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 $\mathbb{Q}[\sqrt{-1}, \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}]$.

8-166	8-172	8-178	8-180	8-182	8-189	8-196
8-198	8-200	8-204	11-160	11-166	11-171	11-173
11-178	11-188	11-195	14-129	14-135	14-139	14-141
14-142	14-148	14-152	20-100	20-104	20-105	22-91
22-95	29-94	36-82	36-86	37-80	37-84	51-73
51-75	103-4	103-10	103-12	103-16	103-18	103-20
103-22	103-24	103-26	103-28	103-35	103-37	103-39
103-41	103-43	103-45	103-49	103-59	103-61	103-66
103-68	103-72	103-77	103-79	103-81	103-89	103-91
103-97	103-100	103-102	103-105	103-107	103-111	103-114
103-116	103-118	103-121	103-122	103-125	103-127	103-128
103-130	104-162	104-168	104-169	104-170	104-175	104-177
104-178	104-179	104-187	104-192	104-193	104-194	105-130
105-136	105-140	105-144	105-150	105-154	106-154	106-155
106-156	106-157	106-164	106-167	106-168	108-100	108-101
108-105	108-106	108-116	109-4	109-10	109-12	109-18
109-20	109-26	109-34	109-36	109-38	109-42	109-44
109-48	109-51	109-53	109-59	109-61	109-63	109-67
109-69	109-73	109-75	109-77	109-79	109-81	109-84
109-87	109-89	109-90	109-95	109-96	110-53	112-85
113-41	113-43	114-40	116-119	116-120	116-121	116-125
116-126	118-106	118-110	129-106	129-109

Pierre Guillot, 2023.