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): 864 Minimal number of generators: 145 Number of equivalence classes of cusps: 48 Genus: 49 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 2/15 3/14 4/13 5/12 1/2 6/11 7/10 8/9 1/1 9/8 19/15 7/5 10/7 3/2 52/33 21/13 11/6 2/1 12/5 5/2 8/3 25/9 3/1 13/4 10/3 105/31 7/2 11/3 67/18 27/7 4/1 13/3 9/2 14/3 5/1 57/11 16/3 11/2 29/5 6/1 13/2 7/1 15/2 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 1/2 -6/13 3/5 -17/37 2/3 -11/24 1/2 -5/11 0/1 -9/20 1/2 -4/9 3/5 -19/43 2/3 -15/34 3/4 -26/59 1/1 -11/25 0/1 -7/16 3/4 -17/39 0/1 -27/62 1/2 -10/23 1/1 -3/7 2/3 -8/19 1/1 -5/12 3/4 -7/17 0/1 -9/22 1/2 -2/5 1/1 -9/23 4/3 -7/18 1/0 -5/13 2/1 -8/21 1/3 -11/29 4/7 -3/8 1/2 -7/19 2/1 -25/68 1/2 -18/49 1/1 -11/30 3/2 -4/11 1/1 -5/14 1/4 -6/17 1/1 -1/3 0/1 -5/16 1/2 -4/13 -1/1 -3/10 1/2 -8/27 3/7 -13/44 1/2 -5/17 2/5 -2/7 1/1 -7/25 4/5 -26/93 1/1 -19/68 3/2 -12/43 1/1 -5/18 1/0 -8/29 1/5 -3/11 2/5 -4/15 1/3 -9/34 5/12 -5/19 4/9 -1/4 1/2 -6/25 1/3 -5/21 6/11 -4/17 1/1 -3/13 0/1 -2/9 3/5 -3/14 1/2 -1/5 2/3 -3/16 1/2 -11/59 2/3 -8/43 1/1 -5/27 0/1 -7/38 1/2 -2/11 3/5 -5/28 3/4 -3/17 0/1 -7/40 1/2 -11/63 10/19 -4/23 3/5 -5/29 2/3 -6/35 11/17 -1/6 3/4 -2/13 1/1 -1/7 0/1 -2/15 1/1 -5/38 1/2 -3/23 2/3 -4/31 7/11 -1/8 3/4 0/1 1/1 1/8 3/4 2/15 1/1 1/7 6/5 1/6 3/2 3/17 2/1 2/11 1/1 3/16 1/0 4/21 -1/1 5/26 1/0 1/5 0/1 3/14 1/1 5/23 10/9 2/9 1/1 1/4 3/2 5/19 2/1 9/34 7/2 4/15 3/1 7/26 1/0 3/11 0/1 2/7 3/1 5/17 0/1 8/27 1/1 3/10 1/0 4/13 1/1 5/16 3/2 1/3 2/1 4/11 7/3 11/30 5/2 7/19 4/1 3/8 1/0 5/13 4/3 7/18 9/4 2/5 3/1 5/12 1/0 8/19 -3/1 3/7 0/1 13/30 3/4 10/23 1/1 17/39 2/1 7/16 1/0 11/25 2/1 15/34 1/0 4/9 1/1 1/2 1/0 6/11 1/0 11/20 1/0 5/9 -4/1 4/7 -1/1 11/19 0/1 7/12 1/2 3/5 -2/1 5/8 1/2 7/11 4/3 2/3 1/1 7/10 1/0 12/17 -5/1 5/7 -2/1 18/25 1/1 31/43 0/1 13/18 1/0 8/11 -3/1 3/4 1/0 10/13 -1/1 17/22 -1/2 7/9 -2/5 4/5 1/3 5/6 1/0 21/25 0/1 16/19 1/1 11/13 2/1 6/7 1/1 13/15 2/1 7/8 1/0 8/9 1/0 9/10 1/0 1/1 0/1 9/8 1/1 8/7 1/1 15/13 2/1 7/6 1/0 13/11 2/1 19/16 1/0 25/21 0/1 31/26 -1/2 6/5 1/1 5/4 -1/2 19/15 0/1 14/11 1/9 9/7 2/7 22/17 1/3 13/10 1/2 4/3 1/1 19/14 1/2 34/25 5/11 15/11 4/7 11/8 3/4 18/13 1/1 7/5 2/3 10/7 1/1 13/9 4/3 3/2 1/0 11/7 4/1 52/33 1/0 93/59 -22/1 41/26 1/0 30/19 -3/1 19/12 1/0 8/5 -1/1 21/13 0/1 34/21 1/9 13/8 1/4 5/3 2/3 32/19 1/1 27/16 7/6 49/29 4/3 22/13 1/1 17/10 1/0 12/7 -1/1 31/18 1/6 19/11 0/1 7/4 1/2 16/9 5/7 9/5 4/5 11/6 1/1 13/7 6/5 2/1 1/1 13/6 1/2 11/5 6/5 9/4 1/0 25/11 2/1 16/7 1/1 23/10 1/0 7/3 0/1 12/5 1/1 17/7 6/5 5/2 3/2 18/7 9/5 49/19 2/1 31/12 13/6 13/5 4/1 21/8 1/0 8/3 1/1 19/7 4/3 11/4 7/4 25/9 2/1 39/14 21/10 14/5 7/3 3/1 2/1 13/4 1/0 23/7 0/1 10/3 1/1 37/11 2/1 27/8 1/0 44/13 -1/1 105/31 0/1 166/49 1/5 61/18 1/2 17/5 0/1 24/7 1/1 79/23 2/1 55/16 1/0 31/9 2/1 38/11 1/1 7/2 3/2 18/5 3/1 11/3 0/1 26/7 1/1 67/18 1/1 108/29 1/1 41/11 10/9 15/4 3/2 49/13 4/3 34/9 7/5 53/14 3/2 19/5 2/1 23/6 11/6 27/7 2/1 31/8 17/8 4/1 3/1 13/3 10/3 9/2 1/0 14/3 1/0 19/4 1/0 5/1 0/1 31/6 1/0 57/11 0/1 26/5 1/1 21/4 1/2 37/7 2/1 16/3 1/1 43/8 1/1 27/5 2/1 11/2 1/0 28/5 1/1 17/3 2/1 23/4 7/4 29/5 2/1 35/6 13/6 6/1 3/1 13/2 7/2 7/1 6/1 15/2 1/0 23/3 -12/1 8/1 -3/1 17/2 1/0 9/1 0/1 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(255,118,188,87) (-1/2,-6/13) -> (4/3,19/14) Hyperbolic Matrix(187,86,-1446,-665) (-6/13,-17/37) -> (-3/23,-4/31) Hyperbolic Matrix(375,172,-2008,-921) (-17/37,-11/24) -> (-3/16,-11/59) Hyperbolic Matrix(437,200,378,173) (-11/24,-5/11) -> (15/13,7/6) Hyperbolic Matrix(309,140,64,29) (-5/11,-9/20) -> (19/4,5/1) Hyperbolic Matrix(557,250,430,193) (-9/20,-4/9) -> (22/17,13/10) Hyperbolic Matrix(303,134,-1766,-781) (-4/9,-19/43) -> (-5/29,-6/35) Hyperbolic Matrix(1037,458,120,53) (-19/43,-15/34) -> (17/2,9/1) Hyperbolic Matrix(2005,884,-7174,-3163) (-15/34,-26/59) -> (-26/93,-19/68) Hyperbolic Matrix(849,374,-4556,-2007) (-26/59,-11/25) -> (-11/59,-8/43) Hyperbolic Matrix(305,134,-1154,-507) (-11/25,-7/16) -> (-9/34,-5/19) Hyperbolic Matrix(1087,474,-2956,-1289) (-7/16,-17/39) -> (-7/19,-25/68) Hyperbolic Matrix(2291,998,1926,839) (-17/39,-27/62) -> (19/16,25/21) Hyperbolic Matrix(239,104,-1804,-785) (-27/62,-10/23) -> (-2/15,-5/38) Hyperbolic Matrix(599,260,182,79) (-10/23,-3/7) -> (23/7,10/3) Hyperbolic Matrix(237,100,410,173) (-3/7,-8/19) -> (4/7,11/19) Hyperbolic Matrix(177,74,232,97) (-8/19,-5/12) -> (3/4,10/13) Hyperbolic Matrix(289,120,118,49) (-5/12,-7/17) -> (17/7,5/2) Hyperbolic Matrix(405,166,344,141) (-7/17,-9/22) -> (7/6,13/11) Hyperbolic Matrix(289,118,-1036,-423) (-9/22,-2/5) -> (-12/43,-5/18) Hyperbolic Matrix(111,44,58,23) (-2/5,-9/23) -> (13/7,2/1) Hyperbolic Matrix(169,66,-950,-371) (-9/23,-7/18) -> (-5/28,-3/17) Hyperbolic Matrix(165,64,446,173) (-7/18,-5/13) -> (7/19,3/8) Hyperbolic Matrix(167,64,-608,-233) (-5/13,-8/21) -> (-8/29,-3/11) Hyperbolic Matrix(221,84,-934,-355) (-8/21,-11/29) -> (-5/21,-4/17) Hyperbolic Matrix(549,208,710,269) (-11/29,-3/8) -> (17/22,7/9) Hyperbolic Matrix(161,60,110,41) (-3/8,-7/19) -> (13/9,3/2) Hyperbolic Matrix(2797,1028,536,197) (-25/68,-18/49) -> (26/5,21/4) Hyperbolic Matrix(1291,474,-4622,-1697) (-18/49,-11/30) -> (-19/68,-12/43) Hyperbolic Matrix(323,118,52,19) (-11/30,-4/11) -> (6/1,13/2) Hyperbolic Matrix(105,38,268,97) (-4/11,-5/14) -> (7/18,2/5) Hyperbolic Matrix(209,74,-788,-279) (-5/14,-6/17) -> (-4/15,-9/34) Hyperbolic Matrix(473,166,208,73) (-6/17,-1/3) -> (25/11,16/7) Hyperbolic Matrix(197,62,448,141) (-1/3,-5/16) -> (7/16,11/25) Hyperbolic Matrix(97,30,-540,-167) (-5/16,-4/13) -> (-2/11,-5/28) Hyperbolic Matrix(245,74,96,29) (-4/13,-3/10) -> (5/2,18/7) Hyperbolic Matrix(337,100,920,273) (-3/10,-8/27) -> (4/11,11/30) Hyperbolic Matrix(291,86,-1208,-357) (-8/27,-13/44) -> (-1/4,-6/25) Hyperbolic Matrix(339,100,-1834,-541) (-13/44,-5/17) -> (-5/27,-7/38) Hyperbolic Matrix(239,70,338,99) (-5/17,-2/7) -> (12/17,5/7) Hyperbolic Matrix(613,172,474,133) (-2/7,-7/25) -> (9/7,22/17) Hyperbolic Matrix(6513,1822,1748,489) (-7/25,-26/93) -> (108/29,41/11) Hyperbolic Matrix(989,274,610,169) (-5/18,-8/29) -> (34/21,13/8) Hyperbolic Matrix(373,100,138,37) (-3/11,-4/15) -> (8/3,19/7) Hyperbolic Matrix(229,60,416,109) (-5/19,-1/4) -> (11/20,5/9) Hyperbolic Matrix(359,86,-2058,-493) (-6/25,-5/21) -> (-11/63,-4/23) Hyperbolic Matrix(265,62,312,73) (-4/17,-3/13) -> (11/13,6/7) Hyperbolic Matrix(131,30,310,71) (-3/13,-2/9) -> (8/19,3/7) Hyperbolic Matrix(173,38,132,29) (-2/9,-3/14) -> (13/10,4/3) Hyperbolic Matrix(301,64,174,37) (-3/14,-1/5) -> (19/11,7/4) Hyperbolic Matrix(41,8,128,25) (-1/5,-3/16) -> (5/16,1/3) Hyperbolic Matrix(1335,248,1588,295) (-8/43,-5/27) -> (21/25,16/19) Hyperbolic Matrix(207,38,1084,199) (-7/38,-2/11) -> (4/21,5/26) Hyperbolic Matrix(205,36,-1566,-275) (-3/17,-7/40) -> (-5/38,-3/23) Hyperbolic Matrix(6303,1102,3998,699) (-7/40,-11/63) -> (93/59,41/26) Hyperbolic Matrix(1315,228,248,43) (-4/23,-5/29) -> (37/7,16/3) Hyperbolic Matrix(1109,190,286,49) (-6/35,-1/6) -> (31/8,4/1) Hyperbolic Matrix(281,44,364,57) (-1/6,-2/13) -> (10/13,17/22) Hyperbolic Matrix(79,12,362,55) (-2/13,-1/7) -> (5/23,2/9) Hyperbolic Matrix(237,32,274,37) (-1/7,-2/15) -> (6/7,13/15) Hyperbolic Matrix(1133,146,194,25) (-4/31,-1/8) -> (35/6,6/1) Hyperbolic Matrix(193,22,114,13) (-1/8,0/1) -> (22/13,17/10) Hyperbolic Matrix(93,-10,214,-23) (0/1,1/8) -> (13/30,10/23) Hyperbolic Matrix(661,-86,392,-51) (1/8,2/15) -> (32/19,27/16) Hyperbolic Matrix(299,-42,178,-25) (2/15,1/7) -> (5/3,32/19) Hyperbolic Matrix(119,-18,324,-49) (1/7,1/6) -> (11/30,7/19) Hyperbolic Matrix(493,-86,86,-15) (1/6,3/17) -> (17/3,23/4) Hyperbolic Matrix(577,-104,172,-31) (3/17,2/11) -> (10/3,37/11) Hyperbolic Matrix(229,-42,518,-95) (2/11,3/16) -> (15/34,4/9) Hyperbolic Matrix(485,-92,58,-11) (3/16,4/21) -> (8/1,17/2) Hyperbolic Matrix(285,-56,56,-11) (5/26,1/5) -> (5/1,31/6) Hyperbolic Matrix(85,-18,392,-83) (1/5,3/14) -> (3/14,5/23) Parabolic Matrix(221,-50,84,-19) (2/9,1/4) -> (21/8,8/3) Hyperbolic Matrix(513,-134,134,-35) (1/4,5/19) -> (19/5,23/6) Hyperbolic Matrix(2179,-576,1290,-341) (5/19,9/34) -> (27/16,49/29) Hyperbolic Matrix(1397,-370,404,-107) (9/34,4/15) -> (38/11,7/2) Hyperbolic Matrix(1395,-374,884,-237) (4/15,7/26) -> (41/26,30/19) Hyperbolic Matrix(1259,-340,374,-101) (7/26,3/11) -> (37/11,27/8) Hyperbolic Matrix(239,-66,134,-37) (3/11,2/7) -> (16/9,9/5) Hyperbolic Matrix(341,-100,474,-139) (2/7,5/17) -> (5/7,18/25) Hyperbolic Matrix(629,-186,1444,-427) (5/17,8/27) -> (10/23,17/39) Hyperbolic Matrix(577,-172,104,-31) (8/27,3/10) -> (11/2,28/5) Hyperbolic Matrix(105,-32,338,-103) (3/10,4/13) -> (4/13,5/16) Parabolic Matrix(75,-26,26,-9) (1/3,4/11) -> (14/5,3/1) Hyperbolic Matrix(221,-84,50,-19) (3/8,5/13) -> (13/3,9/2) Hyperbolic Matrix(637,-246,246,-95) (5/13,7/18) -> (31/12,13/5) Hyperbolic Matrix(121,-50,288,-119) (2/5,5/12) -> (5/12,8/19) Parabolic Matrix(1257,-544,238,-103) (3/7,13/30) -> (21/4,37/7) Hyperbolic Matrix(2651,-1156,782,-341) (17/39,7/16) -> (61/18,17/5) Hyperbolic Matrix(259,-114,284,-125) (11/25,15/34) -> (9/10,1/1) Hyperbolic Matrix(187,-84,118,-53) (4/9,1/2) -> (19/12,8/5) Hyperbolic Matrix(133,-72,242,-131) (1/2,6/11) -> (6/11,11/20) Parabolic Matrix(239,-134,66,-37) (5/9,4/7) -> (18/5,11/3) Hyperbolic Matrix(817,-474,474,-275) (11/19,7/12) -> (31/18,19/11) Hyperbolic Matrix(235,-138,172,-101) (7/12,3/5) -> (15/11,11/8) Hyperbolic Matrix(105,-64,64,-39) (3/5,5/8) -> (13/8,5/3) Hyperbolic Matrix(187,-118,84,-53) (5/8,7/11) -> (11/5,9/4) Hyperbolic Matrix(83,-54,20,-13) (7/11,2/3) -> (4/1,13/3) Hyperbolic Matrix(141,-98,200,-139) (2/3,7/10) -> (7/10,12/17) Parabolic Matrix(2687,-1936,712,-513) (18/25,31/43) -> (49/13,34/9) Hyperbolic Matrix(2923,-2108,850,-613) (31/43,13/18) -> (55/16,31/9) Hyperbolic Matrix(749,-542,474,-343) (13/18,8/11) -> (30/19,19/12) Hyperbolic Matrix(235,-172,138,-101) (8/11,3/4) -> (17/10,12/7) Hyperbolic Matrix(171,-134,134,-105) (7/9,4/5) -> (14/11,9/7) Hyperbolic Matrix(131,-108,74,-61) (4/5,5/6) -> (7/4,16/9) Hyperbolic Matrix(925,-776,776,-651) (5/6,21/25) -> (25/21,31/26) Hyperbolic Matrix(1217,-1026,720,-607) (16/19,11/13) -> (49/29,22/13) Hyperbolic Matrix(617,-538,164,-143) (13/15,7/8) -> (15/4,49/13) Hyperbolic Matrix(145,-128,162,-143) (7/8,8/9) -> (8/9,9/10) Parabolic Matrix(259,-286,48,-53) (1/1,9/8) -> (43/8,27/5) Hyperbolic Matrix(429,-488,80,-91) (9/8,8/7) -> (16/3,43/8) Hyperbolic Matrix(711,-818,206,-237) (8/7,15/13) -> (31/9,38/11) Hyperbolic Matrix(887,-1050,234,-277) (13/11,19/16) -> (53/14,19/5) Hyperbolic Matrix(4621,-5512,1364,-1627) (31/26,6/5) -> (166/49,61/18) Hyperbolic Matrix(107,-132,30,-37) (6/5,5/4) -> (7/2,18/5) Hyperbolic Matrix(661,-834,256,-323) (5/4,19/15) -> (49/19,31/12) Hyperbolic Matrix(809,-1028,314,-399) (19/15,14/11) -> (18/7,49/19) Hyperbolic Matrix(1801,-2448,476,-647) (19/14,34/25) -> (34/9,53/14) Hyperbolic Matrix(663,-902,86,-117) (34/25,15/11) -> (23/3,8/1) Hyperbolic Matrix(185,-256,86,-119) (11/8,18/13) -> (2/1,13/6) Hyperbolic Matrix(341,-474,100,-139) (18/13,7/5) -> (17/5,24/7) Hyperbolic Matrix(141,-200,98,-139) (7/5,10/7) -> (10/7,13/9) Parabolic Matrix(173,-270,66,-103) (3/2,11/7) -> (13/5,21/8) Hyperbolic Matrix(3433,-5408,2178,-3431) (11/7,52/33) -> (52/33,93/59) Parabolic Matrix(547,-882,338,-545) (8/5,21/13) -> (21/13,34/21) Parabolic Matrix(525,-902,188,-323) (12/7,31/18) -> (39/14,14/5) Hyperbolic Matrix(133,-242,72,-131) (9/5,11/6) -> (11/6,13/7) Parabolic Matrix(321,-698,86,-187) (13/6,11/5) -> (41/11,15/4) Hyperbolic Matrix(229,-518,42,-95) (9/4,25/11) -> (27/5,11/2) Hyperbolic Matrix(629,-1444,186,-427) (16/7,23/10) -> (27/8,44/13) Hyperbolic Matrix(93,-214,10,-23) (23/10,7/3) -> (9/1,1/0) Hyperbolic Matrix(121,-288,50,-119) (7/3,12/5) -> (12/5,17/7) Parabolic Matrix(119,-324,18,-49) (19/7,11/4) -> (13/2,7/1) Hyperbolic Matrix(451,-1250,162,-449) (11/4,25/9) -> (25/9,39/14) Parabolic Matrix(105,-338,32,-103) (3/1,13/4) -> (13/4,23/7) Parabolic Matrix(6511,-22050,1922,-6509) (44/13,105/31) -> (105/31,166/49) Parabolic Matrix(997,-3422,192,-659) (24/7,79/23) -> (57/11,26/5) Hyperbolic Matrix(1625,-5584,314,-1079) (79/23,55/16) -> (31/6,57/11) Hyperbolic Matrix(203,-750,36,-133) (11/3,26/7) -> (28/5,17/3) Hyperbolic Matrix(2413,-8978,648,-2411) (26/7,67/18) -> (67/18,108/29) Parabolic Matrix(379,-1458,98,-377) (23/6,27/7) -> (27/7,31/8) Parabolic Matrix(85,-392,18,-83) (9/2,14/3) -> (14/3,19/4) Parabolic Matrix(291,-1682,50,-289) (23/4,29/5) -> (29/5,35/6) Parabolic Matrix(61,-450,8,-59) (7/1,15/2) -> (15/2,23/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(255,118,188,87) -> Matrix(3,-2,8,-5) Matrix(187,86,-1446,-665) -> Matrix(11,-8,18,-13) Matrix(375,172,-2008,-921) -> Matrix(1,0,0,1) Matrix(437,200,378,173) -> Matrix(3,-2,2,-1) Matrix(309,140,64,29) -> Matrix(1,0,-2,1) Matrix(557,250,430,193) -> Matrix(7,-4,16,-9) Matrix(303,134,-1766,-781) -> Matrix(23,-16,36,-25) Matrix(1037,458,120,53) -> Matrix(3,-2,-4,3) Matrix(2005,884,-7174,-3163) -> Matrix(7,-6,6,-5) Matrix(849,374,-4556,-2007) -> Matrix(1,-2,2,-3) Matrix(305,134,-1154,-507) -> Matrix(7,-4,16,-9) Matrix(1087,474,-2956,-1289) -> Matrix(3,-2,2,-1) Matrix(2291,998,1926,839) -> Matrix(1,0,-2,1) Matrix(239,104,-1804,-785) -> Matrix(1,0,0,1) Matrix(599,260,182,79) -> Matrix(3,-2,2,-1) Matrix(237,100,410,173) -> Matrix(3,-2,-4,3) Matrix(177,74,232,97) -> Matrix(3,-2,-4,3) Matrix(289,120,118,49) -> Matrix(7,-6,6,-5) Matrix(405,166,344,141) -> Matrix(3,-2,2,-1) Matrix(289,118,-1036,-423) -> Matrix(3,-2,2,-1) Matrix(111,44,58,23) -> Matrix(3,-2,2,-1) Matrix(169,66,-950,-371) -> Matrix(3,-4,4,-5) Matrix(165,64,446,173) -> Matrix(1,2,0,1) Matrix(167,64,-608,-233) -> Matrix(1,0,2,1) Matrix(221,84,-934,-355) -> Matrix(5,-2,8,-3) Matrix(549,208,710,269) -> Matrix(3,-2,-4,3) Matrix(161,60,110,41) -> Matrix(3,-2,2,-1) Matrix(2797,1028,536,197) -> Matrix(1,0,0,1) Matrix(1291,474,-4622,-1697) -> Matrix(1,0,0,1) Matrix(323,118,52,19) -> Matrix(1,2,0,1) Matrix(105,38,268,97) -> Matrix(1,2,0,1) Matrix(209,74,-788,-279) -> Matrix(3,-2,8,-5) Matrix(473,166,208,73) -> Matrix(3,-2,2,-1) Matrix(197,62,448,141) -> Matrix(3,-2,2,-1) Matrix(97,30,-540,-167) -> Matrix(1,-2,2,-3) Matrix(245,74,96,29) -> Matrix(7,-2,4,-1) Matrix(337,100,920,273) -> Matrix(21,-8,8,-3) Matrix(291,86,-1208,-357) -> Matrix(5,-2,8,-3) Matrix(339,100,-1834,-541) -> Matrix(5,-2,8,-3) Matrix(239,70,338,99) -> Matrix(9,-4,-2,1) Matrix(613,172,474,133) -> Matrix(3,-2,8,-5) Matrix(6513,1822,1748,489) -> Matrix(15,-14,14,-13) Matrix(989,274,610,169) -> Matrix(1,0,4,1) Matrix(373,100,138,37) -> Matrix(7,-2,4,-1) Matrix(229,60,416,109) -> Matrix(17,-8,-2,1) Matrix(359,86,-2058,-493) -> Matrix(9,-4,16,-7) Matrix(265,62,312,73) -> Matrix(3,-2,2,-1) Matrix(131,30,310,71) -> Matrix(1,0,-2,1) Matrix(173,38,132,29) -> Matrix(3,-2,8,-5) Matrix(301,64,174,37) -> Matrix(3,-2,8,-5) Matrix(41,8,128,25) -> Matrix(5,-4,4,-3) Matrix(1335,248,1588,295) -> Matrix(1,0,0,1) Matrix(207,38,1084,199) -> Matrix(3,-2,2,-1) Matrix(205,36,-1566,-275) -> Matrix(5,-2,8,-3) Matrix(6303,1102,3998,699) -> Matrix(25,-12,-2,1) Matrix(1315,228,248,43) -> Matrix(7,-4,2,-1) Matrix(1109,190,286,49) -> Matrix(43,-28,20,-13) Matrix(281,44,364,57) -> Matrix(5,-4,-6,5) Matrix(79,12,362,55) -> Matrix(11,-10,10,-9) Matrix(237,32,274,37) -> Matrix(3,-2,2,-1) Matrix(1133,146,194,25) -> Matrix(31,-20,14,-9) Matrix(193,22,114,13) -> Matrix(5,-4,4,-3) Matrix(93,-10,214,-23) -> Matrix(1,0,0,1) Matrix(661,-86,392,-51) -> Matrix(11,-10,10,-9) Matrix(299,-42,178,-25) -> Matrix(7,-8,8,-9) Matrix(119,-18,324,-49) -> Matrix(11,-14,4,-5) Matrix(493,-86,86,-15) -> Matrix(3,-8,2,-5) Matrix(577,-104,172,-31) -> Matrix(1,0,0,1) Matrix(229,-42,518,-95) -> Matrix(1,0,0,1) Matrix(485,-92,58,-11) -> Matrix(1,-2,0,1) Matrix(285,-56,56,-11) -> Matrix(1,0,0,1) Matrix(85,-18,392,-83) -> Matrix(11,-10,10,-9) Matrix(221,-50,84,-19) -> Matrix(1,-2,2,-3) Matrix(513,-134,134,-35) -> Matrix(7,-16,4,-9) Matrix(2179,-576,1290,-341) -> Matrix(5,-14,4,-11) Matrix(1397,-370,404,-107) -> Matrix(1,-2,0,1) Matrix(1395,-374,884,-237) -> Matrix(1,-6,0,1) Matrix(1259,-340,374,-101) -> Matrix(1,2,0,1) Matrix(239,-66,134,-37) -> Matrix(3,-4,4,-5) Matrix(341,-100,474,-139) -> Matrix(1,-2,0,1) Matrix(629,-186,1444,-427) -> Matrix(3,-2,2,-1) Matrix(577,-172,104,-31) -> Matrix(1,0,0,1) Matrix(105,-32,338,-103) -> Matrix(3,-2,2,-1) Matrix(75,-26,26,-9) -> Matrix(1,0,0,1) Matrix(221,-84,50,-19) -> Matrix(1,2,0,1) Matrix(637,-246,246,-95) -> Matrix(5,-8,2,-3) Matrix(121,-50,288,-119) -> Matrix(1,-6,0,1) Matrix(1257,-544,238,-103) -> Matrix(3,-2,2,-1) Matrix(2651,-1156,782,-341) -> Matrix(1,-2,2,-3) Matrix(259,-114,284,-125) -> Matrix(1,-2,0,1) Matrix(187,-84,118,-53) -> Matrix(1,-2,0,1) Matrix(133,-72,242,-131) -> Matrix(1,-10,0,1) Matrix(239,-134,66,-37) -> Matrix(1,4,0,1) Matrix(817,-474,474,-275) -> Matrix(1,0,4,1) Matrix(235,-138,172,-101) -> Matrix(1,-2,2,-3) Matrix(105,-64,64,-39) -> Matrix(1,0,2,1) Matrix(187,-118,84,-53) -> Matrix(3,-2,2,-1) Matrix(83,-54,20,-13) -> Matrix(1,2,0,1) Matrix(141,-98,200,-139) -> Matrix(1,-6,0,1) Matrix(2687,-1936,712,-513) -> Matrix(3,4,2,3) Matrix(2923,-2108,850,-613) -> Matrix(1,2,0,1) Matrix(749,-542,474,-343) -> Matrix(1,0,0,1) Matrix(235,-172,138,-101) -> Matrix(1,2,0,1) Matrix(171,-134,134,-105) -> Matrix(1,0,6,1) Matrix(131,-108,74,-61) -> Matrix(1,-2,2,-3) Matrix(925,-776,776,-651) -> Matrix(1,0,-2,1) Matrix(1217,-1026,720,-607) -> Matrix(3,-2,2,-1) Matrix(617,-538,164,-143) -> Matrix(3,-2,2,-1) Matrix(145,-128,162,-143) -> Matrix(1,-2,0,1) Matrix(259,-286,48,-53) -> Matrix(3,-2,2,-1) Matrix(429,-488,80,-91) -> Matrix(1,0,0,1) Matrix(711,-818,206,-237) -> Matrix(1,0,0,1) Matrix(887,-1050,234,-277) -> Matrix(3,-8,2,-5) Matrix(4621,-5512,1364,-1627) -> Matrix(1,0,4,1) Matrix(107,-132,30,-37) -> Matrix(1,2,0,1) Matrix(661,-834,256,-323) -> Matrix(17,2,8,1) Matrix(809,-1028,314,-399) -> Matrix(27,-2,14,-1) Matrix(1801,-2448,476,-647) -> Matrix(19,-8,12,-5) Matrix(663,-902,86,-117) -> Matrix(17,-8,-2,1) Matrix(185,-256,86,-119) -> Matrix(3,-2,2,-1) Matrix(341,-474,100,-139) -> Matrix(3,-2,2,-1) Matrix(141,-200,98,-139) -> Matrix(7,-6,6,-5) Matrix(173,-270,66,-103) -> Matrix(1,0,0,1) Matrix(3433,-5408,2178,-3431) -> Matrix(1,-26,0,1) Matrix(547,-882,338,-545) -> Matrix(1,0,10,1) Matrix(525,-902,188,-323) -> Matrix(9,2,4,1) Matrix(133,-242,72,-131) -> Matrix(11,-10,10,-9) Matrix(321,-698,86,-187) -> Matrix(5,-4,4,-3) Matrix(229,-518,42,-95) -> Matrix(1,0,0,1) Matrix(629,-1444,186,-427) -> Matrix(1,-2,0,1) Matrix(93,-214,10,-23) -> Matrix(1,0,0,1) Matrix(121,-288,50,-119) -> Matrix(7,-6,6,-5) Matrix(119,-324,18,-49) -> Matrix(9,-14,2,-3) Matrix(451,-1250,162,-449) -> Matrix(29,-56,14,-27) Matrix(105,-338,32,-103) -> Matrix(1,-2,0,1) Matrix(6511,-22050,1922,-6509) -> Matrix(1,0,6,1) Matrix(997,-3422,192,-659) -> Matrix(1,-2,2,-3) Matrix(1625,-5584,314,-1079) -> Matrix(1,-2,0,1) Matrix(203,-750,36,-133) -> Matrix(3,-2,2,-1) Matrix(2413,-8978,648,-2411) -> Matrix(27,-26,26,-25) Matrix(379,-1458,98,-377) -> Matrix(29,-56,14,-27) Matrix(85,-392,18,-83) -> Matrix(1,-10,0,1) Matrix(291,-1682,50,-289) -> Matrix(21,-40,10,-19) Matrix(61,-450,8,-59) -> Matrix(1,-18,0,1) 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): 48 Degree of the the map X: 48 Degree of the the map Y: 144 ----------------------------------------------------------------------- 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): 432 Minimal number of generators: 73 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: 21 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/1 9/8 19/15 7/5 10/7 21/13 11/6 2/1 12/5 25/9 3/1 13/4 105/31 79/23 11/3 67/18 27/7 4/1 13/3 9/2 14/3 5/1 16/3 11/2 29/5 6/1 7/1 15/2 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 0/1 1/1 1/8 3/4 2/15 1/1 1/7 6/5 1/6 3/2 3/17 2/1 2/11 1/1 3/16 1/0 4/21 -1/1 1/5 0/1 3/14 1/1 2/9 1/1 1/4 3/2 5/19 2/1 9/34 7/2 4/15 3/1 7/26 1/0 3/11 0/1 2/7 3/1 5/17 0/1 8/27 1/1 3/10 1/0 4/13 1/1 1/3 2/1 4/11 7/3 3/8 1/0 5/13 4/3 2/5 3/1 5/12 1/0 3/7 0/1 13/30 3/4 10/23 1/1 17/39 2/1 7/16 1/0 4/9 1/1 1/2 1/0 6/11 1/0 5/9 -4/1 4/7 -1/1 7/12 1/2 3/5 -2/1 5/8 1/2 7/11 4/3 2/3 1/1 7/10 1/0 5/7 -2/1 18/25 1/1 31/43 0/1 13/18 1/0 8/11 -3/1 3/4 1/0 7/9 -2/5 4/5 1/3 5/6 1/0 6/7 1/1 7/8 1/0 8/9 1/0 1/1 0/1 9/8 1/1 8/7 1/1 7/6 1/0 6/5 1/1 5/4 -1/2 19/15 0/1 14/11 1/9 9/7 2/7 4/3 1/1 15/11 4/7 11/8 3/4 18/13 1/1 7/5 2/3 10/7 1/1 3/2 1/0 11/7 4/1 52/33 1/0 41/26 1/0 30/19 -3/1 19/12 1/0 8/5 -1/1 21/13 0/1 13/8 1/4 5/3 2/3 17/10 1/0 12/7 -1/1 7/4 1/2 16/9 5/7 9/5 4/5 11/6 1/1 2/1 1/1 13/6 1/2 11/5 6/5 9/4 1/0 16/7 1/1 23/10 1/0 7/3 0/1 12/5 1/1 5/2 3/2 13/5 4/1 21/8 1/0 8/3 1/1 11/4 7/4 25/9 2/1 14/5 7/3 3/1 2/1 13/4 1/0 10/3 1/1 37/11 2/1 27/8 1/0 44/13 -1/1 105/31 0/1 61/18 1/2 17/5 0/1 24/7 1/1 79/23 2/1 55/16 1/0 31/9 2/1 7/2 3/2 18/5 3/1 11/3 0/1 26/7 1/1 67/18 1/1 41/11 10/9 15/4 3/2 34/9 7/5 19/5 2/1 23/6 11/6 27/7 2/1 4/1 3/1 13/3 10/3 9/2 1/0 14/3 1/0 5/1 0/1 21/4 1/2 16/3 1/1 11/2 1/0 28/5 1/1 17/3 2/1 23/4 7/4 29/5 2/1 6/1 3/1 7/1 6/1 15/2 1/0 8/1 -3/1 9/1 0/1 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(93,-10,214,-23) (0/1,1/8) -> (13/30,10/23) Hyperbolic Matrix(240,-31,31,-4) (1/8,2/15) -> (15/2,8/1) Hyperbolic Matrix(210,-29,29,-4) (2/15,1/7) -> (7/1,15/2) Hyperbolic Matrix(118,-19,87,-14) (1/7,1/6) -> (4/3,15/11) Hyperbolic Matrix(493,-86,86,-15) (1/6,3/17) -> (17/3,23/4) Hyperbolic Matrix(577,-104,172,-31) (3/17,2/11) -> (10/3,37/11) Hyperbolic Matrix(200,-37,173,-32) (2/11,3/16) -> (8/7,7/6) Hyperbolic Matrix(228,-43,859,-162) (3/16,4/21) -> (9/34,4/15) Hyperbolic Matrix(686,-131,199,-38) (4/21,1/5) -> (31/9,7/2) Hyperbolic Matrix(140,-29,29,-6) (1/5,3/14) -> (14/3,5/1) Hyperbolic Matrix(252,-55,55,-12) (3/14,2/9) -> (9/2,14/3) Hyperbolic Matrix(221,-50,84,-19) (2/9,1/4) -> (21/8,8/3) Hyperbolic Matrix(513,-134,134,-35) (1/4,5/19) -> (19/5,23/6) Hyperbolic Matrix(458,-121,53,-14) (5/19,9/34) -> (8/1,9/1) Hyperbolic Matrix(1395,-374,884,-237) (4/15,7/26) -> (41/26,30/19) Hyperbolic Matrix(1259,-340,374,-101) (7/26,3/11) -> (37/11,27/8) Hyperbolic Matrix(239,-66,134,-37) (3/11,2/7) -> (16/9,9/5) Hyperbolic Matrix(341,-100,474,-139) (2/7,5/17) -> (5/7,18/25) Hyperbolic Matrix(629,-186,1444,-427) (5/17,8/27) -> (10/23,17/39) Hyperbolic Matrix(577,-172,104,-31) (8/27,3/10) -> (11/2,28/5) Hyperbolic Matrix(260,-79,79,-24) (3/10,4/13) -> (13/4,10/3) Hyperbolic Matrix(78,-25,25,-8) (4/13,1/3) -> (3/1,13/4) Hyperbolic Matrix(75,-26,26,-9) (1/3,4/11) -> (14/5,3/1) Hyperbolic Matrix(100,-37,173,-64) (4/11,3/8) -> (4/7,7/12) Hyperbolic Matrix(221,-84,50,-19) (3/8,5/13) -> (13/3,9/2) Hyperbolic Matrix(74,-29,97,-38) (5/13,2/5) -> (3/4,7/9) Hyperbolic Matrix(120,-49,49,-20) (2/5,5/12) -> (12/5,5/2) Hyperbolic Matrix(168,-71,71,-30) (5/12,3/7) -> (7/3,12/5) Hyperbolic Matrix(808,-349,213,-92) (3/7,13/30) -> (34/9,19/5) Hyperbolic Matrix(2651,-1156,782,-341) (17/39,7/16) -> (61/18,17/5) Hyperbolic Matrix(166,-73,141,-62) (7/16,4/9) -> (7/6,6/5) Hyperbolic Matrix(187,-84,118,-53) (4/9,1/2) -> (19/12,8/5) Hyperbolic Matrix(44,-23,23,-12) (1/2,6/11) -> (11/6,2/1) Hyperbolic Matrix(198,-109,109,-60) (6/11,5/9) -> (9/5,11/6) Hyperbolic Matrix(239,-134,66,-37) (5/9,4/7) -> (18/5,11/3) Hyperbolic Matrix(235,-138,172,-101) (7/12,3/5) -> (15/11,11/8) Hyperbolic Matrix(105,-64,64,-39) (3/5,5/8) -> (13/8,5/3) Hyperbolic Matrix(187,-118,84,-53) (5/8,7/11) -> (11/5,9/4) Hyperbolic Matrix(83,-54,20,-13) (7/11,2/3) -> (4/1,13/3) Hyperbolic Matrix(60,-41,41,-28) (2/3,7/10) -> (10/7,3/2) Hyperbolic Matrix(140,-99,99,-70) (7/10,5/7) -> (7/5,10/7) Hyperbolic Matrix(1028,-741,197,-142) (18/25,31/43) -> (5/1,21/4) Hyperbolic Matrix(2923,-2108,850,-613) (31/43,13/18) -> (55/16,31/9) Hyperbolic Matrix(749,-542,474,-343) (13/18,8/11) -> (30/19,19/12) Hyperbolic Matrix(235,-172,138,-101) (8/11,3/4) -> (17/10,12/7) Hyperbolic Matrix(171,-134,134,-105) (7/9,4/5) -> (14/11,9/7) Hyperbolic Matrix(131,-108,74,-61) (4/5,5/6) -> (7/4,16/9) Hyperbolic Matrix(166,-141,73,-62) (5/6,6/7) -> (9/4,16/7) Hyperbolic Matrix(200,-173,37,-32) (6/7,7/8) -> (16/3,11/2) Hyperbolic Matrix(144,-127,127,-112) (7/8,8/9) -> (9/8,8/7) Hyperbolic Matrix(18,-17,17,-16) (8/9,1/1) -> (1/1,9/8) Parabolic Matrix(107,-132,30,-37) (6/5,5/4) -> (7/2,18/5) Hyperbolic Matrix(286,-361,225,-284) (5/4,19/15) -> (19/15,14/11) Parabolic Matrix(74,-97,29,-38) (9/7,4/3) -> (5/2,13/5) Hyperbolic Matrix(185,-256,86,-119) (11/8,18/13) -> (2/1,13/6) Hyperbolic Matrix(341,-474,100,-139) (18/13,7/5) -> (17/5,24/7) Hyperbolic Matrix(173,-270,66,-103) (3/2,11/7) -> (13/5,21/8) Hyperbolic Matrix(1822,-2869,489,-770) (11/7,52/33) -> (67/18,41/11) Hyperbolic Matrix(2600,-4099,699,-1102) (52/33,41/26) -> (26/7,67/18) Hyperbolic Matrix(274,-441,169,-272) (8/5,21/13) -> (21/13,13/8) Parabolic Matrix(88,-149,13,-22) (5/3,17/10) -> (6/1,7/1) Hyperbolic Matrix(100,-173,37,-64) (12/7,7/4) -> (8/3,11/4) Hyperbolic Matrix(321,-698,86,-187) (13/6,11/5) -> (41/11,15/4) Hyperbolic Matrix(629,-1444,186,-427) (16/7,23/10) -> (27/8,44/13) Hyperbolic Matrix(93,-214,10,-23) (23/10,7/3) -> (9/1,1/0) Hyperbolic Matrix(226,-625,81,-224) (11/4,25/9) -> (25/9,14/5) Parabolic Matrix(3256,-11025,961,-3254) (44/13,105/31) -> (105/31,61/18) Parabolic Matrix(1818,-6241,529,-1816) (24/7,79/23) -> (79/23,55/16) Parabolic Matrix(203,-750,36,-133) (11/3,26/7) -> (28/5,17/3) Hyperbolic Matrix(228,-859,43,-162) (15/4,34/9) -> (21/4,16/3) Hyperbolic Matrix(190,-729,49,-188) (23/6,27/7) -> (27/7,4/1) Parabolic Matrix(146,-841,25,-144) (23/4,29/5) -> (29/5,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,1,1) Matrix(93,-10,214,-23) -> Matrix(1,0,0,1) Matrix(240,-31,31,-4) -> Matrix(7,-6,-1,1) Matrix(210,-29,29,-4) -> Matrix(11,-12,1,-1) Matrix(118,-19,87,-14) -> Matrix(1,-2,3,-5) Matrix(493,-86,86,-15) -> Matrix(3,-8,2,-5) Matrix(577,-104,172,-31) -> Matrix(1,0,0,1) Matrix(200,-37,173,-32) -> Matrix(1,-2,1,-1) Matrix(228,-43,859,-162) -> Matrix(3,-4,1,-1) Matrix(686,-131,199,-38) -> Matrix(1,-2,1,-1) Matrix(140,-29,29,-6) -> Matrix(1,0,-1,1) Matrix(252,-55,55,-12) -> Matrix(9,-10,1,-1) Matrix(221,-50,84,-19) -> Matrix(1,-2,2,-3) Matrix(513,-134,134,-35) -> Matrix(7,-16,4,-9) Matrix(458,-121,53,-14) -> Matrix(1,-2,-1,3) Matrix(1395,-374,884,-237) -> Matrix(1,-6,0,1) Matrix(1259,-340,374,-101) -> Matrix(1,2,0,1) Matrix(239,-66,134,-37) -> Matrix(3,-4,4,-5) Matrix(341,-100,474,-139) -> Matrix(1,-2,0,1) Matrix(629,-186,1444,-427) -> Matrix(3,-2,2,-1) Matrix(577,-172,104,-31) -> Matrix(1,0,0,1) Matrix(260,-79,79,-24) -> Matrix(1,-2,1,-1) Matrix(78,-25,25,-8) -> Matrix(3,-4,1,-1) Matrix(75,-26,26,-9) -> Matrix(1,0,0,1) Matrix(100,-37,173,-64) -> Matrix(1,-2,-1,3) Matrix(221,-84,50,-19) -> Matrix(1,2,0,1) Matrix(74,-29,97,-38) -> Matrix(1,-2,-1,3) Matrix(120,-49,49,-20) -> Matrix(1,-6,1,-5) Matrix(168,-71,71,-30) -> Matrix(1,0,1,1) Matrix(808,-349,213,-92) -> Matrix(5,-2,3,-1) Matrix(2651,-1156,782,-341) -> Matrix(1,-2,2,-3) Matrix(166,-73,141,-62) -> Matrix(1,-2,1,-1) Matrix(187,-84,118,-53) -> Matrix(1,-2,0,1) Matrix(44,-23,23,-12) -> Matrix(1,-2,1,-1) Matrix(198,-109,109,-60) -> Matrix(1,8,1,9) Matrix(239,-134,66,-37) -> Matrix(1,4,0,1) Matrix(235,-138,172,-101) -> Matrix(1,-2,2,-3) Matrix(105,-64,64,-39) -> Matrix(1,0,2,1) Matrix(187,-118,84,-53) -> Matrix(3,-2,2,-1) Matrix(83,-54,20,-13) -> Matrix(1,2,0,1) Matrix(60,-41,41,-28) -> Matrix(1,-2,1,-1) Matrix(140,-99,99,-70) -> Matrix(1,4,1,5) Matrix(1028,-741,197,-142) -> Matrix(1,0,1,1) Matrix(2923,-2108,850,-613) -> Matrix(1,2,0,1) Matrix(749,-542,474,-343) -> Matrix(1,0,0,1) Matrix(235,-172,138,-101) -> Matrix(1,2,0,1) Matrix(171,-134,134,-105) -> Matrix(1,0,6,1) Matrix(131,-108,74,-61) -> Matrix(1,-2,2,-3) Matrix(166,-141,73,-62) -> Matrix(1,-2,1,-1) Matrix(200,-173,37,-32) -> Matrix(1,-2,1,-1) Matrix(144,-127,127,-112) -> Matrix(1,-2,1,-1) Matrix(18,-17,17,-16) -> Matrix(1,0,1,1) Matrix(107,-132,30,-37) -> Matrix(1,2,0,1) Matrix(286,-361,225,-284) -> Matrix(1,0,11,1) Matrix(74,-97,29,-38) -> Matrix(5,-2,3,-1) Matrix(185,-256,86,-119) -> Matrix(3,-2,2,-1) Matrix(341,-474,100,-139) -> Matrix(3,-2,2,-1) Matrix(173,-270,66,-103) -> Matrix(1,0,0,1) Matrix(1822,-2869,489,-770) -> Matrix(1,-14,1,-13) Matrix(2600,-4099,699,-1102) -> Matrix(1,12,1,13) Matrix(274,-441,169,-272) -> Matrix(1,0,5,1) Matrix(88,-149,13,-22) -> Matrix(3,-4,1,-1) Matrix(100,-173,37,-64) -> Matrix(5,-2,3,-1) Matrix(321,-698,86,-187) -> Matrix(5,-4,4,-3) Matrix(629,-1444,186,-427) -> Matrix(1,-2,0,1) Matrix(93,-214,10,-23) -> Matrix(1,0,0,1) Matrix(226,-625,81,-224) -> Matrix(15,-28,7,-13) Matrix(3256,-11025,961,-3254) -> Matrix(1,0,3,1) Matrix(1818,-6241,529,-1816) -> Matrix(3,-4,1,-1) Matrix(203,-750,36,-133) -> Matrix(3,-2,2,-1) Matrix(228,-859,43,-162) -> Matrix(3,-4,1,-1) Matrix(190,-729,49,-188) -> Matrix(15,-28,7,-13) Matrix(146,-841,25,-144) -> Matrix(11,-20,5,-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 1 1 1/1 0/1 1 17 9/8 1/1 1 1 8/7 1/1 1 17 7/6 1/0 1 17 6/5 1/1 1 17 5/4 -1/2 1 17 19/15 0/1 11 1 9/7 2/7 1 17 4/3 1/1 1 17 11/8 3/4 1 17 18/13 1/1 1 17 7/5 2/3 1 17 10/7 1/1 3 1 3/2 1/0 1 17 11/7 4/1 1 17 8/5 -1/1 1 17 21/13 0/1 5 1 5/3 2/3 1 17 17/10 1/0 1 17 12/7 -1/1 1 17 7/4 1/2 1 17 9/5 4/5 1 17 11/6 1/1 5 1 2/1 1/1 1 17 13/6 1/2 1 17 11/5 6/5 1 17 9/4 1/0 1 17 16/7 1/1 1 17 23/10 1/0 1 17 7/3 0/1 1 17 12/5 1/1 3 1 5/2 3/2 1 17 13/5 4/1 1 17 8/3 1/1 1 17 11/4 7/4 1 17 25/9 2/1 7 1 3/1 2/1 1 17 13/4 1/0 1 1 10/3 1/1 1 17 27/8 1/0 1 17 44/13 -1/1 1 17 105/31 0/1 3 1 17/5 0/1 1 17 24/7 1/1 1 17 79/23 2/1 1 1 31/9 2/1 1 17 7/2 3/2 1 17 18/5 3/1 1 17 11/3 0/1 1 17 26/7 1/1 1 17 67/18 1/1 13 1 41/11 10/9 1 17 15/4 3/2 1 17 34/9 7/5 1 17 19/5 2/1 1 17 27/7 2/1 7 1 4/1 3/1 1 17 13/3 10/3 1 17 9/2 1/0 1 17 14/3 1/0 5 1 5/1 0/1 1 17 21/4 1/2 1 17 16/3 1/1 1 17 11/2 1/0 1 17 28/5 1/1 1 17 17/3 2/1 1 17 29/5 2/1 5 1 6/1 3/1 1 17 7/1 6/1 1 17 15/2 1/0 9 1 8/1 -3/1 1 17 9/1 0/1 1 17 1/0 1/0 1 17 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(17,-18,16,-17) (1/1,9/8) -> (1/1,9/8) Reflection Matrix(127,-144,112,-127) (9/8,8/7) -> (9/8,8/7) Reflection Matrix(173,-200,32,-37) (8/7,7/6) -> (16/3,11/2) Glide Reflection Matrix(141,-166,62,-73) (7/6,6/5) -> (9/4,16/7) Glide Reflection Matrix(107,-132,30,-37) (6/5,5/4) -> (7/2,18/5) Hyperbolic Matrix(151,-190,120,-151) (5/4,19/15) -> (5/4,19/15) Reflection Matrix(134,-171,105,-134) (19/15,9/7) -> (19/15,9/7) Reflection Matrix(74,-97,29,-38) (9/7,4/3) -> (5/2,13/5) Hyperbolic Matrix(172,-235,101,-138) (4/3,11/8) -> (17/10,12/7) Glide Reflection Matrix(185,-256,86,-119) (11/8,18/13) -> (2/1,13/6) Hyperbolic Matrix(341,-474,100,-139) (18/13,7/5) -> (17/5,24/7) Hyperbolic Matrix(99,-140,70,-99) (7/5,10/7) -> (7/5,10/7) Reflection Matrix(41,-60,28,-41) (10/7,3/2) -> (10/7,3/2) Reflection Matrix(54,-83,13,-20) (3/2,11/7) -> (4/1,13/3) Glide Reflection Matrix(118,-187,53,-84) (11/7,8/5) -> (11/5,9/4) Glide Reflection Matrix(209,-336,130,-209) (8/5,21/13) -> (8/5,21/13) Reflection Matrix(64,-105,39,-64) (21/13,5/3) -> (21/13,5/3) Reflection Matrix(88,-149,13,-22) (5/3,17/10) -> (6/1,7/1) Hyperbolic Matrix(100,-173,37,-64) (12/7,7/4) -> (8/3,11/4) Hyperbolic Matrix(134,-239,37,-66) (7/4,9/5) -> (18/5,11/3) Glide Reflection Matrix(109,-198,60,-109) (9/5,11/6) -> (9/5,11/6) Reflection Matrix(23,-44,12,-23) (11/6,2/1) -> (11/6,2/1) Reflection Matrix(321,-698,86,-187) (13/6,11/5) -> (41/11,15/4) Hyperbolic Matrix(629,-1444,186,-427) (16/7,23/10) -> (27/8,44/13) Hyperbolic Matrix(93,-214,10,-23) (23/10,7/3) -> (9/1,1/0) Hyperbolic Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(84,-221,19,-50) (13/5,8/3) -> (13/3,9/2) Glide Reflection Matrix(199,-550,72,-199) (11/4,25/9) -> (11/4,25/9) Reflection Matrix(26,-75,9,-26) (25/9,3/1) -> (25/9,3/1) Reflection Matrix(25,-78,8,-25) (3/1,13/4) -> (3/1,13/4) Reflection Matrix(79,-260,24,-79) (13/4,10/3) -> (13/4,10/3) Reflection Matrix(172,-577,31,-104) (10/3,27/8) -> (11/2,28/5) Glide Reflection Matrix(2729,-9240,806,-2729) (44/13,105/31) -> (44/13,105/31) Reflection Matrix(526,-1785,155,-526) (105/31,17/5) -> (105/31,17/5) Reflection Matrix(1105,-3792,322,-1105) (24/7,79/23) -> (24/7,79/23) Reflection Matrix(712,-2449,207,-712) (79/23,31/9) -> (79/23,31/9) Reflection Matrix(199,-686,38,-131) (31/9,7/2) -> (5/1,21/4) Glide Reflection Matrix(203,-750,36,-133) (11/3,26/7) -> (28/5,17/3) Hyperbolic Matrix(937,-3484,252,-937) (26/7,67/18) -> (26/7,67/18) Reflection Matrix(1475,-5494,396,-1475) (67/18,41/11) -> (67/18,41/11) Reflection Matrix(228,-859,43,-162) (15/4,34/9) -> (21/4,16/3) Hyperbolic Matrix(121,-458,14,-53) (34/9,19/5) -> (8/1,9/1) Glide Reflection Matrix(134,-513,35,-134) (19/5,27/7) -> (19/5,27/7) Reflection Matrix(55,-216,14,-55) (27/7,4/1) -> (27/7,4/1) Reflection Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) Reflection Matrix(29,-140,6,-29) (14/3,5/1) -> (14/3,5/1) Reflection Matrix(86,-493,15,-86) (17/3,29/5) -> (17/3,29/5) Reflection Matrix(59,-348,10,-59) (29/5,6/1) -> (29/5,6/1) Reflection Matrix(29,-210,4,-29) (7/1,15/2) -> (7/1,15/2) Reflection Matrix(31,-240,4,-31) (15/2,8/1) -> (15/2,8/1) 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(1,0,1,-1) (-1/1,1/1) -> (0/1,2/1) Matrix(17,-18,16,-17) -> Matrix(1,0,2,-1) (1/1,9/8) -> (0/1,1/1) Matrix(127,-144,112,-127) -> Matrix(-1,2,0,1) (9/8,8/7) -> (1/1,1/0) Matrix(173,-200,32,-37) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(141,-166,62,-73) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(107,-132,30,-37) -> Matrix(1,2,0,1) 1/0 Matrix(151,-190,120,-151) -> Matrix(-1,0,4,1) (5/4,19/15) -> (-1/2,0/1) Matrix(134,-171,105,-134) -> Matrix(1,0,7,-1) (19/15,9/7) -> (0/1,2/7) Matrix(74,-97,29,-38) -> Matrix(5,-2,3,-1) Matrix(172,-235,101,-138) -> Matrix(3,-2,1,-1) Matrix(185,-256,86,-119) -> Matrix(3,-2,2,-1) 1/1 Matrix(341,-474,100,-139) -> Matrix(3,-2,2,-1) 1/1 Matrix(99,-140,70,-99) -> Matrix(5,-4,6,-5) (7/5,10/7) -> (2/3,1/1) Matrix(41,-60,28,-41) -> Matrix(-1,2,0,1) (10/7,3/2) -> (1/1,1/0) Matrix(54,-83,13,-20) -> Matrix(3,-2,1,-1) Matrix(118,-187,53,-84) -> Matrix(1,2,1,1) Matrix(209,-336,130,-209) -> Matrix(-1,0,2,1) (8/5,21/13) -> (-1/1,0/1) Matrix(64,-105,39,-64) -> Matrix(1,0,3,-1) (21/13,5/3) -> (0/1,2/3) Matrix(88,-149,13,-22) -> Matrix(3,-4,1,-1) 2/1 Matrix(100,-173,37,-64) -> Matrix(5,-2,3,-1) Matrix(134,-239,37,-66) -> Matrix(5,-4,1,-1) Matrix(109,-198,60,-109) -> Matrix(9,-8,10,-9) (9/5,11/6) -> (4/5,1/1) Matrix(23,-44,12,-23) -> Matrix(-1,2,0,1) (11/6,2/1) -> (1/1,1/0) Matrix(321,-698,86,-187) -> Matrix(5,-4,4,-3) 1/1 Matrix(629,-1444,186,-427) -> Matrix(1,-2,0,1) 1/0 Matrix(93,-214,10,-23) -> Matrix(1,0,0,1) Matrix(71,-168,30,-71) -> Matrix(1,0,2,-1) (7/3,12/5) -> (0/1,1/1) Matrix(49,-120,20,-49) -> Matrix(5,-6,4,-5) (12/5,5/2) -> (1/1,3/2) Matrix(84,-221,19,-50) -> Matrix(3,-2,1,-1) Matrix(199,-550,72,-199) -> Matrix(15,-28,8,-15) (11/4,25/9) -> (7/4,2/1) Matrix(26,-75,9,-26) -> Matrix(1,0,1,-1) (25/9,3/1) -> (0/1,2/1) Matrix(25,-78,8,-25) -> Matrix(-1,4,0,1) (3/1,13/4) -> (2/1,1/0) Matrix(79,-260,24,-79) -> Matrix(-1,2,0,1) (13/4,10/3) -> (1/1,1/0) Matrix(172,-577,31,-104) -> Matrix(1,0,1,-1) *** -> (0/1,2/1) Matrix(2729,-9240,806,-2729) -> Matrix(-1,0,2,1) (44/13,105/31) -> (-1/1,0/1) Matrix(526,-1785,155,-526) -> Matrix(1,0,1,-1) (105/31,17/5) -> (0/1,2/1) Matrix(1105,-3792,322,-1105) -> Matrix(3,-4,2,-3) (24/7,79/23) -> (1/1,2/1) Matrix(712,-2449,207,-712) -> Matrix(1,0,1,-1) (79/23,31/9) -> (0/1,2/1) Matrix(199,-686,38,-131) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(203,-750,36,-133) -> Matrix(3,-2,2,-1) 1/1 Matrix(937,-3484,252,-937) -> Matrix(7,-6,8,-7) (26/7,67/18) -> (3/4,1/1) Matrix(1475,-5494,396,-1475) -> Matrix(19,-20,18,-19) (67/18,41/11) -> (1/1,10/9) Matrix(228,-859,43,-162) -> Matrix(3,-4,1,-1) 2/1 Matrix(121,-458,14,-53) -> Matrix(1,-2,-2,3) Matrix(134,-513,35,-134) -> Matrix(9,-16,5,-9) (19/5,27/7) -> (8/5,2/1) Matrix(55,-216,14,-55) -> Matrix(5,-12,2,-5) (27/7,4/1) -> (2/1,3/1) Matrix(55,-252,12,-55) -> Matrix(-1,10,0,1) (9/2,14/3) -> (5/1,1/0) Matrix(29,-140,6,-29) -> Matrix(1,0,0,-1) (14/3,5/1) -> (0/1,1/0) Matrix(86,-493,15,-86) -> Matrix(5,-8,3,-5) (17/3,29/5) -> (4/3,2/1) Matrix(59,-348,10,-59) -> Matrix(5,-12,2,-5) (29/5,6/1) -> (2/1,3/1) Matrix(29,-210,4,-29) -> Matrix(-1,12,0,1) (7/1,15/2) -> (6/1,1/0) Matrix(31,-240,4,-31) -> Matrix(1,6,0,-1) (15/2,8/1) -> (-3/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.