// Example code from paper 4:
// _Studying the Birch and Swinnerton-Dyer conjecture for modular abelian
//  varieties using Magma_ by William Stein
//
print "Examples from: Studying the Birch and Swinnerton-Dyer conjecture for modular abelian varieties using Magma";
ei := GetEchoInput();
SetEchoInput(true);
// Section 5: Modular symbols
M := ModularSymbols(389);
Basis(M);
M ! < 1, [ Cusps() | -1/337, Infinity() ] > ;
S := CuspForms(389);
SetPrecision(S, 40);
Basis(S);
M := ModularSymbols(389);
N := NewSubspace(CuspidalSubspace(M));
NewformDecomposition(N);
M := ModularSymbols(389, 2, +1);    // the plus one quotient


// Section 6: Visibility theory
M := ModularSymbols(389);
N := NewSubspace(CuspidalSubspace(M));
D := SortDecomposition(NewformDecomposition(N));
A := D[5]; B := D[1];
IntersectionGroup(A, B);
TorsionBound(A, 7);
TorsionBound(B, 7);
TamagawaNumber(A, 389);
TamagawaNumber(B, 389);
E := EllipticCurve(B);
Rank(E);
G := ModularKernel(A);
Factorization(#G);


// Section 7: Computing special values of modular L-function
LRatio(A, 1);


// Section 9: Computing the torsion subgroup
RationalCuspidalSubgroup(A);   // subgroup of $A(\Q)$


// Section 11: An element of the Shafarevich-Tate group that becomes
//             visible at a higher level
M := ModularSymbols(551);
N := NewSubspace(CuspidalSubspace(M));
D := SortDecomposition(NewformDecomposition(N));
A := D[8];
TorsionBound(A, 7);
RationalCuspidalSubgroup(A);
TorsionBound(A, 31);
Factorization(#ModularKernel(A));
// This next call intentionally raises an error
TamagawaNumber(A, 19);   // takes over a minute; gives an error
TamagawaNumber(A, 29);
ComponentGroupOrder(A, 19);
AtkinLehnerOperator(A, 19)[1,1];
LRatio(A, 1);

M := ModularSymbols(2*551, 2);
N := NewSubspace(CuspidalSubspace(M));
D := SortDecomposition(NewformDecomposition(N));
M551 := ModularSymbols(M, 551);
N551 := NewSubspace(CuspidalSubspace(M551));
D551 := SortDecomposition(NewformDecomposition(N551));
A551 := D551[#D551];
C := M !! A551;  // sum of images under degeneracy maps
IntersectionGroup(C, D[1]);
B := EllipticCurve(D[1]); B;
TamagawaNumber(B, 2);
TamagawaNumber(B, 19);
TamagawaNumber(B, 29);
MordellWeilGroup(B);
T3 := HeckeOperator(A, 3);
d := Determinant(T3 - 4);
Valuation(d, 3);
SetEchoInput(ei);