I am trying to compute the divisor of $\Delta(z)/\Delta(pz)$ on the modular curve $X_0(p)$ where $p$ is a prime. I know that as a function on the full modular group, the $\Delta$ function has only a simple zero at infinity, but I can't get much further than this. I know that $X_0(p)$ has two cusps, $0$ and $\infty$, so I'm thinking I can compute this by considering the map from $X_0(p)$ to to the half plane modulo the full modular group, but I can't quite figure this out either.
If I'm to believe Gross in his paper on Heegner points, I expect the answer to be $(p-1)\{(0)-(\infty)\}$
Thanks for any insight.