Computing algebraic Belyi functions on Bring's curve

Abstract

In this paper, we explicitly compute two kinds of algebraic Belyi functions on Bring's curve. One is related to a congruence subgroup of ${\rm SL}_2(\mathbb{Z})$ and the other is related to a congruence subgroup of the triangle group $\Delta(2,4,5)\subset \SL_2(\R)$. To carry out the computation, we use elliptic cusp forms of weight 2 for the former case and the automorphism group of Bring's curve for the latter case. We also discuss a suitable base field (a number field) for describing isomorphisms between Hulek-Craig's curve, Bring's curve, and another algebraic model obtained as a modular curve.