Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $N$ be its conductor. By the modularity of elliptic curves over $\mathbb{Q}$, there exists a surjective map $f:X_0(N)\to E$, where $X_0(N)$ is the modular curve.
I want to know about the ramification of this map, is there any results or examples or references?
New Edit: By the way, what do we know about the degree of $f$ ?
The ramification at cusps of $X_0(N)$ is very well understood. For example, if the newform attached to $E$ is twist-minimal Brunault [1] showed that $f$ is unramified at the cusps. This happens for example when $N$ is square-free. This was extended by Corbett-Saha [2] who calculated the ramification degree of $f$ at every cusp.
The ramification of $f$ in the upper-half plane has been studied in the PhD theses of Christophe Delaunay and Hao Chen (arXiv 1412.2827), who manages to compute the critical subgroup of $E $ (the subgroup generated by the images of ramification points on the imaginary axis) for some elliptic curves of rank 2.
Regarding the degree of $f $, Watkins has shown in unpublished work that it is $\gg N^{7/6-\varepsilon} $. Finding a good polynomial upper bound is a hard problem related to the abc conjecture.
