// Magma code for working with elliptic curve 1728.h1


// Define the curve: 
E := EllipticCurve([0, 0, 0, -6, -6]);

// Torsion subgroup: 
TorsionSubgroup(E);

// Integral points: 
IntegralPoints(E);

// Conductor: 
Conductor(E);

// Discriminant: 
Discriminant(E);

// j-invariant: 
jInvariant(E);

// Rank: 
Rank(E);

// Regulator: 
Regulator(E);

// Real Period: 
(Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);

// Tamagawa numbers: 
TamagawaNumbers(E);

// Torsion order: 
Order(TorsionSubgroup(E));

// Order of Sha: 
MordellWeilShaInformation(E);

// Special L-value: 
Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

// q-expansion of modular form: 
ModularForm(E);

// Modular degree: 
ModularDegree(E);

// Manin constant: 
ManinConstant(E);

// Local data: 
[LocalInformation(E,p) : p in BadPrimes(E)];

// mod p Galois image: 
[GaloisRepresentation(E,p): p in PrimesUpTo(20)];