TITLE: What do we know about the ramification of the modularity map $X_0(N)\to E$?
QUESTION [7 upvotes]: 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 $f$ ?

REPLY [3 votes]: 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Regarding the degree of $f $, Watkins has shown in unpublished work that it is $\gg N^{7/6-\varepsilon} $.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.