A computational note about Fricke-Macbeath's curve

Abstract

The well known Hurwitz upper bound states that a closed Riemann surface $S$ of genus $g \geq 2$ has at most $84(g-1)$ conformal automorphisms. If $S$ has exactly $84(g-1)$ conformal automorphisms, then it is called a Hurwitz curve. The first two genera for which there are Hurwitz's curves are $g \in \{3,7\}$. In both situations there is exactly one such curve up to conformal equivalence, in particular, in both cases the field of moduli is ${\mathbb Q}$. As these two curves are quasiplatonic curves, they are definable over ${\mathbb Q}$. The Hurwitz's curve of genus $g=3$ is given by Klein's quartic $x^3y+y^3z+z^3x=0$. The Hurwitz's curve of genus $g=7$ is known as Fricke-Macbeath's curve and equations over ${\mathbb Q}(\rho)$, where $\rho=e^{2 \pi i/7}$, are known due to Macbeath. Unfortunately, explicit equations over ${\mathbb Q}$ are not easy to find for this curve. In this paper we first explain how to construct an explicit model $Z_{2}$ of Fricke-Macbeath's curve over ${\mathbb Q}(\sqrt{-7})$ and an explicit isomorphism $L_{1}:X \to Z_{2}$, defined over ${\mathbb Q}(\rho)$. Next, using that explicit model we construct another explicit isomorphism $L_{2}:Z_{2} \to W$, defined over ${\mathbb Q}(\sqrt{-7})$, where $W$ is some algebraic curve defined over ${\mathbb Q}$. Unfortunately, the equations for $W$ are quite long to write down, but everything is explained in order to perform the computations in a computer.