strongBSDgenus2/TwoPartOfSha.magma at main · TimoKellerMath/strongBSDgenus2 · GitHub

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// This file requires the files from
// https://github.com/TomAFisher/Cassels-Tate-pairing--Genus-2 to be in this folder

Attach("g2minimisation.m");
Attach("genus2ctp.m");

//SetClassGroupBounds("GRH");
SetDebugOnError(true);

SetVerbose("TwoDescent",0);
SetVerbose("CasselsTatePairing",0);
SetVerbose("Minimisation",0);
SetVerbose("PointSearch",0);

load "LMFDB-curves.magma";

function TwoPartOfSha(C)
  J := Jacobian(C);

  time S2,S2map := TwoDescentGenus2(J);
  dimS2 := #Generators(S2);
  printf "dim Sel_2(J/Q) = %o\n", dimS2;
  time MW := MordellWeilGroup(J);
  rank := TorsionFreeRank(MW);
  printf "r = %o\n", rank;
  dim2tors := Valuation(#TwoTorsionSubgroup(J), 2);
  printf "dim J(Q)[2] = %o\n", dim2tors;
  dimSha2 := dimS2 - rank - dim2tors;
  printf "=> dim Sha(J/Q)[2] = %o\n", dimSha2;

  time ctp := CasselsTatePairingGenus2(J,S2map);
  print "The Cassels-Tate pairing on Sel_2(J/Q) is given by";
  print ctp;
  printf "and has rank %o.\n", Rank(ctp);
  if IsInvertible(ctp) then
    return 2^dimSha2;
  else
    return "Sha(2) could not be determined.";
  end if;
end function;

C := HyperellipticCurve(LMFDB_curves[30][2], LMFDB_curves[30][3]);
D := -31;
CK := QuadraticTwist(C, D);
printf "#Sha(J^K/Q)(2) = %o\n\n", TwoPartOfSha(CK);

C := HyperellipticCurve(LMFDB_curves[55][2], LMFDB_curves[55][3]);
D := -19;
CK := QuadraticTwist(C, D);
printf "#Sha(J^K/Q)(2) = %o\n\n", TwoPartOfSha(CK);

C := HyperellipticCurve(LMFDB_curves[57][2], LMFDB_curves[57][3]);
D := -15;
CK := QuadraticTwist(C, D);
printf "#Sha(J^K/Q)(2) = %o\n\n", TwoPartOfSha(CK);

C := HyperellipticCurve(LMFDB_curves[74][2], LMFDB_curves[74][3]);
D := -35;
CK := QuadraticTwist(C, D);
printf "#Sha(J^K/Q)(2) = %o\n\n", TwoPartOfSha(CK);

C := HyperellipticCurve(LMFDB_curves[77][2], LMFDB_curves[77][3]);
D := -11;
CK := QuadraticTwist(C, D);
printf "#Sha(J^K/Q)(2) = %o\n\n", TwoPartOfSha(CK);

C := HyperellipticCurve(LMFDB_curves[82][2], LMFDB_curves[82][3]);
D := -11;
CK := QuadraticTwist(C, D);
printf "#Sha(J^K/Q)(2) = %o\n\n", TwoPartOfSha(CK);