,(1,2,3,4)(5,6),(1,3,6,2)(4,5).png "(1,5,3,6,4)_(1,2,3,4)(5,6)_(1,3,6,2)(4,5)")
A
,(1,4,3,2)(5,6),(1,2,6,3)(4,5).png "(1,4,6,3,5)_(1,4,3,2)(5,6)_(1,2,6,3)(4,5)")
A
,(1,2,3,4)(5,6),(1,5,4,6)(2,3).png "(1,5,4,2,6)_(1,2,3,4)(5,6)_(1,5,4,6)(2,3)")
B
,(1,2,3,4)(5,6),(1,3)(2,4,5,6).png "(1,2,3,6,4)_(1,2,3,4)(5,6)_(1,3)(2,4,5,6)")
B
These dessins correspond to Belyi maps from elliptic curves to the Riemann sphere.
The elliptic curves in this puzzle are defined over the number fields \(\mathbb{Q}(\sqrt{-15})\) and \(\mathbb{Q}(\sqrt{15})\). The curve from orbit A is defined over \(\mathbb{Q}(\sqrt{-15})\) and has conductor norm 11520, while the curve from orbit B, which has LMFDB label 2.2.60.1-90.1-a3, is defined over \(\mathbb{Q}(\sqrt{15})\) and has conductor norm 90.