# SageMath code for working with elliptic curve 2.0.3.1-124.2-a1 # (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. = NumberField(x^2 - x + 1) # Define the curve: E = EllipticCurve(K, [a, a - 1, a, -1301*a + 751, 10550*a - 16530]) # 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()