TITLE: Algebraic equations for modular parameterizations
QUESTION [7 upvotes]: I was wondering if there some place where for some small $N$ I can find explicit modular parameterizations in an algebraic way. 
One type of model for $X_0(N)$ is just given by a single algebraic relation $\phi(j,j')$ between $j(\tau)$ and $j(N\tau)$. So given an elliptic curve $E$ of conductor $N$ I would really like to find $X, Y \in \mathbb Q(j)[j']/\phi$ such that $\mathbb Q(X,Y) \cong \mathbb Q(E)$.
Explicit parameterizations by other models of $X_0(N)$, for example using the weber f function instead of j, would also work for me. And in the light of the remark of David Loeffler below maybe even better. As long as there is a know expression for the $j$-invariant on that model.
I'm especially interested in the case where $N = 121$.

REPLY [2 votes]: There's a similar question on MO at:
Where can I find a comprehensive list of equations for small genus modular curves?
You might also check out
Y. Yang, "Defining equations of modular curves," Advances in Mathematics 204 (2006), 481-508.
Yang surveys the problem and then gives a method for finding defining equations with small coefficients by using explicit properties of some generalized Dedekind $\eta$ functions.  He also gives lists of equations for $X_0(N)$ for $N \leq 50$.  (So not $N=121$, but the techniques should work.)  In his examples the coefficients are usually no more than $2$ digits.