Function: ellanalyticrank
Section: elliptic_curves
C-Name: ellanalyticrank
Prototype: GDGb
Help: ellanalyticrank(E,{eps}): returns the order of vanishing at s=1
 of the L-function of the elliptic curve E and the value of the first
 nonzero derivative. To determine this order, it is assumed that any
 value less than eps is zero. If no value of eps is given, 2^(-bitprecision/2)
 is used.
Doc: returns the order of vanishing at $s=1$ of the $L$-function of the
 elliptic curve $E/\Q$ and the value of the first nonzero derivative. To
 determine this order, it is assumed that any value less than \kbd{eps} is
 zero. If \kbd{eps} is omitted, $2^{-b/2}$ is used, where $b$
 is the current bit precision.
 \bprog
 ? E = ellinit("11a1"); \\ rank 0
 ? ellanalyticrank(E)
 %2 = [0, 0.2538418608559106843377589233]
 ? E = ellinit("37a1"); \\ rank 1
 ? ellanalyticrank(E)
 %4 = [1, 0.3059997738340523018204836835]
 ? E = ellinit("389a1"); \\ rank 2
 ? ellanalyticrank(E)
 %6 = [2, 1.518633000576853540460385214]
 ? E = ellinit("5077a1"); \\ rank 3
 ? ellanalyticrank(E)
 %8 = [3, 10.39109940071580413875185035]
 @eprog\noindent Analogous and more general functionalities for $E$
 defined over general number fields are available through \kbd{lfun}
 and \kbd{lfunorderzero}.
