# SageMath code for working with elliptic curve 3.3.733.1-2.1-c6 # (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([8, -7, -1, 1])) # Define the curve: E = EllipticCurve([K([1,1,0]),K([1,-1,0]),K([-5,0,1]),K([535500577238303301,-255910771809962227,-168562107136250046]),K([194913423901376825030234481,-93147322088742339719223147,-61353841318711367560260837])]) # 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()