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: 20 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -2/1 0/1 1/1 4/3 3/2 12/7 9/5 2/1 12/5 5/2 8/3 3/1 4/1 9/2 24/5 5/1 6/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/2 1/0 -5/1 -1/2 0/1 -9/2 -1/2 -4/1 0/1 -3/1 1/2 1/0 -8/3 1/1 -5/2 1/2 -12/5 1/1 -7/3 1/1 3/2 -16/7 1/1 -9/4 3/2 -11/5 3/2 2/1 -2/1 1/0 -9/5 1/2 1/0 -16/9 1/1 -7/4 1/0 -12/7 1/0 -5/3 0/1 1/0 -13/8 1/2 -8/5 1/1 -3/2 1/0 -4/3 0/1 -9/7 1/2 1/0 -14/11 1/0 -19/15 -1/2 0/1 -24/19 0/1 -5/4 1/2 -11/9 0/1 1/0 -6/5 1/2 1/0 -7/6 1/2 -8/7 1/1 -1/1 1/1 1/0 0/1 1/0 1/1 -1/1 1/0 6/5 -1/2 1/0 5/4 -1/2 9/7 -1/2 1/0 4/3 0/1 3/2 1/0 8/5 -1/1 5/3 0/1 1/0 12/7 1/0 7/4 1/0 16/9 -1/1 9/5 -1/2 1/0 11/6 -1/2 2/1 1/0 9/4 -3/2 16/7 -1/1 7/3 -3/2 -1/1 12/5 -1/1 5/2 -1/2 13/5 0/1 1/0 8/3 -1/1 3/1 -1/2 1/0 4/1 0/1 9/2 1/2 14/3 1/0 19/4 -1/2 24/5 0/1 5/1 0/1 1/2 11/2 1/0 6/1 1/2 1/0 7/1 1/2 1/1 8/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,48,-6,-41) (-6/1,1/0) -> (-6/5,-7/6) Hyperbolic Matrix(17,96,-14,-79) (-6/1,-5/1) -> (-11/9,-6/5) Hyperbolic Matrix(31,144,-14,-65) (-5/1,-9/2) -> (-9/4,-11/5) 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(55,144,-34,-89) (-8/3,-5/2) -> (-13/8,-8/5) 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(31,72,-28,-65) (-7/3,-16/7) -> (-8/7,-1/1) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(89,192,-70,-151) (-11/5,-2/1) -> (-14/11,-19/15) Hyperbolic Matrix(79,144,-62,-113) (-2/1,-9/5) -> (-9/7,-14/11) Hyperbolic Matrix(161,288,90,161) (-9/5,-16/9) -> (16/9,9/5) Hyperbolic Matrix(95,168,-82,-145) (-16/9,-7/4) -> (-7/6,-8/7) 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(103,168,-84,-137) (-5/3,-13/8) -> (-5/4,-11/9) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(55,72,42,55) (-4/3,-9/7) -> (9/7,4/3) Hyperbolic Matrix(455,576,94,119) (-19/15,-24/19) -> (24/5,5/1) Hyperbolic Matrix(457,576,96,121) (-24/19,-5/4) -> (19/4,24/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(41,-48,6,-7) (1/1,6/5) -> (6/1,7/1) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(113,-144,62,-79) (5/4,9/7) -> (9/5,11/6) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(41,-72,4,-7) (7/4,16/9) -> (8/1,1/0) Hyperbolic Matrix(103,-192,22,-41) (11/6,2/1) -> (14/3,19/4) Hyperbolic Matrix(65,-144,14,-31) (2/1,9/4) -> (9/2,14/3) Hyperbolic Matrix(73,-168,10,-23) (16/7,7/3) -> (7/1,8/1) Hyperbolic Matrix(65,-168,12,-31) (5/2,13/5) -> (5/1,11/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,48,-6,-41) -> Matrix(1,0,2,1) Matrix(17,96,-14,-79) -> Matrix(1,0,2,1) Matrix(31,144,-14,-65) -> Matrix(1,2,0,1) Matrix(17,72,4,17) -> Matrix(1,0,4,1) Matrix(7,24,2,7) -> Matrix(1,0,-2,1) Matrix(17,48,6,17) -> Matrix(1,0,-2,1) Matrix(55,144,-34,-89) -> Matrix(1,0,0,1) Matrix(49,120,20,49) -> Matrix(3,-2,-4,3) Matrix(71,168,30,71) -> Matrix(5,-6,-4,5) Matrix(31,72,-28,-65) -> Matrix(1,-2,2,-3) Matrix(127,288,56,127) -> Matrix(5,-6,-4,5) Matrix(89,192,-70,-151) -> Matrix(1,-2,0,1) Matrix(79,144,-62,-113) -> Matrix(1,0,0,1) Matrix(161,288,90,161) -> Matrix(1,0,-2,1) Matrix(95,168,-82,-145) -> Matrix(1,-2,2,-3) Matrix(97,168,56,97) -> Matrix(1,-2,0,1) Matrix(71,120,42,71) -> Matrix(1,0,0,1) Matrix(103,168,-84,-137) -> Matrix(1,0,0,1) Matrix(31,48,20,31) -> Matrix(1,-2,0,1) Matrix(17,24,12,17) -> Matrix(1,0,0,1) Matrix(55,72,42,55) -> Matrix(1,0,-2,1) Matrix(455,576,94,119) -> Matrix(1,0,4,1) Matrix(457,576,96,121) -> Matrix(1,0,-4,1) Matrix(1,0,2,1) -> Matrix(1,-2,0,1) Matrix(41,-48,6,-7) -> Matrix(1,0,2,1) Matrix(79,-96,14,-17) -> Matrix(1,0,2,1) Matrix(113,-144,62,-79) -> Matrix(1,0,0,1) Matrix(89,-144,34,-55) -> Matrix(1,0,0,1) Matrix(41,-72,4,-7) -> Matrix(1,2,0,1) Matrix(103,-192,22,-41) -> Matrix(1,0,0,1) Matrix(65,-144,14,-31) -> Matrix(1,2,0,1) Matrix(73,-168,10,-23) -> Matrix(1,2,0,1) Matrix(65,-168,12,-31) -> Matrix(1,0,2,1) 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)(3,11,26,28,12,4)(5,16,14)(8,9,22)(10,17,13)(15,23)(24,25)(27,29,30); (1,4,14,24,27,11,21,20,30,15,5,2)(3,8,7,10)(6,17,23,22,31,32,26,9,25,13,12,18)(16,19,29,28); (1,2,8,23,30,19,18,12,29,24,9,3)(4,13,6,5)(7,21,11,10,25,14,28,32,31,16,15,17)(20,22,26,27)) ----------------------------------------------------------------------- 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 -1/2 1/0 -5/1 -1/2 0/1 -9/2 -1/2 -4/1 0/1 -3/1 1/2 1/0 -8/3 1/1 -5/2 1/2 -12/5 1/1 -7/3 1/1 3/2 -9/4 3/2 -2/1 1/0 -3/2 1/0 -4/3 0/1 -5/4 1/2 -6/5 1/2 1/0 -1/1 1/1 1/0 0/1 1/0 1/1 -1/1 1/0 6/5 -1/2 1/0 5/4 -1/2 4/3 0/1 3/2 1/0 2/1 1/0 9/4 -3/2 7/3 -3/2 -1/1 12/5 -1/1 5/2 -1/2 8/3 -1/1 3/1 -1/2 1/0 4/1 0/1 9/2 1/2 5/1 0/1 1/2 6/1 1/2 1/0 1/0 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(1,1,0,1) Matrix(7,36,6,31) -> Matrix(1,1,-2,-1) Matrix(23,108,10,47) -> Matrix(1,-1,0,1) Matrix(17,72,4,17) -> Matrix(1,0,4,1) Matrix(7,24,2,7) -> Matrix(1,0,-2,1) Matrix(17,48,6,17) -> Matrix(1,0,-2,1) Matrix(23,60,18,47) -> Matrix(1,-1,0,1) Matrix(49,120,20,49) -> Matrix(3,-2,-4,3) Matrix(71,168,30,71) -> Matrix(5,-6,-4,5) Matrix(47,108,10,23) -> Matrix(1,-1,0,1) Matrix(17,36,8,17) -> Matrix(1,-3,0,1) Matrix(7,12,4,7) -> Matrix(1,-1,0,1) Matrix(17,24,12,17) -> Matrix(1,0,0,1) Matrix(47,60,18,23) -> Matrix(1,-1,0,1) Matrix(49,60,40,49) -> Matrix(1,-1,0,1) Matrix(31,36,6,7) -> Matrix(1,-1,2,-1) Matrix(1,0,2,1) -> Matrix(1,-2,0,1) 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 1/0 1 1 1/1 (-1/1,1/0) 0 12 6/5 (-1/2,1/0) 0 2 5/4 -1/2 1 12 4/3 0/1 1 3 3/2 1/0 1 4 2/1 1/0 1 6 9/4 -3/2 1 4 7/3 (-3/2,-1/1) 0 12 12/5 -1/1 4 1 5/2 -1/2 1 12 8/3 -1/1 1 3 3/1 (-1/1,0/1) 0 4 4/1 0/1 3 3 9/2 1/2 1 4 5/1 (0/1,1/2) 0 12 6/1 (1/2,1/0) 0 2 1/0 1/0 1 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(1,2,0,-1) (0/1,1/1) -> (-1/1,1/0) Matrix(31,-36,6,-7) -> Matrix(1,1,2,1) Matrix(49,-60,40,-49) -> Matrix(1,1,0,-1) (6/5,5/4) -> (-1/2,1/0) Matrix(47,-60,18,-23) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(17,-24,12,-17) -> Matrix(1,0,0,-1) (4/3,3/2) -> (0/1,1/0) Matrix(7,-12,4,-7) -> Matrix(1,1,0,-1) (3/2,2/1) -> (-1/2,1/0) Matrix(17,-36,8,-17) -> Matrix(1,3,0,-1) (2/1,9/4) -> (-3/2,1/0) Matrix(47,-108,10,-23) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(71,-168,30,-71) -> Matrix(5,6,-4,-5) (7/3,12/5) -> (-3/2,-1/1) Matrix(49,-120,20,-49) -> Matrix(3,2,-4,-3) (12/5,5/2) -> (-1/1,-1/2) Matrix(17,-48,6,-17) -> Matrix(-1,0,2,1) (8/3,3/1) -> (-1/1,0/1) Matrix(7,-24,2,-7) -> Matrix(-1,0,2,1) (3/1,4/1) -> (-1/1,0/1) Matrix(17,-72,4,-17) -> Matrix(1,0,4,-1) (4/1,9/2) -> (0/1,1/2) Matrix(-1,12,0,1) -> Matrix(-1,1,0,1) (6/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.