Superelliptic curves with minimal weighted moduli height

For a superelliptic curve $\mathcal X$, defined over $\mathbb Q$, let $\mathfrak p$ denote the corresponding moduli point in the weighted moduli space. We describe a method how to determine a minimal integral model of $\mathcal X$ such that: i) the corresponding moduli point $\mathfrak p$ has minimal weighted height, ii) the equation of the curve has minimal coefficients. Part i) is accomplished by reduction of the moduli point which is equivalent with obtaining a representation of the moduli point $\mathfrak p$ with minimal weighted height, as defined in [5], and part ii) by the classical reduction of the binary forms.