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 1/14 -6/13 1/12 -5/11 0/1 -9/20 0/1 1/12 -4/9 1/12 -7/16 1/11 -3/7 1/12 1/11 -8/19 3/32 -5/12 2/21 1/10 -7/17 0/1 -2/5 1/10 -9/23 1/9 3/26 -7/18 0/1 1/8 -5/13 1/11 1/10 -8/21 3/28 -3/8 1/9 -10/27 1/8 -7/19 2/17 -4/11 1/8 -5/14 1/9 -11/31 2/17 -6/17 1/8 -1/3 1/8 1/7 -5/16 4/27 3/20 -9/29 2/13 -4/13 1/6 -3/10 2/13 1/6 -8/27 3/20 -5/17 2/13 -2/7 1/6 -7/25 4/23 -5/18 5/28 2/11 -3/11 1/6 1/5 -7/26 5/28 2/11 -4/15 3/16 -5/19 1/6 1/5 -1/4 1/5 -3/13 2/9 -2/9 1/4 -5/23 4/17 -3/14 8/33 1/4 -1/5 1/4 3/11 -2/11 3/10 -1/6 1/3 -2/13 3/8 -1/7 2/5 -1/8 6/13 1/2 0/1 1/0 1/7 -7/13 -1/2 2/13 -1/2 1/6 -1/2 -6/13 1/5 -2/5 3/14 -8/21 -3/8 5/23 -3/8 -13/35 2/9 -3/8 1/4 -1/3 3/11 -1/3 -3/10 5/18 -4/13 -3/10 2/7 -3/10 5/17 -1/3 -3/10 3/10 -7/24 -2/7 1/3 -3/11 -1/4 4/11 -1/4 7/19 -1/4 -19/77 3/8 -1/4 -8/33 2/5 -1/4 5/12 -1/4 -4/17 13/31 -3/13 -17/74 8/19 -5/22 3/7 -2/9 7/16 -8/37 -3/14 4/9 -1/4 1/2 -1/5 6/11 -3/16 5/9 -1/5 -1/6 9/16 -1/5 4/7 -3/16 3/5 -1/5 -1/6 14/23 -3/16 11/18 -2/11 -1/6 8/13 -1/6 13/21 -3/16 -5/27 5/8 -2/11 -5/28 2/3 -1/6 7/10 -4/25 -7/44 12/17 -17/108 17/24 -18/115 -5/32 5/7 -2/13 3/4 -1/6 -2/13 7/9 -1/6 -1/7 18/23 -3/20 11/14 -1/7 4/5 -1/6 13/16 -1/6 -2/13 9/11 -2/13 5/6 -3/20 -4/27 6/7 -7/48 7/8 -1/7 1/1 -1/7 -1/8 8/7 -1/8 15/13 -1/8 -13/105 7/6 -1/8 -6/49 6/5 -1/8 11/9 -2/17 5/4 -1/9 14/11 -1/8 23/18 -1/8 -2/17 9/7 -1/9 -1/10 13/10 -1/10 0/1 17/13 0/1 4/3 -1/8 11/8 -1/8 -6/49 29/21 -4/33 18/13 -13/108 43/31 -3/25 -29/242 25/18 -11/92 -8/67 7/5 -2/17 3/2 -1/9 11/7 -2/19 30/19 -3/28 49/31 -2/19 19/12 -2/19 -1/10 46/29 -1/10 27/17 -1/9 -1/10 8/5 -3/28 29/18 -2/19 -13/124 21/13 -5/48 -3/29 13/8 -3/29 31/19 -4/39 49/30 -22/215 -9/88 18/11 -11/108 23/14 -12/119 -1/10 28/17 -1/10 5/3 -1/10 -1/11 12/7 -1/12 19/11 0/1 26/15 -1/4 7/4 -1/8 0/1 9/5 -3/26 -1/9 2/1 -1/10 11/5 -5/54 -1/11 31/14 -1/11 51/23 -1/11 -23/254 20/9 -9/100 9/4 -2/23 -1/12 25/11 0/1 16/7 -1/12 7/3 0/1 19/8 -1/9 31/13 -1/9 -1/10 12/5 -1/8 29/12 -3/28 -2/19 17/7 -2/19 5/2 -1/10 -2/21 13/5 -2/21 34/13 -3/32 55/21 -1/10 -1/11 76/29 -1/10 21/8 -1/10 -2/21 29/11 -2/21 8/3 -3/32 11/4 -1/11 14/5 -5/56 3/1 -1/11 -1/12 13/4 -1/16 0/1 23/7 0/1 33/10 0/1 1/0 10/3 -1/8 17/5 -3/29 -1/10 24/7 -7/72 31/9 -2/21 7/2 -1/11 18/5 -1/12 47/13 0/1 29/8 -1/10 0/1 11/3 -1/10 -1/11 4/1 -1/12 13/3 -3/38 -1/13 9/2 -1/12 0/1 23/5 -5/64 -1/13 37/8 -1/13 14/3 -3/40 19/4 -4/55 -1/14 5/1 0/1 16/3 -1/12 11/2 -1/12 -2/25 6/1 -1/12 13/2 -1/13 20/3 -3/40 7/1 -1/13 -1/14 8/1 -1/14 9/1 0/1 1/0 -1/14 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(1,0,28,1) Matrix(171,80,280,131) -> Matrix(27,-2,-148,11) Matrix(501,230,220,101) -> Matrix(1,0,-24,1) Matrix(659,298,272,123) -> Matrix(27,-2,-256,19) Matrix(487,218,382,171) -> Matrix(25,-2,-212,17) Matrix(269,118,212,93) -> Matrix(1,0,-20,1) Matrix(427,186,264,115) -> Matrix(19,-2,-180,19) Matrix(155,66,54,23) -> Matrix(23,-2,-264,23) Matrix(153,64,-514,-215) -> Matrix(1,0,-4,1) Matrix(155,64,356,147) -> Matrix(17,-2,-76,9) Matrix(151,62,358,147) -> Matrix(25,-2,-112,9) Matrix(249,98,592,233) -> Matrix(75,-8,-328,35) Matrix(691,270,540,211) -> Matrix(17,-2,-144,17) Matrix(145,56,-536,-207) -> Matrix(21,-2,116,-11) Matrix(487,186,144,55) -> Matrix(19,-2,-180,19) Matrix(143,54,-384,-145) -> Matrix(37,-4,324,-35) Matrix(997,368,382,141) -> Matrix(35,-4,-376,43) Matrix(93,34,-424,-155) -> Matrix(15,-2,68,-9) Matrix(329,118,92,33) -> Matrix(1,0,-20,1) Matrix(651,232,188,67) -> Matrix(35,-4,-376,43) Matrix(1763,624,1116,395) -> Matrix(67,-8,-628,75) Matrix(91,32,418,147) -> Matrix(83,-10,-224,27) Matrix(395,124,86,27) -> Matrix(27,-4,-344,51) Matrix(1917,596,1174,365) -> Matrix(197,-30,-1924,293) Matrix(349,108,42,13) -> Matrix(13,-2,-188,29) Matrix(85,26,304,93) -> Matrix(15,-2,-52,7) Matrix(807,238,512,151) -> Matrix(1,0,-16,1) Matrix(83,24,-294,-85) -> Matrix(37,-6,216,-35) Matrix(751,210,540,151) -> Matrix(103,-18,-864,151) Matrix(123,34,416,115) -> Matrix(21,-4,-68,13) Matrix(781,210,450,121) -> Matrix(11,-2,-60,11) Matrix(447,118,572,151) -> Matrix(1,0,-12,1) Matrix(485,126,204,53) -> Matrix(11,-2,-104,19) Matrix(275,64,116,27) -> Matrix(9,-2,-76,17) Matrix(271,62,118,27) -> Matrix(9,-2,-112,25) Matrix(461,100,650,141) -> Matrix(93,-22,-596,141) Matrix(113,24,306,65) -> Matrix(65,-16,-264,65) Matrix(183,34,296,55) -> Matrix(13,-4,-68,21) Matrix(145,26,184,33) -> Matrix(7,-2,-52,15) Matrix(251,40,320,51) -> Matrix(17,-6,-116,41) Matrix(177,26,34,5) -> Matrix(5,-2,-52,21) Matrix(173,24,36,5) -> Matrix(5,-2,-72,29) Matrix(167,18,102,11) -> Matrix(11,-6,-108,59) Matrix(185,-24,54,-7) -> Matrix(7,4,-72,-41) Matrix(261,-38,158,-23) -> Matrix(15,8,-152,-81) Matrix(467,-74,284,-45) -> Matrix(37,18,-368,-179) Matrix(127,-22,52,-9) -> Matrix(9,4,-88,-39) Matrix(395,-84,174,-37) -> Matrix(5,2,-68,-27) Matrix(371,-80,320,-69) -> Matrix(69,26,-560,-211) Matrix(97,-22,172,-39) -> Matrix(17,6,-88,-31) Matrix(119,-32,212,-57) -> Matrix(7,2,-32,-9) Matrix(517,-142,142,-39) -> Matrix(13,4,-140,-43) Matrix(185,-54,24,-7) -> Matrix(7,2,-88,-25) Matrix(69,-22,22,-7) -> Matrix(7,2,-88,-25) Matrix(89,-32,242,-87) -> Matrix(87,22,-352,-89) Matrix(87,-34,64,-25) -> Matrix(9,2,-68,-15) Matrix(127,-52,22,-9) -> Matrix(9,2,-104,-23) Matrix(1291,-540,930,-389) -> Matrix(155,36,-1296,-301) Matrix(395,-174,84,-37) -> Matrix(19,4,-252,-53) Matrix(21,-10,40,-19) -> Matrix(19,4,-100,-21) Matrix(369,-202,232,-127) -> Matrix(1,0,-4,1) Matrix(95,-56,56,-33) -> Matrix(11,2,-116,-21) Matrix(261,-158,38,-23) -> Matrix(1,0,-8,1) Matrix(241,-148,298,-183) -> Matrix(23,4,-144,-25) Matrix(777,-482,482,-299) -> Matrix(87,16,-832,-153) Matrix(37,-24,54,-35) -> Matrix(35,6,-216,-37) Matrix(353,-248,158,-111) -> Matrix(63,10,-712,-113) Matrix(1265,-894,774,-547) -> Matrix(331,52,-3240,-509) Matrix(87,-64,34,-25) -> Matrix(1,0,-4,1) Matrix(153,-118,118,-91) -> Matrix(13,2,-124,-19) Matrix(993,-808,628,-511) -> Matrix(25,4,-244,-39) Matrix(295,-244,214,-177) -> Matrix(93,14,-764,-115) Matrix(275,-234,114,-97) -> Matrix(41,6,-376,-55) Matrix(371,-320,80,-69) -> Matrix(69,10,-904,-131) Matrix(221,-198,48,-43) -> Matrix(43,6,-552,-77) Matrix(113,-128,98,-111) -> Matrix(111,14,-896,-113) Matrix(151,-178,28,-33) -> Matrix(33,4,-388,-47) Matrix(217,-262,82,-99) -> Matrix(67,8,-712,-85) Matrix(241,-298,148,-183) -> Matrix(87,10,-844,-97) Matrix(847,-1104,234,-305) -> Matrix(1,0,0,1) Matrix(375,-494,104,-137) -> Matrix(1,0,-4,1) Matrix(907,-1254,264,-365) -> Matrix(149,18,-1548,-187) Matrix(1283,-1778,578,-801) -> Matrix(333,40,-3688,-443) Matrix(37,-54,24,-35) -> Matrix(35,4,-324,-37) Matrix(1521,-2410,580,-919) -> Matrix(39,4,-400,-41) Matrix(2887,-4582,1102,-1749) -> Matrix(19,2,-200,-21) Matrix(345,-554,104,-167) -> Matrix(19,2,-124,-13) Matrix(419,-722,242,-417) -> Matrix(1,0,8,1) Matrix(97,-172,22,-39) -> Matrix(1,0,-4,1) Matrix(21,-40,10,-19) -> Matrix(19,2,-200,-21) Matrix(869,-1922,392,-867) -> Matrix(307,28,-3388,-309) Matrix(717,-1714,274,-655) -> Matrix(19,2,-200,-21) Matrix(83,-218,8,-21) -> Matrix(21,2,-284,-27) Matrix(89,-242,32,-87) -> Matrix(87,8,-968,-89) Matrix(323,-1058,98,-321) -> Matrix(1,0,16,1) Matrix(25,-96,6,-23) -> Matrix(23,2,-288,-25) Matrix(53,-338,8,-51) -> Matrix(51,4,-676,-53) 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): 24 Degree of the the map X: 48 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 -2/7 -1/4 -1/6 0/1 2/13 1/4 2/7 4/11 3/7 1/2 2/3 4/5 1/1 8/7 5/4 3/2 46/29 2/1 8/3 11/4 3/1 23/7 10/3 4/1 5/1 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 -1/2 0/1 1/14 -4/9 1/12 -7/16 1/11 -3/7 1/12 1/11 -8/19 3/32 -5/12 2/21 1/10 -2/5 1/10 -5/13 1/11 1/10 -3/8 1/9 -1/3 1/8 1/7 -3/10 2/13 1/6 -2/7 1/6 -3/11 1/6 1/5 -4/15 3/16 -1/4 1/5 -1/5 1/4 3/11 -1/6 1/3 -2/13 3/8 -1/7 2/5 0/1 1/0 1/7 -7/13 -1/2 2/13 -1/2 1/6 -1/2 -6/13 1/5 -2/5 2/9 -3/8 1/4 -1/3 3/11 -1/3 -3/10 5/18 -4/13 -3/10 2/7 -3/10 5/17 -1/3 -3/10 3/10 -7/24 -2/7 1/3 -3/11 -1/4 4/11 -1/4 3/8 -1/4 -8/33 2/5 -1/4 5/12 -1/4 -4/17 3/7 -2/9 7/16 -8/37 -3/14 4/9 -1/4 1/2 -1/5 6/11 -3/16 5/9 -1/5 -1/6 9/16 -1/5 4/7 -3/16 3/5 -1/5 -1/6 2/3 -1/6 3/4 -1/6 -2/13 7/9 -1/6 -1/7 18/23 -3/20 11/14 -1/7 4/5 -1/6 9/11 -2/13 5/6 -3/20 -4/27 6/7 -7/48 7/8 -1/7 1/1 -1/7 -1/8 8/7 -1/8 7/6 -1/8 -6/49 6/5 -1/8 5/4 -1/9 14/11 -1/8 9/7 -1/9 -1/10 13/10 -1/10 0/1 17/13 0/1 4/3 -1/8 11/8 -1/8 -6/49 7/5 -2/17 3/2 -1/9 11/7 -2/19 30/19 -3/28 19/12 -2/19 -1/10 46/29 -1/10 27/17 -1/9 -1/10 8/5 -3/28 21/13 -5/48 -3/29 13/8 -3/29 5/3 -1/10 -1/11 2/1 -1/10 5/2 -1/10 -2/21 8/3 -3/32 11/4 -1/11 14/5 -5/56 3/1 -1/11 -1/12 13/4 -1/16 0/1 23/7 0/1 10/3 -1/8 17/5 -3/29 -1/10 24/7 -7/72 7/2 -1/11 4/1 -1/12 5/1 0/1 16/3 -1/12 11/2 -1/12 -2/25 6/1 -1/12 13/2 -1/13 7/1 -1/13 -1/14 1/0 -1/14 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(109,50,85,39) (-1/2,-4/9) -> (14/11,9/7) 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(105,44,-389,-163) (-8/19,-5/12) -> (-3/11,-4/15) Hyperbolic Matrix(49,20,-125,-51) (-5/12,-2/5) -> (-2/5,-5/13) Parabolic Matrix(95,36,219,83) (-5/13,-3/8) -> (3/7,7/16) Hyperbolic Matrix(49,18,117,43) (-3/8,-1/3) -> (5/12,3/7) Hyperbolic Matrix(45,14,151,47) (-1/3,-3/10) -> (5/17,3/10) Hyperbolic Matrix(41,12,-147,-43) (-3/10,-2/7) -> (-2/7,-3/11) Parabolic Matrix(285,74,181,47) (-4/15,-1/4) -> (11/7,30/19) Hyperbolic Matrix(79,18,57,13) (-1/4,-1/5) -> (11/8,7/5) Hyperbolic Matrix(75,14,91,17) (-1/5,-1/6) -> (9/11,5/6) 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(175,24,51,7) (-1/7,0/1) -> (24/7,7/2) Hyperbolic Matrix(185,-24,54,-7) (0/1,1/7) -> (17/5,24/7) Hyperbolic Matrix(27,-4,169,-25) (1/7,2/13) -> (2/13,1/6) Parabolic Matrix(103,-18,63,-11) (1/6,1/5) -> (13/8,5/3) Hyperbolic Matrix(75,-16,61,-13) (1/5,2/9) -> (6/5,5/4) 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(95,-26,11,-3) (3/11,5/18) -> (7/1,1/0) Hyperbolic Matrix(71,-20,245,-69) (5/18,2/7) -> (2/7,5/17) Parabolic Matrix(69,-22,22,-7) (3/10,1/3) -> (3/1,13/4) Hyperbolic Matrix(45,-16,121,-43) (1/3,4/11) -> (4/11,3/8) 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(313,-138,93,-41) (7/16,4/9) -> (10/3,17/5) 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(75,-44,29,-17) (4/7,3/5) -> (5/2,8/3) Hyperbolic Matrix(19,-12,27,-17) (3/5,2/3) -> (2/3,3/4) Parabolic Matrix(153,-118,118,-91) (3/4,7/9) -> (9/7,13/10) Hyperbolic Matrix(707,-552,447,-349) (7/9,18/23) -> (30/19,19/12) Hyperbolic Matrix(101,-80,125,-99) (11/14,4/5) -> (4/5,9/11) Parabolic Matrix(195,-166,121,-103) (5/6,6/7) -> (8/5,21/13) Hyperbolic Matrix(241,-208,73,-63) (6/7,7/8) -> (23/7,10/3) Hyperbolic Matrix(127,-114,39,-35) (7/8,1/1) -> (13/4,23/7) Hyperbolic Matrix(57,-64,49,-55) (1/1,8/7) -> (8/7,7/6) Parabolic Matrix(151,-178,28,-33) (7/6,6/5) -> (16/3,11/2) Hyperbolic Matrix(221,-288,33,-43) (13/10,17/13) -> (13/2,7/1) Hyperbolic Matrix(117,-154,19,-25) (17/13,4/3) -> (6/1,13/2) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(1335,-2116,841,-1333) (19/12,46/29) -> (46/29,27/17) Parabolic Matrix(11,-20,5,-9) (5/3,2/1) -> (2/1,5/2) Parabolic Matrix(89,-242,32,-87) (8/3,11/4) -> (11/4,14/5) Parabolic Matrix(13,-48,3,-11) (7/2,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,28,1) Matrix(109,50,85,39) -> Matrix(13,-1,-116,9) Matrix(269,118,212,93) -> Matrix(1,0,-20,1) Matrix(427,186,264,115) -> Matrix(19,-2,-180,19) Matrix(155,66,54,23) -> Matrix(23,-2,-264,23) Matrix(105,44,-389,-163) -> Matrix(31,-3,176,-17) Matrix(49,20,-125,-51) -> Matrix(11,-1,100,-9) Matrix(95,36,219,83) -> Matrix(47,-5,-216,23) Matrix(49,18,117,43) -> Matrix(25,-3,-108,13) Matrix(45,14,151,47) -> Matrix(33,-5,-112,17) Matrix(41,12,-147,-43) -> Matrix(19,-3,108,-17) Matrix(285,74,181,47) -> Matrix(17,-3,-164,29) Matrix(79,18,57,13) -> Matrix(13,-3,-108,25) Matrix(75,14,91,17) -> Matrix(17,-5,-112,33) Matrix(251,40,320,51) -> Matrix(17,-6,-116,41) Matrix(177,26,34,5) -> Matrix(5,-2,-52,21) Matrix(175,24,51,7) -> Matrix(7,-3,-72,31) Matrix(185,-24,54,-7) -> Matrix(7,4,-72,-41) Matrix(27,-4,169,-25) -> Matrix(25,13,-52,-27) Matrix(103,-18,63,-11) -> Matrix(11,5,-108,-49) Matrix(75,-16,61,-13) -> Matrix(13,5,-112,-43) Matrix(97,-22,172,-39) -> Matrix(17,6,-88,-31) Matrix(119,-32,212,-57) -> Matrix(7,2,-32,-9) Matrix(95,-26,11,-3) -> Matrix(3,1,-52,-17) Matrix(71,-20,245,-69) -> Matrix(29,9,-100,-31) Matrix(69,-22,22,-7) -> Matrix(7,2,-88,-25) Matrix(45,-16,121,-43) -> Matrix(43,11,-176,-45) Matrix(87,-34,64,-25) -> Matrix(9,2,-68,-15) Matrix(127,-52,22,-9) -> Matrix(9,2,-104,-23) Matrix(313,-138,93,-41) -> Matrix(5,1,-36,-7) Matrix(21,-10,40,-19) -> Matrix(19,4,-100,-21) Matrix(369,-202,232,-127) -> Matrix(1,0,-4,1) Matrix(75,-44,29,-17) -> Matrix(17,3,-176,-31) Matrix(19,-12,27,-17) -> Matrix(17,3,-108,-19) Matrix(153,-118,118,-91) -> Matrix(13,2,-124,-19) Matrix(707,-552,447,-349) -> Matrix(19,3,-184,-29) Matrix(101,-80,125,-99) -> Matrix(5,1,-36,-7) Matrix(195,-166,121,-103) -> Matrix(75,11,-716,-105) Matrix(241,-208,73,-63) -> Matrix(7,1,-8,-1) Matrix(127,-114,39,-35) -> Matrix(7,1,-120,-17) Matrix(57,-64,49,-55) -> Matrix(55,7,-448,-57) Matrix(151,-178,28,-33) -> Matrix(33,4,-388,-47) Matrix(221,-288,33,-43) -> Matrix(11,1,-144,-13) Matrix(117,-154,19,-25) -> Matrix(7,1,-92,-13) Matrix(37,-54,24,-35) -> Matrix(35,4,-324,-37) Matrix(1335,-2116,841,-1333) -> Matrix(9,1,-100,-11) Matrix(11,-20,5,-9) -> Matrix(9,1,-100,-11) Matrix(89,-242,32,-87) -> Matrix(87,8,-968,-89) Matrix(13,-48,3,-11) -> Matrix(11,1,-144,-13) 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): 24 ----------------------------------------------------------------------- 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 14 1 0/1 1/0 1 15 2/13 -1/2 13 1 1/6 (-1/2,-6/13) 0 15 1/5 -2/5 2 5 2/9 -3/8 1 15 1/4 -1/3 4 3 3/11 (-1/3,-3/10) 0 15 2/7 -3/10 1 5 1/3 (-3/11,-1/4) 0 15 4/11 -1/4 11 1 2/5 -1/4 1 15 1/2 -1/5 2 5 5/9 (-1/5,-1/6) 0 15 9/16 -1/5 4 3 4/7 -3/16 1 15 3/5 (-1/5,-1/6) 0 15 2/3 -1/6 3 3 3/4 (-1/6,-2/13) 0 15 7/9 (-1/6,-1/7) 0 15 4/5 -1/6 1 5 5/6 (-3/20,-4/27) 0 15 1/1 (-1/7,-1/8) 0 15 8/7 -1/8 7 1 6/5 -1/8 1 15 5/4 -1/9 2 5 4/3 -1/8 1 15 7/5 -2/17 2 5 3/2 -1/9 2 3 11/7 -2/19 2 5 19/12 (-2/19,-1/10) 0 15 46/29 -1/10 1 1 8/5 -3/28 1 15 13/8 -3/29 2 5 5/3 (-1/10,-1/11) 0 15 2/1 -1/10 1 5 5/2 (-1/10,-2/21) 0 15 8/3 -3/32 1 15 11/4 -1/11 4 1 3/1 (-1/11,-1/12) 0 15 13/4 (-1/16,0/1) 0 15 23/7 0/1 8 1 10/3 -1/8 1 15 7/2 -1/11 2 5 4/1 -1/12 1 3 5/1 0/1 2 5 6/1 -1/12 1 15 13/2 -1/13 2 1 7/1 (-1/13,-1/14) 0 15 1/0 (-1/14,0/1) 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(1,0,13,-1) (0/1,2/13) -> (0/1,2/13) Reflection Matrix(25,-4,156,-25) (2/13,1/6) -> (2/13,1/6) Reflection Matrix(103,-18,63,-11) (1/6,1/5) -> (13/8,5/3) Hyperbolic Matrix(75,-16,61,-13) (1/5,2/9) -> (6/5,5/4) 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(93,-26,118,-33) (3/11,2/7) -> (7/9,4/5) Glide Reflection Matrix(47,-14,57,-17) (2/7,1/3) -> (4/5,5/6) Glide Reflection Matrix(23,-8,66,-23) (1/3,4/11) -> (1/3,4/11) Reflection Matrix(21,-8,55,-21) (4/11,2/5) -> (4/11,2/5) Reflection Matrix(43,-18,31,-13) (2/5,1/2) -> (4/3,7/5) Glide Reflection Matrix(137,-74,87,-47) (1/2,5/9) -> (11/7,19/12) Glide Reflection Matrix(75,-44,29,-17) (4/7,3/5) -> (5/2,8/3) Hyperbolic Matrix(19,-12,27,-17) (3/5,2/3) -> (2/3,3/4) Parabolic Matrix(67,-52,9,-7) (3/4,7/9) -> (7/1,1/0) Glide Reflection Matrix(81,-68,25,-21) (5/6,1/1) -> (3/1,13/4) Glide Reflection Matrix(15,-16,14,-15) (1/1,8/7) -> (1/1,8/7) Reflection Matrix(41,-48,35,-41) (8/7,6/5) -> (8/7,6/5) Reflection Matrix(39,-50,7,-9) (5/4,4/3) -> (5/1,6/1) Glide Reflection Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(1103,-1748,696,-1103) (19/12,46/29) -> (19/12,46/29) Reflection Matrix(231,-368,145,-231) (46/29,8/5) -> (46/29,8/5) Reflection Matrix(115,-186,34,-55) (8/5,13/8) -> (10/3,7/2) Glide Reflection Matrix(11,-20,5,-9) (5/3,2/1) -> (2/1,5/2) Parabolic 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(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(13,-48,3,-11) (7/2,4/1) -> (4/1,5/1) 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(-1,0,28,1) (-1/1,1/0) -> (-1/14,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,13,-1) -> Matrix(1,1,0,-1) (0/1,2/13) -> (-1/2,1/0) Matrix(25,-4,156,-25) -> Matrix(25,12,-52,-25) (2/13,1/6) -> (-1/2,-6/13) Matrix(103,-18,63,-11) -> Matrix(11,5,-108,-49) Matrix(75,-16,61,-13) -> Matrix(13,5,-112,-43) Matrix(97,-22,172,-39) -> Matrix(17,6,-88,-31) Matrix(119,-32,212,-57) -> Matrix(7,2,-32,-9) -1/4 Matrix(93,-26,118,-33) -> Matrix(7,2,-52,-15) Matrix(47,-14,57,-17) -> Matrix(17,5,-112,-33) Matrix(23,-8,66,-23) -> Matrix(23,6,-88,-23) (1/3,4/11) -> (-3/11,-1/4) Matrix(21,-8,55,-21) -> Matrix(21,5,-88,-21) (4/11,2/5) -> (-1/4,-5/22) Matrix(43,-18,31,-13) -> Matrix(13,3,-108,-25) Matrix(137,-74,87,-47) -> Matrix(17,3,-164,-29) Matrix(75,-44,29,-17) -> Matrix(17,3,-176,-31) Matrix(19,-12,27,-17) -> Matrix(17,3,-108,-19) -1/6 Matrix(67,-52,9,-7) -> Matrix(7,1,-104,-15) Matrix(81,-68,25,-21) -> Matrix(7,1,-104,-15) Matrix(15,-16,14,-15) -> Matrix(15,2,-112,-15) (1/1,8/7) -> (-1/7,-1/8) Matrix(41,-48,35,-41) -> Matrix(41,5,-336,-41) (8/7,6/5) -> (-1/8,-5/42) Matrix(39,-50,7,-9) -> Matrix(9,1,-116,-13) Matrix(37,-54,24,-35) -> Matrix(35,4,-324,-37) -1/9 Matrix(1103,-1748,696,-1103) -> Matrix(39,4,-380,-39) (19/12,46/29) -> (-2/19,-1/10) Matrix(231,-368,145,-231) -> Matrix(29,3,-280,-29) (46/29,8/5) -> (-3/28,-1/10) Matrix(115,-186,34,-55) -> Matrix(19,2,-180,-19) *** -> (-1/9,-1/10) Matrix(11,-20,5,-9) -> Matrix(9,1,-100,-11) -1/10 Matrix(65,-176,24,-65) -> Matrix(65,6,-704,-65) (8/3,11/4) -> (-3/32,-1/11) Matrix(23,-66,8,-23) -> Matrix(23,2,-264,-23) (11/4,3/1) -> (-1/11,-1/12) Matrix(183,-598,56,-183) -> Matrix(-1,0,32,1) (13/4,23/7) -> (-1/16,0/1) Matrix(139,-460,42,-139) -> Matrix(-1,0,16,1) (23/7,10/3) -> (-1/8,0/1) Matrix(13,-48,3,-11) -> Matrix(11,1,-144,-13) -1/12 Matrix(25,-156,4,-25) -> Matrix(25,2,-312,-25) (6/1,13/2) -> (-1/12,-1/13) Matrix(27,-182,4,-27) -> Matrix(27,2,-364,-27) (13/2,7/1) -> (-1/13,-1/14) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.