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): 1152 Minimal number of generators: 193 Number of equivalence classes of cusps: 80 Genus: 57 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -4/9 -3/7 -7/17 -3/8 -1/3 -2/7 -1/5 -1/6 -1/7 0/1 1/7 3/17 1/4 1/3 3/7 1/2 7/13 3/5 2/3 5/7 9/11 13/15 7/8 1/1 11/9 14/11 9/7 23/17 7/5 3/2 61/39 31/19 5/3 19/11 16/9 9/5 13/7 2/1 41/19 11/5 7/3 17/7 5/2 18/7 13/5 29/11 8/3 117/43 11/4 147/53 3/1 61/19 23/7 10/3 17/5 31/9 7/2 39/11 11/3 15/4 4/1 13/3 9/2 23/5 14/3 33/7 5/1 21/4 16/3 11/2 17/3 6/1 13/2 7/1 22/3 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 -1/2 1/0 -1/2 -1/1 0/1 -7/15 -1/2 1/0 -6/13 -4/3 -1/1 -5/11 -3/4 -9/20 -1/1 -2/3 -13/29 -3/4 -1/2 -4/9 -1/2 -11/25 -1/2 -7/16 -1/1 0/1 -10/23 0/1 1/1 -3/7 -1/1 -14/33 -1/1 -2/3 -11/26 -3/4 -8/19 -1/1 -2/3 -13/31 -1/2 -31/74 -1/1 -4/5 -18/43 -3/4 -5/12 -2/3 -3/5 -12/29 -3/5 -4/7 -7/17 -1/2 -9/22 -1/3 0/1 -11/27 -1/2 -1/4 -2/5 -1/1 0/1 -9/23 -3/4 -7/18 -1/1 -2/3 -5/13 -3/4 -1/2 -8/21 -2/3 -7/11 -19/50 -2/3 -3/5 -11/29 -5/8 -3/8 -1/2 -13/35 -3/8 -10/27 -1/3 0/1 -7/19 0/1 -25/68 -1/1 0/1 -18/49 -1/1 0/1 -11/30 -1/3 0/1 -4/11 -1/1 0/1 -9/25 -1/2 1/0 -14/39 -1/2 -5/14 -1/1 0/1 -6/17 -1/1 -2/3 -7/20 -1/1 -2/3 -8/23 -1/2 -1/3 -1/2 -7/22 -1/2 -6/19 -1/3 0/1 -5/16 -1/3 0/1 -4/13 -1/1 0/1 -11/36 -1/2 -7/23 -1/2 1/0 -10/33 -1/1 -2/3 -3/10 -1/1 0/1 -8/27 -1/1 -2/3 -13/44 -1/1 0/1 -18/61 -1/1 0/1 -5/17 -1/1 -7/24 -1/1 -2/3 -2/7 -1/2 -9/32 -3/7 -2/5 -7/25 -3/8 -5/18 -4/11 -1/3 -3/11 -1/2 -1/4 -4/15 -1/3 0/1 -9/34 -1/4 -5/19 -1/4 -1/4 -1/1 0/1 -5/21 -3/4 -1/2 -9/38 -1/1 -2/3 -4/17 -1/1 -2/3 -3/13 -1/2 -5/22 -3/7 -2/5 -7/31 -1/2 -3/8 -2/9 -2/5 -1/3 -9/41 -3/10 -1/4 -7/32 -1/4 -12/55 -1/5 0/1 -5/23 -1/2 -8/37 -1/3 0/1 -3/14 -1/3 0/1 -7/33 -1/4 -4/19 -1/4 -5/24 -1/3 0/1 -1/5 0/1 -3/16 -2/1 -1/1 -5/27 1/0 -7/38 -1/1 0/1 -2/11 -1/1 0/1 -3/17 -1/2 -1/6 -1/2 -3/19 -1/2 -1/4 -5/32 -1/3 0/1 -2/13 -1/3 0/1 -1/7 -1/4 -1/8 0/1 1/3 -2/17 1/0 -1/9 -1/2 1/0 0/1 -1/1 0/1 1/8 -1/1 0/1 1/7 -1/2 1/0 1/6 -1/5 0/1 3/17 0/1 2/11 0/1 1/9 1/5 1/2 3/14 0/1 1/1 5/23 1/1 2/9 0/1 1/1 3/13 1/2 1/0 1/4 1/0 5/19 -3/2 1/0 9/34 -4/3 -1/1 4/15 -1/1 -2/3 7/26 -1/1 0/1 3/11 1/0 8/29 -4/3 -1/1 5/18 -1/1 -2/3 7/25 -1/2 1/0 9/32 -1/1 0/1 2/7 -1/1 0/1 3/10 -1/1 -2/3 7/23 -1/2 4/13 -1/3 0/1 1/3 0/1 6/17 0/1 1/1 11/31 1/2 5/14 0/1 1/1 4/11 1/2 3/8 0/1 1/1 5/13 1/0 12/31 0/1 1/3 7/18 1/2 2/5 1/1 2/1 5/12 2/1 3/1 3/7 1/0 7/16 -4/1 -3/1 18/41 -4/1 -3/1 11/25 -3/1 4/9 -2/1 -1/1 5/11 -3/2 1/0 6/13 -6/5 -1/1 1/2 -1/1 0/1 7/13 0/1 6/11 0/1 1/5 17/31 1/4 1/2 11/20 0/1 1/3 5/9 1/2 4/7 0/1 1/1 19/33 1/1 15/26 1/1 2/1 11/19 1/0 7/12 0/1 1/1 17/29 -1/2 10/17 0/1 1/5 3/5 1/2 1/0 11/18 0/1 1/1 19/31 1/2 27/44 2/3 1/1 8/13 0/1 1/1 13/21 3/4 18/29 10/11 1/1 5/8 1/1 4/3 12/19 1/1 2/1 7/11 3/2 2/3 1/0 9/13 -5/2 25/36 -23/11 -2/1 41/59 -2/1 16/23 -2/1 -13/7 7/10 -2/1 -1/1 19/27 1/0 12/17 -2/1 -1/1 17/24 -4/3 -1/1 22/31 -2/1 -1/1 5/7 -1/1 28/39 -1/1 -2/3 23/32 -1/2 18/25 -1/1 0/1 13/18 -1/1 0/1 8/11 -2/1 -1/1 27/37 -3/4 -1/2 19/26 -1/2 11/15 -1/2 3/4 -1/1 0/1 7/9 -1/2 1/0 25/32 -1/3 0/1 18/23 0/1 1/3 29/37 1/0 11/14 1/0 15/19 -1/2 4/5 -1/1 0/1 9/11 0/1 14/17 0/1 1/3 5/6 0/1 1/1 6/7 0/1 1/1 13/15 1/1 20/23 1/1 2/1 7/8 1/0 15/17 -1/2 1/0 8/9 0/1 1/1 1/1 1/0 10/9 -2/1 -1/1 9/8 -2/1 -1/1 8/7 1/0 23/20 -3/1 -2/1 15/13 -2/1 7/6 -2/1 -1/1 6/5 -2/1 -1/1 11/9 -1/1 16/13 -1/1 -2/3 5/4 -1/1 0/1 24/19 -1/1 0/1 19/15 -1/2 33/26 0/1 1/1 14/11 1/0 37/29 1/0 60/47 -2/1 -1/1 23/18 -4/3 -1/1 9/7 -1/2 1/0 22/17 -1/5 0/1 35/27 1/4 13/10 0/1 1/1 4/3 -1/1 0/1 23/17 -1/2 1/0 19/14 -1/1 0/1 15/11 -1/2 26/19 -1/2 11/8 0/1 1/1 29/21 1/0 18/13 -1/1 0/1 43/31 -1/2 68/49 -1/1 0/1 93/67 -1/2 1/0 25/18 -1/1 0/1 7/5 0/1 31/22 0/1 1/1 86/61 0/1 1/1 55/39 1/0 24/17 0/1 1/3 41/29 1/2 17/12 0/1 1/1 27/19 1/0 37/26 0/1 1/1 10/7 0/1 1/1 3/2 1/0 14/9 -4/1 -3/1 39/25 -7/2 -13/4 25/16 -22/7 -3/1 61/39 -3/1 36/23 -3/1 -32/11 47/30 -3/1 -14/5 11/7 -5/2 8/5 -7/3 -2/1 37/23 -2/1 29/18 -2/1 -21/11 21/13 -7/4 76/47 -2/1 -5/3 55/34 -2/1 -5/3 34/21 -3/2 13/8 -2/1 -1/1 44/27 -2/1 -5/3 31/19 -3/2 80/49 -4/3 -1/1 49/30 -4/3 -1/1 18/11 -2/1 -1/1 5/3 -3/2 1/0 17/10 -6/5 -1/1 12/7 -2/1 -1/1 31/18 -1/1 0/1 19/11 1/0 45/26 -3/1 -2/1 26/15 -3/1 -2/1 7/4 -2/1 -1/1 16/9 -3/2 25/14 -1/1 0/1 59/33 -1/1 34/19 -2/1 -1/1 9/5 -3/2 20/11 -4/3 -1/1 11/6 -6/5 -1/1 13/7 -1/1 15/8 -1/1 -6/7 2/1 -1/1 0/1 15/7 -1/4 28/13 -1/13 0/1 41/19 0/1 13/6 0/1 1/5 24/11 1/2 11/5 1/2 1/0 20/9 0/1 1/1 9/4 0/1 1/1 7/3 1/0 19/8 -6/1 -5/1 31/13 -4/1 43/18 -4/1 -3/1 98/41 -4/1 -3/1 55/23 1/0 12/5 -4/1 -3/1 41/17 -13/4 29/12 -3/1 -14/5 46/19 -5/2 17/7 -5/2 1/0 5/2 -3/1 -2/1 23/9 -7/4 -3/2 41/16 -5/3 -8/5 18/7 -3/2 67/26 -7/5 -4/3 49/19 -4/3 31/12 -4/3 -1/1 44/17 -10/9 -1/1 101/39 -1/1 57/22 -1/1 -8/9 13/5 1/0 21/8 -2/1 -1/1 29/11 -3/2 1/0 37/14 -2/1 -1/1 8/3 -2/1 -1/1 19/7 -3/2 68/25 -2/1 -1/1 117/43 -3/2 1/0 166/61 -2/1 -1/1 49/18 -2/1 -1/1 30/11 -3/1 -2/1 41/15 -7/4 -3/2 11/4 -3/2 36/13 -4/3 -1/1 61/22 -4/3 -1/1 147/53 -3/2 -5/4 86/31 -4/3 -1/1 25/9 -5/4 14/5 -2/1 -1/1 3/1 -1/1 16/5 -1/1 -2/3 61/19 -3/4 -1/2 106/33 -1/1 -2/3 45/14 -1/1 -2/3 29/9 -1/2 42/13 -1/2 13/4 -1/1 -2/3 23/7 -1/2 10/3 -1/3 0/1 17/5 -1/2 24/7 -1/1 0/1 31/9 -1/2 1/0 38/11 -1/1 0/1 7/2 -1/1 0/1 39/11 -1/2 32/9 -1/1 0/1 25/7 -1/2 1/0 18/5 -1/3 0/1 47/13 0/1 29/8 0/1 1/3 11/3 1/0 37/10 -1/1 0/1 63/17 -1/2 1/0 26/7 -1/1 0/1 15/4 -1/3 0/1 4/1 1/0 17/4 -7/3 -2/1 47/11 -11/6 30/7 -2/1 -5/3 13/3 -3/2 1/0 22/5 -2/1 -1/1 9/2 -2/1 -1/1 32/7 -3/1 -2/1 55/12 -2/1 -1/1 23/5 -2/1 14/3 -2/1 -1/1 33/7 -3/2 1/0 52/11 -2/1 -1/1 19/4 -2/1 -1/1 5/1 -3/2 21/4 -5/4 16/3 -6/5 -1/1 11/2 -10/9 -1/1 17/3 -1/1 23/4 -1/1 -18/19 6/1 -1/1 -4/5 13/2 -1/2 33/5 -1/2 20/3 -1/3 0/1 7/1 -1/2 1/0 22/3 1/0 15/2 -2/1 -1/1 8/1 -1/1 0/1 9/1 1/0 1/0 -1/1 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(149,70,530,249) (-1/2,-7/15) -> (7/25,9/32) Hyperbolic Matrix(147,68,-668,-309) (-7/15,-6/13) -> (-2/9,-9/41) Hyperbolic Matrix(367,168,592,271) (-6/13,-5/11) -> (13/21,18/29) Hyperbolic Matrix(727,328,512,231) (-5/11,-9/20) -> (17/12,27/19) Hyperbolic Matrix(361,162,-1522,-683) (-9/20,-13/29) -> (-5/21,-9/38) Hyperbolic Matrix(795,356,364,163) (-13/29,-4/9) -> (24/11,11/5) Hyperbolic Matrix(785,346,574,253) (-4/9,-11/25) -> (15/11,26/19) Hyperbolic Matrix(355,156,644,283) (-11/25,-7/16) -> (11/20,5/9) Hyperbolic Matrix(565,246,-1486,-647) (-7/16,-10/23) -> (-8/21,-19/50) Hyperbolic Matrix(281,122,638,277) (-10/23,-3/7) -> (11/25,4/9) Hyperbolic Matrix(489,208,1112,473) (-3/7,-14/33) -> (18/41,11/25) Hyperbolic Matrix(1109,470,210,89) (-14/33,-11/26) -> (21/4,16/3) Hyperbolic Matrix(275,116,-1036,-437) (-11/26,-8/19) -> (-4/15,-9/34) Hyperbolic Matrix(343,144,-1584,-665) (-8/19,-13/31) -> (-5/23,-8/37) Hyperbolic Matrix(2271,952,1448,607) (-13/31,-31/74) -> (47/30,11/7) Hyperbolic Matrix(1581,662,1378,577) (-31/74,-18/43) -> (8/7,23/20) Hyperbolic Matrix(67,28,-548,-229) (-18/43,-5/12) -> (-1/8,-2/17) Hyperbolic Matrix(207,86,-686,-285) (-5/12,-12/29) -> (-10/33,-3/10) Hyperbolic Matrix(271,112,888,367) (-12/29,-7/17) -> (7/23,4/13) Hyperbolic Matrix(205,84,676,277) (-7/17,-9/22) -> (3/10,7/23) Hyperbolic Matrix(201,82,-1282,-523) (-9/22,-11/27) -> (-3/19,-5/32) Hyperbolic Matrix(875,356,204,83) (-11/27,-2/5) -> (30/7,13/3) Hyperbolic Matrix(199,78,722,283) (-2/5,-9/23) -> (3/11,8/29) Hyperbolic Matrix(195,76,-916,-357) (-9/23,-7/18) -> (-3/14,-7/33) Hyperbolic Matrix(197,76,324,125) (-7/18,-5/13) -> (3/5,11/18) Hyperbolic Matrix(193,74,326,125) (-5/13,-8/21) -> (10/17,3/5) Hyperbolic Matrix(453,172,-2452,-931) (-19/50,-11/29) -> (-5/27,-7/38) Hyperbolic Matrix(191,72,-512,-193) (-11/29,-3/8) -> (-3/8,-13/35) Parabolic Matrix(1527,566,634,235) (-13/35,-10/27) -> (12/5,41/17) Hyperbolic Matrix(887,328,192,71) (-10/27,-7/19) -> (23/5,14/3) Hyperbolic Matrix(1457,536,1264,465) (-7/19,-25/68) -> (23/20,15/13) Hyperbolic Matrix(5437,1998,1962,721) (-25/68,-18/49) -> (36/13,61/22) Hyperbolic Matrix(1957,718,1202,441) (-18/49,-11/30) -> (13/8,44/27) Hyperbolic Matrix(945,346,254,93) (-11/30,-4/11) -> (26/7,15/4) Hyperbolic Matrix(127,46,-566,-205) (-4/11,-9/25) -> (-7/31,-2/9) Hyperbolic Matrix(1813,652,748,269) (-9/25,-14/39) -> (46/19,17/7) Hyperbolic Matrix(2621,940,1620,581) (-14/39,-5/14) -> (55/34,34/21) Hyperbolic Matrix(309,110,250,89) (-5/14,-6/17) -> (16/13,5/4) Hyperbolic Matrix(125,44,-804,-283) (-6/17,-7/20) -> (-5/32,-2/13) Hyperbolic Matrix(1901,662,738,257) (-7/20,-8/23) -> (18/7,67/26) Hyperbolic Matrix(793,274,246,85) (-8/23,-1/3) -> (29/9,42/13) Hyperbolic Matrix(765,244,116,37) (-1/3,-7/22) -> (13/2,33/5) Hyperbolic Matrix(1053,334,1466,465) (-7/22,-6/19) -> (28/39,23/32) Hyperbolic Matrix(235,74,-994,-313) (-6/19,-5/16) -> (-9/38,-4/17) Hyperbolic Matrix(289,90,350,109) (-5/16,-4/13) -> (14/17,5/6) Hyperbolic Matrix(1439,440,520,159) (-4/13,-11/36) -> (11/4,36/13) Hyperbolic Matrix(1729,528,632,193) (-11/36,-7/23) -> (41/15,11/4) Hyperbolic Matrix(1609,488,1032,313) (-7/23,-10/33) -> (14/9,39/25) Hyperbolic Matrix(229,68,852,253) (-3/10,-8/27) -> (4/15,7/26) Hyperbolic Matrix(1081,320,1760,521) (-8/27,-13/44) -> (27/44,8/13) Hyperbolic Matrix(6699,1978,4142,1223) (-13/44,-18/61) -> (76/47,55/34) Hyperbolic Matrix(1133,334,1306,385) (-18/61,-5/17) -> (13/15,20/23) Hyperbolic Matrix(171,50,790,231) (-5/17,-7/24) -> (3/14,5/23) Hyperbolic Matrix(111,32,-392,-113) (-7/24,-2/7) -> (-2/7,-9/32) Parabolic Matrix(1445,406,1114,313) (-9/32,-7/25) -> (35/27,13/10) Hyperbolic Matrix(165,46,-886,-247) (-7/25,-5/18) -> (-3/16,-5/27) Hyperbolic Matrix(277,76,164,45) (-5/18,-3/11) -> (5/3,17/10) Hyperbolic Matrix(273,74,166,45) (-3/11,-4/15) -> (18/11,5/3) Hyperbolic Matrix(1195,316,1524,403) (-9/34,-5/19) -> (29/37,11/14) Hyperbolic Matrix(595,156,164,43) (-5/19,-1/4) -> (29/8,11/3) Hyperbolic Matrix(209,50,790,189) (-1/4,-5/21) -> (5/19,9/34) Hyperbolic Matrix(521,122,158,37) (-4/17,-3/13) -> (23/7,10/3) Hyperbolic Matrix(675,154,206,47) (-3/13,-5/22) -> (13/4,23/7) Hyperbolic Matrix(929,210,1190,269) (-5/22,-7/31) -> (7/9,25/32) Hyperbolic Matrix(767,168,872,191) (-9/41,-7/32) -> (7/8,15/17) Hyperbolic Matrix(1025,224,1176,257) (-7/32,-12/55) -> (20/23,7/8) Hyperbolic Matrix(3275,714,766,167) (-12/55,-5/23) -> (47/11,30/7) Hyperbolic Matrix(1481,320,560,121) (-8/37,-3/14) -> (37/14,8/3) Hyperbolic Matrix(1165,246,914,193) (-7/33,-4/19) -> (14/11,37/29) Hyperbolic Matrix(963,202,758,159) (-4/19,-5/24) -> (33/26,14/11) Hyperbolic Matrix(959,198,402,83) (-5/24,-1/5) -> (31/13,43/18) Hyperbolic Matrix(591,112,248,47) (-1/5,-3/16) -> (19/8,31/13) Hyperbolic Matrix(1521,280,440,81) (-7/38,-2/11) -> (38/11,7/2) Hyperbolic Matrix(439,78,242,43) (-2/11,-3/17) -> (9/5,20/11) Hyperbolic Matrix(389,68,532,93) (-3/17,-1/6) -> (19/26,11/15) Hyperbolic Matrix(523,84,716,115) (-1/6,-3/19) -> (27/37,19/26) Hyperbolic Matrix(331,50,470,71) (-2/13,-1/7) -> (19/27,12/17) Hyperbolic Matrix(371,50,230,31) (-1/7,-1/8) -> (29/18,21/13) Hyperbolic Matrix(317,36,44,5) (-2/17,-1/9) -> (7/1,22/3) Hyperbolic Matrix(313,32,88,9) (-1/9,0/1) -> (32/9,25/7) Hyperbolic Matrix(253,-26,146,-15) (0/1,1/8) -> (45/26,26/15) Hyperbolic Matrix(213,-28,388,-51) (1/8,1/7) -> (17/31,11/20) Hyperbolic Matrix(173,-28,68,-11) (1/7,1/6) -> (5/2,23/9) Hyperbolic Matrix(715,-124,444,-77) (1/6,3/17) -> (37/23,29/18) Hyperbolic Matrix(543,-98,338,-61) (3/17,2/11) -> (8/5,37/23) Hyperbolic Matrix(237,-44,404,-75) (2/11,1/5) -> (17/29,10/17) Hyperbolic Matrix(165,-34,34,-7) (1/5,3/14) -> (19/4,5/1) Hyperbolic Matrix(263,-58,458,-101) (5/23,2/9) -> (4/7,19/33) Hyperbolic Matrix(359,-82,162,-37) (2/9,3/13) -> (11/5,20/9) Hyperbolic Matrix(33,-8,128,-31) (3/13,1/4) -> (1/4,5/19) Parabolic Matrix(1929,-512,1232,-327) (9/34,4/15) -> (36/23,47/30) Hyperbolic Matrix(693,-188,188,-51) (7/26,3/11) -> (11/3,37/10) Hyperbolic Matrix(471,-130,250,-69) (8/29,5/18) -> (15/8,2/1) Hyperbolic Matrix(1439,-402,562,-157) (5/18,7/25) -> (23/9,41/16) Hyperbolic Matrix(2903,-818,1778,-501) (9/32,2/7) -> (80/49,49/30) Hyperbolic Matrix(155,-46,246,-73) (2/7,3/10) -> (5/8,12/19) Hyperbolic Matrix(31,-10,90,-29) (4/13,1/3) -> (1/3,6/17) Parabolic Matrix(1181,-418,178,-63) (6/17,11/31) -> (33/5,20/3) Hyperbolic Matrix(1801,-640,560,-199) (11/31,5/14) -> (45/14,29/9) Hyperbolic Matrix(619,-222,382,-137) (5/14,4/11) -> (34/21,13/8) Hyperbolic Matrix(205,-76,116,-43) (4/11,3/8) -> (7/4,16/9) Hyperbolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(491,-190,230,-89) (5/13,12/31) -> (2/1,15/7) Hyperbolic Matrix(1037,-402,1442,-559) (12/31,7/18) -> (23/32,18/25) Hyperbolic Matrix(173,-68,28,-11) (7/18,2/5) -> (6/1,13/2) Hyperbolic Matrix(113,-46,86,-35) (2/5,5/12) -> (13/10,4/3) Hyperbolic Matrix(85,-36,196,-83) (5/12,3/7) -> (3/7,7/16) Parabolic Matrix(529,-232,472,-207) (7/16,18/41) -> (10/9,9/8) Hyperbolic Matrix(359,-162,82,-37) (4/9,5/11) -> (13/3,22/5) Hyperbolic Matrix(439,-202,602,-277) (5/11,6/13) -> (8/11,27/37) Hyperbolic Matrix(491,-230,190,-89) (6/13,1/2) -> (31/12,44/17) Hyperbolic Matrix(471,-250,130,-69) (1/2,7/13) -> (47/13,29/8) Hyperbolic Matrix(751,-408,208,-113) (7/13,6/11) -> (18/5,47/13) Hyperbolic Matrix(413,-226,466,-255) (6/11,17/31) -> (15/17,8/9) Hyperbolic Matrix(205,-116,76,-43) (5/9,4/7) -> (8/3,19/7) Hyperbolic Matrix(2359,-1360,1320,-761) (19/33,15/26) -> (25/14,59/33) Hyperbolic Matrix(253,-146,26,-15) (15/26,11/19) -> (9/1,1/0) Hyperbolic Matrix(557,-324,404,-235) (11/19,7/12) -> (11/8,29/21) Hyperbolic Matrix(1185,-694,934,-547) (7/12,17/29) -> (19/15,33/26) Hyperbolic Matrix(1265,-774,894,-547) (11/18,19/31) -> (41/29,17/12) Hyperbolic Matrix(1933,-1186,546,-335) (19/31,27/44) -> (7/2,39/11) Hyperbolic Matrix(619,-382,222,-137) (8/13,13/21) -> (25/9,14/5) Hyperbolic Matrix(715,-444,124,-77) (18/29,5/8) -> (23/4,6/1) Hyperbolic Matrix(587,-372,172,-109) (12/19,7/11) -> (17/5,24/7) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(1913,-1328,448,-311) (9/13,25/36) -> (17/4,47/11) Hyperbolic Matrix(2243,-1558,1038,-721) (25/36,41/59) -> (41/19,13/6) Hyperbolic Matrix(2595,-1804,1204,-837) (41/59,16/23) -> (28/13,41/19) Hyperbolic Matrix(755,-526,966,-673) (16/23,7/10) -> (25/32,18/23) Hyperbolic Matrix(1269,-892,892,-627) (7/10,19/27) -> (27/19,37/26) Hyperbolic Matrix(1265,-894,774,-547) (12/17,17/24) -> (49/30,18/11) Hyperbolic Matrix(2153,-1526,1686,-1195) (17/24,22/31) -> (60/47,23/18) Hyperbolic Matrix(351,-250,490,-349) (22/31,5/7) -> (5/7,28/39) Parabolic Matrix(557,-402,442,-319) (18/25,13/18) -> (5/4,24/19) Hyperbolic Matrix(557,-404,324,-235) (13/18,8/11) -> (12/7,31/18) Hyperbolic Matrix(345,-254,254,-187) (11/15,3/4) -> (19/14,15/11) Hyperbolic Matrix(113,-86,46,-35) (3/4,7/9) -> (17/7,5/2) Hyperbolic Matrix(2153,-1686,1526,-1195) (18/23,29/37) -> (55/39,24/17) Hyperbolic Matrix(469,-370,90,-71) (11/14,15/19) -> (5/1,21/4) Hyperbolic Matrix(557,-442,402,-319) (15/19,4/5) -> (18/13,43/31) Hyperbolic Matrix(199,-162,242,-197) (4/5,9/11) -> (9/11,14/17) Parabolic Matrix(195,-164,44,-37) (5/6,6/7) -> (22/5,9/2) Hyperbolic Matrix(923,-796,516,-445) (6/7,13/15) -> (59/33,34/19) Hyperbolic Matrix(507,-452,212,-189) (8/9,1/1) -> (55/23,12/5) Hyperbolic Matrix(593,-648,248,-271) (1/1,10/9) -> (98/41,55/23) Hyperbolic Matrix(281,-318,38,-43) (9/8,8/7) -> (22/3,15/2) Hyperbolic Matrix(853,-986,186,-215) (15/13,7/6) -> (55/12,23/5) Hyperbolic Matrix(165,-194,74,-87) (7/6,6/5) -> (20/9,9/4) Hyperbolic Matrix(199,-242,162,-197) (6/5,11/9) -> (11/9,16/13) Parabolic Matrix(1837,-2324,1324,-1675) (24/19,19/15) -> (43/31,68/49) Hyperbolic Matrix(5079,-6482,3602,-4597) (37/29,60/47) -> (86/61,55/39) Hyperbolic Matrix(877,-1122,562,-719) (23/18,9/7) -> (39/25,25/16) Hyperbolic Matrix(907,-1172,332,-429) (9/7,22/17) -> (30/11,41/15) Hyperbolic Matrix(1011,-1310,470,-609) (22/17,35/27) -> (15/7,28/13) Hyperbolic Matrix(631,-850,170,-229) (4/3,23/17) -> (63/17,26/7) Hyperbolic Matrix(1511,-2048,408,-553) (23/17,19/14) -> (37/10,63/17) Hyperbolic Matrix(439,-602,202,-277) (26/19,11/8) -> (13/6,24/11) Hyperbolic Matrix(285,-394,34,-47) (29/21,18/13) -> (8/1,9/1) Hyperbolic Matrix(12965,-17994,4674,-6487) (68/49,93/67) -> (147/53,86/31) Hyperbolic Matrix(6733,-9348,2428,-3371) (93/67,25/18) -> (61/22,147/53) Hyperbolic Matrix(1037,-1442,402,-559) (25/18,7/5) -> (49/19,31/12) Hyperbolic Matrix(1413,-1988,548,-771) (7/5,31/22) -> (67/26,49/19) Hyperbolic Matrix(6641,-9360,2440,-3439) (31/22,86/61) -> (166/61,49/18) Hyperbolic Matrix(1591,-2248,448,-633) (24/17,41/29) -> (39/11,32/9) Hyperbolic Matrix(1485,-2114,314,-447) (37/26,10/7) -> (52/11,19/4) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(3839,-6002,1482,-2317) (25/16,61/39) -> (101/39,57/22) Hyperbolic Matrix(4039,-6320,1560,-2441) (61/39,36/23) -> (44/17,101/39) Hyperbolic Matrix(155,-246,46,-73) (11/7,8/5) -> (10/3,17/5) Hyperbolic Matrix(2293,-3706,826,-1335) (21/13,76/47) -> (86/31,25/9) Hyperbolic Matrix(2357,-3844,1444,-2355) (44/27,31/19) -> (31/19,80/49) Parabolic Matrix(237,-404,44,-75) (17/10,12/7) -> (16/3,11/2) Hyperbolic Matrix(837,-1444,484,-835) (31/18,19/11) -> (19/11,45/26) Parabolic Matrix(263,-458,58,-101) (26/15,7/4) -> (9/2,32/7) Hyperbolic Matrix(905,-1614,374,-667) (16/9,25/14) -> (29/12,46/19) Hyperbolic Matrix(701,-1258,258,-463) (34/19,9/5) -> (19/7,68/25) Hyperbolic Matrix(213,-388,28,-51) (20/11,11/6) -> (15/2,8/1) Hyperbolic Matrix(183,-338,98,-181) (11/6,13/7) -> (13/7,15/8) Parabolic Matrix(85,-196,36,-83) (9/4,7/3) -> (7/3,19/8) Parabolic Matrix(3753,-8968,1168,-2791) (43/18,98/41) -> (106/33,45/14) Hyperbolic Matrix(1125,-2714,434,-1047) (41/17,29/12) -> (57/22,13/5) Hyperbolic Matrix(763,-1956,236,-605) (41/16,18/7) -> (42/13,13/4) Hyperbolic Matrix(639,-1682,242,-637) (21/8,29/11) -> (29/11,37/14) Parabolic Matrix(10063,-27378,3698,-10061) (68/25,117/43) -> (117/43,166/61) Parabolic Matrix(1181,-3218,258,-703) (49/18,30/11) -> (32/7,55/12) Hyperbolic Matrix(31,-90,10,-29) (14/5,3/1) -> (3/1,16/5) Parabolic Matrix(2319,-7442,722,-2317) (16/5,61/19) -> (61/19,106/33) Parabolic Matrix(559,-1922,162,-557) (24/7,31/9) -> (31/9,38/11) Parabolic Matrix(175,-626,26,-93) (25/7,18/5) -> (20/3,7/1) Hyperbolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(463,-2178,98,-461) (14/3,33/7) -> (33/7,52/11) Parabolic Matrix(103,-578,18,-101) (11/2,17/3) -> (17/3,23/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,0,1) Matrix(149,70,530,249) -> Matrix(1,0,0,1) Matrix(147,68,-668,-309) -> Matrix(1,2,-4,-7) Matrix(367,168,592,271) -> Matrix(7,6,8,7) Matrix(727,328,512,231) -> Matrix(3,2,4,3) Matrix(361,162,-1522,-683) -> Matrix(1,0,0,1) Matrix(795,356,364,163) -> Matrix(3,2,4,3) Matrix(785,346,574,253) -> Matrix(1,0,0,1) Matrix(355,156,644,283) -> Matrix(1,0,4,1) Matrix(565,246,-1486,-647) -> Matrix(5,2,-8,-3) Matrix(281,122,638,277) -> Matrix(1,-2,0,1) Matrix(489,208,1112,473) -> Matrix(13,10,-4,-3) Matrix(1109,470,210,89) -> Matrix(9,8,-8,-7) Matrix(275,116,-1036,-437) -> Matrix(3,2,-8,-5) Matrix(343,144,-1584,-665) -> Matrix(3,2,-8,-5) Matrix(2271,952,1448,607) -> Matrix(1,-2,0,1) Matrix(1581,662,1378,577) -> Matrix(13,10,-4,-3) Matrix(67,28,-548,-229) -> Matrix(3,2,4,3) Matrix(207,86,-686,-285) -> Matrix(3,2,-8,-5) Matrix(271,112,888,367) -> Matrix(7,4,-16,-9) Matrix(205,84,676,277) -> Matrix(5,2,-8,-3) Matrix(201,82,-1282,-523) -> Matrix(1,0,0,1) Matrix(875,356,204,83) -> Matrix(7,2,-4,-1) Matrix(199,78,722,283) -> Matrix(5,4,-4,-3) Matrix(195,76,-916,-357) -> Matrix(3,2,-8,-5) Matrix(197,76,324,125) -> Matrix(3,2,4,3) Matrix(193,74,326,125) -> Matrix(3,2,4,3) Matrix(453,172,-2452,-931) -> Matrix(3,2,-8,-5) Matrix(191,72,-512,-193) -> Matrix(7,4,-16,-9) Matrix(1527,566,634,235) -> Matrix(15,4,-4,-1) Matrix(887,328,192,71) -> Matrix(7,2,-4,-1) Matrix(1457,536,1264,465) -> Matrix(1,-2,0,1) Matrix(5437,1998,1962,721) -> Matrix(5,4,-4,-3) Matrix(1957,718,1202,441) -> Matrix(7,2,-4,-1) Matrix(945,346,254,93) -> Matrix(1,0,0,1) Matrix(127,46,-566,-205) -> Matrix(3,2,-8,-5) Matrix(1813,652,748,269) -> Matrix(1,-2,0,1) Matrix(2621,940,1620,581) -> Matrix(7,2,-4,-1) Matrix(309,110,250,89) -> Matrix(1,0,0,1) Matrix(125,44,-804,-283) -> Matrix(3,2,-8,-5) Matrix(1901,662,738,257) -> Matrix(17,10,-12,-7) Matrix(793,274,246,85) -> Matrix(1,0,0,1) Matrix(765,244,116,37) -> Matrix(1,0,0,1) Matrix(1053,334,1466,465) -> Matrix(5,2,-8,-3) Matrix(235,74,-994,-313) -> Matrix(5,2,-8,-3) Matrix(289,90,350,109) -> Matrix(1,0,4,1) Matrix(1439,440,520,159) -> Matrix(5,4,-4,-3) Matrix(1729,528,632,193) -> Matrix(7,2,-4,-1) Matrix(1609,488,1032,313) -> Matrix(13,10,-4,-3) Matrix(229,68,852,253) -> Matrix(1,0,0,1) Matrix(1081,320,1760,521) -> Matrix(3,2,4,3) Matrix(6699,1978,4142,1223) -> Matrix(7,2,-4,-1) Matrix(1133,334,1306,385) -> Matrix(1,2,0,1) Matrix(171,50,790,231) -> Matrix(3,2,4,3) Matrix(111,32,-392,-113) -> Matrix(7,4,-16,-9) Matrix(1445,406,1114,313) -> Matrix(5,2,12,5) Matrix(165,46,-886,-247) -> Matrix(5,2,-8,-3) Matrix(277,76,164,45) -> Matrix(7,2,-4,-1) Matrix(273,74,166,45) -> Matrix(7,2,-4,-1) Matrix(1195,316,1524,403) -> Matrix(1,0,4,1) Matrix(595,156,164,43) -> Matrix(1,0,4,1) Matrix(209,50,790,189) -> Matrix(5,4,-4,-3) Matrix(521,122,158,37) -> Matrix(3,2,-8,-5) Matrix(675,154,206,47) -> Matrix(9,4,-16,-7) Matrix(929,210,1190,269) -> Matrix(5,2,-8,-3) Matrix(767,168,872,191) -> Matrix(7,2,-4,-1) Matrix(1025,224,1176,257) -> Matrix(9,2,4,1) Matrix(3275,714,766,167) -> Matrix(15,2,-8,-1) Matrix(1481,320,560,121) -> Matrix(7,2,-4,-1) Matrix(1165,246,914,193) -> Matrix(7,2,-4,-1) Matrix(963,202,758,159) -> Matrix(1,0,4,1) Matrix(959,198,402,83) -> Matrix(15,4,-4,-1) Matrix(591,112,248,47) -> Matrix(1,-4,0,1) Matrix(1521,280,440,81) -> Matrix(1,0,0,1) Matrix(439,78,242,43) -> Matrix(5,4,-4,-3) Matrix(389,68,532,93) -> Matrix(1,0,0,1) Matrix(523,84,716,115) -> Matrix(5,2,-8,-3) Matrix(331,50,470,71) -> Matrix(7,2,-4,-1) Matrix(371,50,230,31) -> Matrix(15,2,-8,-1) Matrix(317,36,44,5) -> Matrix(1,0,0,1) Matrix(313,32,88,9) -> Matrix(1,0,0,1) Matrix(253,-26,146,-15) -> Matrix(1,-2,0,1) Matrix(213,-28,388,-51) -> Matrix(1,0,4,1) Matrix(173,-28,68,-11) -> Matrix(7,2,-4,-1) Matrix(715,-124,444,-77) -> Matrix(31,2,-16,-1) Matrix(543,-98,338,-61) -> Matrix(25,-2,-12,1) Matrix(237,-44,404,-75) -> Matrix(1,0,-4,1) Matrix(165,-34,34,-7) -> Matrix(1,-2,0,1) Matrix(263,-58,458,-101) -> Matrix(1,0,0,1) Matrix(359,-82,162,-37) -> Matrix(1,0,0,1) Matrix(33,-8,128,-31) -> Matrix(1,-2,0,1) Matrix(1929,-512,1232,-327) -> Matrix(23,26,-8,-9) Matrix(693,-188,188,-51) -> Matrix(1,0,0,1) Matrix(471,-130,250,-69) -> Matrix(3,4,-4,-5) Matrix(1439,-402,562,-157) -> Matrix(7,2,-4,-1) Matrix(2903,-818,1778,-501) -> Matrix(5,4,-4,-3) Matrix(155,-46,246,-73) -> Matrix(1,2,0,1) Matrix(31,-10,90,-29) -> Matrix(1,0,4,1) Matrix(1181,-418,178,-63) -> Matrix(1,0,-4,1) Matrix(1801,-640,560,-199) -> Matrix(3,-2,-4,3) Matrix(619,-222,382,-137) -> Matrix(1,-2,0,1) Matrix(205,-76,116,-43) -> Matrix(1,-2,0,1) Matrix(377,-144,144,-55) -> Matrix(1,-2,0,1) Matrix(491,-190,230,-89) -> Matrix(1,0,-4,1) Matrix(1037,-402,1442,-559) -> Matrix(1,0,-4,1) Matrix(173,-68,28,-11) -> Matrix(3,-2,-4,3) Matrix(113,-46,86,-35) -> Matrix(1,-2,0,1) Matrix(85,-36,196,-83) -> Matrix(1,-6,0,1) Matrix(529,-232,472,-207) -> Matrix(1,2,0,1) Matrix(359,-162,82,-37) -> Matrix(1,0,0,1) Matrix(439,-202,602,-277) -> Matrix(3,4,-4,-5) Matrix(491,-230,190,-89) -> Matrix(5,4,-4,-3) Matrix(471,-250,130,-69) -> Matrix(1,0,4,1) Matrix(751,-408,208,-113) -> Matrix(1,0,-8,1) Matrix(413,-226,466,-255) -> Matrix(1,0,-4,1) Matrix(205,-116,76,-43) -> Matrix(1,-2,0,1) Matrix(2359,-1360,1320,-761) -> Matrix(1,-2,0,1) Matrix(253,-146,26,-15) -> Matrix(1,-2,0,1) Matrix(557,-324,404,-235) -> Matrix(1,0,0,1) Matrix(1185,-694,934,-547) -> Matrix(1,0,0,1) Matrix(1265,-774,894,-547) -> Matrix(1,0,0,1) Matrix(1933,-1186,546,-335) -> Matrix(3,-2,-4,3) Matrix(619,-382,222,-137) -> Matrix(1,-2,0,1) Matrix(715,-444,124,-77) -> Matrix(15,-14,-16,15) Matrix(587,-372,172,-109) -> Matrix(1,-2,0,1) Matrix(49,-32,72,-47) -> Matrix(1,-4,0,1) Matrix(1913,-1328,448,-311) -> Matrix(15,32,-8,-17) Matrix(2243,-1558,1038,-721) -> Matrix(1,2,16,33) Matrix(2595,-1804,1204,-837) -> Matrix(1,2,-20,-39) Matrix(755,-526,966,-673) -> Matrix(1,2,-4,-7) Matrix(1269,-892,892,-627) -> Matrix(1,2,0,1) Matrix(1265,-894,774,-547) -> Matrix(1,0,0,1) Matrix(2153,-1526,1686,-1195) -> Matrix(1,0,0,1) Matrix(351,-250,490,-349) -> Matrix(3,4,-4,-5) Matrix(557,-402,442,-319) -> Matrix(1,0,0,1) Matrix(557,-404,324,-235) -> Matrix(1,0,0,1) Matrix(345,-254,254,-187) -> Matrix(1,0,0,1) Matrix(113,-86,46,-35) -> Matrix(1,-2,0,1) Matrix(2153,-1686,1526,-1195) -> Matrix(1,0,0,1) Matrix(469,-370,90,-71) -> Matrix(5,4,-4,-3) Matrix(557,-442,402,-319) -> Matrix(1,0,0,1) Matrix(199,-162,242,-197) -> Matrix(1,0,4,1) Matrix(195,-164,44,-37) -> Matrix(1,-2,0,1) Matrix(923,-796,516,-445) -> Matrix(1,-2,0,1) Matrix(507,-452,212,-189) -> Matrix(1,-4,0,1) Matrix(593,-648,248,-271) -> Matrix(1,-2,0,1) Matrix(281,-318,38,-43) -> Matrix(1,0,0,1) Matrix(853,-986,186,-215) -> Matrix(1,0,0,1) Matrix(165,-194,74,-87) -> Matrix(1,2,0,1) Matrix(199,-242,162,-197) -> Matrix(3,4,-4,-5) Matrix(1837,-2324,1324,-1675) -> Matrix(1,0,0,1) Matrix(5079,-6482,3602,-4597) -> Matrix(1,2,0,1) Matrix(877,-1122,562,-719) -> Matrix(13,10,-4,-3) Matrix(907,-1172,332,-429) -> Matrix(7,2,-4,-1) Matrix(1011,-1310,470,-609) -> Matrix(1,0,-8,1) Matrix(631,-850,170,-229) -> Matrix(1,0,0,1) Matrix(1511,-2048,408,-553) -> Matrix(1,0,0,1) Matrix(439,-602,202,-277) -> Matrix(1,0,4,1) Matrix(285,-394,34,-47) -> Matrix(1,0,0,1) Matrix(12965,-17994,4674,-6487) -> Matrix(5,4,-4,-3) Matrix(6733,-9348,2428,-3371) -> Matrix(5,4,-4,-3) Matrix(1037,-1442,402,-559) -> Matrix(5,4,-4,-3) Matrix(1413,-1988,548,-771) -> Matrix(11,-4,-8,3) Matrix(6641,-9360,2440,-3439) -> Matrix(1,-2,0,1) Matrix(1591,-2248,448,-633) -> Matrix(1,0,-4,1) Matrix(1485,-2114,314,-447) -> Matrix(1,-2,0,1) Matrix(49,-72,32,-47) -> Matrix(1,-4,0,1) Matrix(3839,-6002,1482,-2317) -> Matrix(15,46,-16,-49) Matrix(4039,-6320,1560,-2441) -> Matrix(21,62,-20,-59) Matrix(155,-246,46,-73) -> Matrix(1,2,0,1) Matrix(2293,-3706,826,-1335) -> Matrix(11,18,-8,-13) Matrix(2357,-3844,1444,-2355) -> Matrix(11,18,-8,-13) Matrix(237,-404,44,-75) -> Matrix(5,4,-4,-3) Matrix(837,-1444,484,-835) -> Matrix(1,-2,0,1) Matrix(263,-458,58,-101) -> Matrix(1,0,0,1) Matrix(905,-1614,374,-667) -> Matrix(11,14,-4,-5) Matrix(701,-1258,258,-463) -> Matrix(1,0,0,1) Matrix(213,-388,28,-51) -> Matrix(3,4,-4,-5) Matrix(183,-338,98,-181) -> Matrix(11,12,-12,-13) Matrix(85,-196,36,-83) -> Matrix(1,-6,0,1) Matrix(3753,-8968,1168,-2791) -> Matrix(3,10,-4,-13) Matrix(1125,-2714,434,-1047) -> Matrix(3,10,-4,-13) Matrix(763,-1956,236,-605) -> Matrix(1,2,-4,-7) Matrix(639,-1682,242,-637) -> Matrix(1,0,0,1) Matrix(10063,-27378,3698,-10061) -> Matrix(1,0,0,1) Matrix(1181,-3218,258,-703) -> Matrix(1,0,0,1) Matrix(31,-90,10,-29) -> Matrix(3,4,-4,-5) Matrix(2319,-7442,722,-2317) -> Matrix(1,0,0,1) Matrix(559,-1922,162,-557) -> Matrix(1,0,0,1) Matrix(175,-626,26,-93) -> Matrix(1,0,0,1) Matrix(33,-128,8,-31) -> Matrix(1,-2,0,1) Matrix(463,-2178,98,-461) -> Matrix(1,0,0,1) Matrix(103,-578,18,-101) -> Matrix(27,28,-28,-29) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 24 Degree of the the map X: 48 Degree of the the map Y: 192 ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 288 Minimal number of generators: 49 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/4 1/3 3/7 11/19 3/5 2/3 5/7 1/1 11/9 23/17 7/5 3/2 61/39 5/3 19/11 13/7 2/1 7/3 17/7 13/5 29/11 3/1 7/2 11/3 4/1 13/3 33/7 5/1 17/3 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/2 1/0 0/1 -1/1 0/1 1/6 -1/5 0/1 1/5 1/2 3/14 0/1 1/1 2/9 0/1 1/1 1/4 1/0 4/15 -1/1 -2/3 3/11 1/0 2/7 -1/1 0/1 1/3 0/1 3/8 0/1 1/1 5/13 1/0 2/5 1/1 2/1 3/7 1/0 4/9 -2/1 -1/1 1/2 -1/1 0/1 5/9 1/2 4/7 0/1 1/1 11/19 1/0 7/12 0/1 1/1 3/5 1/2 1/0 8/13 0/1 1/1 5/8 1/1 4/3 7/11 3/2 2/3 1/0 9/13 -5/2 16/23 -2/1 -13/7 7/10 -2/1 -1/1 12/17 -2/1 -1/1 5/7 -1/1 13/18 -1/1 0/1 8/11 -2/1 -1/1 11/15 -1/2 3/4 -1/1 0/1 7/9 -1/2 1/0 18/23 0/1 1/3 11/14 1/0 4/5 -1/1 0/1 1/1 1/0 6/5 -2/1 -1/1 11/9 -1/1 5/4 -1/1 0/1 9/7 -1/2 1/0 22/17 -1/5 0/1 13/10 0/1 1/1 4/3 -1/1 0/1 23/17 -1/2 1/0 19/14 -1/1 0/1 15/11 -1/2 11/8 0/1 1/1 29/21 1/0 18/13 -1/1 0/1 7/5 0/1 17/12 0/1 1/1 10/7 0/1 1/1 3/2 1/0 14/9 -4/1 -3/1 25/16 -22/7 -3/1 61/39 -3/1 36/23 -3/1 -32/11 11/7 -5/2 8/5 -7/3 -2/1 13/8 -2/1 -1/1 5/3 -3/2 1/0 12/7 -2/1 -1/1 19/11 1/0 26/15 -3/1 -2/1 7/4 -2/1 -1/1 9/5 -3/2 11/6 -6/5 -1/1 13/7 -1/1 2/1 -1/1 0/1 7/3 1/0 12/5 -4/1 -3/1 29/12 -3/1 -14/5 17/7 -5/2 1/0 5/2 -3/1 -2/1 13/5 1/0 21/8 -2/1 -1/1 29/11 -3/2 1/0 8/3 -2/1 -1/1 3/1 -1/1 7/2 -1/1 0/1 11/3 1/0 15/4 -1/3 0/1 4/1 1/0 17/4 -7/3 -2/1 13/3 -3/2 1/0 9/2 -2/1 -1/1 14/3 -2/1 -1/1 33/7 -3/2 1/0 19/4 -2/1 -1/1 5/1 -3/2 11/2 -10/9 -1/1 17/3 -1/1 6/1 -1/1 -4/5 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(86,-13,53,-8) (0/1,1/6) -> (8/5,13/8) Hyperbolic Matrix(85,-16,16,-3) (1/6,1/5) -> (5/1,11/2) Hyperbolic Matrix(165,-34,34,-7) (1/5,3/14) -> (19/4,5/1) Hyperbolic Matrix(328,-71,231,-50) (3/14,2/9) -> (17/12,10/7) Hyperbolic Matrix(115,-26,146,-33) (2/9,1/4) -> (11/14,4/5) Hyperbolic Matrix(237,-62,302,-79) (1/4,4/15) -> (18/23,11/14) Hyperbolic Matrix(346,-93,253,-68) (4/15,3/11) -> (15/11,11/8) Hyperbolic Matrix(140,-39,79,-22) (3/11,2/7) -> (7/4,9/5) Hyperbolic Matrix(16,-5,45,-14) (2/7,1/3) -> (1/3,3/8) Parabolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(100,-39,159,-62) (5/13,2/5) -> (5/8,7/11) Hyperbolic Matrix(43,-18,98,-41) (2/5,3/7) -> (3/7,4/9) Parabolic Matrix(125,-56,96,-43) (4/9,1/2) -> (13/10,4/3) Hyperbolic Matrix(117,-64,64,-35) (1/2,5/9) -> (9/5,11/6) Hyperbolic Matrix(140,-79,39,-22) (5/9,4/7) -> (7/2,11/3) Hyperbolic Matrix(545,-314,394,-227) (4/7,11/19) -> (29/21,18/13) Hyperbolic Matrix(557,-324,404,-235) (11/19,7/12) -> (11/8,29/21) Hyperbolic Matrix(76,-45,125,-74) (7/12,3/5) -> (3/5,8/13) Parabolic Matrix(86,-53,13,-8) (8/13,5/8) -> (6/1,1/0) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(721,-500,460,-319) (9/13,16/23) -> (36/23,11/7) Hyperbolic Matrix(566,-395,235,-164) (16/23,7/10) -> (12/5,29/12) Hyperbolic Matrix(328,-231,71,-50) (7/10,12/17) -> (9/2,14/3) Hyperbolic Matrix(176,-125,245,-174) (12/17,5/7) -> (5/7,13/18) Parabolic Matrix(545,-394,314,-227) (13/18,8/11) -> (26/15,7/4) Hyperbolic Matrix(346,-253,93,-68) (8/11,11/15) -> (11/3,15/4) Hyperbolic Matrix(345,-254,254,-187) (11/15,3/4) -> (19/14,15/11) Hyperbolic Matrix(113,-86,46,-35) (3/4,7/9) -> (17/7,5/2) Hyperbolic Matrix(652,-509,269,-210) (7/9,18/23) -> (29/12,17/7) Hyperbolic Matrix(11,-10,10,-9) (4/5,1/1) -> (1/1,6/5) Parabolic Matrix(100,-121,81,-98) (6/5,11/9) -> (11/9,5/4) Parabolic Matrix(115,-146,26,-33) (5/4,9/7) -> (13/3,9/2) Hyperbolic Matrix(340,-439,79,-102) (9/7,22/17) -> (17/4,13/3) Hyperbolic Matrix(488,-633,313,-406) (22/17,13/10) -> (14/9,25/16) Hyperbolic Matrix(392,-529,289,-390) (4/3,23/17) -> (23/17,19/14) Parabolic Matrix(176,-245,125,-174) (18/13,7/5) -> (7/5,17/12) Parabolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(2380,-3721,1521,-2378) (25/16,61/39) -> (61/39,36/23) Parabolic Matrix(100,-159,39,-62) (11/7,8/5) -> (5/2,13/5) Hyperbolic Matrix(76,-125,45,-74) (13/8,5/3) -> (5/3,12/7) Parabolic Matrix(419,-722,242,-417) (12/7,19/11) -> (19/11,26/15) Parabolic Matrix(92,-169,49,-90) (11/6,13/7) -> (13/7,2/1) Parabolic Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(320,-841,121,-318) (21/8,29/11) -> (29/11,8/3) Parabolic Matrix(16,-45,5,-14) (8/3,3/1) -> (3/1,7/2) Parabolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(232,-1089,49,-230) (14/3,33/7) -> (33/7,19/4) Parabolic Matrix(52,-289,9,-50) (11/2,17/3) -> (17/3,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,1,-2,-1) Matrix(86,-13,53,-8) -> Matrix(3,2,-2,-1) Matrix(85,-16,16,-3) -> Matrix(5,-1,-4,1) Matrix(165,-34,34,-7) -> Matrix(1,-2,0,1) Matrix(328,-71,231,-50) -> Matrix(1,-1,2,-1) Matrix(115,-26,146,-33) -> Matrix(1,-1,0,1) Matrix(237,-62,302,-79) -> Matrix(1,1,0,1) Matrix(346,-93,253,-68) -> Matrix(1,1,-2,-1) Matrix(140,-39,79,-22) -> Matrix(3,2,-2,-1) Matrix(16,-5,45,-14) -> Matrix(1,0,2,1) Matrix(377,-144,144,-55) -> Matrix(1,-2,0,1) Matrix(100,-39,159,-62) -> Matrix(3,-2,2,-1) Matrix(43,-18,98,-41) -> Matrix(1,-3,0,1) Matrix(125,-56,96,-43) -> Matrix(1,1,0,1) Matrix(117,-64,64,-35) -> Matrix(5,-1,-4,1) Matrix(140,-79,39,-22) -> Matrix(1,0,-2,1) Matrix(545,-314,394,-227) -> Matrix(1,-1,0,1) Matrix(557,-324,404,-235) -> Matrix(1,0,0,1) Matrix(76,-45,125,-74) -> Matrix(1,-1,2,-1) Matrix(86,-53,13,-8) -> Matrix(1,0,-2,1) Matrix(49,-32,72,-47) -> Matrix(1,-4,0,1) Matrix(721,-500,460,-319) -> Matrix(11,25,-4,-9) Matrix(566,-395,235,-164) -> Matrix(7,11,-2,-3) Matrix(328,-231,71,-50) -> Matrix(3,5,-2,-3) Matrix(176,-125,245,-174) -> Matrix(1,2,-2,-3) Matrix(545,-394,314,-227) -> Matrix(1,-1,0,1) Matrix(346,-253,93,-68) -> Matrix(1,1,-2,-1) Matrix(345,-254,254,-187) -> Matrix(1,0,0,1) Matrix(113,-86,46,-35) -> Matrix(1,-2,0,1) Matrix(652,-509,269,-210) -> Matrix(5,3,-2,-1) Matrix(11,-10,10,-9) -> Matrix(1,-1,0,1) Matrix(100,-121,81,-98) -> Matrix(1,2,-2,-3) Matrix(115,-146,26,-33) -> Matrix(1,-1,0,1) Matrix(340,-439,79,-102) -> Matrix(3,2,-2,-1) Matrix(488,-633,313,-406) -> Matrix(7,-3,-2,1) Matrix(392,-529,289,-390) -> Matrix(1,1,-2,-1) Matrix(176,-245,125,-174) -> Matrix(1,0,2,1) Matrix(49,-72,32,-47) -> Matrix(1,-4,0,1) Matrix(2380,-3721,1521,-2378) -> Matrix(53,162,-18,-55) Matrix(100,-159,39,-62) -> Matrix(3,8,-2,-5) Matrix(76,-125,45,-74) -> Matrix(3,5,-2,-3) Matrix(419,-722,242,-417) -> Matrix(1,-1,0,1) Matrix(92,-169,49,-90) -> Matrix(5,6,-6,-7) Matrix(43,-98,18,-41) -> Matrix(1,-3,0,1) Matrix(320,-841,121,-318) -> Matrix(3,5,-2,-3) Matrix(16,-45,5,-14) -> Matrix(1,2,-2,-3) Matrix(33,-128,8,-31) -> Matrix(1,-2,0,1) Matrix(232,-1089,49,-230) -> Matrix(3,5,-2,-3) Matrix(52,-289,9,-50) -> Matrix(13,14,-14,-15) 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): 24 ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d -1/1 (-1/1,0/1).(-1/2,1/0) 0 1 1/1 1/0 1 10 11/9 -1/1 2 1 5/4 (-1/1,0/1) 0 20 9/7 (-1/1,0/1).(-1/2,1/0) 0 5 4/3 (-1/1,0/1) 0 20 23/17 (-1/1,0/1).(-1/2,1/0) 0 1 15/11 -1/2 1 10 11/8 (0/1,1/1) 0 20 18/13 (-1/1,0/1) 0 20 7/5 0/1 2 5 17/12 (0/1,1/1) 0 20 10/7 (0/1,1/1) 0 20 3/2 1/0 2 4 14/9 (-4/1,-3/1) 0 20 25/16 (-22/7,-3/1) 0 20 61/39 -3/1 18 1 11/7 -5/2 1 10 8/5 (-7/3,-2/1) 0 20 13/8 (-2/1,-1/1) 0 20 5/3 (-2/1,-1/1).(-3/2,1/0) 0 5 12/7 (-2/1,-1/1) 0 20 19/11 1/0 1 2 26/15 (-3/1,-2/1) 0 20 7/4 (-2/1,-1/1) 0 20 9/5 -3/2 1 10 13/7 -1/1 6 1 2/1 (-1/1,0/1) 0 20 7/3 1/0 3 2 12/5 (-4/1,-3/1) 0 20 29/12 (-3/1,-14/5) 0 20 17/7 (-3/1,-2/1).(-5/2,1/0) 0 5 5/2 (-3/1,-2/1) 0 20 13/5 1/0 1 10 29/11 (-2/1,-1/1).(-3/2,1/0) 0 1 8/3 (-2/1,-1/1) 0 20 3/1 -1/1 2 5 7/2 (-1/1,0/1) 0 20 11/3 1/0 1 10 15/4 (-1/3,0/1) 0 20 4/1 1/0 1 4 17/4 (-7/3,-2/1) 0 20 13/3 (-2/1,-1/1).(-3/2,1/0) 0 5 9/2 (-2/1,-1/1) 0 20 14/3 (-2/1,-1/1) 0 20 33/7 (-2/1,-1/1).(-3/2,1/0) 0 1 5/1 -3/2 1 10 17/3 -1/1 14 1 6/1 (-1/1,-4/5) 0 20 1/0 (-1/1,0/1) 0 20 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(10,-11,9,-10) (1/1,11/9) -> (1/1,11/9) Reflection Matrix(89,-110,72,-89) (11/9,5/4) -> (11/9,5/4) Reflection Matrix(115,-146,26,-33) (5/4,9/7) -> (13/3,9/2) Hyperbolic Matrix(86,-113,35,-46) (9/7,4/3) -> (17/7,5/2) Glide Reflection Matrix(137,-184,102,-137) (4/3,23/17) -> (4/3,23/17) Reflection Matrix(254,-345,187,-254) (23/17,15/11) -> (23/17,15/11) Reflection Matrix(253,-346,68,-93) (15/11,11/8) -> (11/3,15/4) Glide Reflection Matrix(394,-545,227,-314) (11/8,18/13) -> (26/15,7/4) Glide Reflection Matrix(176,-245,125,-174) (18/13,7/5) -> (7/5,17/12) Parabolic Matrix(231,-328,50,-71) (17/12,10/7) -> (9/2,14/3) Glide Reflection Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(453,-706,188,-293) (14/9,25/16) -> (12/5,29/12) Glide Reflection Matrix(1951,-3050,1248,-1951) (25/16,61/39) -> (25/16,61/39) Reflection Matrix(428,-671,273,-428) (61/39,11/7) -> (61/39,11/7) Reflection Matrix(100,-159,39,-62) (11/7,8/5) -> (5/2,13/5) Hyperbolic Matrix(53,-86,8,-13) (8/5,13/8) -> (6/1,1/0) Glide Reflection Matrix(76,-125,45,-74) (13/8,5/3) -> (5/3,12/7) Parabolic Matrix(419,-722,242,-417) (12/7,19/11) -> (19/11,26/15) Parabolic Matrix(79,-140,22,-39) (7/4,9/5) -> (7/2,11/3) Glide Reflection Matrix(64,-117,35,-64) (9/5,13/7) -> (9/5,13/7) Reflection Matrix(27,-52,14,-27) (13/7,2/1) -> (13/7,2/1) Reflection Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(275,-666,64,-155) (29/12,17/7) -> (17/4,13/3) Glide Reflection Matrix(144,-377,55,-144) (13/5,29/11) -> (13/5,29/11) Reflection Matrix(175,-464,66,-175) (29/11,8/3) -> (29/11,8/3) Reflection Matrix(16,-45,5,-14) (8/3,3/1) -> (3/1,7/2) Parabolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(197,-924,42,-197) (14/3,33/7) -> (14/3,33/7) Reflection Matrix(34,-165,7,-34) (33/7,5/1) -> (33/7,5/1) Reflection Matrix(16,-85,3,-16) (5/1,17/3) -> (5/1,17/3) Reflection Matrix(35,-204,6,-35) (17/3,6/1) -> (17/3,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(0,1,1,0) -> Matrix(1,1,0,-1) (-1/1,1/1) -> (-1/2,1/0) Matrix(10,-11,9,-10) -> Matrix(1,2,0,-1) (1/1,11/9) -> (-1/1,1/0) Matrix(89,-110,72,-89) -> Matrix(-1,0,2,1) (11/9,5/4) -> (-1/1,0/1) Matrix(115,-146,26,-33) -> Matrix(1,-1,0,1) 1/0 Matrix(86,-113,35,-46) -> Matrix(1,3,0,-1) *** -> (-3/2,1/0) Matrix(137,-184,102,-137) -> Matrix(-1,0,2,1) (4/3,23/17) -> (-1/1,0/1) Matrix(254,-345,187,-254) -> Matrix(1,1,0,-1) (23/17,15/11) -> (-1/2,1/0) Matrix(253,-346,68,-93) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(394,-545,227,-314) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(176,-245,125,-174) -> Matrix(1,0,2,1) 0/1 Matrix(231,-328,50,-71) -> Matrix(3,-2,-2,1) Matrix(49,-72,32,-47) -> Matrix(1,-4,0,1) 1/0 Matrix(453,-706,188,-293) -> Matrix(7,24,-2,-7) *** -> (-4/1,-3/1) Matrix(1951,-3050,1248,-1951) -> Matrix(43,132,-14,-43) (25/16,61/39) -> (-22/7,-3/1) Matrix(428,-671,273,-428) -> Matrix(11,30,-4,-11) (61/39,11/7) -> (-3/1,-5/2) Matrix(100,-159,39,-62) -> Matrix(3,8,-2,-5) -2/1 Matrix(53,-86,8,-13) -> Matrix(1,1,-2,-3) Matrix(76,-125,45,-74) -> Matrix(3,5,-2,-3) (-2/1,-1/1).(-3/2,1/0) Matrix(419,-722,242,-417) -> Matrix(1,-1,0,1) 1/0 Matrix(79,-140,22,-39) -> Matrix(1,1,-2,-3) Matrix(64,-117,35,-64) -> Matrix(5,6,-4,-5) (9/5,13/7) -> (-3/2,-1/1) Matrix(27,-52,14,-27) -> Matrix(-1,0,2,1) (13/7,2/1) -> (-1/1,0/1) Matrix(43,-98,18,-41) -> Matrix(1,-3,0,1) 1/0 Matrix(275,-666,64,-155) -> Matrix(3,7,-2,-5) Matrix(144,-377,55,-144) -> Matrix(1,3,0,-1) (13/5,29/11) -> (-3/2,1/0) Matrix(175,-464,66,-175) -> Matrix(3,4,-2,-3) (29/11,8/3) -> (-2/1,-1/1) Matrix(16,-45,5,-14) -> Matrix(1,2,-2,-3) -1/1 Matrix(33,-128,8,-31) -> Matrix(1,-2,0,1) 1/0 Matrix(197,-924,42,-197) -> Matrix(3,4,-2,-3) (14/3,33/7) -> (-2/1,-1/1) Matrix(34,-165,7,-34) -> Matrix(1,3,0,-1) (33/7,5/1) -> (-3/2,1/0) Matrix(16,-85,3,-16) -> Matrix(5,6,-4,-5) (5/1,17/3) -> (-3/2,-1/1) Matrix(35,-204,6,-35) -> Matrix(9,8,-10,-9) (17/3,6/1) -> (-1/1,-4/5) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.