computing the order of the image of 0 under the modular parametrization map for an elliptic curve 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).   
 A: 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 $\mathbf{Q}$.
A: Hi, there is a nontrivial Manin constant of 5 for 11a3, I think this changes the answer, when comparing the L-value and periods.
