// Magma code for working with elliptic curve 6.6.300125.1-71.4-c2

// (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, -2, 7, 2, -7, -1, 1]);

// Define the curve: 
E := EllipticCurve([K![-11,-38,27,58,2,-8],K![-4,-9,5,8,0,-1],K![-2,-7,5,8,0,-1],K![6,88,-27,-145,-21,24],K![-106,-84,251,101,-92,11]]);

// 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);