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 -2/1 0/1 -5/1 -1/1 -9/2 1/0 -4/1 1/0 -3/1 -1/1 -8/3 0/1 -5/2 1/0 -12/5 -1/1 -7/3 -1/1 -16/7 -2/3 0/1 -9/4 -1/2 -11/5 -1/3 -2/1 0/1 -9/5 1/1 -16/9 0/1 2/1 -7/4 1/0 -12/7 1/0 -5/3 -1/1 -13/8 1/0 -8/5 0/1 -3/2 1/0 -4/3 -1/1 -9/7 -1/1 -14/11 0/1 -19/15 -1/3 -24/19 0/1 -5/4 1/0 -11/9 -1/1 -6/5 -2/1 0/1 -7/6 1/0 -8/7 -2/1 0/1 -1/1 -1/1 0/1 0/1 1/1 1/1 6/5 0/1 2/1 5/4 1/0 9/7 1/1 4/3 1/1 3/2 1/0 8/5 0/1 5/3 1/1 12/7 1/0 7/4 1/0 16/9 -2/1 0/1 9/5 -1/1 11/6 -1/2 2/1 0/1 9/4 1/2 16/7 0/1 2/3 7/3 1/1 12/5 1/1 5/2 1/0 13/5 1/1 8/3 0/1 3/1 1/1 4/1 1/0 9/2 1/0 14/3 0/1 19/4 -1/2 24/5 0/1 5/1 1/1 11/2 1/0 6/1 0/1 2/1 7/1 1/1 8/1 0/1 2/1 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(1,0,0,1) Matrix(17,96,-14,-79) -> Matrix(1,0,0,1) Matrix(31,144,-14,-65) -> Matrix(1,0,-2,1) Matrix(17,72,4,17) -> Matrix(1,0,0,1) Matrix(7,24,2,7) -> Matrix(1,2,0,1) 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(1,2,0,1) Matrix(71,168,30,71) -> Matrix(3,2,4,3) Matrix(31,72,-28,-65) -> Matrix(3,2,-2,-1) Matrix(127,288,56,127) -> Matrix(3,2,4,3) Matrix(89,192,-70,-151) -> Matrix(1,0,0,1) Matrix(79,144,-62,-113) -> Matrix(1,0,-2,1) Matrix(161,288,90,161) -> Matrix(1,-2,0,1) Matrix(95,168,-82,-145) -> Matrix(1,-2,0,1) Matrix(97,168,56,97) -> Matrix(1,-2,0,1) Matrix(71,120,42,71) -> Matrix(1,2,0,1) Matrix(103,168,-84,-137) -> Matrix(1,0,0,1) Matrix(31,48,20,31) -> Matrix(1,0,0,1) Matrix(17,24,12,17) -> Matrix(1,2,0,1) 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,-2,1) Matrix(1,0,2,1) -> Matrix(1,0,2,1) Matrix(41,-48,6,-7) -> Matrix(1,0,0,1) Matrix(79,-96,14,-17) -> Matrix(1,0,0,1) Matrix(113,-144,62,-79) -> Matrix(1,0,-2,1) 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(1,0,-2,1) Matrix(73,-168,10,-23) -> Matrix(3,-2,2,-1) Matrix(65,-168,12,-31) -> 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): 7 Degree of the the map X: 7 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 2/1 12/5 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 0/1 0/1 1/1 1/1 6/5 0/1 2/1 5/4 1/0 9/7 1/1 4/3 1/1 3/2 1/0 8/5 0/1 5/3 1/1 12/7 1/0 7/4 1/0 16/9 -2/1 0/1 9/5 -1/1 11/6 -1/2 2/1 0/1 9/4 1/2 16/7 0/1 2/3 7/3 1/1 12/5 1/1 5/2 1/0 13/5 1/1 8/3 0/1 3/1 1/1 4/1 1/0 9/2 1/0 14/3 0/1 19/4 -1/2 24/5 0/1 5/1 1/1 11/2 1/0 6/1 0/1 2/1 7/1 1/1 8/1 0/1 2/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (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(55,-72,13,-17) (9/7,4/3) -> (4/1,9/2) Hyperbolic Matrix(17,-24,5,-7) (4/3,3/2) -> (3/1,4/1) Hyperbolic Matrix(31,-48,11,-17) (3/2,8/5) -> (8/3,3/1) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(71,-120,29,-49) (5/3,12/7) -> (12/5,5/2) Hyperbolic Matrix(97,-168,41,-71) (12/7,7/4) -> (7/3,12/5) Hyperbolic Matrix(41,-72,4,-7) (7/4,16/9) -> (8/1,1/0) Hyperbolic Matrix(161,-288,71,-127) (16/9,9/5) -> (9/4,16/7) 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 Matrix(121,-576,25,-119) (19/4,24/5) -> (24/5,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,1,1) Matrix(41,-48,6,-7) -> Matrix(1,0,0,1) Matrix(79,-96,14,-17) -> Matrix(1,0,0,1) Matrix(113,-144,62,-79) -> Matrix(1,0,-2,1) Matrix(55,-72,13,-17) -> Matrix(1,0,-1,1) Matrix(17,-24,5,-7) -> Matrix(1,-2,1,-1) Matrix(31,-48,11,-17) -> Matrix(1,0,1,1) Matrix(89,-144,34,-55) -> Matrix(1,0,0,1) Matrix(71,-120,29,-49) -> Matrix(1,-2,1,-1) Matrix(97,-168,41,-71) -> Matrix(1,2,1,3) Matrix(41,-72,4,-7) -> Matrix(1,2,0,1) Matrix(161,-288,71,-127) -> Matrix(1,2,1,3) Matrix(103,-192,22,-41) -> Matrix(1,0,0,1) Matrix(65,-144,14,-31) -> Matrix(1,0,-2,1) Matrix(73,-168,10,-23) -> Matrix(3,-2,2,-1) Matrix(65,-168,12,-31) -> Matrix(1,0,0,1) Matrix(121,-576,25,-119) -> Matrix(1,0,3,1) 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): 7 ----------------------------------------------------------------------- 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 1 1 2/1 0/1 1 6 9/4 1/2 1 4 16/7 (0/1,2/3).(1/2,1/1) 0 3 7/3 1/1 1 12 12/5 1/1 2 1 5/2 1/0 1 12 8/3 0/1 1 3 3/1 1/1 1 4 4/1 1/0 1 3 9/2 1/0 1 4 14/3 0/1 1 6 24/5 0/1 3 1 5/1 1/1 1 12 6/1 (0/1,2/1) 0 2 8/1 (0/1,2/1).(1/1,1/0) 0 3 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(65,-144,14,-31) (2/1,9/4) -> (9/2,14/3) Hyperbolic Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(31,-72,3,-7) (16/7,7/3) -> (8/1,1/0) 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(47,-120,9,-23) (5/2,13/5) -> (5/1,11/2) Glide Reflection Matrix(55,-144,21,-55) (18/7,8/3) -> (18/7,8/3) 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(71,-336,15,-71) (14/3,24/5) -> (14/3,24/5) Reflection Matrix(49,-240,10,-49) (24/5,5/1) -> (24/5,5/1) Reflection Matrix(17,-96,3,-17) (16/3,6/1) -> (16/3,6/1) Reflection Matrix(7,-48,1,-7) (6/1,8/1) -> (6/1,8/1) 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(1,0,1,-1) -> Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Matrix(65,-144,14,-31) -> Matrix(1,0,-2,1) 0/1 Matrix(127,-288,56,-127) -> Matrix(3,-2,4,-3) (9/4,16/7) -> (1/2,1/1) Matrix(31,-72,3,-7) -> Matrix(3,-2,1,-1) Matrix(71,-168,30,-71) -> Matrix(3,-2,4,-3) (7/3,12/5) -> (1/2,1/1) Matrix(49,-120,20,-49) -> Matrix(-1,2,0,1) (12/5,5/2) -> (1/1,1/0) Matrix(47,-120,9,-23) -> Matrix(1,0,1,-1) *** -> (0/1,2/1) Matrix(55,-144,21,-55) -> Matrix(1,0,1,-1) (18/7,8/3) -> (0/1,2/1) Matrix(17,-48,6,-17) -> Matrix(1,0,2,-1) (8/3,3/1) -> (0/1,1/1) Matrix(7,-24,2,-7) -> Matrix(-1,2,0,1) (3/1,4/1) -> (1/1,1/0) Matrix(17,-72,4,-17) -> Matrix(1,0,0,-1) (4/1,9/2) -> (0/1,1/0) Matrix(71,-336,15,-71) -> Matrix(-1,0,1,1) (14/3,24/5) -> (-2/1,0/1) Matrix(49,-240,10,-49) -> Matrix(1,0,2,-1) (24/5,5/1) -> (0/1,1/1) Matrix(17,-96,3,-17) -> Matrix(1,0,1,-1) (16/3,6/1) -> (0/1,2/1) Matrix(7,-48,1,-7) -> Matrix(1,0,1,-1) (6/1,8/1) -> (0/1,2/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.