# SageMath code for working with elliptic curve 2304.a2


# Define the curve: 
E = EllipticCurve([0, 0, 0, 18, 0])

# Simplified equation: 
E.short_weierstrass_model()

# Mordell-Weil generators: 
E.gens()

# 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()

# Potential complex multiplication: 
E.has_cm()

# Mordell-Weil rank: 
E.rank()

# Analytic rank: 
E.analytic_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()

# BSD formula: 
# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve([0, 0, 0, 18, 0]); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

# 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()]