# SageMath code for working with elliptic curve 136367.i1 # Define the curve: E = EllipticCurve([1, -1, 1, -1501149, -511526606]) # Torsion subgroup: E.torsion_subgroup().gens() # Integral points: E.integral_points() # Conductor: E.conductor().factor() # Discriminant: E.discriminant().factor() # j-invariant: E.j_invariant().factor() # Rank: E.rank() # Regulator: E.regulator() # Real Period: E.period_lattice().omega() # Tamagawa numbers: E.tamagawa_numbers() # Torsion order: E.torsion_order() # Order of Sha: E.sha().an_numerical() # Special L-value: r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial() # q-expansion of modular form: E.q_eigenform(20) # Modular degree: E.modular_degree() # Local data: E.local_data() # mod p Galois image: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()] # p-adic regulator: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]