The Belyi degree is computable

TL;DR

This paper presents an algorithm that computes the minimal degree of a Belyi map for a given algebraic curve over a number field, advancing computational methods in algebraic geometry.

Contribution

It introduces the first known algorithm to explicitly compute the Belyi degree of a curve over a number field.

Findings

Algorithm successfully computes Belyi degrees for various curves.

Provides a constructive method for Belyi map degree calculation.

Enhances computational tools in algebraic geometry.

Abstract

We exhibit an algorithm that, given input a curve $X$ over a number field, computes as output the minimal degree of a Belyi map $X \to \mathbb{P}^1$.