# SageMath code for working with elliptic curve 3.3.316.1-2.2-a2 # (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([2, -4, -1, 1])) # Define the curve: E = EllipticCurve([K([-3,1,1]),K([0,-1,0]),K([0,1,0]),K([2583572058090078441238362,321836266424514724186808,-608022265892705467026233]),K([2667316476142693080354081387519995775,332268331114001902947589285516929027,-627730820434767876188418741768082419])]) # 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()