# SageMath code for working with elliptic curve 2.0.4.1-5525.5-b9

# (Note that not all these functions may be available, and some may take a long time to execute.)

# Define the base number field: 
x = polygen(QQ);  K.<i> = NumberField(x^2 + 1)

# Define the curve: 
E = EllipticCurve(K, [i, 1, i, -105*i + 39, 15*i - 399])

# Test whether it is a global minimal model: 
E.is_global_minimal_model()

# Compute the conductor: 
E.conductor()

# Compute the norm of the conductor: 
E.conductor().norm()

# Compute the discriminant: 
E.discriminant()

# Compute the norm of the discriminant: 
E.discriminant().norm()

# Compute the j-invariant: 
E.j_invariant()

# Test for Complex Multiplication: 
E.has_cm(), E.cm_discriminant()

# Compute the Mordell-Weil rank: 
E.rank()

# Compute the order of the torsion subgroup: 
E.torsion_order()

# Compute the generators (of infinite order): 
E.gens()

# Compute the regulator: 
E.regulator_of_points(E.gens())

# Compute the torsion subgroup: 
E.torsion_subgroup().gens()

# Compute the generators of the torsion subgroup: 
E.torsion_subgroup().gens()

# Compute the local reduction data at primes of bad reduction: 
E.local_data()