changes for 353

Grp.m

+intrinsic OuterAutomorphismRepresentatives(G :: Grp) -> []
+  { representatives for Out(G)/Inn(G) }
+  aut := AutomorphismGroup(G);
+  // m1 : autfp -> aut
+  autfp, m1 := FPGroup(aut);
+  // m2 : autfp -> outfp
+  outfp, m2 := OuterFPGroup(aut);
+  // m3 : outfp -> out
+  out, m3 := PermutationGroup(outfp);
+
+  return [g @@ m3 @@ m2 @ m1 : g in out];
+end intrinsic;
+
+intrinsic OuterAutomorphismGenerators(G :: Grp) -> []
+  { generators for Out(G)/Inn(G) }
+  aut := AutomorphismGroup(G);
+  // m1 : autfp -> aut
+  autfp, m1 := FPGroup(aut);
+  // m2 : autfp -> outfp
+  outfp, m2 := OuterFPGroup(aut);
+
+  return [g @@ m2 @ m1 : g in Generators(outfp)];
+end intrinsic;
+
+intrinsic IsIsomorphicPerm(G :: GrpPerm, H :: GrpPerm) -> BoolElt, Map
+  { Assuming G, H have the same degree, return an isomorphism which
+    preserves the cycle decomposition, if such isomorphism exists }
+
+    n := Degree(G);
+    if Degree(H) ne n then
+        return false;
+    end if;
+
+    b, x := IsConjugate(Sym(n), G, H);
+
+    if not b then
+        return false;
+    end if;
+
+    gens := OrderedGenerators(G);
+    isom := Isomorphism(G, H, gens, [x^-1*g*x : g in gens]);
+
+    return true, isom;
+end intrinsic;
+

ex-353.m

+// 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)]);
+

run-277.m

 Attach("Modularity.m");
 Attach("GrpExt.m");
+Attach("Grp.m");
 
 load "ex.m";
 load "ex-277.m";
@@ -7,10 +8,10 @@ load "ex-277.m";
 m := ModNew(R1, rho);
 PS := Multiset([]);
 time for i := 1 to #qexts do
-  R2:=qexts[i];
-  phis := ModPhis(m, R2);
+  R2:=NumberField(qexts[i]);
+  proj, phis := ModPhis(m, R2);
   dim := Integers() ! Log(2, #Domain(phis[1])/#G1);
-  ps := [ModPrimes(m, R2, phi) : phi in phis];
+  ps := [ModPrimes(m, R2, proj, phi) : phi in phis];
   for p in ps do 
       Include(~PS, p);
   end for;

run-353.m

+Attach("Modularity.m");
+Attach("GrpExt.m");
+Attach("Grp.m");
+
+//load "ex.m";
+load "ex-353.m";
+
+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) : phi in phis];
+  for p in ps do 
+      Include(~PS, p);
+  end for;
+  printf "%4o: dim =%2o, %o\n", i, dim, Sort(ps);
+
+  if i mod 10 eq 0 then
+      printf "\n%o\n\n", PS;
+  end if;
+
+end for;
+
+printf "\n\nFINAL: %o\n\n", PS;