Let $N$ be a positive integer, and let $f$ be a newform for $S_2(\Gamma_0(N))$. Then by Shimura's construction, the variety $J_0(N)$ has a quotient $A_f$ which is an abelian variety attaced to $f$. $$\pi:J_0(N) \to A_f$$ Dualizing this map we get a map $A_f^\vee \to J_0(N)^\vee$, and since the latter is isomorphic to $J_0(N)$ (it is a Jacobian), we get a map $$A_f^\vee \to A_f$$ This map is an isogeny of degree $d^2$ for some positive integer $d$, and the modular degree is defined precisely as $d$. The curve $X_0(N)$ has a natural inclusion into its Jacobian, so it make sence to look at the map $$ X_0(N) \hookrightarrow J_0(N) \to A_f$$ Question: is any relation between the degree of the map $X_0(N) \to A_f$ (which I mean the degree of $X_0(N)$ onto its image) and $d$? I would guess that to prove an inequality should not be that hard, but is it an equality? (in the case $A_f$ has dimension $1$ it is)