Some orbits of the Galois action on dessins d'enfants
In the summer of 2023, I have computed a little database of dessins d'enfants, in fact bipartite planar trees, each with a corresponding polynomial $P \in \bar{\mathbb{Q}}[X]$ such that the tree can be recovered as $P^{-1}([0,1])$. Moreover, each polynomial is a Belyi map ($P'(z) = 0 \Rightarrow P(z) \in \{0,1\}$). This allows the computation of the action of $Gal(\bar{\mathbb{Q}} / \mathbb{Q})$ on the trees.
At present the database contains 12,170 trees (pairwise non-isomorphic). Among these, 6,040 are fixed by the Galois group, while 5,506 belong to an orbit of size 2, and 608 belong to an orbit of size 4 ; the remaining 16 fall into two orbits of size 8.
On this page I have included the orbits of size 4, just as an illustration. (Ignore the weird numbering.) I have drawn each dessin, and my original algorithm for this was improved by following a suggestion on ask.sagemath made by Samuel Lelièvre. I am indebted to Frédéric Chapoton for pointing this out. I have used floating numbers with 100 bits of precision, and in some cases, such as 8-189, it was not enough.
Some personal favourites are 20-105, 103-49, 104-178, 109-61, 113-43.
In each case I have indicated the moduli field of (each dessin in) the orbit. This is the extension of $\mathbb{Q}$ which is the fixed field of the stabilizer of the dessin. By construction, for each dessin in the database, the moduli field is a subfield of $\mathbb{Q}[\sqrt{-1}, \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}]$.
Pierre Guillot, 2023.