### Edmonds maps on Fricke-Macbeath curve

#### Abstract

In 1985, L. D. James and G. A. Jones proved that the complete graph

*K*_{n}defines a clean dessin d’enfant (the bipartite graph is given by taking as the black vertices the vertices of*K*_{n}and the white vertices as middle points of edges) if and only if*n*=*p*^{e}, where*p*is a prime. Later, in 2010, G. A. Jones, M. Streit and J. Wolfart computed the minimal field of definition of them. The minimal genus*g*> 1 of these types of clean dessins d’enfant is*g*= 7, obtained for*p*= 2 and*e*= 3. In that case, there are exactly two such clean dessins d’enfant (previously known as Edmonds maps), both of them defining the Fricke-Macbeath curve (the only Hurwitz curve of genus 7) and both forming a chiral pair. The uniqueness of the Fricke-Macbeath curve asserts that it is definable over**Q**, but both Edmonds maps cannot be defined over**Q**; in fact they have as minimal field of definition the quadratic field**Q**(sqrt( − 7)). It seems that no explicit models for the Edmonds maps over**Q**(sqrt( − 7)) are written in the literature. In this paper we start with an explicit model*X*for the Fricke-Macbeath curve provided by Macbeath, which is defined over**Q**(*e*^{2πi / 7}), and we construct an explicit birational isomorphism*L*:*X*→*Z*, where*Z*is defined over**Q**(sqrt( − 7)), so that both Edmonds maps are also defined over that field.#### Keywords

Riemann surface, algebraic curve, dessin d’enfant

DOI: https://doi.org/10.26493/1855-3974.496.61a

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications