Is there some way of quantifying ramification of a cover up to Mobius transformations? i.e. If $K$ is an algebraically closed field of characteristic $p>0$, we can consider the Artin-Schreier extensions of $K(x)$ given by $y^p - y - f(x)$ and we can apply Mobius transformations to the coordinates x and y.
For example, both $y^p-y-x$ and $y^p-y-x^2$ are ramified over the point $x=\infty$, but I would expect the ramification to somehow differ between the two cases as there isn't an automorphism between them. But the former case should be "the same" as the cover given by $y^p-y-x+r$ for some $r\in K$.
What, if anything, is an invariant I could use to quantify the ramification?