(3,4,5),(1,5,3,7)(2,6,4),(1,2,3)(4,5,6).png "(1,7,2,6)(3,4,5)_(1,5,3,7)(2,6,4)_(1,2,3)(4,5,6)")
A
(5,6,7),(1,6,5)(2,3,7,4),(1,2,7)(3,4,5).png "(1,4,2,3)(5,6,7)_(1,6,5)(2,3,7,4)_(1,2,7)(3,4,5)")
B
(3,4,5),(1,5,3)(2,7,6,4),(1,2,3)(4,5,6).png "(1,2,6,7)(3,4,5)_(1,5,3)(2,7,6,4)_(1,2,3)(4,5,6)")
B
(3,5,4),(1,3,5)(2,4,6,7),(1,3,2)(4,6,5).png "(1,7,6,2)(3,5,4)_(1,3,5)(2,4,6,7)_(1,3,2)(4,6,5)")
B
Just as in the first puzzle, one of the maps stands alone and is defined over the rational numbers, while the other three are related by Galois symmetries.
The distinguished map is defined on the elliptic curve with LMFDB label 294.c1.