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 1/26 -6/13 3/71 -5/11 2/47 -9/20 1/22 -4/9 1/23 -7/16 1/23 -3/7 0/1 -8/19 1/23 -5/12 1/24 -7/17 2/45 -2/5 1/22 -9/23 6/131 -7/18 5/108 -5/13 4/85 -8/21 1/21 -3/8 1/21 -10/27 1/21 -7/19 4/83 -4/11 5/103 -5/14 3/61 -11/31 4/81 -6/17 5/101 -1/3 2/39 -5/16 5/94 -9/29 4/75 -4/13 3/56 -3/10 5/92 -8/27 3/55 -5/17 4/73 -2/7 1/18 -7/25 2/35 -5/18 1/18 -3/11 4/71 -7/26 9/158 -4/15 5/87 -5/19 6/103 -1/4 1/17 -3/13 2/33 -2/9 1/15 -5/23 2/33 -3/14 1/16 -1/5 0/1 -2/11 1/16 -1/6 1/16 -2/13 1/17 -1/7 2/31 -1/8 3/46 0/1 1/13 1/7 4/45 2/13 1/11 1/6 3/32 1/5 2/21 3/14 1/10 5/23 8/79 2/9 1/9 1/4 1/10 3/11 2/17 5/18 1/14 2/7 1/10 5/17 2/19 3/10 1/10 1/3 0/1 4/11 1/10 7/19 2/19 3/8 1/10 2/5 1/11 5/12 1/10 13/31 8/79 8/19 1/10 3/7 2/19 7/16 5/46 4/9 1/9 1/2 1/9 6/11 1/9 5/9 6/53 9/16 3/26 4/7 5/43 3/5 4/33 14/23 3/23 11/18 1/10 8/13 1/8 13/21 2/17 5/8 1/8 2/3 1/8 7/10 1/8 12/17 9/71 17/24 1/8 5/7 4/31 3/4 5/38 7/9 12/89 18/23 11/81 11/14 3/22 4/5 3/22 13/16 17/124 9/11 4/29 5/6 5/36 6/7 1/7 7/8 1/7 1/1 2/13 8/7 1/6 15/13 24/143 7/6 1/6 6/5 5/29 11/9 4/23 5/4 3/17 14/11 3/17 23/18 11/62 9/7 12/67 13/10 17/94 17/13 2/11 4/3 5/27 11/8 3/16 29/21 2/11 18/13 11/59 43/31 22/117 25/18 3/16 7/5 4/21 3/2 1/5 11/7 2/9 30/19 1/5 49/31 0/1 19/12 3/16 46/29 1/5 27/17 8/39 8/5 1/5 29/18 3/14 21/13 2/9 13/8 1/5 31/19 2/9 49/30 1/4 18/11 1/3 23/14 3/16 28/17 1/5 5/3 4/19 12/7 9/41 19/11 2/9 26/15 19/85 7/4 5/22 9/5 6/25 2/1 1/4 11/5 4/17 31/14 1/4 51/23 18/71 20/9 7/27 9/4 1/4 25/11 4/15 16/7 5/19 7/3 2/7 19/8 1/3 31/13 8/25 12/5 1/3 29/12 3/8 17/7 0/1 5/2 1/2 13/5 2/7 34/13 1/3 55/21 8/25 76/29 1/3 21/8 3/8 29/11 0/1 8/3 1/3 11/4 1/3 14/5 1/3 3/1 0/1 13/4 1/4 23/7 2/7 33/10 3/10 10/3 1/3 17/5 2/7 24/7 3/7 31/9 0/1 7/2 1/3 18/5 -1/1 47/13 0/1 29/8 1/10 11/3 2/9 4/1 1/3 13/3 4/9 9/2 1/4 23/5 8/25 37/8 1/3 14/3 1/3 19/4 1/2 5/1 2/5 16/3 5/11 11/2 1/2 6/1 3/7 13/2 1/2 20/3 11/21 7/1 4/7 8/1 1/2 9/1 2/3 1/0 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(1,0,26,1) Matrix(171,80,280,131) -> Matrix(1,0,-16,1) Matrix(501,230,220,101) -> Matrix(49,-2,172,-7) Matrix(659,298,272,123) -> Matrix(47,-2,118,-5) Matrix(487,218,382,171) -> Matrix(187,-8,1052,-45) Matrix(269,118,212,93) -> Matrix(49,-2,270,-11) Matrix(427,186,264,115) -> Matrix(47,-2,212,-9) Matrix(155,66,54,23) -> Matrix(1,0,-20,1) Matrix(153,64,-514,-215) -> Matrix(43,-2,796,-37) Matrix(155,64,356,147) -> Matrix(91,-4,842,-37) Matrix(151,62,358,147) -> Matrix(89,-4,868,-39) Matrix(249,98,592,233) -> Matrix(45,-2,428,-19) Matrix(691,270,540,211) -> Matrix(391,-18,2194,-101) Matrix(145,56,-536,-207) -> Matrix(171,-8,3014,-141) Matrix(487,186,144,55) -> Matrix(43,-2,108,-5) Matrix(143,54,-384,-145) -> Matrix(127,-6,2646,-125) Matrix(997,368,382,141) -> Matrix(125,-6,396,-19) Matrix(93,34,-424,-155) -> Matrix(41,-2,718,-35) Matrix(329,118,92,33) -> Matrix(41,-2,62,-3) Matrix(651,232,188,67) -> Matrix(81,-4,304,-15) Matrix(1763,624,1116,395) -> Matrix(81,-4,304,-15) Matrix(91,32,418,147) -> Matrix(121,-6,1190,-59) Matrix(395,124,86,27) -> Matrix(113,-6,358,-19) Matrix(1917,596,1174,365) -> Matrix(263,-14,1146,-61) Matrix(349,108,42,13) -> Matrix(187,-10,318,-17) Matrix(85,26,304,93) -> Matrix(37,-2,426,-23) Matrix(807,238,512,151) -> Matrix(37,-2,130,-7) Matrix(83,24,-294,-85) -> Matrix(109,-6,1944,-107) Matrix(751,210,540,151) -> Matrix(33,-2,182,-11) Matrix(123,34,416,115) -> Matrix(35,-2,368,-21) Matrix(781,210,450,121) -> Matrix(491,-28,2192,-125) Matrix(447,118,572,151) -> Matrix(311,-18,2298,-133) Matrix(485,126,204,53) -> Matrix(33,-2,116,-7) Matrix(275,64,116,27) -> Matrix(67,-4,218,-13) Matrix(271,62,118,27) -> Matrix(65,-4,244,-15) Matrix(461,100,650,141) -> Matrix(97,-6,760,-47) Matrix(113,24,306,65) -> Matrix(33,-2,314,-19) Matrix(183,34,296,55) -> Matrix(31,-2,264,-17) Matrix(145,26,184,33) -> Matrix(29,-2,218,-15) Matrix(251,40,320,51) -> Matrix(125,-8,922,-59) Matrix(177,26,34,5) -> Matrix(63,-4,142,-9) Matrix(173,24,36,5) -> Matrix(61,-4,168,-11) Matrix(167,18,102,11) -> Matrix(1,0,-10,1) Matrix(185,-24,54,-7) -> Matrix(23,-2,58,-5) Matrix(261,-38,158,-23) -> Matrix(89,-8,434,-39) Matrix(467,-74,284,-45) -> Matrix(65,-6,336,-31) Matrix(127,-22,52,-9) -> Matrix(21,-2,74,-7) Matrix(395,-84,174,-37) -> Matrix(61,-6,234,-23) Matrix(371,-80,320,-69) -> Matrix(161,-16,956,-95) Matrix(97,-22,172,-39) -> Matrix(23,-2,196,-17) Matrix(119,-32,212,-57) -> Matrix(37,-4,324,-35) Matrix(517,-142,142,-39) -> Matrix(1,0,-4,1) Matrix(185,-54,24,-7) -> Matrix(21,-2,32,-3) Matrix(69,-22,22,-7) -> Matrix(1,0,-6,1) Matrix(89,-32,242,-87) -> Matrix(21,-2,200,-19) Matrix(87,-34,64,-25) -> Matrix(17,-2,94,-11) Matrix(127,-52,22,-9) -> Matrix(19,-2,48,-5) Matrix(1291,-540,930,-389) -> Matrix(17,-2,94,-11) Matrix(395,-174,84,-37) -> Matrix(55,-6,156,-17) Matrix(21,-10,40,-19) -> Matrix(19,-2,162,-17) Matrix(369,-202,232,-127) -> Matrix(19,-2,86,-9) Matrix(95,-56,56,-33) -> Matrix(67,-8,310,-37) Matrix(261,-158,38,-23) -> Matrix(65,-8,122,-15) Matrix(241,-148,298,-183) -> Matrix(83,-10,606,-73) Matrix(777,-482,482,-299) -> Matrix(67,-8,310,-37) Matrix(37,-24,54,-35) -> Matrix(49,-6,384,-47) Matrix(353,-248,158,-111) -> Matrix(15,-2,68,-9) Matrix(1265,-894,774,-547) -> Matrix(63,-8,260,-33) Matrix(87,-64,34,-25) -> Matrix(15,-2,68,-9) Matrix(153,-118,118,-91) -> Matrix(179,-24,992,-133) Matrix(993,-808,628,-511) -> Matrix(29,-4,196,-27) Matrix(295,-244,214,-177) -> Matrix(15,-2,68,-9) Matrix(275,-234,114,-97) -> Matrix(57,-8,164,-23) Matrix(371,-320,80,-69) -> Matrix(113,-16,332,-47) Matrix(221,-198,48,-43) -> Matrix(69,-10,214,-31) Matrix(113,-128,98,-111) -> Matrix(157,-26,936,-155) Matrix(151,-178,28,-33) -> Matrix(59,-10,124,-21) Matrix(217,-262,82,-99) -> Matrix(23,-4,98,-17) Matrix(241,-298,148,-183) -> Matrix(57,-10,268,-47) Matrix(847,-1104,234,-305) -> Matrix(11,-2,204,-37) Matrix(375,-494,104,-137) -> Matrix(11,-2,-38,7) Matrix(907,-1254,264,-365) -> Matrix(11,-2,6,-1) Matrix(1283,-1778,578,-801) -> Matrix(171,-32,668,-125) Matrix(37,-54,24,-35) -> Matrix(31,-6,150,-29) Matrix(1521,-2410,580,-919) -> Matrix(31,-6,88,-17) Matrix(2887,-4582,1102,-1749) -> Matrix(79,-16,242,-49) Matrix(345,-554,104,-167) -> Matrix(29,-6,92,-19) Matrix(419,-722,242,-417) -> Matrix(253,-56,1134,-251) Matrix(97,-172,22,-39) -> Matrix(9,-2,14,-3) Matrix(21,-40,10,-19) -> Matrix(9,-2,32,-7) Matrix(869,-1922,392,-867) -> Matrix(89,-22,352,-87) Matrix(717,-1714,274,-655) -> Matrix(1,0,0,1) Matrix(83,-218,8,-21) -> Matrix(5,-2,8,-3) Matrix(89,-242,32,-87) -> Matrix(7,-2,18,-5) Matrix(323,-1058,98,-321) -> Matrix(29,-8,98,-27) Matrix(25,-96,6,-23) -> Matrix(7,-2,18,-5) Matrix(53,-338,8,-51) -> Matrix(29,-14,56,-27) 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): 48 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, lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z. ----------------------------------------------------------------------- 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 1/13 1/7 4/45 2/13 1/11 1/6 3/32 1/5 2/21 3/14 1/10 2/9 1/9 1/4 1/10 3/11 2/17 2/7 1/10 3/10 1/10 1/3 0/1 4/11 1/10 3/8 1/10 2/5 1/11 5/12 1/10 3/7 2/19 4/9 1/9 1/2 1/9 6/11 1/9 5/9 6/53 9/16 3/26 4/7 5/43 3/5 4/33 11/18 1/10 8/13 1/8 5/8 1/8 2/3 1/8 7/10 1/8 12/17 9/71 5/7 4/31 3/4 5/38 7/9 12/89 4/5 3/22 13/16 17/124 9/11 4/29 5/6 5/36 6/7 1/7 7/8 1/7 1/1 2/13 8/7 1/6 7/6 1/6 6/5 5/29 11/9 4/23 5/4 3/17 9/7 12/67 13/10 17/94 17/13 2/11 4/3 5/27 11/8 3/16 29/21 2/11 18/13 11/59 7/5 4/21 3/2 1/5 11/7 2/9 19/12 3/16 46/29 1/5 27/17 8/39 8/5 1/5 13/8 1/5 31/19 2/9 18/11 1/3 5/3 4/19 12/7 9/41 19/11 2/9 7/4 5/22 9/5 6/25 2/1 1/4 11/5 4/17 31/14 1/4 20/9 7/27 9/4 1/4 7/3 2/7 12/5 1/3 17/7 0/1 5/2 1/2 13/5 2/7 21/8 3/8 29/11 0/1 8/3 1/3 11/4 1/3 3/1 0/1 13/4 1/4 23/7 2/7 10/3 1/3 7/2 1/3 11/3 2/9 4/1 1/3 13/3 4/9 9/2 1/4 14/3 1/3 5/1 2/5 11/2 1/2 6/1 3/7 13/2 1/2 7/1 4/7 1/0 1/0 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(1,0,13,1) Matrix(80,-11,131,-18) -> Matrix(1,0,-3,1) Matrix(182,-27,27,-4) -> Matrix(89,-8,167,-15) Matrix(156,-25,25,-4) -> Matrix(65,-6,141,-13) Matrix(127,-22,52,-9) -> Matrix(21,-2,74,-7) Matrix(298,-63,123,-26) -> Matrix(21,-2,53,-5) Matrix(124,-27,147,-32) -> Matrix(59,-6,423,-43) Matrix(97,-22,172,-39) -> Matrix(23,-2,196,-17) Matrix(119,-32,212,-57) -> Matrix(37,-4,324,-35) Matrix(118,-33,93,-26) -> Matrix(23,-2,127,-11) Matrix(186,-55,115,-34) -> Matrix(21,-2,95,-9) Matrix(69,-22,22,-7) -> Matrix(1,0,-6,1) Matrix(66,-23,23,-8) -> Matrix(1,0,-7,1) Matrix(176,-65,65,-24) -> Matrix(19,-2,67,-7) Matrix(87,-34,64,-25) -> Matrix(17,-2,94,-11) Matrix(127,-52,22,-9) -> Matrix(19,-2,48,-5) Matrix(64,-27,147,-62) -> Matrix(39,-4,361,-37) Matrix(21,-10,40,-19) -> Matrix(19,-2,162,-17) Matrix(369,-202,232,-127) -> Matrix(19,-2,86,-9) Matrix(95,-56,56,-33) -> Matrix(67,-8,310,-37) Matrix(241,-148,298,-183) -> Matrix(83,-10,606,-73) Matrix(186,-115,55,-34) -> Matrix(17,-2,43,-5) Matrix(37,-24,54,-35) -> Matrix(49,-6,384,-47) Matrix(353,-248,158,-111) -> Matrix(15,-2,68,-9) Matrix(368,-261,141,-100) -> Matrix(47,-6,149,-19) Matrix(87,-64,34,-25) -> Matrix(15,-2,68,-9) Matrix(153,-118,118,-91) -> Matrix(179,-24,992,-133) Matrix(118,-93,33,-26) -> Matrix(15,-2,23,-3) Matrix(662,-539,479,-390) -> Matrix(189,-26,1025,-141) Matrix(295,-244,214,-177) -> Matrix(15,-2,68,-9) Matrix(112,-97,97,-84) -> Matrix(155,-22,923,-131) Matrix(16,-15,15,-14) -> Matrix(27,-4,169,-25) Matrix(124,-147,27,-32) -> Matrix(35,-6,111,-19) Matrix(217,-262,82,-99) -> Matrix(23,-4,98,-17) Matrix(241,-298,148,-183) -> Matrix(57,-10,268,-47) Matrix(222,-289,169,-220) -> Matrix(243,-44,1331,-241) Matrix(238,-331,151,-210) -> Matrix(11,-2,39,-7) Matrix(37,-54,24,-35) -> Matrix(31,-6,150,-29) Matrix(952,-1509,429,-680) -> Matrix(51,-10,199,-39) Matrix(846,-1343,383,-608) -> Matrix(59,-12,241,-49) Matrix(718,-1173,273,-446) -> Matrix(9,-2,23,-5) Matrix(80,-131,11,-18) -> Matrix(1,0,-3,1) Matrix(210,-361,121,-208) -> Matrix(127,-28,567,-125) Matrix(97,-172,22,-39) -> Matrix(9,-2,14,-3) Matrix(21,-40,10,-19) -> Matrix(9,-2,32,-7) Matrix(64,-147,27,-62) -> Matrix(15,-4,49,-13) Matrix(162,-529,49,-160) -> Matrix(15,-4,49,-13) Matrix(25,-96,6,-23) -> Matrix(7,-2,18,-5) Matrix(26,-125,5,-24) -> Matrix(11,-4,25,-9) 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): 48 ----------------------------------------------------------------------- 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 13 1 1/1 2/13 1 15 8/7 1/6 13 1 7/6 1/6 1 15 6/5 5/29 1 15 11/9 4/23 3 3 5/4 3/17 1 5 9/7 12/67 1 15 17/13 2/11 11 1 4/3 5/27 1 15 7/5 4/21 1 5 3/2 1/5 3 3 11/7 2/9 1 5 19/12 3/16 1 15 8/5 1/5 1 15 13/8 1/5 1 5 31/19 2/9 3 3 18/11 1/3 1 15 5/3 4/19 1 15 19/11 2/9 7 1 7/4 5/22 1 15 9/5 6/25 1 15 2/1 1/4 1 5 11/5 4/17 1 15 31/14 1/4 11 1 20/9 7/27 1 15 9/4 1/4 1 15 7/3 2/7 1 3 12/5 1/3 1 15 5/2 1/2 1 15 13/5 2/7 1 5 21/8 3/8 1 15 29/11 0/1 3 3 8/3 1/3 1 15 11/4 1/3 1 1 3/1 0/1 1 15 23/7 2/7 1 1 10/3 1/3 1 15 7/2 1/3 1 5 11/3 2/9 1 15 4/1 1/3 1 3 13/3 4/9 1 15 9/2 1/4 1 15 14/3 1/3 1 15 5/1 2/5 1 5 11/2 1/2 1 15 6/1 3/7 1 15 13/2 1/2 7 1 7/1 4/7 1 15 1/0 1/0 1 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(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(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(0,1,1,0) -> Matrix(1,0,13,-1) (-1/1,1/1) -> (0/1,2/13) Matrix(15,-16,14,-15) -> Matrix(25,-4,156,-25) (1/1,8/7) -> (2/13,1/6) Matrix(97,-112,84,-97) -> Matrix(131,-22,780,-131) (8/7,7/6) -> (1/6,11/65) Matrix(124,-147,27,-32) -> Matrix(35,-6,111,-19) Matrix(217,-262,82,-99) -> Matrix(23,-4,98,-17) Matrix(241,-298,148,-183) -> Matrix(57,-10,268,-47) Matrix(93,-118,26,-33) -> Matrix(11,-2,16,-3) Matrix(118,-153,91,-118) -> Matrix(133,-24,737,-133) (9/7,17/13) -> (12/67,2/11) Matrix(103,-136,78,-103) -> Matrix(109,-20,594,-109) (17/13,4/3) -> (2/11,5/27) Matrix(64,-87,25,-34) -> Matrix(11,-2,49,-9) Matrix(37,-54,24,-35) -> Matrix(31,-6,150,-29) 1/5 Matrix(303,-478,116,-183) -> Matrix(1,0,8,-1) *** -> (0/1,1/4) Matrix(208,-331,93,-148) -> Matrix(19,-4,71,-15) Matrix(115,-186,34,-55) -> Matrix(9,-2,22,-5) Matrix(718,-1173,273,-446) -> Matrix(9,-2,23,-5) Matrix(80,-131,11,-18) -> Matrix(1,0,-3,1) 0/1 Matrix(56,-95,33,-56) -> Matrix(37,-8,171,-37) (5/3,19/11) -> (4/19,2/9) Matrix(153,-266,88,-153) -> Matrix(89,-20,396,-89) (19/11,7/4) -> (2/9,5/22) Matrix(97,-172,22,-39) -> Matrix(9,-2,14,-3) Matrix(21,-40,10,-19) -> Matrix(9,-2,32,-7) 1/4 Matrix(309,-682,140,-309) -> Matrix(33,-8,136,-33) (11/5,31/14) -> (4/17,1/4) Matrix(559,-1240,252,-559) -> Matrix(55,-14,216,-55) (31/14,20/9) -> (1/4,7/27) Matrix(64,-147,27,-62) -> Matrix(15,-4,49,-13) 2/7 Matrix(52,-127,9,-22) -> Matrix(7,-2,17,-5) Matrix(65,-176,24,-65) -> Matrix(7,-2,24,-7) (8/3,11/4) -> (1/4,1/3) Matrix(23,-66,8,-23) -> Matrix(1,0,6,-1) (11/4,3/1) -> (0/1,1/3) Matrix(22,-69,7,-22) -> Matrix(1,0,7,-1) (3/1,23/7) -> (0/1,2/7) Matrix(139,-460,42,-139) -> Matrix(13,-4,42,-13) (23/7,10/3) -> (2/7,1/3) Matrix(25,-96,6,-23) -> Matrix(7,-2,18,-5) 1/3 Matrix(26,-125,5,-24) -> Matrix(11,-4,25,-9) 2/5 Matrix(25,-156,4,-25) -> Matrix(13,-6,28,-13) (6/1,13/2) -> (3/7,1/2) Matrix(27,-182,4,-27) -> Matrix(15,-8,28,-15) (13/2,7/1) -> (1/2,4/7) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.