// Magma code for working with elliptic curve 435.d1 // Define the curve: E := EllipticCurve([1, 0, 1, -258, -1589]); // Torsion subgroup: TorsionSubgroup(E); // Integral points: IntegralPoints(E); // Conductor: Conductor(E); // Discriminant: Discriminant(E); // j-invariant: jInvariant(E); // Rank: Rank(E); // Regulator: Regulator(E); // Real Period: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); // Tamagawa numbers: TamagawaNumbers(E); // Torsion order: Order(TorsionSubgroup(E)); // Order of Sha: MordellWeilShaInformation(E); // Special L-value: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); // q-expansion of modular form: ModularForm(E); // Modular degree: ModularDegree(E); // Manin constant: ManinConstant(E); // Local data: [LocalInformation(E,p) : p in BadPrimes(E)]; // mod p Galois image: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];