If the modular curve $X_{0}(N)$ has genus $0$, its function field is generated by a single element $j_{N}(z)$ with pole of order $1$ at the cusp $\infty$ and holomorphic at all other cusps (this is called a Hauptmodul for $X_{0}(N)$). That's because $X_{0}(N)$ is equivalent to the Riemann sphere, whose function field is generated by $z \mapsto z$, having order $1$ pole at $\infty$.
If instead $X_{0}(N)$ has higher genus, say genus $1$, there is more than one generator for its function field.
Is there a lower bound on the order of the poles at $\infty$ of these generators?
Is it possible to have a generator with order $1$ pole at $\infty$ for this space?
If the answer to (2) were "yes", call the function $f$, then I could build any modular function on $X_{0}(N)$ with pole of any order. If $N$ is such that there is $M \mid N$ with $\text{genus}X_{0}(M) = 0$, then $f - j_{M}(z) = :g$ is a cusp form of level $M$, of weight $0$. Thus $g = 0$, whence $f = j_{M}(z)$. But that seems fishy...