# Oscar code for working with elliptic curve 50.a3


# Define the curve: 
E = elliptic_curve([1, 0, 1, -1, -2])

# Torsion subgroup: 
torsion_structure(E)

# Conductor: 
conductor(E)

# Discriminant: 
discriminant(E)

# j-invariant: 
j_invariant(E)

# Tamagawa numbers: 
tamagawa_numbers(E)

# Torsion order: 
prod(torsion_structure(E)[1])

# Local data: 
[(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]