Deciding superellipticity and computing the Weierstrass normal form

Abstract: Let ( \mathcal{S}{g,n} \subset \mathcal{M}_g ) be the locus of curves of genus ( g \geq 2 ) admitting a model ( y^n = h(x) ) with ( h ) separable; such curves $C$ have a cyclic group ( C_n \leq \operatorname{Aut}(C) ) of order ( n ) with ( C/C_n \cong \mathbb{P}^1 ). % We give an algorithm which, given an absolutely irreducible plane model ( F(x,y) = 0 ) of a curve ( C ) over a field ( k_0 ) of characteristic zero, decides for which ( n ) the curve lies in ( \mathcal{S}{g,n} ) and returns a model ( y^n = h(x) ) together with the birational transformation to it.