How to determine the genus and the number of poles in a ring $\mathbb{C}[t,t^{-1},u]/\langle u^m-p(t)\rangle$?

A pole of the finite polynomial $ p(t)=\sum_{i\in\mathbb{Z}}a_it^i$ is a point $ p_0$ such that $ p(p_0)=\infty$ .

I already know that when $ m=2$ , the genus $ g$ depends on the defining polynomial $ p(t)$ with degree $ d$ according to the formula $ $ g=\frac{d-1}{2}, $ $ equivalently $ $ 2g=\begin{cases} d-2 \textrm{ if $ d$ is even} \d-1 \textrm{ if $ d$ is odd.}\end{cases} $ $

After that, I have that the number $ n$ where poles are allowed depends on $ p(t)$ according to the formula $ n=4-r$ where $ r$ is the number of ramified points in $ \{0,\infty \}$ : $ 0$ is ramified exactly when the constant term $ a_0=0$ , and $ \infty$ is ramified exactly when the degree $ d$ is odd. Combining this information gives $ $ 2g+n-1=\begin{cases} d+1 \textrm{ if } a_0\neq0, \d-1 \textrm{ if } a_0=0. \end{cases} $ $

As I said, it is the case when $ m=2$ . When $ m$ is arbitrary, what could I say?

How to determine the genus and the number of poles in a ring $ \mathbb{C}[t,t^{-1},u]/\langle u^m-p(t)\rangle$ ?