# SageMath code for working with elliptic curve 2.2.12.1-3072.1-bt8 # (Note that not all these functions may be available, and some may take a long time to execute.) # Define the base number field: R. = PolynomialRing(QQ); K. = NumberField(R([-3, 0, 1])) # Define the curve: E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-3105,1770]),K([121131,-69998])]) # 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 generators (of infinite order): gens = E.gens(); gens # Compute the heights of the generators (of infinite order): [P.height() for P in gens] # Compute the regulator: E.regulator_of_points(gens) # Compute the torsion subgroup: T = E.torsion_subgroup(); T.invariants() # Compute the order of the torsion subgroup: T.order() # Compute the generators of the torsion subgroup: T.gens() # Compute the local reduction data at primes of bad reduction: E.local_data()