// Magma code for working with elliptic curve 5.5.24217.1-43.1-b1 // (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, 3, -1, -5, 0, 1]); // Define the curve: E := EllipticCurve([K![4,3,-9,-1,2],K![7,6,-19,-2,4],K![-2,-3,5,1,-1],K![-4,-13,9,3,-2],K![5,13,-1,-3,0]]); // 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);