Let $E/\mathbb{Q}$ be an elliptic curve of rank 0, with modular parametrization $\phi: X\_0(N) \to E$. Let $\Omega\_0$ be the least positive real period. A paper I'm reading (Yoshida, Some variants of the congruent number problem I) seems to use the following line of reasoning. Say $\Omega\_0/L(E/\mathbb{Q}, 1) = c$. We have that $\Omega\_0/c = L(E/\mathbb{Q}, 1) = -I(0)$, so $I(0)$ must be a torsion point of exact order $c$. 


I'm confused because from Cremona's tables, there are elliptic curves (for example 11a3) with torsion subgroup of order 5 but $c = 25$. So something is wrong here... Any idea what it is? I'm most likely just misinterpreting the paper or incorrectly generalizing (he was only doing this for a few specific elliptic curves). 



computing the order of the image of 0 under the modular parametrization map for an elliptic curve
Here is an expanded version of g6hq's answer. Your question is indeed sensitive to the Manin constant of the modular parametrization $X\_0(N) \to E$. This in turns depends on whether the elliptic curve $E$ is the so-called "strong Weil curve" in its isogeny class, which by definition means that the kernel of $\phi\_\* : J\_0(N) \to E$ is connected.


For example when $E=11a1=X\_0(11)$, then $\phi$ is an isomorphism so everything is fine. But when $E=11a3=X\_1(11)$, the modular parametrization $\phi : X\_0(11) \to X\_1(11)$ of least degree has degree 5 and in fact satisfies $\phi^\* \omega\_E = c\_E \omega\_f$ where $f=f\_E \in S\_2(\Gamma\_0(11))$ is the newform associated to $E$, with a non-trivial Manin constant $c\_E=5$. Thus the torsion point $I(0)$ has, in fact, order $c/c\_E=5$.


In general, given any elliptic curve $E/\mathbf{Q}$, there is an optimal parametrization $\phi : X\_1(N) \to E$. Such a parametrization conjecturally satisfies $\phi^\* \omega\_E = \omega\_f$ (Stevens' conjecture). But the class of the divisor $[0]-[\infty]$ in the Jacobian of $X\_1(N)$ is defined only over $\mathbf{Q}(\mu\_N)^+$, not over $\mathbf{Q}$.


