  We study families of superelliptic curves with fixed automorphism groups.
Such families are parametrized with invariants expressed in terms of the
coefficients of the curves. Algebraic relations among such invariants determine
the lattice of inclusions among the loci of superelliptic curves and their
field of moduli. We give a Maple package of how to compute the normal form of
an superelliptic curve and its invariants. A complete list of all superelliptic
curves of genus $g \leq 10$ defined over any field of characteristic $\neq 2$
is given in a subsequent paper.
