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Commit 22c45fa9 authored by Gonzalo Tornaría's avatar Gonzalo Tornaría
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test 353 with precomputed fields and galois groups

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// Galois theory for C353
// faster but conditional
// SetClassGroupBounds("GRH");
load "ex.m";
//load "polys.m";
ZZ := IntegerRing();
ZZx<x> := PolynomialRing(ZZ);
load "all-353.m";
f18 := DefiningPolynomial(R1);
gal, r, s := GaloisGroup(f18);
b, rho := IsIsomorphic(gal, G5);
assert b;
// we know Frob5
Frob5cycle :=
Reverse(Sort([<x,Multiplicity(s, x)> : x in Set(s)]))
where s is ([ Degree(f[1])
: f in Factorization(ChangeRing(f18,GF(5))) ]);
Frob5 := [c[3] : c in ConjugacyClasses(gal)
| CycleStructure(c[3]) eq Frob5cycle ];
assert(#Frob5 eq 1);
Frob5 := Frob5[1];
// we expect Tr(Frob5) = 1
if Trace(rho(Frob5)) ne 1 then
// fix it by composing rho with the outer automorphism
rho := rho * a
where a is autg`FpGroup[2](outg.1 @@ m)
where outg, m is OuterFPGroup(autg)
where autg is AutomorphismGroup(G5);
print ".";
end if;
// We expect Tr(Frob5) = 1
// Make sure this is right
assert Trace(rho(Frob5)) eq 1;
f353_R1 := GaloisSubgroup(s, H5 @@ rho);
//R1 := NumberField(f353_R1);
P2 := Radical(2*Integers(R1));
S0 := Radical(353*Integers(R1));
S5 := P2^5 * S0;
//gal, r, s := GaloisGroup(f353_R1);
function quad_ext(S, inf)
ray, m := RayClassGroup(S, inf);
return
[ AbelianExtension(Inverse(mq)*m)
where _, mq is quo<ray | Q>
where Q is QQ`subgroup
: QQ in Subgroups(ray : Quot:=[2])];
end function;
//qexts := quad_ext(S5, [1..#RealPlaces(R1)]);
ex.m deleted 100644 → 0
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//AttachSpec("mod/mod.spec");
//////////////////////////////////////////////////
GSp4 := Sp(4,2);
ex_skip := false;
// S5(B)
if not ex_skip or not assigned(G1) then
// printf ".";
G1 := [H`subgroup
: H in Subgroups(GSp4 : OrderEqual:=120)
| &and [Trace(x) eq 1 : x in order36]
where order36 := [x : x in H`subgroup | Order(x) in [3,6]]
];
assert #G1 eq 1; G1 := G1[1];
H1 := [H`subgroup
: H in Subgroups(G1 : OrderEqual:=12)
| #ConjugacyClasses(H`subgroup) eq 6 ];
assert #H1 eq 1; H1 := H1[1];
assert IsIsomorphic(H1, DihedralGroup(6));
assert IsTrivial(&meet Conjugates(G1,H1));
end if;
// S5(A)
if not ex_skip or not assigned(G2) then
// printf ".";
G2 := [H`subgroup
: H in Subgroups(GSp4 : OrderEqual:=120)
| &and [Trace(x) eq 0 : x in order36]
where order36 := [x : x in H`subgroup | Order(x) in [3,6]]
];
assert #G2 eq 1; G2 := G2[1];
H2 := [H`subgroup
: H in Subgroups(G2 : OrderEqual:=6)
| &+ [#Conjugates(G2, x) : x in H`subgroup] eq 86];
assert #H2 eq 1; H2 := H2[1];
assert IsIsomorphic(H2, SymmetricGroup(3));
assert #Conjugates(G2,ConjugacyClasses(H2)[2][3]) eq 15;
assert IsTrivial(&meet Conjugates(G2,H2));
end if;
// S6
if not ex_skip or not assigned(G3) then
// printf ".";
G3 := GSp4;
H3 := [H`subgroup
: H in Subgroups(G3 : OrderEqual:=36)
| not IsEmpty(order6)
and &and [Trace(x) eq 1 : x in order6]
where order6 := [x : x in H`subgroup | Order(x) eq 6]
];
assert #H3 eq 1; H3 := H3[1];
assert GroupName(H3) eq "S3^2";
assert IsTrivial(&meet Conjugates(G3,H3));
end if;
// A6
if not ex_skip or not assigned(G4) then
// printf ".";
G4 := Subgroups(GSp4:OrderEqual:=360);
assert #G4 eq 1; G4 := G4[1]`subgroup;
H4 := [H`subgroup
: H in Subgroups(G4:OrderEqual:=18)];
assert #H4 eq 1; H4 := H4[1];
assert GroupName(H4) eq "C3:S3";
assert IsTrivial(&meet Conjugates(G4,H4));
end if;
// S3wrC2
if not ex_skip or not assigned(G5) then
// printf ".";
G5 := Subgroups(GSp4:OrderEqual:=72);
assert #G5 eq 1; G5 := G5[1]`subgroup;
H5 := [H`subgroup
: H in Subgroups(G5:OrderEqual:=4)
| #&meet [ Kernel( hom<gl4->gl4 |
[h*a-a*h : a in OrderedGenerators(gl4)]> )
: h in Generators(H`subgroup) ] eq 64 ]
where gl4 is RMatrixSpace(GF(2), 4, 4);
assert #H5 eq 1; H5 := H5[1];
assert GroupName(H5) eq "C2^2";
assert IsTrivial(&meet Conjugates(G5,H5));
end if;
// A5(B)
if not ex_skip or not assigned(G6) then
// printf ".";
G6 := [H`subgroup
: H in Subgroups(GSp4:OrderEqual:=60)
| IsAbsolutelyIrreducible(H`subgroup)];
assert #G6 eq 1; G6 := G6[1];
H6 := Subgroups(G6:OrderEqual:=6);
assert #H6 eq 1; H6 := H6[1]`subgroup;
assert IsTrivial(&meet Conjugates(G6,H6));
end if;
// S3^2(A)
if not ex_skip or not assigned(G7) then
// printf ".";
G7 := [H`subgroup
: H in Subgroups(GSp4 : OrderEqual:=36)
| not IsEmpty(order6)
and &and [Trace(x) eq 0 : x in order6]
where order6 := [x : x in H`subgroup | Order(x) eq 6]
];
assert #G7 eq 1; G7 := G7[1];
H7 := [H`subgroup
: H in Subgroups(G7:OrderEqual:=2)
| #Centralizer(G7, H`subgroup) eq 4];
assert #H7 eq 1; H7 := H7[1];
assert IsTrivial(&meet Conjugates(G7,H7));
end if;
// C3:S3.C2
if not ex_skip or not assigned(G8) then
// printf ".";
G8 := [H`subgroup
: H in Subgroups(GSp4 : OrderEqual:=36)
| IsEmpty(order6)
where order6 := [x : x in H`subgroup | Order(x) eq 6]
];
assert #G8 eq 1; G8 := G8[1];
H8 := Subgroups(G8 : OrderEqual:=2);
assert #H8 eq 1; H8 := H8[1]`subgroup;
assert IsTrivial(&meet Conjugates(G8,H8));
end if;
// F5
if not ex_skip or not assigned(G9) then
// printf ".";
G9 := Subgroups(GSp4 : OrderEqual:=20);
assert #G9 eq 1; G9 := G9[1]`subgroup;
H9 := Subgroups(G9 : OrderEqual:=2);
assert #H9 eq 1; H9 := H9[1]`subgroup;
assert IsTrivial(&meet Conjugates(G9,H9));
end if;
//print "";
Gs := [G1, G2, G3, G4, G5, G6, G7, G8, G9];
Hs := [H1, H2, H3, H4, H5, H6, H7, H8, H9];
run-353.m
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AttachSpec("mod.spec");
ZZ := IntegerRing();
ZZx<x> := PolynomialRing(ZZ);
//load "ex.m";
load "ex-353.m";
/// DATA for C353
load "../Brumer_G72/TR36v";
load "../Brumer_G72/test353Gal36";
m := ModNew(R1, rho);
PS := Multiset([]);
time for i := 1 to #all do
R2:=NumberField(Polynomial(R1, [-all[i], 0, 1]));
proj, phis := ModPhis(m, R2);
dim := Integers() ! Log(2, #Domain(phis[1])/#G5);
ps := [ ModPrimes( m, R2, proj, phi
: check_residual:=[<5,1>] ) : phi in phis ];
for p in ps do
Include(~PS, p);
end for;
printf "%4o: dim =%2o, %o\n", i, dim, Sort(ps);
K0 := NumberField(CAND353_18[1,3]);
residual_traces := [<5,1>];
if i mod 10 eq 0 then
printf "\n%o\n\n", PS;
end if;
num_fields := #CAND353_18[2];
fields := func<i | NumberField(CAND353_18[2,i])>;
gals := func<i | TR36v[CAND353_18[3,i]]>;
end for;
printf "\n\nFINAL: %o\n\n", PS;
load "run-test.m";
run-test.m
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View file @ 22c45fa9
......@@ -33,8 +33,6 @@ end for;
assert #rhos eq 1;
rho := rhos[1];
assert IsConjugate(Sp(4,2), Image(rho), G1);
tr := func<g | g[1,5]+g[2,6]+g[3,7]+g[4,8]>;
tr1 := func<g | g[1,1]+g[2,2]+g[3,3]+g[4,4]>;
......@@ -62,7 +60,7 @@ time for i := 1 to num_fields do
/////
dim := Integers() ! Log(2, #Domain(phis[1])/#G1);
dim := Integers() ! Log(2, #gal2/#gal);
ps := [];
for phi in phis do
......@@ -96,6 +94,10 @@ time for i := 1 to num_fields do
continue;
end if;
pat := <m`frobR1[p], FrobeniusStructure(R2, p)>;
if not IsDefined(fc, pat) then
Append(~ps, -p);
continue phi;
end if;
all_classes := fc[pat];
bad_classes := [ c : c in all_classes | tr(phi(c)) eq 0 ];
if bad_classes eq [] then
......
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