On the curves of genus g with automorphisms of prime order \(2g+1\) (Q793804)

Let k be an algebraically closed field of characteristic p. Let \(q=5\) be a prime different from p. For a pair of positive numbers r, s such that r, s and \(r+s\) are coprime to q, let C(r,s) be a non-singular model of the irreducible equation \(y^ r(y-1)^ s=x^ q\) over k. Then C(r,s) has an automorphism of order q. In this paper a condition is given for two such curves to be isomorphic in terms of r and s. It is shown that the cardinality of the set of isomorphism classes is \((q+5)/6\) or \((q+1)/6\) according as q is congruent 1 or 2 mod 3. Finally, the order of the automorphism group is determined in characteristic 0.