Visualizing Toroidal \Belyi Pairs

A \Belyi map $\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ is a rational function with at most three critical values; we may assume these values are $\{ 0, \, 1, \, \infty \}$.  A Dessin d'Enfant is a planar bipartite graph obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1.  Such graphs can be drawn on the sphere by composing with stereographic projection: $\beta^{-1} \bigl( [0,1] \bigr) \subseteq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R)$.  

Replacing $\mathbb P^1$ with an elliptic curve $E$, there is a similar definition of a Bely\u{\i} map $\beta: E(\mathbb C) \to \mathbb P^1(\mathbb C)$.  The corresponding Dessin d'Enfant can be drawn on the torus by composing with an elliptic logarithm: $\beta^{-1} \bigl( [0,1] \bigr) \subseteq E(\mathbb C) \simeq \mathbb T^2(\mathbb R)$.  In this project, we use the open source \texttt{Sage} to write code which takes an elliptic curve $E$ and a \Belyi map $\beta$ to return the Dessin d'Enfant of this map -- both in two and three dimensions.  We focus on several examples of \Belyi maps which appear in the $L$-Series and Modular Forms Database (LMFDB).