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A superelliptic curve over a discrete valuation ring 
$\mathscr{O}$ of residual characteristic p is a curve given by an equation 
$\mathscr{C}\;:\; y^n=\,f(x)$, with 
$\textrm{Disc}(\,f)\neq 0$. 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 
$\mathscr{O}$ and that 
$p \nmid n$. 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.).
Given 
$k\geqslant 2$ , we show that there are at most finitely many rational numbers 
$x$ and 
$y\neq 0$ and integers 
$\ell \geqslant 2$ (with 
$(k,\ell )\neq (2,2)$ ) for which In particular, if we assume that 
$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$
$\ell$ is prime, then all such triples 
$(x,y,\ell )$ satisfy either 
$y=0$ or 
$\ell <\exp (3^{k})$ .
In this paper, we obtain all rational points (x,y) on the superelliptic curves 
