 12 6 10 2  9 5  8;
@@ -253,7 +253,7 @@
 [0   -1   -1    2   -1    0    0]
 [0    0    0   -1    2   -1    0]
 [0    0    0    0   -1    2   -1]
-[0    0    0    0    0   -1    2]
source
ADE_type Method
julia
ADE_type(G::MatrixElem) -> Tuple{Symbol,Int64}
Return the type of the irreducible root lattice with gram matrix G.
See also root_lattice_recognition.
Examples
julia
julia> Hecke.ADE_type(gram_matrix(root_lattice(:A,3)))
+[0    0    0    0    0   -1    2]
source
ADE_type Method
julia
ADE_type(G::MatrixElem) -> Tuple{Symbol,Int64}
Return the type of the irreducible root lattice with gram matrix G.
See also root_lattice_recognition.
Examples
julia
julia> Hecke.ADE_type(gram_matrix(root_lattice(:A,3)))
 (:A, 3)
source
coxeter_number Method
julia
coxeter_number(ADE::Symbol, n) -> Int
Return the Coxeter number of the corresponding ADE root lattice.
If L is a root lattice and R its set of roots, then the Coxeter number h is |R|/n where n is the rank of L.
Examples
julia
julia> coxeter_number(:D, 4)
 6
source
highest_root Method
julia
highest_root(ADE::Symbol, n) -> ZZMatrix
Return coordinates of the highest root of root_lattice(ADE, n).
Examples
julia
julia> highest_root(:E, 6)
 [1   2   3   2   1   2]
source
Module operations ​
Most module operations assume that the lattices live in the same ambient space. For instance only lattices in the same ambient space compare.
== Method
Return true if both lattices have the same ambient quadratic space and the same underlying module.
source
is_sublattice Method
julia
is_sublattice(L::AbstractLat, M::AbstractLat) -> Bool
Return whether M is a sublattice of the lattice L.
source
is_sublattice_with_relations Method
julia
is_sublattice_with_relations(M::ZZLat, N::ZZLat) -> Bool, QQMatrix
Returns whether N is a sublattice of M. In this case, the second return value is a matrix B such that BBM=BN, where BM and BN are the basis matrices of M and N respectively.
source
+ Method
julia
+(L::AbstractLat, M::AbstractLat) -> AbstractLat
Return the sum of the lattices L and M.
The lattices L and M must have the same ambient space.
source
* Method
julia
*(a::RationalUnion, L::ZZLat) -> ZZLat
Return the lattice aM inside the ambient space of M.
source
intersect Method
julia
intersect(L::AbstractLat, M::AbstractLat) -> AbstractLat
Return the intersection of the lattices L and M.
The lattices L and M must have the same ambient space.
source
in Method
julia
Base.in(v::Vector, L::ZZLat) -> Bool
Return whether the vector v lies in the lattice L.
source
in Method
julia
Base.in(v::QQMatrix, L::ZZLat) -> Bool
Return whether the row span of v lies in the lattice L.
source
primitive_closure Method
julia
primitive_closure(M::ZZLat, N::ZZLat) -> ZZLat
Given two Z-lattices M and N with N⊆QM, return the primitive closure M∩QN of N in M.
Examples
julia
julia> M = root_lattice(:D, 6);
@@ -302,7 +302,7 @@
 true
source
is_primitive Method
julia
is_primitive(L::ZZLat, v::Union{Vector, QQMatrix}) -> Bool
Return whether the vector v is primitive in L.
A vector v in a Z-lattice L is called primitive if for all w in L such that v=dw for some integer d, then d=±1.
source
divisibility Method
julia
divisibility(L::ZZLat, v::Union{Vector, QQMatrix}) -> QQFieldElem
Return the divisibility of v with respect to L.
For a vector v in the ambient quadratic space (V,Φ) of L, we call the divisibility of v with the respect to L the non-negative generator of the fractional Z-ideal Φ(v,L).
source
Embeddings ​
Categorical constructions ​
direct_sum Method
julia
direct_sum(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
 direct_sum(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
Given a collection of Z-lattices L1,…,Ln, return their direct sum L:=L1⊕…⊕Ln, together with the injections Li→L. (seen as maps between the corresponding ambient spaces).
For objects of type ZZLat, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain L as a direct product with the projections L→Li, one should call direct_product(x). If one wants to obtain L as a biproduct with the injections Li→L and the projections L→Li, one should call biproduct(x).
source
direct_product Method
julia
direct_product(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
 direct_product(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
Given a collection of Z-lattices L1,…,Ln, return their direct product L:=L1×…×Ln, together with the projections L→Li. (seen as maps between the corresponding ambient spaces).
For objects of type ZZLat, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain L as a direct sum with the injections Li→L, one should call direct_sum(x). If one wants to obtain L as a biproduct with the injections Li→L and the projections L→Li, one should call biproduct(x).
source
biproduct Method
julia
biproduct(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
-biproduct(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
Given a collection of Z-lattices L1,…,Ln, return their biproduct L:=L1⊕…⊕Ln, together with the injections Li→L and the projections L→Li. (seen as maps between the corresponding ambient spaces).
For objects of type ZZLat, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain L as a direct sum with the injections Li→L, one should call direct_sum(x). If one wants to obtain L as a direct product with the projections L→Li, one should call direct_product(x).
source
Orthogonal sublattices ​
orthogonal_submodule Method
julia
orthogonal_submodule(L::ZZLat, S::ZZLat) -> ZZLat
Return the largest submodule of L orthogonal to S.
source
irreducible_components Method
julia
irreducible_components(L::ZZLat) -> Vector{ZZLat}
Return the irreducible components Li of the positive definite lattice L.
This yields a maximal orthogonal splitting of L as
L=⨁iLi.
source
Dual lattice ​
dual Method
julia
dual(L::AbstractLat) -> AbstractLat
Return the dual lattice of the lattice L.
source
Discriminant group ​
See discriminant_group(L::ZZLat).
Overlattices ​
glue_map Method
julia
glue_map(L::ZZLat, S::ZZLat, R::ZZLat; check=true)
+biproduct(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
Given a collection of Z-lattices L1,…,Ln, return their biproduct L:=L1⊕…⊕Ln, together with the injections Li→L and the projections L→Li. (seen as maps between the corresponding ambient spaces).
For objects of type ZZLat, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain L as a direct sum with the injections Li→L, one should call direct_sum(x). If one wants to obtain L as a direct product with the projections L→Li, one should call direct_product(x).
source
Orthogonal sublattices ​
orthogonal_submodule Method
julia
orthogonal_submodule(L::ZZLat, S::ZZLat) -> ZZLat
Return the largest submodule of L orthogonal to S.
source
irreducible_components Method
julia
irreducible_components(L::ZZLat) -> Vector{ZZLat}
Return the irreducible components Li of the positive definite lattice L.
This yields a maximal orthogonal splitting of L as
L=⨁iLi.
source
Dual lattice ​
dual Method
julia
dual(L::AbstractLat) -> AbstractLat
Return the dual lattice of the lattice L.
source
Discriminant group ​
See discriminant_group(L::ZZLat).
Overlattices ​
glue_map Method
julia
glue_map(L::ZZLat, S::ZZLat, R::ZZLat; check=true)
                        -> Tuple{TorQuadModuleMap, TorQuadModuleMap, TorQuadModuleMap}
Given three integral Z-lattices L, S and R, with S and R primitive sublattices of L and such that the sum of the ranks of S and R is equal to the rank of L, return the glue map γ of the primitive extension S+R⊆L, as well as the inclusion maps of the domain and codomain of γ into the respective discriminant groups of S and R.
Example
julia
julia> M = root_lattice(:E,8);
 
 julia> f = matrix(QQ, 8, 8, [-1 -1  0  0  0  0  0  0;
@@ -371,7 +371,7 @@
                   ambient_representation::Bool = true) -> ZZLat
Given a Z-lattice L and a list of matrices G inducing endomorphisms of L (or just one matrix G), return the lattice LG, consisting on elements fixed by G.
If ambient_representation is true (the default), the endomorphism is represented with respect to the ambient space of L. Otherwise, the endomorphism is represented with respect to the basis of L.
source
coinvariant_lattice Method
julia
coinvariant_lattice(L::ZZLat, G::Vector{MatElem};
                     ambient_representation::Bool = true) -> ZZLat
 coinvariant_lattice(L::ZZLat, G::MatElem;
-                    ambient_representation::Bool = true) -> ZZLat
Given a Z-lattice L and a list of matrices G inducing endomorphisms of L (or just one matrix G), return the orthogonal complement LG in L of the fixed lattice LG (see invariant_lattice).
If ambient_representation is true (the default), the endomorphism is represented with respect to the ambient space of L. Otherwise, the endomorphism is represented with respect to the basis of L.
source
Computing embeddings ​
embed Method
julia
embed(S::ZZLat, G::Genus, primitive::Bool=true) -> Bool, embedding
Return a (primitive) embedding of the integral lattice S into some lattice in the genus of G.
julia
julia> G = integer_genera((8,0), 1, even=true)[1];
+                    ambient_representation::Bool = true) -> ZZLat
Given a Z-lattice L and a list of matrices G inducing endomorphisms of L (or just one matrix G), return the orthogonal complement LG in L of the fixed lattice LG (see invariant_lattice).
If ambient_representation is true (the default), the endomorphism is represented with respect to the ambient space of L. Otherwise, the endomorphism is represented with respect to the basis of L.
source
Computing embeddings ​
embed Method
julia
embed(S::ZZLat, G::Genus, primitive::Bool=true) -> Bool, embedding
Return a (primitive) embedding of the integral lattice S into some lattice in the genus of G.
julia
julia> G = integer_genera((8,0), 1, even=true)[1];
 
 julia> L, S, i = embed(root_lattice(:A,5), G);
source
embed_in_unimodular Method
julia
embed_in_unimodular(S::ZZLat, pos::Int, neg::Int, primitive=true, even=true) -> Bool, L, S', iS, iR
Return a (primitive) embedding of the integral lattice S into some (even) unimodular lattice of signature (pos, neg).
For now this works only for even lattices.
julia
julia> NS = direct_sum(integer_lattice(:U), rescale(root_lattice(:A, 16), -1))[1];
 
@@ -407,10 +407,10 @@
 
 julia> is_primitive(LK3, iNS)
 true
source
LLL, Short and Close Vectors ​
LLL and indefinite LLL ​
lll Method
julia
lll(L::ZZLat, same_ambient::Bool = true) -> ZZLat
Given an integral Z-lattice L with basis matrix B, compute a basis C of L such that the gram matrix GC of L with respect to C is LLL-reduced.
By default, it creates the lattice in the same ambient space as L. This can be disabled by setting same_ambient = false. Works with both definite and indefinite lattices.
source
Short Vectors ​
short_vectors Function
julia
short_vectors(L::ZZLat, [lb = 0], ub, [elem_type = ZZRingElem]; check::Bool = true)
-                                   -> Vector{Tuple{Vector{elem_type}, QQFieldElem}}
Return all tuples (v, n) such that n=|vGvt| satisfies lb <= n <= ub, where G is the Gram matrix of L and v is non-zero.
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors_iterator for an iterator version.
source
shortest_vectors Function
julia
shortest_vectors(L::ZZLat, [elem_type = ZZRingElem]; check::Bool = true)
-                                           -> QQFieldElem, Vector{elem_type}, QQFieldElem}
Return the list of shortest non-zero vectors in absolute value. Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also minimum.
source
short_vectors_iterator Function
julia
short_vectors_iterator(L::ZZLat, [lb = 0], ub,
+                                   -> Vector{Tuple{Vector{elem_type}, QQFieldElem}}
Return all tuples (v, n) such that n=|vGvt| satisfies lb <= n <= ub, where G is the Gram matrix of L and v is non-zero.
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors_iterator for an iterator version.
source
shortest_vectors Function
julia
shortest_vectors(L::ZZLat, [elem_type = ZZRingElem]; check::Bool = true)
+                                           -> QQFieldElem, Vector{elem_type}, QQFieldElem}
Return the list of shortest non-zero vectors in absolute value. Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also minimum.
source
short_vectors_iterator Function
julia
short_vectors_iterator(L::ZZLat, [lb = 0], ub,
                        [elem_type = ZZRingElem]; check::Bool = true)
-                                -> Tuple{Vector{elem_type}, QQFieldElem} (iterator)
Return an iterator for all tuples (v, n) such that n=|vGvt| satisfies lb <= n <= ub, where G is the Gram matrix of L and v is non-zero.
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors.
source
minimum Method
julia
minimum(L::ZZLat) -> QQFieldElem
Return the minimum absolute squared length among the non-zero vectors in L.
source
kissing_number Method
julia
kissing_number(L::ZZLat) -> Int
Return the Kissing number of the sphere packing defined by L.
This is the number of non-overlapping spheres touching any other given sphere.
source
Close Vectors ​
close_vectors Method
julia
close_vectors(L:ZZLat, v:Vector, [lb,], ub; check::Bool = false)
+                                -> Tuple{Vector{elem_type}, QQFieldElem} (iterator)
Return an iterator for all tuples (v, n) such that n=|vGvt| satisfies lb <= n <= ub, where G is the Gram matrix of L and v is non-zero.
Note that the vectors are computed up to sign (so only one of v and -v appears).
It is assumed and checked that L is definite.
See also short_vectors.
source
minimum Method
julia
minimum(L::ZZLat) -> QQFieldElem
Return the minimum absolute squared length among the non-zero vectors in L.
source
kissing_number Method
julia
kissing_number(L::ZZLat) -> Int
Return the Kissing number of the sphere packing defined by L.
This is the number of non-overlapping spheres touching any other given sphere.
source
Close Vectors ​
close_vectors Method
julia
close_vectors(L:ZZLat, v:Vector, [lb,], ub; check::Bool = false)
                                         -> Vector{Tuple{Vector{Int}}, QQFieldElem}
Return all tuples (x, d) where x is an element of L such that d = b(v - x, v - x) <= ub. If lb is provided, then also lb <= d.
If filter is not nothing, then only those x with filter(x) evaluating to true are returned.
By default, it will be checked whether L is positive definite. This can be disabled setting check = false.
Both input and output are with respect to the basis matrix of L.
Examples
julia
julia> L = integer_lattice(matrix(QQ, 2, 2, [1, 0, 0, 2]));
 
 julia> close_vectors(L, [1, 1], 1)
@@ -422,8 +422,8 @@
 julia> close_vectors(L, [1, 1], 1, 1)
 2-element Vector{Tuple{Vector{ZZRingElem}, QQFieldElem}}:
  ([2, 1], 1)
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     Introduction | Hecke
     
     
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Introduction ​
This chapter deals with quadratic and hermitian spaces, and lattices there of. Note that even though quadratic spaces/lattices are theoretically a special case of hermitian spaces/lattices, a particular distinction is made here. As a note for knowledgeable users, only methods regarding hermitian spaces/lattices over degree 1 and degree 2 extensions of number fields are implemented up to now.
Definitions and vocabulary ​
We begin by collecting the necessary definitions and vocabulary. The terminology follows mainly [Kir16]
Quadratic and hermitian spaces ​
Let K be a number field and let E be a finitely generated etale algebra over K of dimension 1 or 2, i.e. E=K or E is a separable extension of K of degree 2. In both cases, E/K is endowed with an K-linear involution x―:E→E for which K is the fixed field (in the case E=K, this is simply the identity of K).
A hermitian space V over E/K is a finite-dimensional E-vector space, together with a sesquilinear (with respect to the involution of E/K) morphism Φ:V×V→E. In the trivial case E=K, Φ is therefore a K-bilinear morphism and we called (V,Φ) a quadratic hermitian space over K.
We will always work with an implicit canonical basis e1,…,en of V. In view of this, hermitian spaces over E/K are in bijection with hermitian matrices with entries in E, with respect to the involution x―. In particular, there is a bijection between quadratic hermitian spaces over K and symmetric matrices with entries in K. For any basis B=(v1,…,vn) of (V,Φ), we call the matrix GB=(Φ(vi,vj))1≤i,j≤n∈En×n the Gram matrix of (V,Φ) associated to B. If B is the implicit fixed canonical basis of (V,Φ), we simply talk about the Gram matrix of (V,Φ).
For a hermitian space V, we refer to the field E as the base ring of V and to x― as the involution of V. Meanwhile, the field K is referred to as the fixed field of V.
By abuse of language, non-quadratic hermitian spaces are sometimes simply called hermitian spaces and, in contrast, quadratic hermitian spaces are called quadratic spaces. In a general context, an arbitrary space (quadratic or hermitian) is referred to as a space throughout this chapter.
Quadratic and hermitian lattices ​
Let V be a space over E/K. A finitely generated OE-submodule L of V is called a hermitian lattice. By extension of vocabulary if V is quadratic (i.e. E=K), L is called a quadratic hermitian lattice. We call V the ambient space of L and L⊗OEE the rational span of L.
For a hermitian lattice L, we refer to E as the base field of L and to the ring OE as the base ring of L. We also call x―:E→E the involution of L. Finally, we refer to the field K fixed by this involution as the fixed field of L and to OK as the fixed ring of L.
Once again by abuse of language, non-quadratic hermitian lattices are sometimes simply called hermitian lattices and quadratic lattices refer to quadratic hermitian lattices. Therefore, in a general context, an arbitrary lattice is referred to as a lattice in this chapter.
References ​
Many of the implemented algorithms for computing with quadratic and hermitian lattices over number fields are based on the Magma implementation of Markus Kirschmer, which can be found here.
Most of the definitions and results are taken from:
[Kir16] : Definite quadratic and hermitian forms with small class number. Habilitationsschrift. RWTH Aachen University, 2016. pdf
[Kir19] : Determinant groups of hermitian lattices over local fields, Archiv der Mathematik, 113 (2019), no. 4, 337–347. pdf

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Introduction ​
This chapter deals with quadratic and hermitian spaces, and lattices there of. Note that even though quadratic spaces/lattices are theoretically a special case of hermitian spaces/lattices, a particular distinction is made here. As a note for knowledgeable users, only methods regarding hermitian spaces/lattices over degree 1 and degree 2 extensions of number fields are implemented up to now.
Definitions and vocabulary ​
We begin by collecting the necessary definitions and vocabulary. The terminology follows mainly [Kir16]
Quadratic and hermitian spaces ​
Let K be a number field and let E be a finitely generated etale algebra over K of dimension 1 or 2, i.e. E=K or E is a separable extension of K of degree 2. In both cases, E/K is endowed with an K-linear involution x―:E→E for which K is the fixed field (in the case E=K, this is simply the identity of K).
A hermitian space V over E/K is a finite-dimensional E-vector space, together with a sesquilinear (with respect to the involution of E/K) morphism Φ:V×V→E. In the trivial case E=K, Φ is therefore a K-bilinear morphism and we called (V,Φ) a quadratic hermitian space over K.
We will always work with an implicit canonical basis e1,…,en of V. In view of this, hermitian spaces over E/K are in bijection with hermitian matrices with entries in E, with respect to the involution x―. In particular, there is a bijection between quadratic hermitian spaces over K and symmetric matrices with entries in K. For any basis B=(v1,…,vn) of (V,Φ), we call the matrix GB=(Φ(vi,vj))1≤i,j≤n∈En×n the Gram matrix of (V,Φ) associated to B. If B is the implicit fixed canonical basis of (V,Φ), we simply talk about the Gram matrix of (V,Φ).
For a hermitian space V, we refer to the field E as the base ring of V and to x― as the involution of V. Meanwhile, the field K is referred to as the fixed field of V.
By abuse of language, non-quadratic hermitian spaces are sometimes simply called hermitian spaces and, in contrast, quadratic hermitian spaces are called quadratic spaces. In a general context, an arbitrary space (quadratic or hermitian) is referred to as a space throughout this chapter.
Quadratic and hermitian lattices ​
Let V be a space over E/K. A finitely generated OE-submodule L of V is called a hermitian lattice. By extension of vocabulary if V is quadratic (i.e. E=K), L is called a quadratic hermitian lattice. We call V the ambient space of L and L⊗OEE the rational span of L.
For a hermitian lattice L, we refer to E as the base field of L and to the ring OE as the base ring of L. We also call x―:E→E the involution of L. Finally, we refer to the field K fixed by this involution as the fixed field of L and to OK as the fixed ring of L.
Once again by abuse of language, non-quadratic hermitian lattices are sometimes simply called hermitian lattices and quadratic lattices refer to quadratic hermitian lattices. Therefore, in a general context, an arbitrary lattice is referred to as a lattice in this chapter.
References ​
Many of the implemented algorithms for computing with quadratic and hermitian lattices over number fields are based on the Magma implementation of Markus Kirschmer, which can be found here.
Most of the definitions and results are taken from:
[Kir16] : Definite quadratic and hermitian forms with small class number. Habilitationsschrift. RWTH Aachen University, 2016. pdf
[Kir19] : Determinant groups of hermitian lattices over local fields, Archiv der Mathematik, 113 (2019), no. 4, 337–347. pdf

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     Lattices | Hecke
     
     
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Lattices ​
Creation of lattices ​
Inside a given ambient space ​
lattice Method
julia
lattice(V::AbstractSpace) -> AbstractLat
Given an ambient space V, return the lattice with the standard basis matrix. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.
source
lattice Method
julia
lattice(V::AbstractSpace, B::PMat ; check::Bool = true) -> AbstractLat
Given an ambient space V and a pseudo-matrix B, return the lattice spanned by the pseudo-matrix B inside V. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.
By default, B is checked to be of full rank. This test can be disabled by setting check to false.
source
lattice Method
julia
lattice(V::AbstractSpace, basis::MatElem ; check::Bool = true) -> AbstractLat
Given an ambient space V and a matrix basis, return the lattice spanned by the rows of basis inside V. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.
By default, basis is checked to be of full rank. This test can be disabled by setting check to false.
source
lattice Method
julia
lattice(V::AbstractSpace, gens::Vector) -> AbstractLat
Given an ambient space V and a list of generators gens, return the lattice spanned by gens in V. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.
If gens is empty, the function returns the zero lattice in V.
source
Quadratic lattice over a number field ​
quadratic_lattice Method
julia
quadratic_lattice(K::Field ; gram::MatElem) -> Union{ZZLat, QuadLat}
Given a matrix gram and a field K, return the free quadratic lattice inside the quadratic space over K with Gram matrix gram.
If K=Q, then the output lattice is of type ZZLat, seen as a lattice over the ring Z.
source
quadratic_lattice Method
julia
quadratic_lattice(K::Field, B::PMat ; gram = nothing,
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Hecke
Appearance
Lattices ​
Creation of lattices ​
Inside a given ambient space ​
lattice Method
julia
lattice(V::AbstractSpace) -> AbstractLat
Given an ambient space V, return the lattice with the standard basis matrix. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.
source
lattice Method
julia
lattice(V::AbstractSpace, B::PMat ; check::Bool = true) -> AbstractLat
Given an ambient space V and a pseudo-matrix B, return the lattice spanned by the pseudo-matrix B inside V. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.
By default, B is checked to be of full rank. This test can be disabled by setting check to false.
source
lattice Method
julia
lattice(V::AbstractSpace, basis::MatElem ; check::Bool = true) -> AbstractLat
Given an ambient space V and a matrix basis, return the lattice spanned by the rows of basis inside V. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.
By default, basis is checked to be of full rank. This test can be disabled by setting check to false.
source
lattice Method
julia
lattice(V::AbstractSpace, gens::Vector) -> AbstractLat
Given an ambient space V and a list of generators gens, return the lattice spanned by gens in V. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.
If gens is empty, the function returns the zero lattice in V.
source
Quadratic lattice over a number field ​
quadratic_lattice Method
julia
quadratic_lattice(K::Field ; gram::MatElem) -> Union{ZZLat, QuadLat}
Given a matrix gram and a field K, return the free quadratic lattice inside the quadratic space over K with Gram matrix gram.
If K=Q, then the output lattice is of type ZZLat, seen as a lattice over the ring Z.
source
quadratic_lattice Method
julia
quadratic_lattice(K::Field, B::PMat ; gram = nothing,
                                       check:::Bool = true) -> QuadLat
Given a pseudo-matrix B with entries in a field K return the quadratic lattice spanned by the pseudo-matrix B inside the quadratic space over K with Gram matrix gram.
If gram is not supplied, the Gram matrix of the ambient space will be the identity matrix over K of size the number of columns of B.
By default, B is checked to be of full rank. This test can be disabled by setting check to false.
source
quadratic_lattice Method
julia
quadratic_lattice(K::Field, basis::MatElem ; gram = nothing,
                                              check::Bool = true)
                                                       -> Union{ZZLat, QuadLat}
Given a matrix basis and a field K, return the quadratic lattice spanned by the rows of basis inside the quadratic space over K with Gram matrix gram.
If gram is not supplied, the Gram matrix of the ambient space will be the identity matrix over K of size the number of columns of basis.
By default, basis is checked to be of full rank. This test can be disabled by setting check to false.
If K=Q, then the output lattice is of type ZZLat, seen as a lattice over the ring Z.
source
quadratic_lattice Method
julia
quadratic_lattice(K::Field, gens::Vector ; gram = nothing) -> Union{ZZLat, QuadLat}
Given a list of vectors gens and a field K, return the quadratic lattice spanned by the elements of gens inside the quadratic space over K with Gram matrix gram.
If gram is not supplied, the Gram matrix of the ambient space will be the identity matrix over K of size the length of the elements of gens.
If gens is empty, gram must be supplied and the function returns the zero lattice in the quadratic space over K with gram matrix gram.
If K=Q, then the output lattice is of type ZZLat, seen as a lattice over the ring Z.
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Hermitian lattice over a degree 2 extension ​
hermitian_lattice Method
julia
hermitian_lattice(E::NumField; gram::MatElem) -> HermLat
Given a matrix gram and a number field E of degree 2, return the free hermitian lattice inside the hermitian space over E with Gram matrix gram.
source
hermitian_lattice Method
julia
hermitian_lattice(E::NumField, B::PMat; gram = nothing,
@@ -627,8 +627,8 @@
 Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
 with basis pseudo-matrix
 (1//1 * <1, 1>) * [1 0]
-(1//14 * <1, 1>) * [6 1])

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     Bibliography | Hecke
     
     
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Bibliography ​
  1. H. Cohen. Advanced topics in computational number theory. Vol. 193 of Graduate Texts in Mathematics (Springer-Verlag, New York, 2000); p. xvi+578.
  2. H. Cohen. A course in computational algebraic number theory. Vol. 138 of Graduate Texts in Mathematics (Springer-Verlag, Berlin, 1993); p. xii+534.
  3. M. Pohst and H. Zassenhaus. Algorithmic algebraic number theory. Vol. 30 of Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 1997); p. xiv+499.
  4. D. A. Marcus. Number fields. Universitext (Springer, Cham, 2018); p. xviii+203.
  5. J. H. Conway and N. J. Sloane. Sphere packings, lattices and groups. Third Edition, Vol. 290 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, New York, 1999); p. lxxiv+703.
  6. V. V. Nikulin. Integer symmetric bilinear forms and some of their geometric applications. Izv. Akad. Nauk SSSR Ser. Mat. 43, 111–177, 238 (1979).

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Bibliography ​
  1. H. Cohen. Advanced topics in computational number theory. Vol. 193 of Graduate Texts in Mathematics (Springer-Verlag, New York, 2000); p. xvi+578.
  2. H. Cohen. A course in computational algebraic number theory. Vol. 138 of Graduate Texts in Mathematics (Springer-Verlag, Berlin, 1993); p. xii+534.
  3. M. Pohst and H. Zassenhaus. Algorithmic algebraic number theory. Vol. 30 of Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 1997); p. xiv+499.
  4. D. A. Marcus. Number fields. Universitext (Springer, Cham, 2018); p. xviii+203.
  5. J. H. Conway and N. J. Sloane. Sphere packings, lattices and groups. Third Edition, Vol. 290 of Grundlehren der mathematischen Wissenschaften (Springer-Verlag, New York, 1999); p. lxxiv+703.
  6. V. V. Nikulin. Integer symmetric bilinear forms and some of their geometric applications. Izv. Akad. Nauk SSSR Ser. Mat. 43, 111–177, 238 (1979).

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     Getting started | Hecke
     
     
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Getting started ​
To use Hecke, a julia version of 1.0 is necessary (the latest stable julia version will do). Please see https://julialang.org/downloads/ for instructions on how to obtain julia for your system. Once a suitable julia version is installed, use the following steps at the julia prompt to install Hecke:
julia
julia> using Pkg
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Getting started ​
To use Hecke, a julia version of 1.0 is necessary (the latest stable julia version will do). Please see https://julialang.org/downloads/ for instructions on how to obtain julia for your system. Once a suitable julia version is installed, use the following steps at the julia prompt to install Hecke:
julia
julia> using Pkg
 
 julia> Pkg.add("Hecke")
Here is a quick example of using Hecke to define a number field and compute its class group::
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julia> using Hecke
 
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 julia> C
 (Z/2)^2

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     Tutorials | Hecke
     
     
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Tutorials ​

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Tutorials ​

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