// Magma code for working with elliptic curve 5.5.65657.1-45.1-b2 // (Note that not all these functions may be available, and some may take a long time to execute.) // Define the base number field: R := PolynomialRing(Rationals()); K := NumberField(R![1, 5, 2, -5, -1, 1]); // Define the curve: E := EllipticCurve([K![0,1,0,0,0],K![3,7,-4,-2,1],K![-4,-7,5,2,-1],K![-314,-1577,-1529,154,330],K![-6558,-38029,-41530,3425,9329]]); // Compute the conductor: Conductor(E); // Compute the norm of the conductor: Norm(Conductor(E)); // Compute the discriminant: Discriminant(E); // Compute the norm of the discriminant: Norm(Discriminant(E)); // Compute the j-invariant: jInvariant(E); // Test for Complex Multiplication: HasComplexMultiplication(E); // Compute the Mordell-Weil rank: Rank(E); // Compute the generators (of infinite order): gens := [P:P in Generators(E)|Order(P) eq 0]; gens; // Compute the heights of the generators (of infinite order): [Height(P):P in gens]; // Compute the regulator: Regulator(gens); // Compute the torsion subgroup: T,piT := TorsionSubgroup(E); Invariants(T); // Compute the order of the torsion subgroup: Order(T); // Compute the generators of the torsion subgroup: [piT(P) : P in Generators(T)]; // Compute the local reduction data at primes of bad reduction: LocalInformation(E);