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### Member "pari-2.13.1/src/functions/elliptic_curves/elltaniyama" (14 Nov 2020, 1085 Bytes) of package /linux/misc/pari-2.13.1.tar.gz:

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1 Function: elltaniyama
2 Section: elliptic_curves
3 C-Name: elltaniyama
4 Prototype: GDP
5 Help: elltaniyama(E, {n = seriesprecision}): modular parametrization of
6 elliptic curve E/Q.
7 Doc:
8 computes the modular parametrization of the elliptic curve $E/\Q$,
9 where $E$ is an \kbd{ell} structure as output by \kbd{ellinit}. This returns
10 a two-component vector $[u,v]$ of power series, given to $n$ significant
11 terms (\tet{seriesprecision} by default), characterized by the following two
12 properties. First the point $(u,v)$ satisfies the equation of the elliptic
13 curve. Second, let $N$ be the conductor of $E$ and $\Phi: X_0(N)\to E$
14 be a modular parametrization; the pullback by $\Phi$ of the
15 N\'eron differential $du/(2v+a_1u+a_3)$ is equal to $2i\pi
16 f(z)dz$, a holomorphic differential form. The variable used in the power
17 series for $u$ and $v$ is $x$, which is implicitly understood to be equal to
18 $\exp(2i\pi z)$.
19
20 The algorithm assumes that $E$ is a \emph{strong} \idx{Weil curve}
21 and that the Manin constant is equal to 1: in fact, $f(x) = \sum_{n > 0}
22 \kbd{ellak}(E, n) x^n$.