Deciding superellipticity and computing the Weierstrass normal form

Source 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.