  Let $K$ be a field with a discrete valuation, and let $p$ be a prime. It is
known that if $\Gamma \lhd \Gamma_0 < \mathrm{PGL}_2(K)$ is a Schottky group
normally contained in a larger group which is generated by order-$p$ elements
each fixing $2$ points $a_i, b_i \in \mathbb{P}_K^1$, then the quotient of a
certain subset of the projective line $\mathbb{P}_K^1$ by the action of
$\Gamma$ can be algebraized as a superelliptic curve $C : y^p = f(x) / K$. The
subset $S \subset K \cup \{\infty\}$ consisting of these pairs $a_i, b_i$ of
fixed points is mapped modulo $\Gamma$ to the set of branch points of the
superelliptic map $x : C \to \mathbb{P}_K^1$. We produce an algorithm for
determining whether an input even-cardinality subset $S \subset K \cup
\{\infty\}$ consists of fixed points of generators of such a group $\Gamma_0$
and which, in the case of a positive answer, modifies $S$ into a subset
$S^{\mathrm{min}} \subset K \cup \{\infty\}$ with particularly nice properties.
Our results do not involve any restrictions on the prime $p$ or on the residue
characteristic of $K$ and allow these to be the same.
