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.