Consider the a modular curve $X_0(l)$. I know that for $2 \leq l \leq 10$, $X_0(l)$ is genus 0, and so has a single generate for the function field $\mathbb{C}(X_0(l))$.
Now I am trying to study how Dedekind $\eta$ functions relate to $X_0(l)$. I know that for $l=2,3,5,7$ $\mathbb{C}(X_0(l))$ can be generated by Dedikind $\eta$ quotients.
Is it true that $X_0(l)$ has genus 0 $\iff$ it can be generated by Dedekind $\eta$ quotients? Or are they non-related? What exactly does allow for the function field of $X_0(l)$ to be generated by Dedekind $\eta$ quotients?