INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF PURE MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -9/2 -4/1 -3/1 -2/1 -3/2 -4/3 0/1 1/1 6/5 4/3 3/2 12/7 2/1 12/5 5/2 3/1 36/11 7/2 4/1 9/2 5/1 11/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 0/1 -5/1 1/1 1/0 -9/2 1/0 -4/1 -1/1 0/1 1/0 -15/4 1/0 -11/3 -2/1 1/0 -7/2 -1/1 1/0 -3/1 -1/2 1/0 -11/4 -1/2 0/1 -8/3 -1/1 -1/2 0/1 -5/2 -1/1 -1/2 -12/5 -1/2 -7/3 -1/2 -1/3 -9/4 -1/2 -11/5 -1/2 0/1 -2/1 -1/2 -1/3 0/1 -13/7 -1/2 0/1 -24/13 -1/2 -11/6 -1/2 -2/5 -9/5 -1/2 -3/8 -7/4 -3/8 -1/3 -12/7 -1/3 -5/3 -1/3 -3/10 -3/2 -1/4 -7/5 -1/5 -1/6 -18/13 0/1 -11/8 -1/4 0/1 -4/3 -1/4 -1/5 0/1 -5/4 -1/4 -1/5 -6/5 -1/5 -7/6 -1/5 -1/6 -1/1 -1/6 0/1 0/1 0/1 1/1 0/1 1/6 6/5 1/5 5/4 1/5 1/4 4/3 0/1 1/5 1/4 3/2 1/4 8/5 1/4 2/7 1/3 13/8 1/4 2/7 5/3 3/10 1/3 12/7 1/3 7/4 1/3 3/8 2/1 0/1 1/3 1/2 11/5 0/1 1/2 9/4 1/2 7/3 1/3 1/2 12/5 1/2 5/2 1/2 1/1 8/3 0/1 1/2 1/1 11/4 0/1 1/2 3/1 1/2 1/0 13/4 0/1 1/2 36/11 1/2 23/7 1/2 2/3 10/3 1/2 2/3 1/1 7/2 1/1 1/0 18/5 1/1 11/3 2/1 1/0 4/1 0/1 1/1 1/0 13/3 2/1 1/0 9/2 1/0 5/1 -1/1 1/0 11/2 -1/2 0/1 6/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(25,132,-18,-95) (-6/1,-5/1) -> (-7/5,-18/13) Hyperbolic Matrix(23,108,10,47) (-5/1,-9/2) -> (9/4,7/3) Hyperbolic Matrix(23,96,-6,-25) (-9/2,-4/1) -> (-4/1,-15/4) Parabolic Matrix(71,264,32,119) (-15/4,-11/3) -> (11/5,9/4) Hyperbolic Matrix(23,84,-20,-73) (-11/3,-7/2) -> (-7/6,-1/1) Hyperbolic Matrix(25,84,-14,-47) (-7/2,-3/1) -> (-9/5,-7/4) Hyperbolic Matrix(47,132,-26,-73) (-3/1,-11/4) -> (-11/6,-9/5) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(47,108,10,23) (-7/3,-9/4) -> (9/2,5/1) Hyperbolic Matrix(97,216,22,49) (-9/4,-11/5) -> (13/3,9/2) Hyperbolic Matrix(23,48,-12,-25) (-11/5,-2/1) -> (-2/1,-13/7) Parabolic Matrix(551,1020,168,311) (-13/7,-24/13) -> (36/11,23/7) Hyperbolic Matrix(385,708,118,217) (-24/13,-11/6) -> (13/4,36/11) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,42,71) (-12/7,-5/3) -> (5/3,12/7) Hyperbolic Matrix(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(191,264,34,47) (-18/13,-11/8) -> (11/2,6/1) Hyperbolic Matrix(97,132,36,49) (-11/8,-4/3) -> (8/3,11/4) Hyperbolic Matrix(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(143,168,40,47) (-6/5,-7/6) -> (7/2,18/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(73,-84,20,-23) (1/1,6/5) -> (18/5,11/3) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(73,-120,14,-23) (13/8,5/3) -> (5/1,11/2) Hyperbolic Matrix(47,-84,14,-25) (7/4,2/1) -> (10/3,7/2) Hyperbolic Matrix(73,-156,22,-47) (2/1,11/5) -> (23/7,10/3) Hyperbolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(25,-96,6,-23) (11/3,4/1) -> (4/1,13/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,0,0,1) Matrix(25,132,-18,-95) -> Matrix(1,0,-6,1) Matrix(23,108,10,47) -> Matrix(1,0,2,1) Matrix(23,96,-6,-25) -> Matrix(1,0,0,1) Matrix(71,264,32,119) -> Matrix(1,2,2,5) Matrix(23,84,-20,-73) -> Matrix(1,2,-6,-11) Matrix(25,84,-14,-47) -> Matrix(3,2,-8,-5) Matrix(47,132,-26,-73) -> Matrix(3,2,-8,-5) Matrix(71,192,44,119) -> Matrix(3,2,10,7) Matrix(23,60,18,47) -> Matrix(1,0,6,1) Matrix(49,120,20,49) -> Matrix(3,2,4,3) Matrix(71,168,30,71) -> Matrix(5,2,12,5) Matrix(47,108,10,23) -> Matrix(1,0,2,1) Matrix(97,216,22,49) -> Matrix(5,2,2,1) Matrix(23,48,-12,-25) -> Matrix(1,0,0,1) Matrix(551,1020,168,311) -> Matrix(3,2,4,3) Matrix(385,708,118,217) -> Matrix(5,2,12,5) Matrix(97,168,56,97) -> Matrix(17,6,48,17) Matrix(71,120,42,71) -> Matrix(19,6,60,19) Matrix(23,36,-16,-25) -> Matrix(7,2,-32,-9) Matrix(191,264,34,47) -> Matrix(1,0,2,1) Matrix(97,132,36,49) -> Matrix(1,0,6,1) Matrix(47,60,18,23) -> Matrix(1,0,6,1) Matrix(49,60,40,49) -> Matrix(9,2,40,9) Matrix(143,168,40,47) -> Matrix(1,0,6,1) Matrix(1,0,2,1) -> Matrix(1,0,12,1) Matrix(73,-84,20,-23) -> Matrix(11,-2,6,-1) Matrix(25,-36,16,-23) -> Matrix(9,-2,32,-7) Matrix(73,-120,14,-23) -> Matrix(7,-2,-10,3) Matrix(47,-84,14,-25) -> Matrix(5,-2,8,-3) Matrix(73,-156,22,-47) -> Matrix(5,-2,8,-3) Matrix(25,-72,8,-23) -> Matrix(1,0,0,1) Matrix(25,-96,6,-23) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 8 Degree of the the map X: 8 Degree of the the map Y: 32 Permutation triple for Y: ((2,6,19,31,20,7)(3,12,30,29,13,4)(5,11,17)(8,23,24)(9,27,10)(14,22,15)(16,28)(25,26); (1,4,15,28,27,31,32,30,23,16,5,2)(3,10,29,11)(6,17,25,24,12,21,20,9,26,14,13,18)(7,22,19,8); (1,2,8,25,9,3)(4,14)(5,6)(7,21,12,11,16,15)(10,28,23,19,18,13)(17,29,32,31,22,26)(20,27)(24,30)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 96 Minimal number of generators: 17 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 12 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 4/3 3/2 2/1 12/5 5/2 3/1 4/1 9/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 0/1 -5/1 1/1 1/0 -9/2 1/0 -4/1 -1/1 0/1 1/0 -3/1 -1/2 1/0 -8/3 -1/1 -1/2 0/1 -5/2 -1/1 -1/2 -12/5 -1/2 -7/3 -1/2 -1/3 -9/4 -1/2 -2/1 -1/2 -1/3 0/1 -3/2 -1/4 -4/3 -1/4 -1/5 0/1 -5/4 -1/4 -1/5 -6/5 -1/5 -1/1 -1/6 0/1 0/1 0/1 1/1 0/1 1/6 6/5 1/5 5/4 1/5 1/4 4/3 0/1 1/5 1/4 3/2 1/4 2/1 0/1 1/3 1/2 9/4 1/2 7/3 1/3 1/2 12/5 1/2 5/2 1/2 1/1 8/3 0/1 1/2 1/1 3/1 1/2 1/0 4/1 0/1 1/1 1/0 9/2 1/0 5/1 -1/1 1/0 6/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(7,36,6,31) (-6/1,-5/1) -> (1/1,6/5) Hyperbolic Matrix(23,108,10,47) (-5/1,-9/2) -> (9/4,7/3) Hyperbolic Matrix(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(7,24,2,7) (-4/1,-3/1) -> (3/1,4/1) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(47,108,10,23) (-7/3,-9/4) -> (9/2,5/1) Hyperbolic Matrix(17,36,8,17) (-9/4,-2/1) -> (2/1,9/4) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(31,36,6,7) (-6/5,-1/1) -> (5/1,6/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,0,0,1) Matrix(7,36,6,31) -> Matrix(1,-1,6,-5) Matrix(23,108,10,47) -> Matrix(1,0,2,1) Matrix(17,72,4,17) -> Matrix(1,1,0,1) Matrix(7,24,2,7) -> Matrix(1,1,0,1) Matrix(17,48,6,17) -> Matrix(1,1,0,1) Matrix(23,60,18,47) -> Matrix(1,0,6,1) Matrix(49,120,20,49) -> Matrix(3,2,4,3) Matrix(71,168,30,71) -> Matrix(5,2,12,5) Matrix(47,108,10,23) -> Matrix(1,0,2,1) Matrix(17,36,8,17) -> Matrix(3,1,8,3) Matrix(7,12,4,7) -> Matrix(3,1,8,3) Matrix(17,24,12,17) -> Matrix(5,1,24,5) Matrix(47,60,18,23) -> Matrix(1,0,6,1) Matrix(49,60,40,49) -> Matrix(9,2,40,9) Matrix(31,36,6,7) -> Matrix(5,1,-6,-1) Matrix(1,0,2,1) -> Matrix(1,0,12,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 8 ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 0/1 6 1 1/1 (0/1,1/6) 0 12 6/5 1/5 2 2 5/4 (1/5,1/4) 0 12 4/3 (1/6,1/4) 0 3 3/2 1/4 2 4 2/1 (1/4,1/2) 0 6 9/4 1/2 2 4 7/3 (1/3,1/2) 0 12 12/5 1/2 2 1 5/2 (1/2,1/1) 0 12 8/3 (1/2,1/0) 0 3 3/1 (1/2,1/0) 0 4 4/1 (1/2,1/0) 0 3 9/2 1/0 2 4 5/1 (-1/1,1/0) 0 12 6/1 0/1 2 2 1/0 (0/1,1/0) 0 12 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(31,-36,6,-7) (1/1,6/5) -> (5/1,6/1) Glide Reflection Matrix(49,-60,40,-49) (6/5,5/4) -> (6/5,5/4) Reflection Matrix(47,-60,18,-23) (5/4,4/3) -> (5/2,8/3) Glide Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(17,-36,8,-17) (2/1,9/4) -> (2/1,9/4) Reflection Matrix(47,-108,10,-23) (9/4,7/3) -> (9/2,5/1) Glide Reflection Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,12,-1) (0/1,1/1) -> (0/1,1/6) Matrix(31,-36,6,-7) -> Matrix(5,-1,-6,1) Matrix(49,-60,40,-49) -> Matrix(9,-2,40,-9) (6/5,5/4) -> (1/5,1/4) Matrix(47,-60,18,-23) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(17,-24,12,-17) -> Matrix(5,-1,24,-5) (4/3,3/2) -> (1/6,1/4) Matrix(7,-12,4,-7) -> Matrix(3,-1,8,-3) (3/2,2/1) -> (1/4,1/2) Matrix(17,-36,8,-17) -> Matrix(3,-1,8,-3) (2/1,9/4) -> (1/4,1/2) Matrix(47,-108,10,-23) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(71,-168,30,-71) -> Matrix(5,-2,12,-5) (7/3,12/5) -> (1/3,1/2) Matrix(49,-120,20,-49) -> Matrix(3,-2,4,-3) (12/5,5/2) -> (1/2,1/1) Matrix(17,-48,6,-17) -> Matrix(-1,1,0,1) (8/3,3/1) -> (1/2,1/0) Matrix(7,-24,2,-7) -> Matrix(-1,1,0,1) (3/1,4/1) -> (1/2,1/0) Matrix(17,-72,4,-17) -> Matrix(-1,1,0,1) (4/1,9/2) -> (1/2,1/0) Matrix(-1,12,0,1) -> Matrix(1,0,0,-1) (6/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.