An explicit quasiplatonic curve with non-abelian moduli field

TL;DR

This paper presents explicit examples of regular dessins d'enfant with non-abelian fields of moduli, including a curve with a field generated by a cubic root of 2, addressing open questions in the field.

Contribution

It provides the first explicit example of a regular dessin d'enfant with a non-abelian field of moduli, specifically a curve with a non-abelian moduli field, and discusses related examples from literature.

Findings

Constructed a regular dessin with field of moduli generated by a cubic root of 2.

Proved the underlying curve has a non-abelian field of moduli.

Identified existing examples that can be used to find more such dessins.

Abstract

We give an example of a regular dessin d'enfant whose field of moduli is not an abelian extension of the rational numbers, namely it is the field generated by a cubic root of 2. This answers a previous question. We also prove that the underlying curve has non-abelian field of moduli itself, giving an explicit example of a quasiplatonic curve with non-abelian field of moduli. In the last section, we note that two examples in previous literature can be used to find other examples of regular dessins d'enfants with non-abelian field of moduli.