Function: ellak
Section: elliptic_curves
C-Name: akell
Prototype: GG
Help: ellak(E,n): computes the n-th Fourier coefficient of the L-function of
 the elliptic curve E (assumed E is an integral model).
Doc:
 computes the coefficient $a_n$ of the $L$-function of the elliptic curve
 $E/\Q$, i.e.~coefficients of a newform of weight 2 by the modularity theorem
 (\idx{Taniyama-Shimura-Weil conjecture}). $E$ must be an \var{ell} structure
 over $\Q$ as output by \kbd{ellinit}. $E$ must be given by an integral model,
 not necessarily minimal, although a minimal model will make the function
 faster.
 \bprog
 ? E = ellinit([0,1]);
 ? ellak(E, 10)
 %2 = 0
 ? e = ellinit([5^4,5^6]); \\ not minimal at 5
 ? ellak(e, 5) \\ wasteful but works
 %3 = -3
 ? E = ellminimalmodel(e); \\ now minimal
 ? ellak(E, 5)
 %5 = -3
 @eprog\noindent If the model is not minimal at a number of bad primes, then
 the function will be slower on those $n$ divisible by the bad primes.
 The speed should be comparable for other $n$:
 \bprog
 ? for(i=1,10^6, ellak(E,5))
 time = 820 ms.
 ? for(i=1,10^6, ellak(e,5)) \\ 5 is bad, markedly slower
 time = 1,249 ms.

 ? for(i=1,10^5,ellak(E,5*i))
 time = 977 ms.
 ? for(i=1,10^5,ellak(e,5*i)) \\ still slower but not so much on average
 time = 1,008 ms.
 @eprog