Galois representations of superelliptic curves

A superelliptic curve over a discrete valuation ring O of residual characteristic p is a curve given by an equation T : y(n) = f (x), with Disc(f) &NOTEQUexpressionL; 0. The purpose of this 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 O and that p sic n. Our results are inspired on the algorithm given in Bouw and WewersGlasg (Math. J. 59(1) (2017), 77-108.) but our description is given in terms of a cluster picture as defined in Dokchitser et al. (Algebraic curves and their applications, Contemporary Mathematics, vol. 724 (American Mathematical Society, Providence, RI, 2019), 73-135.).