Function: ellmoddegree
Section: elliptic_curves
C-Name: ellmoddegree
Prototype: G
Help: ellmoddegree(e): e being an elliptic curve defined over Q output by
 ellinit, compute the modular degree of e divided by the square of the
 Manin constant.
Doc: $e$ being an elliptic curve defined over $\Q$ output by \kbd{ellinit},
 compute the modular degree of $e$ divided by the square of
 the Manin constant $c$. It is conjectured that $c = 1$ for the strong Weil
 curve in the isogeny class (optimal quotient of $J_{0}(N)$) and this can be
 proven using \kbd{ellweilcurve} when the conductor $N$ is moderate.
 \bprog
 ? E = ellinit("11a1"); \\ from Cremona table: strong Weil curve and c = 1
 ? [v,smith] = ellweilcurve(E); smith \\ proof of the above
 %2 = [[1, 1], [5, 1], [1, 1/5]]
 ? ellmoddegree(E)
 %3 = 1
 ? [ellidentify(e)[1][1] | e<-v]
 %4 = ["11a1", "11a2", "11a3"]
 ? ellmoddegree(ellinit("11a2"))
 %5 = 5
 ? ellmoddegree(ellinit("11a3"))
 %6 = 1/5
 @eprog\noindent The modular degree of \kbd{11a1} is $1$ (because
 \kbd{ellweilcurve} or Cremona's table prove that the Manin constant
 is $1$ for this curve); the output of \kbd{ellweilcurve} also proves
 that the Manin constants of \kbd{11a2} and \kbd{11a3} are 1 and 5
 respectively, so the actual modular degree of both \kbd{11a2} and \kbd{11a3}
 is 5.
