Galois action on families of generalised Fermat curves

It is well known that the complete bipartite graphs Kn,n occur as dessins dʹenfants on the Fermat curves of exponent n. However, there are many more curves having Kn,n as the underlying graph of their dessins, even if we require the strongest regularity condition that the graphs define regular maps on the underlying Riemann surfaces. For odd prime powers n these maps have recently been classified [G.A. Jones, R. Nedela, M. Škoviera, Regular embeddings of Kn,n where n is an odd prime power, European J. Combin., in press]; they fall into certain families characterised by their automorphism groups. In the present paper we show that these families form Galois orbits. We determine the minimal field of definition of the corresponding curves, and in easier cases also their defining equations.