,(1,2,3,4)(5,6),(1,2)(3,5,4)(6,7).png "(1,5,7,6,3)_(1,2,3,4)(5,6)_(1,2)(3,5,4)(6,7)")
A
,(1,2,3,4)(5,6),(1,3,2)(4,5)(6,7).png "(1,5,7,6,4)_(1,2,3,4)(5,6)_(1,3,2)(4,5)(6,7)")
B
,(1,2,3,4)(5,6),(1,7)(2,3)(4,5,6).png "(1,6,4,2,7)_(1,2,3,4)(5,6)_(1,7)(2,3)(4,5,6)")
B
,(1,4,3,2)(5,6),(1,7)(2,3)(4,6,5).png "(1,7,2,4,6)_(1,4,3,2)(5,6)_(1,7)(2,3)(4,6,5)")
B
This is the puzzle from the introduction.
The Belyi map in orbit A stands alone because we can describe it with rational numbers, which have no symmetries besides the identity. To describe the maps in orbit B, we also need a root of the polynomial \(x^3 - x^2 + 2x - 38\). These three maps are related by the symmetries that exchange the roots of the polynomial.