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 0/1 1/2 -6/13 1/0 -17/37 1/2 -11/24 0/1 1/1 -5/11 1/0 -9/20 0/1 1/6 -4/9 1/4 -19/43 3/8 -15/34 1/3 2/5 -26/59 3/8 -11/25 1/2 -7/16 1/3 1/2 -17/39 3/8 -27/62 2/5 1/2 -10/23 1/2 -3/7 1/2 -8/19 1/2 -5/12 0/1 1/0 -7/17 1/4 -9/22 0/1 1/3 -2/5 1/2 -9/23 1/2 -7/18 1/2 2/3 -5/13 1/2 -8/21 3/4 -11/29 3/4 -3/8 2/3 1/1 -7/19 5/4 -25/68 1/1 3/2 -18/49 3/2 -11/30 1/1 2/1 -4/11 1/0 -5/14 -1/1 1/0 -6/17 -1/2 -1/3 1/2 -5/16 2/1 1/0 -4/13 1/0 -3/10 0/1 1/0 -8/27 -1/2 -13/44 -1/1 0/1 -5/17 -1/2 -2/7 1/2 -7/25 1/2 -26/93 1/0 -19/68 0/1 1/1 -12/43 1/0 -5/18 0/1 1/1 -8/29 1/4 -3/11 1/2 -4/15 1/0 -9/34 1/2 1/1 -5/19 1/2 -1/4 0/1 1/1 -6/25 1/0 -5/21 1/2 -4/17 1/2 -3/13 1/2 -2/9 1/0 -3/14 0/1 1/2 -1/5 1/0 -3/16 0/1 1/3 -11/59 1/4 -8/43 1/2 -5/27 1/2 -7/38 0/1 1/3 -2/11 1/2 -5/28 0/1 1/2 -3/17 1/2 -7/40 1/2 2/3 -11/63 3/4 -4/23 1/2 -5/29 3/4 -6/35 7/8 -1/6 0/1 1/1 -2/13 3/4 -1/7 3/2 -2/15 3/2 -5/38 3/2 2/1 -3/23 3/2 -4/31 7/4 -1/8 2/1 1/0 0/1 1/0 1/8 3/1 1/0 2/15 1/0 1/7 1/0 1/6 -2/1 1/0 3/17 -3/2 2/11 -3/2 3/16 -1/1 0/1 4/21 1/0 5/26 -2/1 -1/1 1/5 -3/2 3/14 -1/1 5/23 -9/10 2/9 -3/4 1/4 -1/1 0/1 5/19 -3/4 9/34 -3/4 -5/7 4/15 -1/2 7/26 -2/3 -1/2 3/11 -1/2 2/7 -1/2 5/17 -1/2 8/27 -1/2 3/10 -1/3 0/1 4/13 0/1 5/16 0/1 1/3 1/3 1/0 4/11 -3/4 11/30 -2/3 -1/2 7/19 -1/2 3/8 -1/2 0/1 5/13 1/0 7/18 -3/1 1/0 2/5 1/0 5/12 -1/1 8/19 -3/4 3/7 -1/2 13/30 -3/5 -1/2 10/23 -1/2 17/39 -1/2 7/16 0/1 1/0 11/25 -3/2 15/34 -1/1 -4/5 4/9 -1/2 1/2 -1/1 0/1 6/11 -1/1 11/20 -1/1 -2/3 5/9 -1/2 4/7 1/0 11/19 1/0 7/12 -1/1 -2/3 3/5 -1/2 5/8 -1/1 0/1 7/11 -1/2 2/3 -1/2 7/10 0/1 12/17 1/6 5/7 1/2 18/25 1/2 31/43 1/0 13/18 0/1 1/1 8/11 1/2 3/4 0/1 1/0 10/13 1/2 17/22 1/1 2/1 7/9 1/0 4/5 1/0 5/6 -2/1 1/0 21/25 -3/2 16/19 -3/2 11/13 -3/2 6/7 -3/2 13/15 -3/2 7/8 -6/5 -1/1 8/9 -1/1 9/10 -1/1 -8/9 1/1 -1/2 9/8 0/1 8/7 1/4 15/13 1/0 7/6 0/1 1/1 13/11 1/2 19/16 0/1 1/0 25/21 -1/2 31/26 -1/4 0/1 6/5 1/2 5/4 1/1 1/0 19/15 1/0 14/11 1/0 9/7 1/0 22/17 1/0 13/10 -2/1 -3/2 4/3 1/0 19/14 -2/1 -3/2 34/25 -3/2 15/11 -3/2 11/8 -2/1 -1/1 18/13 -3/2 7/5 -5/4 10/7 -1/1 13/9 -9/10 3/2 -1/1 -2/3 11/7 -3/4 52/33 -2/3 93/59 -5/8 41/26 -2/3 -1/2 30/19 -1/2 19/12 -1/1 -2/3 8/5 -3/4 21/13 -2/3 34/21 -9/14 13/8 -2/3 -3/5 5/3 -1/2 32/19 -1/2 27/16 -1/2 -5/11 49/29 -1/2 22/13 -1/2 17/10 -1/2 0/1 12/7 1/0 31/18 -1/1 -6/7 19/11 -3/4 7/4 -2/3 -1/2 16/9 -1/2 9/5 -1/2 11/6 -1/2 13/7 -1/2 2/1 -1/2 13/6 -1/3 0/1 11/5 -1/4 9/4 -1/3 0/1 25/11 1/0 16/7 -1/4 23/10 -1/2 0/1 7/3 -1/4 12/5 0/1 17/7 1/4 5/2 0/1 1/0 18/7 1/0 49/19 1/0 31/12 -5/1 1/0 13/5 1/0 21/8 -2/1 -1/1 8/3 -1/2 19/7 1/0 11/4 -2/1 -1/1 25/9 -1/1 39/14 -1/1 -12/13 14/5 -5/6 3/1 -1/2 13/4 -1/2 23/7 -1/2 10/3 -1/2 37/11 -1/2 27/8 -1/2 -2/5 44/13 -5/12 105/31 -2/5 166/49 -19/48 61/18 -2/5 -7/18 17/5 -3/8 24/7 -1/2 79/23 -1/3 55/16 -1/3 0/1 31/9 -1/4 38/11 1/0 7/2 -1/2 -1/3 18/5 -3/8 11/3 -1/2 26/7 -3/8 67/18 -1/3 108/29 -1/4 41/11 -1/2 15/4 -2/5 -1/3 49/13 -1/4 34/9 -1/2 53/14 -1/2 -2/5 19/5 -3/8 23/6 -6/17 -1/3 27/7 -1/3 31/8 -1/3 -8/25 4/1 -1/4 13/3 -1/4 9/2 -1/6 0/1 14/3 0/1 19/4 0/1 1/6 5/1 1/0 31/6 -1/1 0/1 57/11 -1/1 26/5 -1/2 21/4 -1/1 -1/2 37/7 -1/2 16/3 -1/4 43/8 0/1 27/5 1/2 11/2 -1/1 0/1 28/5 1/0 17/3 -1/2 23/4 -1/4 0/1 29/5 0/1 35/6 0/1 1/4 6/1 1/0 13/2 -1/1 0/1 7/1 -1/2 15/2 -1/2 23/3 -1/2 8/1 -1/2 17/2 -1/3 0/1 9/1 -1/4 1/0 -1/2 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(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(1,0,4,1) Matrix(255,118,188,87) -> Matrix(1,-2,0,1) Matrix(187,86,-1446,-665) -> Matrix(7,-2,4,-1) Matrix(375,172,-2008,-921) -> Matrix(1,0,2,1) Matrix(437,200,378,173) -> Matrix(1,0,0,1) Matrix(309,140,64,29) -> Matrix(1,0,0,1) Matrix(557,250,430,193) -> Matrix(9,-2,-4,1) Matrix(303,134,-1766,-781) -> Matrix(17,-6,20,-7) Matrix(1037,458,120,53) -> Matrix(5,-2,-12,5) Matrix(2005,884,-7174,-3163) -> Matrix(5,-2,8,-3) Matrix(849,374,-4556,-2007) -> Matrix(5,-2,18,-7) Matrix(305,134,-1154,-507) -> Matrix(5,-2,8,-3) Matrix(1087,474,-2956,-1289) -> Matrix(7,-2,4,-1) Matrix(2291,998,1926,839) -> Matrix(5,-2,-2,1) Matrix(239,104,-1804,-785) -> Matrix(19,-8,12,-5) Matrix(599,260,182,79) -> Matrix(13,-6,-28,13) Matrix(237,100,410,173) -> Matrix(1,0,-2,1) Matrix(177,74,232,97) -> Matrix(1,0,0,1) Matrix(289,120,118,49) -> Matrix(1,0,0,1) Matrix(405,166,344,141) -> Matrix(1,0,-2,1) Matrix(289,118,-1036,-423) -> Matrix(1,0,-2,1) Matrix(111,44,58,23) -> Matrix(1,0,-4,1) Matrix(169,66,-950,-371) -> Matrix(3,-2,8,-5) Matrix(165,64,446,173) -> Matrix(3,-2,-4,3) Matrix(167,64,-608,-233) -> Matrix(3,-2,8,-5) Matrix(221,84,-934,-355) -> Matrix(3,-2,2,-1) Matrix(549,208,710,269) -> Matrix(5,-4,4,-3) Matrix(161,60,110,41) -> Matrix(5,-4,-6,5) Matrix(2797,1028,536,197) -> Matrix(1,-2,0,1) Matrix(1291,474,-4622,-1697) -> Matrix(1,-2,2,-3) 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(1,0,2,1) Matrix(473,166,208,73) -> Matrix(1,0,-2,1) Matrix(197,62,448,141) -> Matrix(1,-2,0,1) Matrix(97,30,-540,-167) -> Matrix(1,-2,2,-3) Matrix(245,74,96,29) -> Matrix(1,0,0,1) Matrix(337,100,920,273) -> Matrix(1,2,-2,-3) Matrix(291,86,-1208,-357) -> Matrix(1,0,2,1) Matrix(339,100,-1834,-541) -> Matrix(1,0,4,1) Matrix(239,70,338,99) -> Matrix(1,0,4,1) Matrix(613,172,474,133) -> Matrix(5,-2,-2,1) Matrix(6513,1822,1748,489) -> Matrix(1,0,-4,1) Matrix(989,274,610,169) -> Matrix(1,2,-2,-3) Matrix(373,100,138,37) -> Matrix(1,0,-2,1) Matrix(229,60,416,109) -> Matrix(3,-2,-4,3) Matrix(359,86,-2058,-493) -> Matrix(1,-2,2,-3) Matrix(265,62,312,73) -> Matrix(1,-2,0,1) Matrix(131,30,310,71) -> Matrix(3,-2,-4,3) Matrix(173,38,132,29) -> Matrix(1,-2,0,1) Matrix(301,64,174,37) -> Matrix(3,-2,-4,3) Matrix(41,8,128,25) -> Matrix(1,0,0,1) Matrix(1335,248,1588,295) -> Matrix(1,-2,0,1) Matrix(207,38,1084,199) -> Matrix(5,-2,-2,1) Matrix(205,36,-1566,-275) -> Matrix(5,-4,4,-3) Matrix(6303,1102,3998,699) -> Matrix(7,-4,-12,7) Matrix(1315,228,248,43) -> Matrix(3,-2,-10,7) Matrix(1109,190,286,49) -> Matrix(9,-8,-28,25) Matrix(281,44,364,57) -> Matrix(3,-2,2,-1) Matrix(79,12,362,55) -> Matrix(7,-6,-8,7) Matrix(237,32,274,37) -> Matrix(5,-6,-4,5) Matrix(1133,146,194,25) -> Matrix(1,-2,4,-7) Matrix(193,22,114,13) -> Matrix(1,-2,-2,5) Matrix(93,-10,214,-23) -> Matrix(1,0,-2,1) Matrix(661,-86,392,-51) -> Matrix(1,-8,-2,17) Matrix(299,-42,178,-25) -> Matrix(1,6,-2,-11) Matrix(119,-18,324,-49) -> Matrix(1,4,-2,-7) Matrix(493,-86,86,-15) -> Matrix(1,2,-4,-7) Matrix(577,-104,172,-31) -> Matrix(5,8,-12,-19) Matrix(229,-42,518,-95) -> Matrix(3,4,-4,-5) Matrix(485,-92,58,-11) -> Matrix(1,0,-2,1) Matrix(285,-56,56,-11) -> Matrix(1,2,-2,-3) Matrix(85,-18,392,-83) -> Matrix(11,12,-12,-13) Matrix(221,-50,84,-19) -> Matrix(3,2,-2,-1) Matrix(513,-134,134,-35) -> Matrix(7,6,-20,-17) Matrix(2179,-576,1290,-341) -> Matrix(13,10,-30,-23) Matrix(1397,-370,404,-107) -> Matrix(3,2,-2,-1) Matrix(1395,-374,884,-237) -> Matrix(1,0,0,1) Matrix(1259,-340,374,-101) -> Matrix(7,4,-16,-9) Matrix(239,-66,134,-37) -> Matrix(5,2,-8,-3) Matrix(341,-100,474,-139) -> Matrix(1,0,4,1) Matrix(629,-186,1444,-427) -> Matrix(1,0,0,1) Matrix(577,-172,104,-31) -> Matrix(1,0,2,1) Matrix(105,-32,338,-103) -> Matrix(1,0,6,1) Matrix(75,-26,26,-9) -> Matrix(1,2,-2,-3) Matrix(221,-84,50,-19) -> Matrix(1,0,-4,1) Matrix(637,-246,246,-95) -> Matrix(1,-2,0,1) Matrix(121,-50,288,-119) -> Matrix(3,4,-4,-5) Matrix(1257,-544,238,-103) -> Matrix(3,2,-8,-5) Matrix(2651,-1156,782,-341) -> Matrix(7,2,-18,-5) Matrix(259,-114,284,-125) -> Matrix(3,4,-4,-5) Matrix(187,-84,118,-53) -> Matrix(1,2,-2,-3) Matrix(133,-72,242,-131) -> Matrix(1,2,-2,-3) Matrix(239,-134,66,-37) -> Matrix(3,2,-8,-5) Matrix(817,-474,474,-275) -> Matrix(3,4,-4,-5) Matrix(235,-138,172,-101) -> Matrix(5,4,-4,-3) Matrix(105,-64,64,-39) -> Matrix(5,2,-8,-3) Matrix(187,-118,84,-53) -> Matrix(1,0,-2,1) Matrix(83,-54,20,-13) -> Matrix(1,0,-2,1) Matrix(141,-98,200,-139) -> Matrix(1,0,8,1) Matrix(2687,-1936,712,-513) -> Matrix(1,0,-4,1) Matrix(2923,-2108,850,-613) -> Matrix(1,0,-4,1) Matrix(749,-542,474,-343) -> Matrix(3,-2,-4,3) Matrix(235,-172,138,-101) -> Matrix(1,0,-2,1) Matrix(171,-134,134,-105) -> Matrix(1,-4,0,1) Matrix(131,-108,74,-61) -> Matrix(1,4,-2,-7) Matrix(925,-776,776,-651) -> Matrix(1,2,-4,-7) Matrix(1217,-1026,720,-607) -> Matrix(5,8,-12,-19) Matrix(617,-538,164,-143) -> Matrix(3,4,-10,-13) Matrix(145,-128,162,-143) -> Matrix(13,14,-14,-15) Matrix(259,-286,48,-53) -> Matrix(1,0,4,1) Matrix(429,-488,80,-91) -> Matrix(1,0,-8,1) Matrix(711,-818,206,-237) -> Matrix(1,0,-4,1) Matrix(887,-1050,234,-277) -> Matrix(1,-2,-2,5) Matrix(4621,-5512,1364,-1627) -> Matrix(15,2,-38,-5) Matrix(107,-132,30,-37) -> Matrix(1,-2,-2,5) Matrix(661,-834,256,-323) -> Matrix(1,-6,0,1) Matrix(809,-1028,314,-399) -> Matrix(1,8,0,1) Matrix(1801,-2448,476,-647) -> Matrix(5,8,-12,-19) Matrix(663,-902,86,-117) -> Matrix(5,8,-12,-19) Matrix(185,-256,86,-119) -> Matrix(1,2,-4,-7) Matrix(341,-474,100,-139) -> Matrix(1,2,-4,-7) Matrix(141,-200,98,-139) -> Matrix(13,14,-14,-15) Matrix(173,-270,66,-103) -> Matrix(5,4,-4,-3) Matrix(3433,-5408,2178,-3431) -> Matrix(23,16,-36,-25) Matrix(547,-882,338,-545) -> Matrix(35,24,-54,-37) Matrix(525,-902,188,-323) -> Matrix(5,6,-6,-7) Matrix(133,-242,72,-131) -> Matrix(15,8,-32,-17) Matrix(321,-698,86,-187) -> Matrix(7,2,-18,-5) Matrix(229,-518,42,-95) -> Matrix(1,0,2,1) Matrix(629,-1444,186,-427) -> Matrix(3,2,-8,-5) Matrix(93,-214,10,-23) -> Matrix(1,0,0,1) Matrix(121,-288,50,-119) -> Matrix(1,0,8,1) Matrix(119,-324,18,-49) -> Matrix(1,2,-2,-3) Matrix(451,-1250,162,-449) -> Matrix(13,14,-14,-15) Matrix(105,-338,32,-103) -> Matrix(15,8,-32,-17) Matrix(6511,-22050,1922,-6509) -> Matrix(119,48,-300,-121) Matrix(997,-3422,192,-659) -> Matrix(5,2,-8,-3) Matrix(1625,-5584,314,-1079) -> Matrix(1,0,2,1) Matrix(203,-750,36,-133) -> Matrix(5,2,-8,-3) Matrix(2413,-8978,648,-2411) -> Matrix(11,4,-36,-13) Matrix(379,-1458,98,-377) -> Matrix(41,14,-126,-43) Matrix(85,-392,18,-83) -> Matrix(1,0,12,1) Matrix(291,-1682,50,-289) -> Matrix(1,0,8,1) Matrix(61,-450,8,-59) -> Matrix(7,4,-16,-9) 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): 40 Degree of the the map X: 40 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 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): 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 0/1 3/14 4/13 5/12 1/2 6/11 7/10 1/1 9/8 19/15 10/7 3/2 11/6 2/1 12/5 5/2 25/9 3/1 13/4 7/2 67/18 4/1 9/2 14/3 5/1 11/2 6/1 13/2 7/1 15/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/2 -5/11 1/0 -9/20 0/1 1/6 -4/9 1/4 -3/7 1/2 -5/12 0/1 1/0 -7/17 1/4 -2/5 1/2 -3/8 2/3 1/1 -7/19 5/4 -4/11 1/0 -5/14 -1/1 1/0 -6/17 -1/2 -1/3 1/2 -2/7 1/2 -3/11 1/2 -4/15 1/0 -5/19 1/2 -1/4 0/1 1/1 -1/5 1/0 0/1 1/0 1/6 -2/1 1/0 2/11 -3/2 1/5 -3/2 3/14 -1/1 2/9 -3/4 1/4 -1/1 0/1 3/11 -1/2 2/7 -1/2 5/17 -1/2 8/27 -1/2 3/10 -1/3 0/1 4/13 0/1 1/3 1/0 4/11 -3/4 3/8 -1/2 0/1 5/13 1/0 7/18 -3/1 1/0 2/5 1/0 5/12 -1/1 3/7 -1/2 7/16 0/1 1/0 4/9 -1/2 1/2 -1/1 0/1 6/11 -1/1 11/20 -1/1 -2/3 5/9 -1/2 4/7 1/0 11/19 1/0 7/12 -1/1 -2/3 3/5 -1/2 5/8 -1/1 0/1 7/11 -1/2 2/3 -1/2 7/10 0/1 5/7 1/2 8/11 1/2 3/4 0/1 1/0 7/9 1/0 4/5 1/0 5/6 -2/1 1/0 1/1 -1/2 9/8 0/1 8/7 1/4 7/6 0/1 1/1 6/5 1/2 5/4 1/1 1/0 19/15 1/0 14/11 1/0 9/7 1/0 22/17 1/0 13/10 -2/1 -3/2 4/3 1/0 19/14 -2/1 -3/2 15/11 -3/2 11/8 -2/1 -1/1 18/13 -3/2 7/5 -5/4 10/7 -1/1 13/9 -9/10 3/2 -1/1 -2/3 11/7 -3/4 8/5 -3/4 21/13 -2/3 13/8 -2/3 -3/5 5/3 -1/2 17/10 -1/2 0/1 12/7 1/0 31/18 -1/1 -6/7 19/11 -3/4 7/4 -2/3 -1/2 16/9 -1/2 9/5 -1/2 11/6 -1/2 2/1 -1/2 13/6 -1/3 0/1 11/5 -1/4 9/4 -1/3 0/1 25/11 1/0 16/7 -1/4 23/10 -1/2 0/1 7/3 -1/4 12/5 0/1 17/7 1/4 5/2 0/1 1/0 18/7 1/0 49/19 1/0 31/12 -5/1 1/0 13/5 1/0 21/8 -2/1 -1/1 8/3 -1/2 19/7 1/0 11/4 -2/1 -1/1 25/9 -1/1 39/14 -1/1 -12/13 14/5 -5/6 3/1 -1/2 13/4 -1/2 10/3 -1/2 17/5 -3/8 24/7 -1/2 31/9 -1/4 7/2 -1/2 -1/3 18/5 -3/8 11/3 -1/2 26/7 -3/8 67/18 -1/3 41/11 -1/2 15/4 -2/5 -1/3 19/5 -3/8 4/1 -1/4 13/3 -1/4 9/2 -1/6 0/1 14/3 0/1 19/4 0/1 1/6 5/1 1/0 11/2 -1/1 0/1 28/5 1/0 17/3 -1/2 6/1 1/0 13/2 -1/1 0/1 7/1 -1/2 15/2 -1/2 8/1 -1/2 1/0 -1/2 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(285,131,124,57) (-1/2,-5/11) -> (16/7,23/10) 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(209,91,62,27) (-4/9,-3/7) -> (10/3,17/5) Hyperbolic Matrix(31,13,174,73) (-3/7,-5/12) -> (1/6,2/11) Hyperbolic Matrix(289,120,118,49) (-5/12,-7/17) -> (17/7,5/2) Hyperbolic Matrix(319,131,56,23) (-7/17,-2/5) -> (17/3,6/1) Hyperbolic Matrix(29,11,108,41) (-2/5,-3/8) -> (1/4,3/11) Hyperbolic Matrix(161,60,110,41) (-3/8,-7/19) -> (13/9,3/2) Hyperbolic Matrix(405,149,106,39) (-7/19,-4/11) -> (19/5,4/1) Hyperbolic Matrix(105,38,268,97) (-4/11,-5/14) -> (7/18,2/5) Hyperbolic Matrix(553,197,160,57) (-5/14,-6/17) -> (31/9,7/2) Hyperbolic Matrix(473,166,208,73) (-6/17,-1/3) -> (25/11,16/7) Hyperbolic Matrix(71,21,98,29) (-1/3,-2/7) -> (5/7,8/11) Hyperbolic Matrix(69,19,236,65) (-2/7,-3/11) -> (2/7,5/17) Hyperbolic Matrix(373,100,138,37) (-3/11,-4/15) -> (8/3,19/7) Hyperbolic Matrix(949,251,276,73) (-4/15,-5/19) -> (24/7,31/9) Hyperbolic Matrix(229,60,416,109) (-5/19,-1/4) -> (11/20,5/9) Hyperbolic Matrix(21,5,46,11) (-1/4,-1/5) -> (4/9,1/2) Hyperbolic Matrix(63,11,40,7) (-1/5,0/1) -> (11/7,8/5) Hyperbolic Matrix(45,-7,58,-9) (0/1,1/6) -> (3/4,7/9) Hyperbolic Matrix(99,-19,172,-33) (2/11,1/5) -> (4/7,11/19) Hyperbolic Matrix(43,-9,196,-41) (1/5,3/14) -> (3/14,2/9) Parabolic Matrix(221,-50,84,-19) (2/9,1/4) -> (21/8,8/3) Hyperbolic Matrix(239,-66,134,-37) (3/11,2/7) -> (16/9,9/5) Hyperbolic Matrix(747,-221,578,-171) (5/17,8/27) -> (9/7,22/17) Hyperbolic Matrix(577,-172,104,-31) (8/27,3/10) -> (11/2,28/5) Hyperbolic Matrix(379,-115,234,-71) (3/10,4/13) -> (21/13,13/8) Hyperbolic Matrix(167,-53,104,-33) (4/13,1/3) -> (8/5,21/13) Hyperbolic Matrix(75,-26,26,-9) (1/3,4/11) -> (14/5,3/1) Hyperbolic Matrix(63,-23,74,-27) (4/11,3/8) -> (5/6,1/1) 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(61,-25,144,-59) (2/5,5/12) -> (5/12,3/7) Parabolic Matrix(155,-67,118,-51) (3/7,7/16) -> (13/10,4/3) Hyperbolic Matrix(367,-161,212,-93) (7/16,4/9) -> (19/11,7/4) 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(71,-49,100,-69) (2/3,7/10) -> (7/10,5/7) Parabolic 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(73,-81,64,-71) (1/1,9/8) -> (9/8,8/7) Parabolic Matrix(213,-247,94,-109) (8/7,7/6) -> (9/4,25/11) Hyperbolic Matrix(85,-101,16,-19) (7/6,6/5) -> (5/1,11/2) 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(167,-225,72,-97) (4/3,19/14) -> (23/10,7/3) Hyperbolic Matrix(123,-167,14,-19) (19/14,15/11) -> (8/1,1/0) 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(195,-331,76,-129) (5/3,17/10) -> (5/2,18/7) Hyperbolic Matrix(525,-902,188,-323) (12/7,31/18) -> (39/14,14/5) Hyperbolic Matrix(67,-121,36,-65) (9/5,11/6) -> (11/6,2/1) Parabolic Matrix(321,-698,86,-187) (13/6,11/5) -> (41/11,15/4) 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(53,-169,16,-51) (3/1,13/4) -> (13/4,10/3) Parabolic Matrix(203,-750,36,-133) (11/3,26/7) -> (28/5,17/3) Hyperbolic Matrix(1207,-4489,324,-1205) (26/7,67/18) -> (67/18,41/11) Parabolic Matrix(89,-337,14,-53) (15/4,19/5) -> (6/1,13/2) Hyperbolic Matrix(85,-392,18,-83) (9/2,14/3) -> (14/3,19/4) Parabolic Matrix(31,-225,4,-29) (7/1,15/2) -> (15/2,8/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(285,131,124,57) -> Matrix(1,0,-4,1) Matrix(309,140,64,29) -> Matrix(1,0,0,1) Matrix(557,250,430,193) -> Matrix(9,-2,-4,1) Matrix(209,91,62,27) -> Matrix(5,-2,-12,5) Matrix(31,13,174,73) -> Matrix(1,-2,0,1) Matrix(289,120,118,49) -> Matrix(1,0,0,1) Matrix(319,131,56,23) -> Matrix(1,0,-4,1) Matrix(29,11,108,41) -> Matrix(3,-2,-4,3) Matrix(161,60,110,41) -> Matrix(5,-4,-6,5) Matrix(405,149,106,39) -> Matrix(3,-4,-8,11) Matrix(105,38,268,97) -> Matrix(1,-2,0,1) Matrix(553,197,160,57) -> Matrix(1,0,-2,1) Matrix(473,166,208,73) -> Matrix(1,0,-2,1) Matrix(71,21,98,29) -> Matrix(1,0,0,1) Matrix(69,19,236,65) -> Matrix(1,0,-4,1) Matrix(373,100,138,37) -> Matrix(1,0,-2,1) Matrix(949,251,276,73) -> Matrix(1,0,-4,1) Matrix(229,60,416,109) -> Matrix(3,-2,-4,3) Matrix(21,5,46,11) -> Matrix(1,0,-2,1) Matrix(63,11,40,7) -> Matrix(3,-2,-4,3) Matrix(45,-7,58,-9) -> Matrix(1,2,0,1) Matrix(99,-19,172,-33) -> Matrix(1,2,-2,-3) Matrix(43,-9,196,-41) -> Matrix(5,6,-6,-7) Matrix(221,-50,84,-19) -> Matrix(3,2,-2,-1) Matrix(239,-66,134,-37) -> Matrix(5,2,-8,-3) Matrix(747,-221,578,-171) -> Matrix(3,2,-2,-1) Matrix(577,-172,104,-31) -> Matrix(1,0,2,1) Matrix(379,-115,234,-71) -> Matrix(9,2,-14,-3) Matrix(167,-53,104,-33) -> Matrix(3,-2,-4,3) Matrix(75,-26,26,-9) -> Matrix(1,2,-2,-3) Matrix(63,-23,74,-27) -> Matrix(3,2,-2,-1) Matrix(221,-84,50,-19) -> Matrix(1,0,-4,1) Matrix(637,-246,246,-95) -> Matrix(1,-2,0,1) Matrix(61,-25,144,-59) -> Matrix(1,2,-2,-3) Matrix(155,-67,118,-51) -> Matrix(3,2,-2,-1) Matrix(367,-161,212,-93) -> Matrix(1,2,-2,-3) Matrix(133,-72,242,-131) -> Matrix(1,2,-2,-3) Matrix(239,-134,66,-37) -> Matrix(3,2,-8,-5) Matrix(817,-474,474,-275) -> Matrix(3,4,-4,-5) Matrix(235,-138,172,-101) -> Matrix(5,4,-4,-3) Matrix(105,-64,64,-39) -> Matrix(5,2,-8,-3) Matrix(187,-118,84,-53) -> Matrix(1,0,-2,1) Matrix(83,-54,20,-13) -> Matrix(1,0,-2,1) Matrix(71,-49,100,-69) -> Matrix(1,0,4,1) Matrix(235,-172,138,-101) -> Matrix(1,0,-2,1) Matrix(171,-134,134,-105) -> Matrix(1,-4,0,1) Matrix(131,-108,74,-61) -> Matrix(1,4,-2,-7) Matrix(73,-81,64,-71) -> Matrix(1,0,6,1) Matrix(213,-247,94,-109) -> Matrix(1,0,-4,1) Matrix(85,-101,16,-19) -> Matrix(1,0,-2,1) Matrix(107,-132,30,-37) -> Matrix(1,-2,-2,5) Matrix(661,-834,256,-323) -> Matrix(1,-6,0,1) Matrix(809,-1028,314,-399) -> Matrix(1,8,0,1) Matrix(167,-225,72,-97) -> Matrix(1,2,-4,-7) Matrix(123,-167,14,-19) -> Matrix(1,2,-4,-7) Matrix(185,-256,86,-119) -> Matrix(1,2,-4,-7) Matrix(341,-474,100,-139) -> Matrix(1,2,-4,-7) Matrix(141,-200,98,-139) -> Matrix(13,14,-14,-15) Matrix(173,-270,66,-103) -> Matrix(5,4,-4,-3) Matrix(195,-331,76,-129) -> Matrix(1,0,2,1) Matrix(525,-902,188,-323) -> Matrix(5,6,-6,-7) Matrix(67,-121,36,-65) -> Matrix(7,4,-16,-9) Matrix(321,-698,86,-187) -> Matrix(7,2,-18,-5) Matrix(121,-288,50,-119) -> Matrix(1,0,8,1) Matrix(119,-324,18,-49) -> Matrix(1,2,-2,-3) Matrix(451,-1250,162,-449) -> Matrix(13,14,-14,-15) Matrix(53,-169,16,-51) -> Matrix(7,4,-16,-9) Matrix(203,-750,36,-133) -> Matrix(5,2,-8,-3) Matrix(1207,-4489,324,-1205) -> Matrix(5,2,-18,-7) Matrix(89,-337,14,-53) -> Matrix(5,2,-8,-3) Matrix(85,-392,18,-83) -> Matrix(1,0,12,1) Matrix(31,-225,4,-29) -> Matrix(3,2,-8,-5) 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): 20 ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d -1/1 0/1 2 1 0/1 1/0 1 17 1/5 -3/2 1 17 3/14 -1/1 6 1 1/4 (-1/1,0/1) 0 17 1/3 1/0 1 17 4/11 -3/4 1 17 3/8 (-1/2,0/1) 0 17 5/13 1/0 1 17 2/5 1/0 1 17 5/12 -1/1 2 1 1/2 (-1/1,0/1) 0 17 6/11 -1/1 1 1 5/9 -1/2 1 17 4/7 1/0 1 17 7/12 (-1/1,-2/3) 0 17 3/5 -1/2 1 17 5/8 (-1/1,0/1) 0 17 7/11 -1/2 1 17 2/3 -1/2 1 17 7/10 0/1 4 1 3/4 (0/1,1/0) 0 17 7/9 1/0 1 17 4/5 1/0 1 17 1/1 -1/2 1 17 9/8 0/1 6 1 7/6 (0/1,1/1) 0 17 6/5 1/2 1 17 5/4 (1/1,1/0) 0 17 19/15 1/0 7 1 14/11 1/0 1 17 9/7 1/0 1 17 4/3 1/0 1 17 15/11 -3/2 1 17 11/8 (-2/1,-1/1) 0 17 7/5 -5/4 1 17 10/7 -1/1 7 1 3/2 (-1/1,-2/3) 0 17 8/5 -3/4 1 17 21/13 -2/3 3 1 13/8 (-2/3,-3/5) 0 17 5/3 -1/2 1 17 12/7 1/0 1 17 7/4 (-2/3,-1/2) 0 17 11/6 -1/2 4 1 2/1 -1/2 1 17 11/5 -1/4 1 17 9/4 (-1/3,0/1) 0 17 7/3 -1/4 1 17 12/5 0/1 4 1 5/2 (0/1,1/0) 0 17 13/5 1/0 1 17 8/3 -1/2 1 17 11/4 (-2/1,-1/1) 0 17 25/9 -1/1 7 1 14/5 -5/6 1 17 3/1 -1/2 1 17 13/4 -1/2 4 1 7/2 (-1/2,-1/3) 0 17 18/5 -3/8 1 17 11/3 -1/2 1 17 26/7 -3/8 1 17 67/18 -1/3 2 1 15/4 (-2/5,-1/3) 0 17 4/1 -1/4 1 17 13/3 -1/4 1 17 9/2 (-1/6,0/1) 0 17 14/3 0/1 6 1 5/1 1/0 1 17 11/2 (-1/1,0/1) 0 17 6/1 1/0 1 17 7/1 -1/2 1 17 15/2 -1/2 2 1 1/0 (-1/2,0/1) 0 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(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(73,-13,28,-5) (0/1,1/5) -> (13/5,8/3) Glide Reflection Matrix(29,-6,140,-29) (1/5,3/14) -> (1/5,3/14) Reflection Matrix(13,-3,56,-13) (3/14,1/4) -> (3/14,1/4) Reflection Matrix(41,-11,26,-7) (1/4,1/3) -> (3/2,8/5) Glide Reflection Matrix(75,-26,26,-9) (1/3,4/11) -> (14/5,3/1) Hyperbolic Matrix(173,-64,100,-37) (4/11,3/8) -> (12/7,7/4) Glide Reflection Matrix(221,-84,50,-19) (3/8,5/13) -> (13/3,9/2) Hyperbolic Matrix(97,-38,74,-29) (5/13,2/5) -> (9/7,4/3) Glide Reflection Matrix(49,-20,120,-49) (2/5,5/12) -> (2/5,5/12) Reflection Matrix(11,-5,24,-11) (5/12,1/2) -> (5/12,1/2) Reflection Matrix(23,-12,44,-23) (1/2,6/11) -> (1/2,6/11) Reflection Matrix(109,-60,198,-109) (6/11,5/9) -> (6/11,5/9) Reflection Matrix(239,-134,66,-37) (5/9,4/7) -> (18/5,11/3) Hyperbolic Matrix(173,-100,64,-37) (4/7,7/12) -> (8/3,11/4) Glide Reflection 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(41,-28,60,-41) (2/3,7/10) -> (2/3,7/10) Reflection Matrix(29,-21,40,-29) (7/10,3/4) -> (7/10,3/4) Reflection Matrix(97,-74,38,-29) (3/4,7/9) -> (5/2,13/5) Glide Reflection Matrix(171,-134,134,-105) (7/9,4/5) -> (14/11,9/7) Hyperbolic Matrix(65,-53,38,-31) (4/5,1/1) -> (5/3,12/7) Glide Reflection Matrix(17,-18,16,-17) (1/1,9/8) -> (1/1,9/8) Reflection Matrix(55,-63,48,-55) (9/8,7/6) -> (9/8,7/6) Reflection Matrix(85,-101,16,-19) (7/6,6/5) -> (5/1,11/2) Hyperbolic 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(419,-532,330,-419) (19/15,14/11) -> (19/15,14/11) Reflection Matrix(87,-118,14,-19) (4/3,15/11) -> (6/1,7/1) Glide Reflection Matrix(107,-149,28,-39) (11/8,7/5) -> (15/4,4/1) Glide Reflection 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(209,-336,130,-209) (8/5,21/13) -> (8/5,21/13) Reflection Matrix(337,-546,208,-337) (21/13,13/8) -> (21/13,13/8) Reflection Matrix(43,-77,24,-43) (7/4,11/6) -> (7/4,11/6) Reflection Matrix(23,-44,12,-23) (11/6,2/1) -> (11/6,2/1) Reflection Matrix(81,-173,22,-47) (2/1,11/5) -> (11/3,26/7) Glide Reflection Matrix(57,-131,10,-23) (9/4,7/3) -> (11/2,6/1) Glide Reflection 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(199,-550,72,-199) (11/4,25/9) -> (11/4,25/9) Reflection Matrix(251,-700,90,-251) (25/9,14/5) -> (25/9,14/5) Reflection Matrix(25,-78,8,-25) (3/1,13/4) -> (3/1,13/4) Reflection Matrix(27,-91,8,-27) (13/4,7/2) -> (13/4,7/2) Reflection Matrix(937,-3484,252,-937) (26/7,67/18) -> (26/7,67/18) Reflection Matrix(269,-1005,72,-269) (67/18,15/4) -> (67/18,15/4) 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(29,-210,4,-29) (7/1,15/2) -> (7/1,15/2) Reflection Matrix(-1,15,0,1) (15/2,1/0) -> (15/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,4,1) (-1/1,1/0) -> (-1/2,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(73,-13,28,-5) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(29,-6,140,-29) -> Matrix(5,6,-4,-5) (1/5,3/14) -> (-3/2,-1/1) Matrix(13,-3,56,-13) -> Matrix(-1,0,2,1) (3/14,1/4) -> (-1/1,0/1) Matrix(41,-11,26,-7) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(75,-26,26,-9) -> Matrix(1,2,-2,-3) -1/1 Matrix(173,-64,100,-37) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(221,-84,50,-19) -> Matrix(1,0,-4,1) 0/1 Matrix(97,-38,74,-29) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(49,-20,120,-49) -> Matrix(1,2,0,-1) (2/5,5/12) -> (-1/1,1/0) Matrix(11,-5,24,-11) -> Matrix(-1,0,2,1) (5/12,1/2) -> (-1/1,0/1) Matrix(23,-12,44,-23) -> Matrix(-1,0,2,1) (1/2,6/11) -> (-1/1,0/1) Matrix(109,-60,198,-109) -> Matrix(3,2,-4,-3) (6/11,5/9) -> (-1/1,-1/2) Matrix(239,-134,66,-37) -> Matrix(3,2,-8,-5) -1/2 Matrix(173,-100,64,-37) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(235,-138,172,-101) -> Matrix(5,4,-4,-3) -1/1 Matrix(105,-64,64,-39) -> Matrix(5,2,-8,-3) -1/2 Matrix(187,-118,84,-53) -> Matrix(1,0,-2,1) 0/1 Matrix(83,-54,20,-13) -> Matrix(1,0,-2,1) 0/1 Matrix(41,-28,60,-41) -> Matrix(-1,0,4,1) (2/3,7/10) -> (-1/2,0/1) Matrix(29,-21,40,-29) -> Matrix(1,0,0,-1) (7/10,3/4) -> (0/1,1/0) Matrix(97,-74,38,-29) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(171,-134,134,-105) -> Matrix(1,-4,0,1) 1/0 Matrix(65,-53,38,-31) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(17,-18,16,-17) -> Matrix(-1,0,4,1) (1/1,9/8) -> (-1/2,0/1) Matrix(55,-63,48,-55) -> Matrix(1,0,2,-1) (9/8,7/6) -> (0/1,1/1) Matrix(85,-101,16,-19) -> Matrix(1,0,-2,1) 0/1 Matrix(107,-132,30,-37) -> Matrix(1,-2,-2,5) Matrix(151,-190,120,-151) -> Matrix(-1,2,0,1) (5/4,19/15) -> (1/1,1/0) Matrix(419,-532,330,-419) -> Matrix(1,12,0,-1) (19/15,14/11) -> (-6/1,1/0) Matrix(87,-118,14,-19) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(107,-149,28,-39) -> Matrix(3,4,-8,-11) Matrix(99,-140,70,-99) -> Matrix(9,10,-8,-9) (7/5,10/7) -> (-5/4,-1/1) Matrix(41,-60,28,-41) -> Matrix(5,4,-6,-5) (10/7,3/2) -> (-1/1,-2/3) Matrix(209,-336,130,-209) -> Matrix(17,12,-24,-17) (8/5,21/13) -> (-3/4,-2/3) Matrix(337,-546,208,-337) -> Matrix(19,12,-30,-19) (21/13,13/8) -> (-2/3,-3/5) Matrix(43,-77,24,-43) -> Matrix(7,4,-12,-7) (7/4,11/6) -> (-2/3,-1/2) Matrix(23,-44,12,-23) -> Matrix(-1,0,4,1) (11/6,2/1) -> (-1/2,0/1) Matrix(81,-173,22,-47) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(57,-131,10,-23) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(71,-168,30,-71) -> Matrix(-1,0,8,1) (7/3,12/5) -> (-1/4,0/1) Matrix(49,-120,20,-49) -> Matrix(1,0,0,-1) (12/5,5/2) -> (0/1,1/0) Matrix(199,-550,72,-199) -> Matrix(3,4,-2,-3) (11/4,25/9) -> (-2/1,-1/1) Matrix(251,-700,90,-251) -> Matrix(11,10,-12,-11) (25/9,14/5) -> (-1/1,-5/6) Matrix(25,-78,8,-25) -> Matrix(3,2,-4,-3) (3/1,13/4) -> (-1/1,-1/2) Matrix(27,-91,8,-27) -> Matrix(5,2,-12,-5) (13/4,7/2) -> (-1/2,-1/3) Matrix(937,-3484,252,-937) -> Matrix(17,6,-48,-17) (26/7,67/18) -> (-3/8,-1/3) Matrix(269,-1005,72,-269) -> Matrix(11,4,-30,-11) (67/18,15/4) -> (-2/5,-1/3) Matrix(55,-252,12,-55) -> Matrix(-1,0,12,1) (9/2,14/3) -> (-1/6,0/1) Matrix(29,-140,6,-29) -> Matrix(1,0,0,-1) (14/3,5/1) -> (0/1,1/0) Matrix(29,-210,4,-29) -> Matrix(3,2,-4,-3) (7/1,15/2) -> (-1/1,-1/2) Matrix(-1,15,0,1) -> Matrix(-1,0,4,1) (15/2,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.