# What do we know about the ramification of the modularity map $X_0(N)\to E$?

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Regarding the degree of $$f$$, Watkins has shown in unpublished work that it is $$\gg N^{7/6-\varepsilon}$$.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.