On Galois representations of superelliptic curves

A superelliptic curve over a DVR ${\mathcal O}$ of residual characteristic $p$ is a curve given by an equation $C:y^n=f(x)$. The purpose of the present article is to describe the Galois representation attached to such a curve under the hypothesis that $f(x)$ has all its roots in the fraction field of ${\mathcal O}$ and that $p \nmid n$. Our results are inspired on the algorithm given in [BW17] but our description is given in terms of a cluster picture as defined in [DDMM18].