Function: elltaniyama
Section: elliptic_curves
C-Name: elltaniyama
Prototype: GDP
Help: elltaniyama(E, {d = seriesprecision}): modular parametrization of
 elliptic curve E/Q.
Doc:
 computes the modular parametrization of the elliptic curve $E/\Q$,
 where $E$ is an \var{ell} structure as output by \kbd{ellinit}. This returns
 a two-component vector $[u,v]$ of power series, given to $d$ significant
 terms (\tet{seriesprecision} by default), characterized by the following two
 properties. First the point $(u,v)$ satisfies the equation of the elliptic
 curve. Second, let $N$ be the conductor of $E$ and $\Phi: X_0(N)\to E$
 be a modular parametrization; the pullback by $\Phi$ of the
 N\'eron differential $du/(2v+a_1u+a_3)$ is equal to $2i\pi
 f(z)dz$, a holomorphic differential form. The variable used in the power
 series for $u$ and $v$ is $x$, which is implicitly understood to be equal to
 $\exp(2i\pi z)$.

 The algorithm assumes that $E$ is a \emph{strong} \idx{Weil curve}
 and that the Manin constant is equal to 1: in fact, $f(x) = \sum_{n > 0}
 \kbd{ellan}(E, n) x^n$.