Earlier this year I completed my Masters dissertation on Andrew Wiles's proof of The Modularity Theorem for semistable elliptic curves as a precursor to the accepted proof of Fermat's Last Theorem. However, there was one step in the proof which has so far eluded my understanding.

Given a semistable elliptic curve $E$ over $\mathbb{Q}$ with $3$-adic representation $\rho_{E,3}$ we know that this representation is modular due to Wiles's work on deformations and the modularity of the mod-$3$ representation, $\bar{\rho}_{E,3}$, which in turn is due to the Langlands-Tunnell theorem.

Now, the version of modularity Wiles uses in his paper is, essentially,

$E$ is modular if there is a non-constant morphism $X_0(N) \rightarrow E$ for some natural number $N$.

In Wiles's proof (http://math.stanford.edu/~lekheng/flt/wiles.pdf) it states on page 542,

By Serre's isogeny theorem, $E$ is modular (in the sense of being a factor of the Jacobian of a modular curve).

This is the step which confuses me. The question I am asking is: can anyone explain in more detail why Serre's (/Faltings's) isogeny theorem tells us that since $\rho_{E,3}$ is modular, there is a non-constant morphism $X_{0}(N) \rightarrow E$?