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 $f$ ?