Non-isomorphic maximal function fields of genus q − 1

The classification of maximal function fields over a finite field is a difficult open problem, and even determining isomorphism classes among known function fields is challenging in general. We study a particular family of maximal function fields defined over a finite field with q 2 elements, where q is the power of an odd prime. When d : = ( q + 1 ) / 2 is a prime, this family is known to contain a large number of non-isomorphic function fields of the same genus and with the same automorphism group. We compute the automorphism group and isomorphism classes also in the case where d is not a prime.