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 1/2 -6/13 1/2 1/1 -5/11 0/1 -9/20 1/4 1/3 -4/9 1/2 -7/16 1/0 -3/7 1/3 -8/19 1/2 3/5 -5/12 1/1 1/0 -7/17 0/1 -2/5 1/2 -9/23 1/1 -7/18 3/4 1/1 -5/13 1/1 -8/21 0/1 1/4 -3/8 1/2 -10/27 2/3 3/4 -7/19 2/3 -4/11 5/6 1/1 -5/14 1/0 -11/31 0/1 -6/17 1/2 1/1 -1/3 1/1 -5/16 1/1 1/0 -9/29 0/1 -4/13 1/2 -3/10 1/1 5/4 -8/27 3/2 5/3 -5/17 2/1 -2/7 1/0 -7/25 0/1 -5/18 -1/2 0/1 -3/11 1/1 -7/26 3/4 1/1 -4/15 1/1 3/2 -5/19 1/1 -1/4 1/0 -3/13 0/1 -2/9 1/2 1/1 -5/23 2/1 -3/14 3/1 1/0 -1/5 -1/1 -2/11 1/2 -1/6 1/0 -2/13 -1/1 -1/2 -1/7 0/1 -1/8 1/1 1/0 0/1 0/1 1/0 1/7 1/1 2/13 1/0 1/6 -1/1 1/0 1/5 0/1 3/14 0/1 1/2 5/23 1/1 2/9 1/2 1/1 1/4 1/0 3/11 -1/1 5/18 -1/1 -7/8 2/7 -1/2 5/17 -1/3 3/10 -1/6 0/1 1/3 1/1 4/11 1/0 7/19 -7/1 3/8 -3/1 1/0 2/5 -1/1 -1/2 5/12 -1/2 0/1 13/31 -1/1 8/19 -1/2 3/7 0/1 7/16 1/1 1/0 4/9 0/1 1/0 1/2 1/0 6/11 -2/1 1/0 5/9 -1/1 9/16 1/0 4/7 -3/2 -1/1 3/5 -1/1 14/23 -1/2 -1/3 11/18 -1/2 0/1 8/13 -1/2 13/21 -1/5 5/8 0/1 1/2 2/3 1/0 7/10 -2/1 -3/2 12/17 -2/1 1/0 17/24 -1/1 1/0 5/7 -2/1 3/4 -5/4 -1/1 7/9 -1/1 18/23 -1/1 -5/6 11/14 -3/4 4/5 -1/2 13/16 -1/2 0/1 9/11 0/1 5/6 -1/1 1/0 6/7 0/1 1/0 7/8 1/0 1/1 -1/1 8/7 -1/2 15/13 -1/3 7/6 -1/2 0/1 6/5 -1/1 -1/2 11/9 0/1 5/4 1/0 14/11 -3/2 23/18 -5/4 -1/1 9/7 -1/1 13/10 -13/12 -1/1 17/13 -1/1 4/3 -1/1 -5/6 11/8 -3/4 -5/7 29/21 -2/3 18/13 -3/4 -2/3 43/31 -5/7 25/18 -7/10 -2/3 7/5 -2/3 3/2 -1/2 11/7 0/1 30/19 -1/2 -1/3 49/31 0/1 19/12 -1/2 0/1 46/29 -1/2 27/17 -1/3 8/5 -1/4 0/1 29/18 0/1 1/10 21/13 1/3 13/8 1/0 31/19 0/1 49/30 1/1 1/0 18/11 0/1 1/0 23/14 1/1 1/0 28/17 1/0 5/3 -1/1 12/7 -3/2 -1/1 19/11 -1/1 26/15 -1/1 -9/10 7/4 -1/1 -3/4 9/5 -1/1 2/1 -1/2 11/5 -1/3 31/14 -1/4 51/23 -1/5 20/9 -1/4 0/1 9/4 -1/2 0/1 25/11 0/1 16/7 -1/2 -1/3 7/3 0/1 19/8 1/0 31/13 -1/1 12/5 0/1 1/0 29/12 -1/1 1/0 17/7 0/1 5/2 -1/1 1/0 13/5 -2/3 34/13 -2/3 -5/8 55/21 -3/5 76/29 -1/2 21/8 -2/3 -1/2 29/11 -2/3 8/3 -3/5 -1/2 11/4 -1/2 14/5 -1/2 -5/11 3/1 -1/3 13/4 -1/10 0/1 23/7 0/1 33/10 0/1 1/18 10/3 0/1 1/4 17/5 1/1 24/7 0/1 1/0 31/9 0/1 7/2 1/0 18/5 -7/6 -1/1 47/13 -1/1 29/8 -1/1 -11/12 11/3 -1/1 4/1 -1/2 13/3 -1/3 9/2 -1/3 -1/4 23/5 -1/3 37/8 -1/4 14/3 -1/4 0/1 19/4 -1/3 -1/4 5/1 0/1 16/3 -1/1 -1/2 11/2 -1/2 0/1 6/1 -1/1 -1/2 13/2 -1/2 20/3 -1/2 -3/7 7/1 -1/3 8/1 -1/2 9/1 0/1 1/0 -1/2 0/1 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,-4,1) Matrix(501,230,220,101) -> Matrix(1,0,-4,1) Matrix(659,298,272,123) -> Matrix(1,0,-4,1) Matrix(487,218,382,171) -> Matrix(11,-4,-8,3) Matrix(269,118,212,93) -> Matrix(1,-2,0,1) Matrix(427,186,264,115) -> Matrix(1,0,0,1) Matrix(155,66,54,23) -> Matrix(5,-2,-12,5) Matrix(153,64,-514,-215) -> Matrix(5,-4,4,-3) Matrix(155,64,356,147) -> Matrix(1,0,0,1) Matrix(151,62,358,147) -> Matrix(1,0,-4,1) Matrix(249,98,592,233) -> Matrix(3,-2,-4,3) Matrix(691,270,540,211) -> Matrix(1,-2,0,1) Matrix(145,56,-536,-207) -> Matrix(1,0,0,1) Matrix(487,186,144,55) -> Matrix(1,0,0,1) Matrix(143,54,-384,-145) -> Matrix(5,-2,8,-3) Matrix(997,368,382,141) -> Matrix(7,-4,-12,7) Matrix(93,34,-424,-155) -> Matrix(5,-4,4,-3) Matrix(329,118,92,33) -> Matrix(1,-2,0,1) Matrix(651,232,188,67) -> Matrix(1,0,0,1) Matrix(1763,624,1116,395) -> Matrix(1,0,-4,1) Matrix(91,32,418,147) -> Matrix(1,0,0,1) Matrix(395,124,86,27) -> Matrix(1,0,-4,1) Matrix(1917,596,1174,365) -> Matrix(1,0,0,1) Matrix(349,108,42,13) -> Matrix(1,0,-4,1) Matrix(85,26,304,93) -> Matrix(3,-2,-4,3) Matrix(807,238,512,151) -> Matrix(1,-2,0,1) Matrix(83,24,-294,-85) -> Matrix(1,-2,0,1) Matrix(751,210,540,151) -> Matrix(3,-2,-4,3) Matrix(123,34,416,115) -> Matrix(1,0,-4,1) Matrix(781,210,450,121) -> Matrix(7,-6,-8,7) 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(1,0,0,1) Matrix(271,62,118,27) -> Matrix(1,0,-4,1) Matrix(461,100,650,141) -> Matrix(1,-4,0,1) Matrix(113,24,306,65) -> Matrix(1,-6,0,1) Matrix(183,34,296,55) -> Matrix(1,0,-4,1) Matrix(145,26,184,33) -> Matrix(3,-2,-4,3) Matrix(251,40,320,51) -> Matrix(3,4,-4,-5) Matrix(177,26,34,5) -> Matrix(1,0,0,1) Matrix(173,24,36,5) -> Matrix(1,0,-4,1) Matrix(167,18,102,11) -> Matrix(1,0,0,1) Matrix(185,-24,54,-7) -> Matrix(1,0,0,1) Matrix(261,-38,158,-23) -> Matrix(1,-2,0,1) Matrix(467,-74,284,-45) -> Matrix(1,2,0,1) Matrix(127,-22,52,-9) -> Matrix(1,0,0,1) Matrix(395,-84,174,-37) -> Matrix(1,0,-4,1) Matrix(371,-80,320,-69) -> Matrix(1,0,-4,1) Matrix(97,-22,172,-39) -> Matrix(1,-2,0,1) Matrix(119,-32,212,-57) -> Matrix(1,0,0,1) Matrix(517,-142,142,-39) -> Matrix(3,4,-4,-5) Matrix(185,-54,24,-7) -> Matrix(1,0,0,1) Matrix(69,-22,22,-7) -> Matrix(1,0,-4,1) Matrix(89,-32,242,-87) -> Matrix(1,-8,0,1) Matrix(87,-34,64,-25) -> Matrix(3,4,-4,-5) Matrix(127,-52,22,-9) -> Matrix(1,0,0,1) Matrix(1291,-540,930,-389) -> Matrix(3,-2,-4,3) Matrix(395,-174,84,-37) -> Matrix(1,0,-4,1) Matrix(21,-10,40,-19) -> Matrix(1,-2,0,1) Matrix(369,-202,232,-127) -> Matrix(1,2,-4,-7) Matrix(95,-56,56,-33) -> Matrix(1,0,0,1) Matrix(261,-158,38,-23) -> Matrix(3,2,-8,-5) Matrix(241,-148,298,-183) -> Matrix(1,0,0,1) Matrix(777,-482,482,-299) -> Matrix(1,0,8,1) Matrix(37,-24,54,-35) -> Matrix(1,-2,0,1) Matrix(353,-248,158,-111) -> Matrix(1,2,-4,-7) Matrix(1265,-894,774,-547) -> Matrix(1,2,0,1) Matrix(87,-64,34,-25) -> Matrix(3,4,-4,-5) Matrix(153,-118,118,-91) -> Matrix(9,8,-8,-7) Matrix(993,-808,628,-511) -> Matrix(1,0,0,1) Matrix(295,-244,214,-177) -> Matrix(3,-2,-4,3) Matrix(275,-234,114,-97) -> Matrix(1,0,0,1) Matrix(371,-320,80,-69) -> Matrix(1,0,-4,1) Matrix(221,-198,48,-43) -> Matrix(1,2,-4,-7) Matrix(113,-128,98,-111) -> Matrix(3,2,-8,-5) Matrix(151,-178,28,-33) -> Matrix(1,0,0,1) Matrix(217,-262,82,-99) -> Matrix(5,2,-8,-3) Matrix(241,-298,148,-183) -> Matrix(1,0,0,1) Matrix(847,-1104,234,-305) -> Matrix(23,24,-24,-25) Matrix(375,-494,104,-137) -> Matrix(13,12,-12,-11) Matrix(907,-1254,264,-365) -> Matrix(3,2,4,3) Matrix(1283,-1778,578,-801) -> Matrix(3,2,-8,-5) Matrix(37,-54,24,-35) -> Matrix(3,2,-8,-5) Matrix(1521,-2410,580,-919) -> Matrix(5,2,-8,-3) Matrix(2887,-4582,1102,-1749) -> Matrix(9,4,-16,-7) Matrix(345,-554,104,-167) -> Matrix(1,0,8,1) Matrix(419,-722,242,-417) -> Matrix(11,12,-12,-13) Matrix(97,-172,22,-39) -> Matrix(3,2,-8,-5) Matrix(21,-40,10,-19) -> Matrix(3,2,-8,-5) Matrix(869,-1922,392,-867) -> Matrix(7,2,-32,-9) Matrix(717,-1714,274,-655) -> Matrix(5,2,-8,-3) Matrix(83,-218,8,-21) -> Matrix(3,2,-8,-5) Matrix(89,-242,32,-87) -> Matrix(15,8,-32,-17) Matrix(323,-1058,98,-321) -> Matrix(1,0,28,1) Matrix(25,-96,6,-23) -> Matrix(3,2,-8,-5) Matrix(53,-338,8,-51) -> Matrix(7,4,-16,-9) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 16 Degree of the the map X: 32 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 1/5 1/4 3/7 1/2 2/3 9/11 1/1 8/7 11/9 5/4 17/13 7/5 3/2 19/11 2/1 31/14 7/3 13/5 11/4 3/1 23/7 7/2 11/3 4/1 9/2 5/1 11/2 6/1 13/2 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 0/1 1/0 1/7 1/1 2/13 1/0 1/6 -1/1 1/0 1/5 0/1 3/14 0/1 1/2 2/9 1/2 1/1 1/4 1/0 3/11 -1/1 2/7 -1/2 3/10 -1/6 0/1 1/3 1/1 4/11 1/0 3/8 -3/1 1/0 2/5 -1/1 -1/2 5/12 -1/2 0/1 3/7 0/1 4/9 0/1 1/0 1/2 1/0 6/11 -2/1 1/0 5/9 -1/1 9/16 1/0 4/7 -3/2 -1/1 3/5 -1/1 11/18 -1/2 0/1 8/13 -1/2 5/8 0/1 1/2 2/3 1/0 7/10 -2/1 -3/2 12/17 -2/1 1/0 5/7 -2/1 3/4 -5/4 -1/1 7/9 -1/1 4/5 -1/2 13/16 -1/2 0/1 9/11 0/1 5/6 -1/1 1/0 6/7 0/1 1/0 7/8 1/0 1/1 -1/1 8/7 -1/2 7/6 -1/2 0/1 6/5 -1/1 -1/2 11/9 0/1 5/4 1/0 9/7 -1/1 13/10 -13/12 -1/1 17/13 -1/1 4/3 -1/1 -5/6 11/8 -3/4 -5/7 29/21 -2/3 18/13 -3/4 -2/3 7/5 -2/3 3/2 -1/2 11/7 0/1 19/12 -1/2 0/1 46/29 -1/2 27/17 -1/3 8/5 -1/4 0/1 13/8 1/0 31/19 0/1 18/11 0/1 1/0 5/3 -1/1 12/7 -3/2 -1/1 19/11 -1/1 7/4 -1/1 -3/4 9/5 -1/1 2/1 -1/2 11/5 -1/3 31/14 -1/4 20/9 -1/4 0/1 9/4 -1/2 0/1 7/3 0/1 12/5 0/1 1/0 17/7 0/1 5/2 -1/1 1/0 13/5 -2/3 21/8 -2/3 -1/2 29/11 -2/3 8/3 -3/5 -1/2 11/4 -1/2 3/1 -1/3 13/4 -1/10 0/1 23/7 0/1 10/3 0/1 1/4 7/2 1/0 11/3 -1/1 4/1 -1/2 13/3 -1/3 9/2 -1/3 -1/4 14/3 -1/4 0/1 5/1 0/1 11/2 -1/2 0/1 6/1 -1/1 -1/2 13/2 -1/2 7/1 -1/3 1/0 -1/2 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(80,-11,131,-18) (0/1,1/7) -> (3/5,11/18) Hyperbolic Matrix(182,-27,27,-4) (1/7,2/13) -> (13/2,7/1) Hyperbolic Matrix(156,-25,25,-4) (2/13,1/6) -> (6/1,13/2) Hyperbolic Matrix(127,-22,52,-9) (1/6,1/5) -> (17/7,5/2) Hyperbolic Matrix(298,-63,123,-26) (1/5,3/14) -> (12/5,17/7) Hyperbolic Matrix(124,-27,147,-32) (3/14,2/9) -> (5/6,6/7) 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(118,-33,93,-26) (3/11,2/7) -> (5/4,9/7) Hyperbolic Matrix(186,-55,115,-34) (2/7,3/10) -> (8/5,13/8) Hyperbolic Matrix(69,-22,22,-7) (3/10,1/3) -> (3/1,13/4) Hyperbolic Matrix(66,-23,23,-8) (1/3,4/11) -> (11/4,3/1) Hyperbolic Matrix(176,-65,65,-24) (4/11,3/8) -> (8/3,11/4) Hyperbolic 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(64,-27,147,-62) (5/12,3/7) -> (3/7,4/9) Parabolic 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(241,-148,298,-183) (11/18,8/13) -> (4/5,13/16) Hyperbolic Matrix(186,-115,55,-34) (8/13,5/8) -> (10/3,7/2) 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(368,-261,141,-100) (12/17,5/7) -> (13/5,21/8) 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(118,-93,33,-26) (7/9,4/5) -> (7/2,11/3) Hyperbolic Matrix(662,-539,479,-390) (13/16,9/11) -> (29/21,18/13) Hyperbolic Matrix(295,-244,214,-177) (9/11,5/6) -> (11/8,29/21) Hyperbolic Matrix(112,-97,97,-84) (6/7,7/8) -> (8/7,7/6) Hyperbolic Matrix(16,-15,15,-14) (7/8,1/1) -> (1/1,8/7) Parabolic Matrix(124,-147,27,-32) (7/6,6/5) -> (9/2,14/3) 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(222,-289,169,-220) (13/10,17/13) -> (17/13,4/3) Parabolic Matrix(238,-331,151,-210) (18/13,7/5) -> (11/7,19/12) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(952,-1509,429,-680) (19/12,46/29) -> (31/14,20/9) Hyperbolic Matrix(846,-1343,383,-608) (46/29,27/17) -> (11/5,31/14) Hyperbolic Matrix(718,-1173,273,-446) (31/19,18/11) -> (21/8,29/11) Hyperbolic Matrix(80,-131,11,-18) (18/11,5/3) -> (7/1,1/0) Hyperbolic Matrix(210,-361,121,-208) (12/7,19/11) -> (19/11,7/4) 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(64,-147,27,-62) (9/4,7/3) -> (7/3,12/5) Parabolic Matrix(162,-529,49,-160) (13/4,23/7) -> (23/7,10/3) Parabolic Matrix(25,-96,6,-23) (11/3,4/1) -> (4/1,13/3) Parabolic Matrix(26,-125,5,-24) (14/3,5/1) -> (5/1,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,2,1) Matrix(80,-11,131,-18) -> Matrix(1,0,-2,1) Matrix(182,-27,27,-4) -> Matrix(1,-2,-2,5) Matrix(156,-25,25,-4) -> Matrix(1,2,-2,-3) Matrix(127,-22,52,-9) -> Matrix(1,0,0,1) Matrix(298,-63,123,-26) -> Matrix(1,0,-2,1) Matrix(124,-27,147,-32) -> Matrix(1,0,-2,1) Matrix(97,-22,172,-39) -> Matrix(1,-2,0,1) Matrix(119,-32,212,-57) -> Matrix(1,0,0,1) Matrix(118,-33,93,-26) -> Matrix(3,2,-2,-1) Matrix(186,-55,115,-34) -> Matrix(1,0,2,1) Matrix(69,-22,22,-7) -> Matrix(1,0,-4,1) Matrix(66,-23,23,-8) -> Matrix(1,-2,-2,5) Matrix(176,-65,65,-24) -> Matrix(1,6,-2,-11) Matrix(87,-34,64,-25) -> Matrix(3,4,-4,-5) Matrix(127,-52,22,-9) -> Matrix(1,0,0,1) Matrix(64,-27,147,-62) -> Matrix(1,0,2,1) Matrix(21,-10,40,-19) -> Matrix(1,-2,0,1) Matrix(369,-202,232,-127) -> Matrix(1,2,-4,-7) Matrix(95,-56,56,-33) -> Matrix(1,0,0,1) Matrix(241,-148,298,-183) -> Matrix(1,0,0,1) Matrix(186,-115,55,-34) -> Matrix(1,0,2,1) Matrix(37,-24,54,-35) -> Matrix(1,-2,0,1) Matrix(353,-248,158,-111) -> Matrix(1,2,-4,-7) Matrix(368,-261,141,-100) -> Matrix(1,4,-2,-7) Matrix(87,-64,34,-25) -> Matrix(3,4,-4,-5) Matrix(153,-118,118,-91) -> Matrix(9,8,-8,-7) Matrix(118,-93,33,-26) -> Matrix(3,2,-2,-1) Matrix(662,-539,479,-390) -> Matrix(1,2,-2,-3) Matrix(295,-244,214,-177) -> Matrix(3,-2,-4,3) Matrix(112,-97,97,-84) -> Matrix(1,0,-2,1) Matrix(16,-15,15,-14) -> Matrix(1,2,-2,-3) Matrix(124,-147,27,-32) -> Matrix(1,0,-2,1) Matrix(217,-262,82,-99) -> Matrix(5,2,-8,-3) Matrix(241,-298,148,-183) -> Matrix(1,0,0,1) Matrix(222,-289,169,-220) -> Matrix(17,18,-18,-19) Matrix(238,-331,151,-210) -> Matrix(3,2,-2,-1) Matrix(37,-54,24,-35) -> Matrix(3,2,-8,-5) Matrix(952,-1509,429,-680) -> Matrix(1,0,-2,1) Matrix(846,-1343,383,-608) -> Matrix(5,2,-18,-7) Matrix(718,-1173,273,-446) -> Matrix(1,2,-2,-3) Matrix(80,-131,11,-18) -> Matrix(1,0,-2,1) Matrix(210,-361,121,-208) -> Matrix(5,6,-6,-7) Matrix(97,-172,22,-39) -> Matrix(3,2,-8,-5) Matrix(21,-40,10,-19) -> Matrix(3,2,-8,-5) Matrix(64,-147,27,-62) -> Matrix(1,0,2,1) Matrix(162,-529,49,-160) -> Matrix(1,0,14,1) Matrix(25,-96,6,-23) -> Matrix(3,2,-8,-5) Matrix(26,-125,5,-24) -> 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 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): 16 ----------------------------------------------------------------------- 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 1/1 -1/1 2 15 8/7 -1/2 2 2 7/6 (-1/2,0/1) 0 30 6/5 (-1/1,-1/2) 0 30 11/9 0/1 2 3 5/4 1/0 2 10 9/7 -1/1 2 15 17/13 -1/1 18 1 4/3 (-1/1,-5/6) 0 30 7/5 -2/3 2 5 3/2 -1/2 2 6 11/7 0/1 2 5 19/12 (-1/2,0/1) 0 30 8/5 (-1/4,0/1) 0 30 13/8 1/0 2 10 31/19 0/1 2 3 18/11 (0/1,1/0) 0 30 5/3 -1/1 2 15 19/11 -1/1 6 1 7/4 (-1/1,-3/4) 0 30 9/5 -1/1 2 15 2/1 -1/2 2 10 11/5 -1/3 2 15 31/14 -1/4 2 2 20/9 (-1/4,0/1) 0 30 9/4 (-1/2,0/1) 0 30 7/3 0/1 2 3 12/5 (0/1,1/0) 0 30 5/2 (-1/1,1/0) 0 30 13/5 -2/3 2 5 21/8 (-2/3,-1/2) 0 30 29/11 -2/3 2 3 8/3 (-3/5,-1/2) 0 30 11/4 -1/2 8 2 3/1 -1/3 2 15 23/7 0/1 14 1 10/3 (0/1,1/4) 0 30 7/2 1/0 2 10 11/3 -1/1 2 15 4/1 -1/2 2 6 13/3 -1/3 2 15 9/2 (-1/3,-1/4) 0 30 14/3 (-1/4,0/1) 0 30 5/1 0/1 2 5 11/2 (-1/2,0/1) 0 30 6/1 (-1/1,-1/2) 0 30 13/2 -1/2 4 2 7/1 -1/3 2 15 1/0 (-1/2,0/1) 0 30 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(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(15,-16,14,-15) (1/1,8/7) -> (1/1,8/7) Reflection Matrix(97,-112,84,-97) (8/7,7/6) -> (8/7,7/6) Reflection Matrix(124,-147,27,-32) (7/6,6/5) -> (9/2,14/3) 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(93,-118,26,-33) (5/4,9/7) -> (7/2,11/3) Glide Reflection Matrix(118,-153,91,-118) (9/7,17/13) -> (9/7,17/13) Reflection Matrix(103,-136,78,-103) (17/13,4/3) -> (17/13,4/3) Reflection Matrix(64,-87,25,-34) (4/3,7/5) -> (5/2,13/5) Glide Reflection Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(303,-478,116,-183) (11/7,19/12) -> (13/5,21/8) Glide Reflection Matrix(208,-331,93,-148) (19/12,8/5) -> (20/9,9/4) Glide Reflection Matrix(115,-186,34,-55) (8/5,13/8) -> (10/3,7/2) Glide Reflection Matrix(718,-1173,273,-446) (31/19,18/11) -> (21/8,29/11) Hyperbolic Matrix(80,-131,11,-18) (18/11,5/3) -> (7/1,1/0) Hyperbolic Matrix(56,-95,33,-56) (5/3,19/11) -> (5/3,19/11) Reflection Matrix(153,-266,88,-153) (19/11,7/4) -> (19/11,7/4) Reflection 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(559,-1240,252,-559) (31/14,20/9) -> (31/14,20/9) Reflection Matrix(64,-147,27,-62) (9/4,7/3) -> (7/3,12/5) Parabolic Matrix(52,-127,9,-22) (12/5,5/2) -> (11/2,6/1) Glide Reflection Matrix(65,-176,24,-65) (8/3,11/4) -> (8/3,11/4) Reflection Matrix(23,-66,8,-23) (11/4,3/1) -> (11/4,3/1) Reflection Matrix(22,-69,7,-22) (3/1,23/7) -> (3/1,23/7) 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(26,-125,5,-24) (14/3,5/1) -> (5/1,11/2) Parabolic Matrix(25,-156,4,-25) (6/1,13/2) -> (6/1,13/2) Reflection Matrix(27,-182,4,-27) (13/2,7/1) -> (13/2,7/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,4,1) (-1/1,1/0) -> (-1/2,0/1) Matrix(0,1,1,0) -> Matrix(-1,0,2,1) (-1/1,1/1) -> (-1/1,0/1) Matrix(15,-16,14,-15) -> Matrix(3,2,-4,-3) (1/1,8/7) -> (-1/1,-1/2) Matrix(97,-112,84,-97) -> Matrix(-1,0,4,1) (8/7,7/6) -> (-1/2,0/1) Matrix(124,-147,27,-32) -> Matrix(1,0,-2,1) 0/1 Matrix(217,-262,82,-99) -> Matrix(5,2,-8,-3) -1/2 Matrix(241,-298,148,-183) -> Matrix(1,0,0,1) Matrix(93,-118,26,-33) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(118,-153,91,-118) -> Matrix(7,8,-6,-7) (9/7,17/13) -> (-4/3,-1/1) Matrix(103,-136,78,-103) -> Matrix(11,10,-12,-11) (17/13,4/3) -> (-1/1,-5/6) Matrix(64,-87,25,-34) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(37,-54,24,-35) -> Matrix(3,2,-8,-5) -1/2 Matrix(303,-478,116,-183) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(208,-331,93,-148) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(115,-186,34,-55) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(718,-1173,273,-446) -> Matrix(1,2,-2,-3) -1/1 Matrix(80,-131,11,-18) -> Matrix(1,0,-2,1) 0/1 Matrix(56,-95,33,-56) -> Matrix(-1,0,2,1) (5/3,19/11) -> (-1/1,0/1) Matrix(153,-266,88,-153) -> Matrix(7,6,-8,-7) (19/11,7/4) -> (-1/1,-3/4) Matrix(97,-172,22,-39) -> Matrix(3,2,-8,-5) -1/2 Matrix(21,-40,10,-19) -> Matrix(3,2,-8,-5) -1/2 Matrix(309,-682,140,-309) -> Matrix(7,2,-24,-7) (11/5,31/14) -> (-1/3,-1/4) Matrix(559,-1240,252,-559) -> Matrix(-1,0,8,1) (31/14,20/9) -> (-1/4,0/1) Matrix(64,-147,27,-62) -> Matrix(1,0,2,1) 0/1 Matrix(52,-127,9,-22) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(65,-176,24,-65) -> Matrix(11,6,-20,-11) (8/3,11/4) -> (-3/5,-1/2) Matrix(23,-66,8,-23) -> Matrix(5,2,-12,-5) (11/4,3/1) -> (-1/2,-1/3) Matrix(22,-69,7,-22) -> Matrix(-1,0,6,1) (3/1,23/7) -> (-1/3,0/1) Matrix(139,-460,42,-139) -> Matrix(1,0,8,-1) (23/7,10/3) -> (0/1,1/4) Matrix(25,-96,6,-23) -> Matrix(3,2,-8,-5) -1/2 Matrix(26,-125,5,-24) -> Matrix(1,0,2,1) 0/1 Matrix(25,-156,4,-25) -> Matrix(3,2,-4,-3) (6/1,13/2) -> (-1/1,-1/2) Matrix(27,-182,4,-27) -> Matrix(5,2,-12,-5) (13/2,7/1) -> (-1/2,-1/3) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.