// Magma code for working with elliptic curve 4.4.1125.1-31.1-a3 // (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, 4, -4, -1, 1]); // Define the curve: E := EllipticCurve([K![-1,-3,1,1],K![1,-3,-1,1],K![1,0,0,0],K![-13,105,-21,-37],K![63,-17,-25,2]]); // 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);