Belyi functions of the special weighted trees of $(2,3)$-type

Abstract

In this paper we calculate the Belyi functions of the weighted trees of $(2,3)$-type with primitive special edge rotation groups. There are 21 Galois orbits of these trees: 6 rational orbits, 12 orbits over quadratic fields, and three orbits over fields of degree 3, 4 and 6. The highest degree of the calculated Belyi function is 32. The calculations are performed using modular functions techniques.

Figures (22)

• Figure 1: "Alien": a dessin with passport $(3^6 1^6 \,|\, 2^{12} \,|\, 21^1 3^1)$ and the edge rotation group $\mathrm{\bf M}_{24}$.
• Figure 2: Group $\mathrm{\bf PSL}_2(5)$: orbit 6.1 and a drawing by F. Klein.
• Figure 3: Group $\mathrm{\bf AGL}_1(7)$: orbit 7.1 of size 2.
• Figure 4: Group $\mathrm{\bf PSL}_3(2)$: orbit 7.2 of size 2 and a drawing by F. Klein.
• Figure 5: Group $\mathrm{\bf PSL}_2(7)$: orbit 8.8 of size 1 and a drawing by F. Klein.