These dessins correspond to Belyi maps from elliptic curves to the Riemann sphere.
The elliptic curve in orbit A is defined over the biquadratic field \(\mathbb{Q}(\sqrt{-6}, \sqrt{10})\) and has conductor norm 40500, while the curve from orbit B is defined over a \(D_4\) number field and has conductor norm 36.
,(1,3,5,6,4),(1,2,3,4)(5,6).png "(1,3,4,5,2)_(1,3,5,6,4)_(1,2,3,4)(5,6)")
,(1,4,6,5,3),(1,4,3,2)(5,6).png "(1,2,5,4,3)_(1,4,6,5,3)_(1,4,3,2)(5,6)")
,(1,2,6,4,3),(1,2,3,4)(5,6).png "(1,2,4,5,6)_(1,2,6,4,3)_(1,2,3,4)(5,6)")
,(1,3,4,6,2),(1,4,3,2)(5,6).png "(1,6,5,4,2)_(1,3,4,6,2)_(1,4,3,2)(5,6)")
,(1,4,2,6,3),(1,2,3,4)(5,6).png "(1,2,3,5,6)_(1,4,2,6,3)_(1,2,3,4)(5,6)")
,(1,2,5,6,4),(1,2,3,4)(5,6).png "(1,3,2,4,5)_(1,2,5,6,4)_(1,2,3,4)(5,6)")
,(1,5,4,3,6),(1,2,3,4)(5,6).png "(1,5,4,6,2)_(1,5,4,3,6)_(1,2,3,4)(5,6)")
,(1,6,3,4,5),(1,4,3,2)(5,6).png "(1,2,6,4,5)_(1,6,3,4,5)_(1,4,3,2)(5,6)")