\\ Pari/GP code for working with elliptic curve 5408.i1


\\ Define the curve: 
E = ellinit([0, 1, 0, -56, -8644])

\\ Torsion subgroup: 
elltors(E)

\\ Conductor: 
ellglobalred(E)[1]

\\ Discriminant: 
E.disc

\\ j-invariant: 
E.j

\\ Rank: 
[lower,upper] = ellrank(E)

\\ Regulator: 
G = E.gen \\ if available
matdet(ellheightmatrix(E,G))

\\ Real Period: 
if(E.disc>0,2,1)*E.omega[1]

\\ Tamagawa numbers: 
gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]

\\ Torsion order: 
elltors(E)[1]

\\ Special L-value: 
[r,L1r] = ellanalyticrank(E); L1r/r!

\\ q-expansion of modular form: 
\\ actual modular form, use for small N
[mf,F] = mffromell(E)
Ser(mfcoefs(mf,20),q)
\\ or just the series
Ser(ellan(E,20),q)*q

\\ Modular degree: 
ellmoddegree(E)

\\ Local data: 
ellglobalred(E)[5]

\\ p-adic regulator: 
G = E.gen \\ if available
[ellpadicregulator(E,p,10,G) | p <- primes([5,20])]