On superelliptic curves of level $n$ and their quotients, I.

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 such invariants, a full list of superelliptic curves and their corresponding equations for any genus $g > 2$.