For a superelliptic curve
X
\mathcal {X}
, defined over
Q
\mathbb {Q}
, let
p
\mathfrak {p}
denote the corresponding moduli point in the weighted moduli space. We describe a method how to determine a minimal integral model of
X
\mathcal {X}
such that: i) the corresponding moduli point
p
\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
p
\mathfrak {p}
with minimal weighted height, as defined in \cite{b-g-sh}, and part ii) by the classical reduction of the binary forms.