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): 576 Minimal number of generators: 97 Number of equivalence classes of cusps: 48 Genus: 25 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -4/9 -2/5 -3/8 -2/7 -1/4 -1/6 0/1 2/13 1/5 1/4 4/11 3/7 1/2 2/3 4/5 7/8 1/1 8/7 5/4 17/13 7/5 3/2 46/29 19/11 2/1 31/14 7/3 5/2 13/5 8/3 11/4 3/1 23/7 10/3 31/9 7/2 11/3 4/1 9/2 5/1 11/2 6/1 13/2 7/1 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/4 -6/13 1/4 2/7 -5/11 0/1 2/7 -9/20 0/1 1/3 -4/9 1/4 1/2 -7/16 0/1 -3/7 0/1 1/3 -8/19 0/1 1/5 -5/12 0/1 1/4 -7/17 1/4 -2/5 0/1 2/7 -9/23 1/4 2/7 -7/18 2/7 1/3 -5/13 0/1 1/2 -8/21 1/4 1/3 -3/8 1/4 1/2 -10/27 1/4 1/3 -7/19 0/1 2/7 -4/11 0/1 1/2 -5/14 0/1 -11/31 1/6 1/4 -6/17 0/1 1/5 -1/3 1/4 1/3 -5/16 1/3 4/11 -9/29 5/14 3/8 -4/13 4/11 2/5 -3/10 2/5 1/2 -8/27 2/5 3/7 -5/17 2/5 4/9 -2/7 1/2 -7/25 0/1 2/3 -5/18 1/2 1/1 -3/11 1/2 2/3 -7/26 4/5 1/1 -4/15 1/1 2/1 -5/19 0/1 1/0 -1/4 0/1 -3/13 1/4 -2/9 1/4 2/7 -5/23 2/7 4/13 -3/14 4/13 1/3 -1/5 1/3 2/5 -2/11 0/1 2/5 -1/6 1/2 -2/13 0/1 1/1 -1/7 0/1 2/3 -1/8 0/1 1/0 0/1 0/1 1/2 1/7 0/1 1/0 2/13 0/1 1/6 0/1 1/4 1/5 0/1 2/5 3/14 1/3 1/2 5/23 1/2 1/1 2/9 0/1 1/3 1/4 1/2 3/11 1/2 2/3 5/18 2/3 3/4 2/7 0/1 2/3 5/17 1/2 2/3 3/10 1/2 1/1 1/3 2/3 1/1 4/11 1/1 7/19 1/1 10/9 3/8 1/1 4/3 2/5 2/1 1/0 5/12 1/1 1/0 13/31 0/1 1/0 8/19 0/1 2/1 3/7 1/0 7/16 -2/1 1/0 4/9 -1/1 -1/2 1/2 0/1 6/11 1/3 1/2 5/9 0/1 1/4 9/16 1/4 4/7 2/7 1/3 3/5 2/5 1/2 14/23 1/2 2/3 11/18 0/1 1/2 8/13 0/1 2/5 13/21 1/3 2/5 5/8 1/3 1/2 2/3 1/2 7/10 1/2 3/5 12/17 1/2 4/7 17/24 4/7 3/5 5/7 4/7 2/3 3/4 1/2 2/3 7/9 2/3 3/4 18/23 5/7 8/11 11/14 3/4 4/5 2/3 4/5 13/16 3/4 4/5 9/11 3/4 5/6 5/6 4/5 1/1 6/7 9/10 1/1 7/8 1/1 1/1 1/1 1/0 8/7 1/0 15/13 -3/1 1/0 7/6 -1/1 1/0 6/5 -1/1 0/1 11/9 -1/2 1/0 5/4 0/1 14/11 1/2 1/0 23/18 0/1 1/1 9/7 0/1 1/0 13/10 -1/2 0/1 17/13 0/1 4/3 0/1 1/2 11/8 1/1 2/1 29/21 3/2 1/0 18/13 2/1 1/0 43/31 0/1 1/0 25/18 1/1 1/0 7/5 0/1 2/1 3/2 1/2 1/0 11/7 0/1 2/1 30/19 0/1 1/1 49/31 1/2 1/0 19/12 0/1 1/0 46/29 0/1 27/17 0/1 1/2 8/5 1/1 1/0 29/18 -1/1 1/0 21/13 -1/1 0/1 13/8 0/1 31/19 1/2 1/0 49/30 0/1 1/1 18/11 0/1 1/0 23/14 -1/2 0/1 28/17 0/1 5/3 0/1 1/2 12/7 4/5 1/1 19/11 1/1 26/15 1/1 8/7 7/4 1/1 2/1 9/5 2/1 1/0 2/1 0/1 2/1 11/5 2/1 1/0 31/14 1/0 51/23 -2/1 1/0 20/9 0/1 1/0 9/4 1/1 1/0 25/11 0/1 2/1 16/7 0/1 1/0 7/3 1/0 19/8 -2/1 31/13 -2/1 1/0 12/5 -3/2 -1/1 29/12 -1/1 -2/3 17/7 -2/3 0/1 5/2 0/1 1/0 13/5 -2/1 0/1 34/13 -1/1 1/0 55/21 -1/2 0/1 76/29 0/1 21/8 0/1 1/0 29/11 -1/2 1/0 8/3 -1/1 0/1 11/4 0/1 14/5 0/1 1/3 3/1 0/1 1/1 13/4 1/1 1/0 23/7 1/0 33/10 -3/1 1/0 10/3 -1/1 1/0 17/5 -1/2 0/1 24/7 0/1 1/0 31/9 -1/2 1/0 7/2 0/1 18/5 0/1 1/0 47/13 0/1 29/8 0/1 1/4 11/3 0/1 1/2 4/1 1/2 1/0 13/3 0/1 1/2 9/2 0/1 1/1 23/5 5/6 1/1 37/8 1/1 14/3 1/1 5/4 19/4 3/2 2/1 5/1 0/1 2/1 16/3 1/1 2/1 11/2 1/1 1/0 6/1 2/1 1/0 13/2 1/0 20/3 -6/1 1/0 7/1 -2/1 1/0 8/1 -2/1 0/1 9/1 -1/2 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(171,80,280,131) (-1/2,-6/13) -> (14/23,11/18) Hyperbolic Matrix(501,230,220,101) (-6/13,-5/11) -> (25/11,16/7) Hyperbolic Matrix(659,298,272,123) (-5/11,-9/20) -> (29/12,17/7) Hyperbolic Matrix(487,218,382,171) (-9/20,-4/9) -> (14/11,23/18) Hyperbolic Matrix(269,118,212,93) (-4/9,-7/16) -> (5/4,14/11) Hyperbolic Matrix(427,186,264,115) (-7/16,-3/7) -> (21/13,13/8) Hyperbolic Matrix(155,66,54,23) (-3/7,-8/19) -> (14/5,3/1) Hyperbolic Matrix(153,64,-514,-215) (-8/19,-5/12) -> (-3/10,-8/27) Hyperbolic Matrix(155,64,356,147) (-5/12,-7/17) -> (3/7,7/16) Hyperbolic Matrix(151,62,358,147) (-7/17,-2/5) -> (8/19,3/7) Hyperbolic Matrix(249,98,592,233) (-2/5,-9/23) -> (13/31,8/19) Hyperbolic Matrix(691,270,540,211) (-9/23,-7/18) -> (23/18,9/7) Hyperbolic Matrix(145,56,-536,-207) (-7/18,-5/13) -> (-3/11,-7/26) Hyperbolic Matrix(487,186,144,55) (-5/13,-8/21) -> (10/3,17/5) Hyperbolic Matrix(143,54,-384,-145) (-8/21,-3/8) -> (-3/8,-10/27) Parabolic Matrix(997,368,382,141) (-10/27,-7/19) -> (13/5,34/13) Hyperbolic Matrix(93,34,-424,-155) (-7/19,-4/11) -> (-2/9,-5/23) Hyperbolic Matrix(329,118,92,33) (-4/11,-5/14) -> (7/2,18/5) Hyperbolic Matrix(651,232,188,67) (-5/14,-11/31) -> (31/9,7/2) Hyperbolic Matrix(1763,624,1116,395) (-11/31,-6/17) -> (30/19,49/31) Hyperbolic Matrix(91,32,418,147) (-6/17,-1/3) -> (5/23,2/9) Hyperbolic Matrix(395,124,86,27) (-1/3,-5/16) -> (9/2,23/5) Hyperbolic Matrix(1917,596,1174,365) (-5/16,-9/29) -> (31/19,49/30) Hyperbolic Matrix(349,108,42,13) (-9/29,-4/13) -> (8/1,9/1) Hyperbolic Matrix(85,26,304,93) (-4/13,-3/10) -> (5/18,2/7) Hyperbolic Matrix(807,238,512,151) (-8/27,-5/17) -> (11/7,30/19) Hyperbolic Matrix(83,24,-294,-85) (-5/17,-2/7) -> (-2/7,-7/25) Parabolic Matrix(751,210,540,151) (-7/25,-5/18) -> (25/18,7/5) Hyperbolic Matrix(123,34,416,115) (-5/18,-3/11) -> (5/17,3/10) Hyperbolic Matrix(781,210,450,121) (-7/26,-4/15) -> (26/15,7/4) Hyperbolic Matrix(447,118,572,151) (-4/15,-5/19) -> (7/9,18/23) Hyperbolic Matrix(485,126,204,53) (-5/19,-1/4) -> (19/8,31/13) Hyperbolic Matrix(275,64,116,27) (-1/4,-3/13) -> (7/3,19/8) Hyperbolic Matrix(271,62,118,27) (-3/13,-2/9) -> (16/7,7/3) Hyperbolic Matrix(461,100,650,141) (-5/23,-3/14) -> (17/24,5/7) Hyperbolic Matrix(113,24,306,65) (-3/14,-1/5) -> (7/19,3/8) Hyperbolic Matrix(183,34,296,55) (-1/5,-2/11) -> (8/13,13/21) Hyperbolic Matrix(145,26,184,33) (-2/11,-1/6) -> (11/14,4/5) Hyperbolic Matrix(251,40,320,51) (-1/6,-2/13) -> (18/23,11/14) Hyperbolic Matrix(177,26,34,5) (-2/13,-1/7) -> (5/1,16/3) Hyperbolic Matrix(173,24,36,5) (-1/7,-1/8) -> (19/4,5/1) Hyperbolic Matrix(167,18,102,11) (-1/8,0/1) -> (18/11,23/14) Hyperbolic Matrix(185,-24,54,-7) (0/1,1/7) -> (17/5,24/7) Hyperbolic Matrix(261,-38,158,-23) (1/7,2/13) -> (28/17,5/3) Hyperbolic Matrix(467,-74,284,-45) (2/13,1/6) -> (23/14,28/17) Hyperbolic Matrix(127,-22,52,-9) (1/6,1/5) -> (17/7,5/2) Hyperbolic Matrix(395,-84,174,-37) (1/5,3/14) -> (9/4,25/11) Hyperbolic Matrix(371,-80,320,-69) (3/14,5/23) -> (15/13,7/6) Hyperbolic Matrix(97,-22,172,-39) (2/9,1/4) -> (9/16,4/7) Hyperbolic Matrix(119,-32,212,-57) (1/4,3/11) -> (5/9,9/16) Hyperbolic Matrix(517,-142,142,-39) (3/11,5/18) -> (29/8,11/3) Hyperbolic Matrix(185,-54,24,-7) (2/7,5/17) -> (7/1,8/1) Hyperbolic Matrix(69,-22,22,-7) (3/10,1/3) -> (3/1,13/4) Hyperbolic Matrix(89,-32,242,-87) (1/3,4/11) -> (4/11,7/19) Parabolic Matrix(87,-34,64,-25) (3/8,2/5) -> (4/3,11/8) Hyperbolic Matrix(127,-52,22,-9) (2/5,5/12) -> (11/2,6/1) Hyperbolic Matrix(1291,-540,930,-389) (5/12,13/31) -> (43/31,25/18) Hyperbolic Matrix(395,-174,84,-37) (7/16,4/9) -> (14/3,19/4) Hyperbolic Matrix(21,-10,40,-19) (4/9,1/2) -> (1/2,6/11) Parabolic Matrix(369,-202,232,-127) (6/11,5/9) -> (27/17,8/5) Hyperbolic Matrix(95,-56,56,-33) (4/7,3/5) -> (5/3,12/7) Hyperbolic Matrix(261,-158,38,-23) (3/5,14/23) -> (20/3,7/1) Hyperbolic Matrix(241,-148,298,-183) (11/18,8/13) -> (4/5,13/16) Hyperbolic Matrix(777,-482,482,-299) (13/21,5/8) -> (29/18,21/13) Hyperbolic Matrix(37,-24,54,-35) (5/8,2/3) -> (2/3,7/10) Parabolic Matrix(353,-248,158,-111) (7/10,12/17) -> (20/9,9/4) Hyperbolic Matrix(1265,-894,774,-547) (12/17,17/24) -> (49/30,18/11) Hyperbolic Matrix(87,-64,34,-25) (5/7,3/4) -> (5/2,13/5) Hyperbolic Matrix(153,-118,118,-91) (3/4,7/9) -> (9/7,13/10) Hyperbolic Matrix(993,-808,628,-511) (13/16,9/11) -> (49/31,19/12) Hyperbolic Matrix(295,-244,214,-177) (9/11,5/6) -> (11/8,29/21) Hyperbolic Matrix(275,-234,114,-97) (5/6,6/7) -> (12/5,29/12) Hyperbolic Matrix(371,-320,80,-69) (6/7,7/8) -> (37/8,14/3) Hyperbolic Matrix(221,-198,48,-43) (7/8,1/1) -> (23/5,37/8) Hyperbolic Matrix(113,-128,98,-111) (1/1,8/7) -> (8/7,15/13) Parabolic Matrix(151,-178,28,-33) (7/6,6/5) -> (16/3,11/2) Hyperbolic Matrix(217,-262,82,-99) (6/5,11/9) -> (29/11,8/3) Hyperbolic Matrix(241,-298,148,-183) (11/9,5/4) -> (13/8,31/19) Hyperbolic Matrix(847,-1104,234,-305) (13/10,17/13) -> (47/13,29/8) Hyperbolic Matrix(375,-494,104,-137) (17/13,4/3) -> (18/5,47/13) Hyperbolic Matrix(907,-1254,264,-365) (29/21,18/13) -> (24/7,31/9) Hyperbolic Matrix(1283,-1778,578,-801) (18/13,43/31) -> (51/23,20/9) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(1521,-2410,580,-919) (19/12,46/29) -> (76/29,21/8) Hyperbolic Matrix(2887,-4582,1102,-1749) (46/29,27/17) -> (55/21,76/29) Hyperbolic Matrix(345,-554,104,-167) (8/5,29/18) -> (33/10,10/3) Hyperbolic Matrix(419,-722,242,-417) (12/7,19/11) -> (19/11,26/15) Parabolic Matrix(97,-172,22,-39) (7/4,9/5) -> (13/3,9/2) Hyperbolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(869,-1922,392,-867) (11/5,31/14) -> (31/14,51/23) Parabolic Matrix(717,-1714,274,-655) (31/13,12/5) -> (34/13,55/21) Hyperbolic Matrix(83,-218,8,-21) (21/8,29/11) -> (9/1,1/0) Hyperbolic Matrix(89,-242,32,-87) (8/3,11/4) -> (11/4,14/5) Parabolic Matrix(323,-1058,98,-321) (13/4,23/7) -> (23/7,33/10) Parabolic Matrix(25,-96,6,-23) (11/3,4/1) -> (4/1,13/3) Parabolic Matrix(53,-338,8,-51) (6/1,13/2) -> (13/2,20/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(171,80,280,131) -> Matrix(1,0,-2,1) Matrix(501,230,220,101) -> Matrix(7,-2,4,-1) Matrix(659,298,272,123) -> Matrix(7,-2,-10,3) Matrix(487,218,382,171) -> Matrix(1,0,-2,1) Matrix(269,118,212,93) -> Matrix(1,0,-2,1) Matrix(427,186,264,115) -> Matrix(1,0,-4,1) Matrix(155,66,54,23) -> Matrix(1,0,-2,1) Matrix(153,64,-514,-215) -> Matrix(7,-2,18,-5) Matrix(155,64,356,147) -> Matrix(9,-2,-4,1) Matrix(151,62,358,147) -> Matrix(7,-2,4,-1) Matrix(249,98,592,233) -> Matrix(7,-2,4,-1) Matrix(691,270,540,211) -> Matrix(7,-2,4,-1) Matrix(145,56,-536,-207) -> Matrix(5,-2,8,-3) Matrix(487,186,144,55) -> Matrix(1,0,-4,1) Matrix(143,54,-384,-145) -> Matrix(1,0,0,1) Matrix(997,368,382,141) -> Matrix(1,0,-4,1) Matrix(93,34,-424,-155) -> Matrix(5,-2,18,-7) Matrix(329,118,92,33) -> Matrix(1,0,-2,1) Matrix(651,232,188,67) -> Matrix(1,0,-6,1) Matrix(1763,624,1116,395) -> Matrix(1,0,-4,1) Matrix(91,32,418,147) -> Matrix(1,0,-2,1) Matrix(395,124,86,27) -> Matrix(11,-4,14,-5) Matrix(1917,596,1174,365) -> Matrix(11,-4,14,-5) Matrix(349,108,42,13) -> Matrix(11,-4,-8,3) Matrix(85,26,304,93) -> Matrix(11,-4,14,-5) Matrix(807,238,512,151) -> Matrix(5,-2,-2,1) Matrix(83,24,-294,-85) -> Matrix(9,-4,16,-7) Matrix(751,210,540,151) -> Matrix(3,-2,2,-1) Matrix(123,34,416,115) -> Matrix(1,0,0,1) Matrix(781,210,450,121) -> Matrix(7,-6,6,-5) Matrix(447,118,572,151) -> Matrix(3,2,4,3) Matrix(485,126,204,53) -> Matrix(1,-2,0,1) Matrix(275,64,116,27) -> Matrix(9,-2,-4,1) Matrix(271,62,118,27) -> Matrix(7,-2,4,-1) Matrix(461,100,650,141) -> Matrix(27,-8,44,-13) Matrix(113,24,306,65) -> Matrix(25,-8,22,-7) Matrix(183,34,296,55) -> Matrix(1,0,0,1) Matrix(145,26,184,33) -> Matrix(11,-4,14,-5) Matrix(251,40,320,51) -> Matrix(13,-8,18,-11) Matrix(177,26,34,5) -> Matrix(3,-2,2,-1) Matrix(173,24,36,5) -> Matrix(3,-2,2,-1) Matrix(167,18,102,11) -> Matrix(1,0,-2,1) Matrix(185,-24,54,-7) -> Matrix(1,0,-2,1) Matrix(261,-38,158,-23) -> Matrix(1,0,2,1) Matrix(467,-74,284,-45) -> Matrix(1,0,-6,1) Matrix(127,-22,52,-9) -> Matrix(1,0,-4,1) Matrix(395,-84,174,-37) -> Matrix(1,0,-2,1) Matrix(371,-80,320,-69) -> Matrix(5,-2,-2,1) Matrix(97,-22,172,-39) -> Matrix(5,-2,18,-7) Matrix(119,-32,212,-57) -> Matrix(3,-2,14,-9) Matrix(517,-142,142,-39) -> Matrix(3,-2,8,-5) Matrix(185,-54,24,-7) -> Matrix(1,0,-2,1) Matrix(69,-22,22,-7) -> Matrix(3,-2,2,-1) Matrix(89,-32,242,-87) -> Matrix(13,-12,12,-11) Matrix(87,-34,64,-25) -> Matrix(1,-2,2,-3) Matrix(127,-52,22,-9) -> Matrix(1,0,0,1) Matrix(1291,-540,930,-389) -> Matrix(1,0,0,1) Matrix(395,-174,84,-37) -> Matrix(3,4,2,3) Matrix(21,-10,40,-19) -> Matrix(1,0,4,1) Matrix(369,-202,232,-127) -> Matrix(1,0,-2,1) Matrix(95,-56,56,-33) -> Matrix(5,-2,8,-3) Matrix(261,-158,38,-23) -> Matrix(9,-4,-2,1) Matrix(241,-148,298,-183) -> Matrix(11,-4,14,-5) Matrix(777,-482,482,-299) -> Matrix(5,-2,-2,1) Matrix(37,-24,54,-35) -> Matrix(9,-4,16,-7) Matrix(353,-248,158,-111) -> Matrix(7,-4,2,-1) Matrix(1265,-894,774,-547) -> Matrix(7,-4,2,-1) Matrix(87,-64,34,-25) -> Matrix(3,-2,2,-1) Matrix(153,-118,118,-91) -> Matrix(3,-2,-4,3) Matrix(993,-808,628,-511) -> Matrix(5,-4,4,-3) Matrix(295,-244,214,-177) -> Matrix(7,-6,6,-5) Matrix(275,-234,114,-97) -> Matrix(7,-6,-8,7) Matrix(371,-320,80,-69) -> Matrix(15,-14,14,-13) Matrix(221,-198,48,-43) -> Matrix(5,-6,6,-7) Matrix(113,-128,98,-111) -> Matrix(1,-4,0,1) Matrix(151,-178,28,-33) -> Matrix(1,2,0,1) Matrix(217,-262,82,-99) -> Matrix(1,0,0,1) Matrix(241,-298,148,-183) -> Matrix(1,0,2,1) Matrix(847,-1104,234,-305) -> Matrix(1,0,6,1) Matrix(375,-494,104,-137) -> Matrix(1,0,-2,1) Matrix(907,-1254,264,-365) -> Matrix(1,-2,0,1) Matrix(1283,-1778,578,-801) -> Matrix(1,-2,0,1) Matrix(37,-54,24,-35) -> Matrix(1,0,0,1) Matrix(1521,-2410,580,-919) -> Matrix(1,0,0,1) Matrix(2887,-4582,1102,-1749) -> Matrix(1,0,-4,1) Matrix(345,-554,104,-167) -> Matrix(1,-2,0,1) Matrix(419,-722,242,-417) -> Matrix(13,-12,12,-11) Matrix(97,-172,22,-39) -> Matrix(1,-2,2,-3) Matrix(21,-40,10,-19) -> Matrix(1,0,0,1) Matrix(869,-1922,392,-867) -> Matrix(1,-4,0,1) Matrix(717,-1714,274,-655) -> Matrix(1,2,-2,-3) Matrix(83,-218,8,-21) -> Matrix(1,0,0,1) Matrix(89,-242,32,-87) -> Matrix(1,0,4,1) Matrix(323,-1058,98,-321) -> Matrix(1,-4,0,1) Matrix(25,-96,6,-23) -> Matrix(1,0,0,1) Matrix(53,-338,8,-51) -> Matrix(1,-8,0,1) 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): 22 Degree of the the map X: 22 Degree of the the map Y: 96 ----------------------------------------------------------------------- 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): 288 Minimal number of generators: 49 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: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -4/9 -2/5 -3/8 -1/4 -1/6 0/1 2/13 1/4 4/11 1/2 2/3 7/8 1/1 8/7 5/4 3/2 2/1 31/14 9/4 7/3 5/2 11/4 3/1 7/2 4/1 9/2 5/1 6/1 13/2 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/4 -5/11 0/1 2/7 -9/20 0/1 1/3 -4/9 1/4 1/2 -3/7 0/1 1/3 -2/5 0/1 2/7 -7/18 2/7 1/3 -5/13 0/1 1/2 -3/8 1/4 1/2 -4/11 0/1 1/2 -1/3 1/4 1/3 -2/7 1/2 -3/11 1/2 2/3 -1/4 0/1 -2/9 1/4 2/7 -3/14 4/13 1/3 -1/5 1/3 2/5 -2/11 0/1 2/5 -1/6 1/2 0/1 0/1 1/2 1/7 0/1 1/0 2/13 0/1 1/6 0/1 1/4 1/5 0/1 2/5 1/4 1/2 2/7 0/1 2/3 5/17 1/2 2/3 3/10 1/2 1/1 1/3 2/3 1/1 4/11 1/1 7/19 1/1 10/9 3/8 1/1 4/3 2/5 2/1 1/0 1/2 0/1 3/5 2/5 1/2 8/13 0/1 2/5 13/21 1/3 2/5 5/8 1/3 1/2 2/3 1/2 7/10 1/2 3/5 12/17 1/2 4/7 17/24 4/7 3/5 5/7 4/7 2/3 3/4 1/2 2/3 7/9 2/3 3/4 11/14 3/4 4/5 2/3 4/5 5/6 4/5 1/1 6/7 9/10 1/1 7/8 1/1 1/1 1/1 1/0 8/7 1/0 7/6 -1/1 1/0 6/5 -1/1 0/1 11/9 -1/2 1/0 5/4 0/1 14/11 1/2 1/0 23/18 0/1 1/1 9/7 0/1 1/0 13/10 -1/2 0/1 17/13 0/1 4/3 0/1 1/2 3/2 1/2 1/0 5/3 0/1 1/2 7/4 1/1 2/1 9/5 2/1 1/0 2/1 0/1 2/1 11/5 2/1 1/0 31/14 1/0 20/9 0/1 1/0 9/4 1/1 1/0 16/7 0/1 1/0 7/3 1/0 12/5 -3/2 -1/1 29/12 -1/1 -2/3 17/7 -2/3 0/1 5/2 0/1 1/0 13/5 -2/1 0/1 8/3 -1/1 0/1 11/4 0/1 3/1 0/1 1/1 13/4 1/1 1/0 23/7 1/0 10/3 -1/1 1/0 17/5 -1/2 0/1 7/2 0/1 18/5 0/1 1/0 11/3 0/1 1/2 4/1 1/2 1/0 13/3 0/1 1/2 9/2 0/1 1/1 5/1 0/1 2/1 6/1 2/1 1/0 13/2 1/0 7/1 -2/1 1/0 8/1 -2/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(27,13,2,1) (-1/2,-5/11) -> (8/1,1/0) Hyperbolic Matrix(659,298,272,123) (-5/11,-9/20) -> (29/12,17/7) Hyperbolic Matrix(487,218,382,171) (-9/20,-4/9) -> (14/11,23/18) Hyperbolic Matrix(131,57,108,47) (-4/9,-3/7) -> (6/5,11/9) Hyperbolic Matrix(131,55,50,21) (-3/7,-2/5) -> (13/5,8/3) Hyperbolic Matrix(175,69,246,97) (-2/5,-7/18) -> (17/24,5/7) Hyperbolic Matrix(75,29,-344,-133) (-7/18,-5/13) -> (-2/9,-3/14) Hyperbolic Matrix(71,27,-192,-73) (-5/13,-3/8) -> (-3/8,-4/11) Parabolic Matrix(161,57,48,17) (-4/11,-1/3) -> (10/3,17/5) Hyperbolic Matrix(105,31,44,13) (-1/3,-2/7) -> (7/3,12/5) Hyperbolic Matrix(189,53,82,23) (-2/7,-3/11) -> (16/7,7/3) Hyperbolic Matrix(19,5,-80,-21) (-3/11,-1/4) -> (-1/4,-2/9) Parabolic Matrix(113,24,306,65) (-3/14,-1/5) -> (7/19,3/8) Hyperbolic Matrix(183,34,296,55) (-1/5,-2/11) -> (8/13,13/21) Hyperbolic Matrix(145,26,184,33) (-2/11,-1/6) -> (11/14,4/5) Hyperbolic Matrix(53,7,68,9) (-1/6,0/1) -> (7/9,11/14) Hyperbolic Matrix(95,-11,26,-3) (0/1,1/7) -> (18/5,11/3) Hyperbolic Matrix(171,-25,130,-19) (1/7,2/13) -> (17/13,4/3) Hyperbolic Matrix(271,-43,208,-33) (2/13,1/6) -> (13/10,17/13) Hyperbolic Matrix(127,-22,52,-9) (1/6,1/5) -> (17/7,5/2) Hyperbolic Matrix(13,-3,48,-11) (1/5,1/4) -> (1/4,2/7) Parabolic Matrix(185,-54,24,-7) (2/7,5/17) -> (7/1,8/1) Hyperbolic Matrix(313,-93,138,-41) (5/17,3/10) -> (9/4,16/7) Hyperbolic Matrix(69,-22,22,-7) (3/10,1/3) -> (3/1,13/4) Hyperbolic Matrix(89,-32,242,-87) (1/3,4/11) -> (4/11,7/19) Parabolic Matrix(75,-29,44,-17) (3/8,2/5) -> (5/3,7/4) Hyperbolic Matrix(11,-5,20,-9) (2/5,1/2) -> (1/2,3/5) Parabolic Matrix(103,-63,18,-11) (3/5,8/13) -> (5/1,6/1) Hyperbolic Matrix(195,-121,166,-103) (13/21,5/8) -> (7/6,6/5) Hyperbolic Matrix(37,-24,54,-35) (5/8,2/3) -> (2/3,7/10) Parabolic Matrix(353,-248,158,-111) (7/10,12/17) -> (20/9,9/4) Hyperbolic Matrix(607,-429,474,-335) (12/17,17/24) -> (23/18,9/7) Hyperbolic Matrix(87,-64,34,-25) (5/7,3/4) -> (5/2,13/5) Hyperbolic Matrix(153,-118,118,-91) (3/4,7/9) -> (9/7,13/10) Hyperbolic Matrix(75,-61,16,-13) (4/5,5/6) -> (9/2,5/1) Hyperbolic Matrix(275,-234,114,-97) (5/6,6/7) -> (12/5,29/12) Hyperbolic Matrix(57,-49,64,-55) (6/7,7/8) -> (7/8,1/1) Parabolic Matrix(93,-103,28,-31) (1/1,8/7) -> (23/7,10/3) Hyperbolic Matrix(229,-265,70,-81) (8/7,7/6) -> (13/4,23/7) Hyperbolic Matrix(101,-125,80,-99) (11/9,5/4) -> (5/4,14/11) Parabolic Matrix(19,-27,12,-17) (4/3,3/2) -> (3/2,5/3) Parabolic Matrix(97,-172,22,-39) (7/4,9/5) -> (13/3,9/2) Hyperbolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(435,-961,196,-433) (11/5,31/14) -> (31/14,20/9) Parabolic Matrix(45,-121,16,-43) (8/3,11/4) -> (11/4,3/1) Parabolic Matrix(71,-245,20,-69) (17/5,7/2) -> (7/2,18/5) Parabolic Matrix(25,-96,6,-23) (11/3,4/1) -> (4/1,13/3) Parabolic Matrix(27,-169,4,-25) (6/1,13/2) -> (13/2,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(27,13,2,1) -> Matrix(1,0,-4,1) Matrix(659,298,272,123) -> Matrix(7,-2,-10,3) Matrix(487,218,382,171) -> Matrix(1,0,-2,1) Matrix(131,57,108,47) -> Matrix(1,0,-4,1) Matrix(131,55,50,21) -> Matrix(1,0,-4,1) Matrix(175,69,246,97) -> Matrix(9,-2,14,-3) Matrix(75,29,-344,-133) -> Matrix(5,-2,18,-7) Matrix(71,27,-192,-73) -> Matrix(1,0,0,1) Matrix(161,57,48,17) -> Matrix(1,0,-4,1) Matrix(105,31,44,13) -> Matrix(5,-2,-2,1) Matrix(189,53,82,23) -> Matrix(3,-2,2,-1) Matrix(19,5,-80,-21) -> Matrix(1,0,2,1) Matrix(113,24,306,65) -> Matrix(25,-8,22,-7) Matrix(183,34,296,55) -> Matrix(1,0,0,1) Matrix(145,26,184,33) -> Matrix(11,-4,14,-5) Matrix(53,7,68,9) -> Matrix(1,-2,2,-3) Matrix(95,-11,26,-3) -> Matrix(1,0,0,1) Matrix(171,-25,130,-19) -> Matrix(1,0,2,1) Matrix(271,-43,208,-33) -> Matrix(1,0,-6,1) Matrix(127,-22,52,-9) -> Matrix(1,0,-4,1) Matrix(13,-3,48,-11) -> Matrix(5,-2,8,-3) Matrix(185,-54,24,-7) -> Matrix(1,0,-2,1) Matrix(313,-93,138,-41) -> Matrix(3,-2,2,-1) Matrix(69,-22,22,-7) -> Matrix(3,-2,2,-1) Matrix(89,-32,242,-87) -> Matrix(13,-12,12,-11) Matrix(75,-29,44,-17) -> Matrix(1,-2,2,-3) Matrix(11,-5,20,-9) -> Matrix(1,0,2,1) Matrix(103,-63,18,-11) -> Matrix(1,0,-2,1) Matrix(195,-121,166,-103) -> Matrix(5,-2,-2,1) Matrix(37,-24,54,-35) -> Matrix(9,-4,16,-7) Matrix(353,-248,158,-111) -> Matrix(7,-4,2,-1) Matrix(607,-429,474,-335) -> Matrix(7,-4,2,-1) Matrix(87,-64,34,-25) -> Matrix(3,-2,2,-1) Matrix(153,-118,118,-91) -> Matrix(3,-2,-4,3) Matrix(75,-61,16,-13) -> Matrix(5,-4,4,-3) Matrix(275,-234,114,-97) -> Matrix(7,-6,-8,7) Matrix(57,-49,64,-55) -> Matrix(11,-10,10,-9) Matrix(93,-103,28,-31) -> Matrix(1,-2,0,1) Matrix(229,-265,70,-81) -> Matrix(1,2,0,1) Matrix(101,-125,80,-99) -> Matrix(1,0,2,1) Matrix(19,-27,12,-17) -> Matrix(1,0,0,1) Matrix(97,-172,22,-39) -> Matrix(1,-2,2,-3) Matrix(21,-40,10,-19) -> Matrix(1,0,0,1) Matrix(435,-961,196,-433) -> Matrix(1,-2,0,1) Matrix(45,-121,16,-43) -> Matrix(1,0,2,1) Matrix(71,-245,20,-69) -> Matrix(1,0,2,1) Matrix(25,-96,6,-23) -> Matrix(1,0,0,1) Matrix(27,-169,4,-25) -> Matrix(1,-4,0,1) 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 11 ----------------------------------------------------------------------- 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 -1/1 0/1 2 1 0/1 (0/1,1/2) 0 15 1/5 0 5 1/4 1/2 2 3 2/7 0 5 3/10 (1/2,1/1) 0 15 1/3 (2/3,1/1) 0 15 4/11 1/1 6 1 3/8 (1/1,4/3) 0 15 2/5 (2/1,1/0) 0 15 1/2 0/1 2 5 3/5 (2/5,1/2) 0 15 8/13 0 5 5/8 (1/3,1/2) 0 15 2/3 1/2 2 3 3/4 (1/2,2/3) 0 15 7/9 (2/3,3/4) 0 15 4/5 0 5 5/6 (4/5,1/1) 0 15 7/8 1/1 10 1 1/1 (1/1,1/0) 0 15 5/4 0/1 2 5 9/7 (0/1,1/0) 0 15 13/10 (-1/2,0/1) 0 15 17/13 0/1 4 1 4/3 (0/1,1/2) 0 15 3/2 0 3 5/3 (0/1,1/2) 0 15 7/4 (1/1,2/1) 0 15 9/5 (2/1,1/0) 0 15 2/1 0 5 11/5 (2/1,1/0) 0 15 31/14 1/0 2 1 9/4 (1/1,1/0) 0 15 7/3 1/0 2 3 5/2 (0/1,1/0) 0 15 11/4 0/1 2 1 3/1 (0/1,1/1) 0 15 13/4 (1/1,1/0) 0 15 23/7 1/0 2 1 10/3 (-1/1,1/0) 0 15 7/2 0/1 2 5 11/3 (0/1,1/2) 0 15 4/1 0 3 13/3 (0/1,1/2) 0 15 9/2 (0/1,1/1) 0 15 5/1 0 5 6/1 (2/1,1/0) 0 15 13/2 1/0 4 1 1/0 (0/1,1/0) 0 15 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(39,-7,50,-9) (0/1,1/5) -> (7/9,4/5) Glide Reflection Matrix(13,-3,48,-11) (1/5,1/4) -> (1/4,2/7) Parabolic Matrix(115,-34,186,-55) (2/7,3/10) -> (8/13,5/8) Glide Reflection Matrix(69,-22,22,-7) (3/10,1/3) -> (3/1,13/4) Hyperbolic Matrix(23,-8,66,-23) (1/3,4/11) -> (1/3,4/11) Reflection Matrix(65,-24,176,-65) (4/11,3/8) -> (4/11,3/8) Reflection Matrix(75,-29,44,-17) (3/8,2/5) -> (5/3,7/4) Hyperbolic Matrix(11,-5,20,-9) (2/5,1/2) -> (1/2,3/5) Parabolic Matrix(103,-63,18,-11) (3/5,8/13) -> (5/1,6/1) Hyperbolic Matrix(83,-53,36,-23) (5/8,2/3) -> (9/4,7/3) Glide Reflection Matrix(43,-31,18,-13) (2/3,3/4) -> (7/3,5/2) Glide Reflection Matrix(153,-118,118,-91) (3/4,7/9) -> (9/7,13/10) Hyperbolic Matrix(75,-61,16,-13) (4/5,5/6) -> (9/2,5/1) Hyperbolic Matrix(41,-35,48,-41) (5/6,7/8) -> (5/6,7/8) Reflection Matrix(15,-14,16,-15) (7/8,1/1) -> (7/8,1/1) Reflection Matrix(47,-57,14,-17) (1/1,5/4) -> (10/3,7/2) Glide Reflection Matrix(93,-118,26,-33) (5/4,9/7) -> (7/2,11/3) Glide Reflection Matrix(339,-442,260,-339) (13/10,17/13) -> (13/10,17/13) Reflection Matrix(103,-136,78,-103) (17/13,4/3) -> (17/13,4/3) Reflection Matrix(19,-27,12,-17) (4/3,3/2) -> (3/2,5/3) Parabolic Matrix(97,-172,22,-39) (7/4,9/5) -> (13/3,9/2) Hyperbolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(309,-682,140,-309) (11/5,31/14) -> (11/5,31/14) Reflection Matrix(125,-279,56,-125) (31/14,9/4) -> (31/14,9/4) Reflection Matrix(21,-55,8,-21) (5/2,11/4) -> (5/2,11/4) Reflection Matrix(23,-66,8,-23) (11/4,3/1) -> (11/4,3/1) Reflection Matrix(183,-598,56,-183) (13/4,23/7) -> (13/4,23/7) Reflection Matrix(139,-460,42,-139) (23/7,10/3) -> (23/7,10/3) Reflection Matrix(25,-96,6,-23) (11/3,4/1) -> (4/1,13/3) Parabolic Matrix(25,-156,4,-25) (6/1,13/2) -> (6/1,13/2) Reflection Matrix(-1,13,0,1) (13/2,1/0) -> (13/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(-1,0,2,1) -> Matrix(1,0,4,-1) (-1/1,0/1) -> (0/1,1/2) Matrix(39,-7,50,-9) -> Matrix(7,-2,10,-3) Matrix(13,-3,48,-11) -> Matrix(5,-2,8,-3) 1/2 Matrix(115,-34,186,-55) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(69,-22,22,-7) -> Matrix(3,-2,2,-1) 1/1 Matrix(23,-8,66,-23) -> Matrix(5,-4,6,-5) (1/3,4/11) -> (2/3,1/1) Matrix(65,-24,176,-65) -> Matrix(7,-8,6,-7) (4/11,3/8) -> (1/1,4/3) Matrix(75,-29,44,-17) -> Matrix(1,-2,2,-3) 1/1 Matrix(11,-5,20,-9) -> Matrix(1,0,2,1) 0/1 Matrix(103,-63,18,-11) -> Matrix(1,0,-2,1) 0/1 Matrix(83,-53,36,-23) -> Matrix(5,-2,2,-1) Matrix(43,-31,18,-13) -> Matrix(3,-2,-2,1) Matrix(153,-118,118,-91) -> Matrix(3,-2,-4,3) Matrix(75,-61,16,-13) -> Matrix(5,-4,4,-3) 1/1 Matrix(41,-35,48,-41) -> Matrix(9,-8,10,-9) (5/6,7/8) -> (4/5,1/1) Matrix(15,-14,16,-15) -> Matrix(-1,2,0,1) (7/8,1/1) -> (1/1,1/0) Matrix(47,-57,14,-17) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(93,-118,26,-33) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(339,-442,260,-339) -> Matrix(-1,0,4,1) (13/10,17/13) -> (-1/2,0/1) Matrix(103,-136,78,-103) -> Matrix(1,0,4,-1) (17/13,4/3) -> (0/1,1/2) Matrix(19,-27,12,-17) -> Matrix(1,0,0,1) Matrix(97,-172,22,-39) -> Matrix(1,-2,2,-3) 1/1 Matrix(21,-40,10,-19) -> Matrix(1,0,0,1) Matrix(309,-682,140,-309) -> Matrix(-1,4,0,1) (11/5,31/14) -> (2/1,1/0) Matrix(125,-279,56,-125) -> Matrix(-1,2,0,1) (31/14,9/4) -> (1/1,1/0) Matrix(21,-55,8,-21) -> Matrix(1,0,0,-1) (5/2,11/4) -> (0/1,1/0) Matrix(23,-66,8,-23) -> Matrix(1,0,2,-1) (11/4,3/1) -> (0/1,1/1) Matrix(183,-598,56,-183) -> Matrix(-1,2,0,1) (13/4,23/7) -> (1/1,1/0) Matrix(139,-460,42,-139) -> Matrix(1,2,0,-1) (23/7,10/3) -> (-1/1,1/0) Matrix(25,-96,6,-23) -> Matrix(1,0,0,1) Matrix(25,-156,4,-25) -> Matrix(-1,4,0,1) (6/1,13/2) -> (2/1,1/0) Matrix(-1,13,0,1) -> Matrix(1,0,0,-1) (13/2,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.