// Magma code for working with elliptic curve 2.0.3.1-171.1-a4 // (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, -1, 1]); // Define the curve: E := ChangeRing(EllipticCurve([1, -1, a + 1, 5*a + 5, -11*a + 11]),K); // 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 order of the torsion subgroup: Order(TorsionSubgroup(E)); // Compute the generators (of infinite order): Generators(E); // includes torsion // Compute the regulator: Regulator(Generators(E)); // Compute the torsion subgroup: TorsionSubgroup(E); // Compute the generators of the torsion subgroup: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E); // Compute the local reduction data at primes of bad reduction: LocalInformation(E);