// 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<x> := PolynomialRing(Rationals()); K<a> := 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);