Field of definition and Galois orbits for the Macbeath-Hurwitz curves (Q1572838)

Hurwitz curves of genus \(g \geq 2\) are those for which the automorphisms group has the maximal order, that is \(84(g-1)\). In this paper the author considers an infinite series of Hurwitz curves, the Macbeath-Hurwitz curves. The main result asserts that a minimal field of definition for these curves are either \(\mathbb Q\) or \({\mathbb Q}(\zeta_{7}+\zeta_{7}^{-1})\), where \(\zeta_{7}=\exp(2 \pi i/7)\).