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 \(\textrm{SL}_2({\mathbb {Z}})\) and the other is related to a congruence subgroup of the triangle group \(\Delta (2,4,5)\subset \textrm{SL}_2({\mathbb {R}}).\) 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.