\\ Pari/GP code for working with elliptic curve 2.2.205.1-45.1-b2
\\ (Note that not all these functions may be available, and some may take a long time to execute.)
\\ Define the base number field:
K = nfinit(Pol(Vecrev([-51, -1, 1])));
\\ Define the curve:
E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([52,-4])),Pol(Vecrev([7923,-1032]))], K);
\\ Compute the conductor:
ellglobalred(E)[1]
\\ Compute the norm of the conductor:
idealnorm(ellglobalred(E)[1])
\\ Compute the discriminant:
E.disc
\\ Compute the norm of the discriminant:
norm(E.disc)
\\ Compute the j-invariant:
E.j
\\ Compute the torsion subgroup:
T = elltors(E); T[2]
\\ Compute the order of the torsion subgroup:
T[1]
\\ Compute the generators of the torsion subgroup:
T[3]