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: 64 Genus: 41 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/6 -7/9 -11/15 -2/3 -35/54 -5/9 -1/2 -11/24 -4/9 -5/12 -41/108 -13/36 -1/3 -17/54 -11/36 -3/10 -29/108 -4/15 -19/72 -1/4 -2/9 -3/14 -3/16 -3/17 -1/6 -1/8 -1/9 0/1 1/8 1/7 2/13 1/6 2/11 3/16 1/5 3/14 2/9 3/13 4/17 1/4 4/15 3/11 5/18 2/7 8/27 3/10 1/3 7/18 2/5 5/12 4/9 11/24 1/2 5/9 11/18 17/27 2/3 13/18 11/15 7/9 5/6 8/9 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/2 -7/8 -3/7 -6/7 -1/3 0/1 -11/13 -1/4 -5/6 0/1 -9/11 1/0 -13/16 -1/1 -30/37 -1/1 -6/7 -17/21 -1/2 -4/5 -1/1 -2/3 -11/14 -4/7 -7/9 -1/2 -17/22 -6/13 -27/35 -1/2 -10/13 -4/9 -3/7 -13/17 -9/22 -3/4 -1/3 -14/19 -1/7 0/1 -11/15 -1/2 -30/41 -2/5 -1/3 -19/26 -2/7 -27/37 -1/4 -8/11 -1/5 0/1 -13/18 0/1 -5/7 1/0 -12/17 -1/1 0/1 -19/27 1/0 -26/37 -2/1 -1/1 -7/10 0/1 -9/13 -1/2 -11/16 -1/1 -2/3 -1/2 -13/20 -1/3 -24/37 -10/29 -1/3 -35/54 -1/3 -11/17 -5/16 -9/14 0/1 -7/11 -1/2 -19/30 0/1 -12/19 -1/1 0/1 -17/27 -1/2 -5/8 -1/3 -18/29 -2/9 -1/5 -31/50 0/1 -13/21 -1/6 -34/55 -1/11 0/1 -21/34 0/1 -8/13 -1/7 0/1 -11/18 0/1 -3/5 1/0 -10/17 -1/3 0/1 -7/12 0/1 -11/19 3/2 -4/7 -1/1 0/1 -13/23 1/0 -9/16 -1/1 -5/9 -1/2 -11/20 -1/3 -6/11 -1/3 0/1 -19/35 -1/2 -13/24 -1/3 -7/13 -1/4 -1/2 0/1 -6/13 0/1 1/3 -17/37 1/2 -11/24 1/1 -5/11 1/0 -4/9 1/0 -11/25 1/0 -7/16 -1/1 -10/23 -1/1 0/1 -3/7 -1/2 -8/19 -1/1 0/1 -5/12 0/1 -7/17 3/4 -9/22 2/1 -2/5 -1/1 0/1 -7/18 0/1 -5/13 1/2 -13/34 0/1 -8/21 1/2 -27/71 1/2 -19/50 2/3 -30/79 4/5 1/1 -41/108 1/1 -11/29 1/0 -3/8 1/1 -13/35 1/0 -10/27 1/0 -7/19 -5/2 -4/11 -1/1 0/1 -13/36 0/1 -9/25 1/2 -5/14 0/1 -6/17 0/1 1/1 -1/3 1/0 -6/19 -2/1 -1/1 -17/54 -2/1 -11/35 -3/2 -5/16 -1/1 -4/13 -2/1 -1/1 -11/36 -1/1 -7/23 -1/2 -3/10 0/1 -11/37 1/0 -8/27 1/0 -5/17 -7/2 -2/7 -2/1 -1/1 -5/18 -1/1 -3/11 -1/2 -13/48 -1/1 -10/37 -1/1 -2/3 -7/26 0/1 -18/67 -4/3 -1/1 -29/108 -1/1 -11/41 -3/4 -4/15 -1/2 -9/34 0/1 -14/53 -1/11 0/1 -19/72 0/1 -5/19 1/4 -1/4 -1/1 -4/17 0/1 1/3 -7/30 0/1 -3/13 1/0 -2/9 1/0 -5/23 1/0 -13/60 -3/1 -8/37 -10/3 -3/1 -11/51 -5/2 -3/14 -2/1 -4/19 -3/1 -2/1 -1/5 -3/2 -4/21 1/0 -7/37 1/0 -3/16 -1/1 -2/11 -2/1 -1/1 -3/17 -11/8 -1/6 -1/1 -3/19 -7/10 -2/13 -1/1 0/1 -1/7 1/0 -1/8 -1/1 -1/9 1/0 0/1 -1/1 0/1 1/8 -1/1 1/7 -1/2 2/13 -1/1 0/1 1/6 -1/1 2/11 -1/1 -2/3 3/16 -1/1 7/37 -1/2 4/21 -1/2 1/5 -3/4 3/14 -2/3 2/9 -1/2 5/22 -2/5 8/35 -1/3 -2/7 3/13 -1/2 4/17 -1/5 0/1 1/4 -1/1 5/19 -1/6 4/15 1/0 11/41 -3/2 7/26 0/1 10/37 -2/1 -1/1 3/11 1/0 5/18 -1/1 2/7 -1/1 -2/3 5/17 -7/12 8/27 -1/2 11/37 -1/2 3/10 0/1 4/13 -1/1 -2/3 5/16 -1/1 1/3 -1/2 7/20 -1/3 13/37 -1/4 19/54 0/1 6/17 -1/3 0/1 5/14 0/1 4/11 -1/1 0/1 11/30 -1/1 7/19 -5/8 10/27 -1/2 3/8 -1/3 11/29 -1/2 19/50 -2/7 8/21 -1/4 21/55 -1/10 13/34 0/1 5/13 -1/4 7/18 0/1 2/5 -1/1 0/1 7/17 -3/10 5/12 0/1 8/19 -1/1 0/1 3/7 1/0 10/23 -1/1 0/1 7/16 -1/1 4/9 -1/2 9/20 -1/3 5/11 -1/2 16/35 -2/5 -1/3 11/24 -1/3 6/13 -1/5 0/1 1/2 0/1 7/13 1/2 20/37 1/1 2/1 13/24 1/1 6/11 0/1 1/1 5/9 1/0 14/25 -2/1 -1/1 9/16 -1/1 13/23 -1/2 4/7 -1/1 0/1 11/19 -3/8 7/12 0/1 10/17 0/1 1/1 13/22 0/1 3/5 -1/2 11/18 0/1 8/13 0/1 1/5 21/34 0/1 13/21 1/4 44/71 1/3 4/11 31/50 0/1 49/79 3/10 67/108 1/3 18/29 1/3 2/5 5/8 1/1 22/35 0/1 1/1 17/27 1/0 12/19 -1/1 0/1 7/11 1/0 23/36 0/1 16/25 0/1 1/3 9/14 0/1 11/17 5/6 2/3 1/0 13/19 -7/6 37/54 -1/1 24/35 -1/1 -8/9 11/16 -1/1 9/13 1/0 25/36 -1/1 16/23 -1/1 -2/3 7/10 0/1 26/37 -1/1 -2/3 19/27 -1/2 12/17 -1/1 0/1 5/7 -1/2 13/18 0/1 8/11 0/1 1/3 35/48 1/3 27/37 1/2 19/26 2/3 49/67 5/6 79/108 1/1 30/41 1/1 2/1 11/15 1/0 25/34 0/1 39/53 -1/10 53/72 0/1 14/19 0/1 1/5 3/4 1/1 13/17 9/4 23/30 3/1 10/13 3/1 4/1 7/9 1/0 18/23 -7/1 -6/1 47/60 -5/1 29/37 1/0 40/51 1/0 11/14 -4/1 15/19 -11/4 4/5 -2/1 -1/1 17/21 1/0 30/37 -6/5 -1/1 13/16 -1/1 9/11 -1/2 14/17 -1/3 0/1 5/6 0/1 16/19 -1/1 0/1 11/13 1/2 6/7 0/1 1/1 7/8 3/1 8/9 1/0 1/1 1/0 1/0 -1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(107,94,-288,-253) (-1/1,-7/8) -> (-3/8,-13/35) Hyperbolic Matrix(179,156,288,251) (-7/8,-6/7) -> (18/29,5/8) Hyperbolic Matrix(143,122,252,215) (-6/7,-11/13) -> (13/23,4/7) Hyperbolic Matrix(109,92,-468,-395) (-11/13,-5/6) -> (-7/30,-3/13) Hyperbolic Matrix(251,206,-396,-325) (-5/6,-9/11) -> (-7/11,-19/30) Hyperbolic Matrix(179,146,396,323) (-9/11,-13/16) -> (9/20,5/11) Hyperbolic Matrix(791,642,1152,935) (-13/16,-30/37) -> (24/35,11/16) Hyperbolic Matrix(1405,1138,2268,1837) (-30/37,-17/21) -> (13/21,44/71) Hyperbolic Matrix(685,554,936,757) (-17/21,-4/5) -> (30/41,11/15) Hyperbolic Matrix(73,58,-180,-143) (-4/5,-11/14) -> (-9/22,-2/5) Hyperbolic Matrix(251,196,-324,-253) (-11/14,-7/9) -> (-7/9,-17/22) Parabolic Matrix(1259,972,-3312,-2557) (-17/22,-27/35) -> (-27/71,-19/50) Hyperbolic Matrix(431,332,-936,-721) (-27/35,-10/13) -> (-6/13,-17/37) Hyperbolic Matrix(73,56,-468,-359) (-10/13,-13/17) -> (-3/19,-2/13) Hyperbolic Matrix(37,28,144,109) (-13/17,-3/4) -> (1/4,5/19) Hyperbolic Matrix(35,26,144,107) (-3/4,-14/19) -> (4/17,1/4) Hyperbolic Matrix(757,556,-1224,-899) (-14/19,-11/15) -> (-13/21,-34/55) Hyperbolic Matrix(757,554,936,685) (-11/15,-30/41) -> (4/5,17/21) Hyperbolic Matrix(755,552,-2808,-2053) (-30/41,-19/26) -> (-7/26,-18/67) Hyperbolic Matrix(397,290,1332,973) (-19/26,-27/37) -> (11/37,3/10) Hyperbolic Matrix(431,314,-792,-577) (-27/37,-8/11) -> (-6/11,-19/35) Hyperbolic Matrix(287,208,396,287) (-8/11,-13/18) -> (13/18,8/11) Hyperbolic Matrix(181,130,252,181) (-13/18,-5/7) -> (5/7,13/18) Hyperbolic Matrix(107,76,252,179) (-5/7,-12/17) -> (8/19,3/7) Hyperbolic Matrix(613,432,972,685) (-12/17,-19/27) -> (17/27,12/19) Hyperbolic Matrix(1223,860,1944,1367) (-19/27,-26/37) -> (22/35,17/27) Hyperbolic Matrix(359,252,1332,935) (-26/37,-7/10) -> (7/26,10/37) Hyperbolic Matrix(109,76,-360,-251) (-7/10,-9/13) -> (-7/23,-3/10) Hyperbolic Matrix(325,224,576,397) (-9/13,-11/16) -> (9/16,13/23) Hyperbolic Matrix(71,48,-108,-73) (-11/16,-2/3) -> (-2/3,-13/20) Parabolic Matrix(1081,702,1332,865) (-13/20,-24/37) -> (30/37,13/16) Hyperbolic Matrix(2665,1728,3888,2521) (-24/37,-35/54) -> (37/54,24/35) Hyperbolic Matrix(1331,862,1944,1259) (-35/54,-11/17) -> (13/19,37/54) Hyperbolic Matrix(397,256,504,325) (-11/17,-9/14) -> (11/14,15/19) Hyperbolic Matrix(181,116,-504,-323) (-9/14,-7/11) -> (-9/25,-5/14) Hyperbolic Matrix(253,160,-1080,-683) (-19/30,-12/19) -> (-4/17,-7/30) Hyperbolic Matrix(685,432,972,613) (-12/19,-17/27) -> (19/27,12/17) Hyperbolic Matrix(35,22,-288,-181) (-17/27,-5/8) -> (-1/8,-1/9) Hyperbolic Matrix(251,156,288,179) (-5/8,-18/29) -> (6/7,7/8) Hyperbolic Matrix(2051,1272,-5400,-3349) (-18/29,-31/50) -> (-19/50,-30/79) Hyperbolic Matrix(613,380,-2844,-1763) (-31/50,-13/21) -> (-11/51,-3/14) Hyperbolic Matrix(971,600,-3672,-2269) (-34/55,-21/34) -> (-9/34,-14/53) Hyperbolic Matrix(217,134,468,289) (-21/34,-8/13) -> (6/13,1/2) Hyperbolic Matrix(287,176,468,287) (-8/13,-11/18) -> (11/18,8/13) Hyperbolic Matrix(109,66,180,109) (-11/18,-3/5) -> (3/5,11/18) Hyperbolic Matrix(37,22,-180,-107) (-3/5,-10/17) -> (-4/19,-1/5) Hyperbolic Matrix(181,106,432,253) (-10/17,-7/12) -> (5/12,8/19) Hyperbolic Matrix(179,104,432,251) (-7/12,-11/19) -> (7/17,5/12) Hyperbolic Matrix(73,42,252,145) (-11/19,-4/7) -> (2/7,5/17) Hyperbolic Matrix(215,122,252,143) (-4/7,-13/23) -> (11/13,6/7) Hyperbolic Matrix(397,224,576,325) (-13/23,-9/16) -> (11/16,9/13) Hyperbolic Matrix(179,100,-324,-181) (-9/16,-5/9) -> (-5/9,-11/20) Parabolic Matrix(73,40,396,217) (-11/20,-6/11) -> (2/11,3/16) Hyperbolic Matrix(2341,1270,2988,1621) (-19/35,-13/24) -> (47/60,29/37) Hyperbolic Matrix(289,156,-1332,-719) (-13/24,-7/13) -> (-5/23,-13/60) Hyperbolic Matrix(179,96,468,251) (-7/13,-1/2) -> (13/34,5/13) Hyperbolic Matrix(289,134,468,217) (-1/2,-6/13) -> (8/13,21/34) Hyperbolic Matrix(1943,892,2664,1223) (-17/37,-11/24) -> (35/48,27/37) Hyperbolic Matrix(215,98,-792,-361) (-11/24,-5/11) -> (-3/11,-13/48) Hyperbolic Matrix(143,64,-324,-145) (-5/11,-4/9) -> (-4/9,-11/25) Parabolic Matrix(469,206,576,253) (-11/25,-7/16) -> (13/16,9/11) Hyperbolic Matrix(179,78,576,251) (-7/16,-10/23) -> (4/13,5/16) Hyperbolic Matrix(37,16,252,109) (-10/23,-3/7) -> (1/7,2/13) Hyperbolic Matrix(179,76,252,107) (-3/7,-8/19) -> (12/17,5/7) Hyperbolic Matrix(253,106,432,181) (-8/19,-5/12) -> (7/12,10/17) Hyperbolic Matrix(251,104,432,179) (-5/12,-7/17) -> (11/19,7/12) Hyperbolic Matrix(395,162,612,251) (-7/17,-9/22) -> (9/14,11/17) Hyperbolic Matrix(71,28,180,71) (-2/5,-7/18) -> (7/18,2/5) Hyperbolic Matrix(181,70,468,181) (-7/18,-5/13) -> (5/13,7/18) Hyperbolic Matrix(251,96,468,179) (-5/13,-13/34) -> (1/2,7/13) Hyperbolic Matrix(325,124,-1224,-467) (-13/34,-8/21) -> (-4/15,-9/34) Hyperbolic Matrix(431,164,2268,863) (-8/21,-27/71) -> (7/37,4/21) Hyperbolic Matrix(7237,2748,11664,4429) (-30/79,-41/108) -> (67/108,18/29) Hyperbolic Matrix(7235,2746,11664,4427) (-41/108,-11/29) -> (49/79,67/108) Hyperbolic Matrix(37,14,288,109) (-11/29,-3/8) -> (1/8,1/7) Hyperbolic Matrix(577,214,1944,721) (-13/35,-10/27) -> (8/27,11/37) Hyperbolic Matrix(287,106,972,359) (-10/27,-7/19) -> (5/17,8/27) Hyperbolic Matrix(71,26,-396,-145) (-7/19,-4/11) -> (-2/11,-3/17) Hyperbolic Matrix(829,300,1296,469) (-4/11,-13/36) -> (23/36,16/25) Hyperbolic Matrix(827,298,1296,467) (-13/36,-9/25) -> (7/11,23/36) Hyperbolic Matrix(361,128,612,217) (-5/14,-6/17) -> (10/17,13/22) Hyperbolic Matrix(35,12,-108,-37) (-6/17,-1/3) -> (-1/3,-6/19) Parabolic Matrix(685,216,1944,613) (-6/19,-17/54) -> (19/54,6/17) Hyperbolic Matrix(1367,430,3888,1223) (-17/54,-11/35) -> (13/37,19/54) Hyperbolic Matrix(217,68,1152,361) (-11/35,-5/16) -> (3/16,7/37) Hyperbolic Matrix(251,78,576,179) (-5/16,-4/13) -> (10/23,7/16) Hyperbolic Matrix(901,276,1296,397) (-4/13,-11/36) -> (25/36,16/23) Hyperbolic Matrix(899,274,1296,395) (-11/36,-7/23) -> (9/13,25/36) Hyperbolic Matrix(973,290,1332,397) (-3/10,-11/37) -> (27/37,19/26) Hyperbolic Matrix(323,96,360,107) (-11/37,-8/27) -> (8/9,1/1) Hyperbolic Matrix(359,106,972,287) (-8/27,-5/17) -> (7/19,10/27) Hyperbolic Matrix(145,42,252,73) (-5/17,-2/7) -> (4/7,11/19) Hyperbolic Matrix(71,20,252,71) (-2/7,-5/18) -> (5/18,2/7) Hyperbolic Matrix(109,30,396,109) (-5/18,-3/11) -> (3/11,5/18) Hyperbolic Matrix(1441,390,2664,721) (-13/48,-10/37) -> (20/37,13/24) Hyperbolic Matrix(935,252,1332,359) (-10/37,-7/26) -> (7/10,26/37) Hyperbolic Matrix(8533,2292,11664,3133) (-18/67,-29/108) -> (79/108,30/41) Hyperbolic Matrix(8531,2290,11664,3131) (-29/108,-11/41) -> (49/67,79/108) Hyperbolic Matrix(179,48,936,251) (-11/41,-4/15) -> (4/21,1/5) Hyperbolic Matrix(3817,1008,5184,1369) (-14/53,-19/72) -> (53/72,14/19) Hyperbolic Matrix(3815,1006,5184,1367) (-19/72,-5/19) -> (39/53,53/72) Hyperbolic Matrix(109,28,144,37) (-5/19,-1/4) -> (3/4,13/17) Hyperbolic Matrix(107,26,144,35) (-1/4,-4/17) -> (14/19,3/4) Hyperbolic Matrix(71,16,-324,-73) (-3/13,-2/9) -> (-2/9,-5/23) Parabolic Matrix(1367,296,2988,647) (-13/60,-8/37) -> (16/35,11/24) Hyperbolic Matrix(2159,466,2664,575) (-8/37,-11/51) -> (17/21,30/37) Hyperbolic Matrix(179,38,504,107) (-3/14,-4/19) -> (6/17,5/14) Hyperbolic Matrix(251,48,936,179) (-1/5,-4/21) -> (4/15,11/41) Hyperbolic Matrix(2089,396,2664,505) (-4/21,-7/37) -> (29/37,40/51) Hyperbolic Matrix(467,88,1332,251) (-7/37,-3/16) -> (7/20,13/37) Hyperbolic Matrix(323,60,576,107) (-3/16,-2/11) -> (14/25,9/16) Hyperbolic Matrix(35,6,-216,-37) (-3/17,-1/6) -> (-1/6,-3/19) Parabolic Matrix(109,16,252,37) (-2/13,-1/7) -> (3/7,10/23) Hyperbolic Matrix(109,14,288,37) (-1/7,-1/8) -> (3/8,11/29) Hyperbolic Matrix(253,26,360,37) (-1/9,0/1) -> (26/37,19/27) Hyperbolic Matrix(181,-22,288,-35) (0/1,1/8) -> (5/8,22/35) Hyperbolic Matrix(359,-56,468,-73) (2/13,1/6) -> (23/30,10/13) Hyperbolic Matrix(145,-26,396,-71) (1/6,2/11) -> (4/11,11/30) Hyperbolic Matrix(107,-22,180,-37) (1/5,3/14) -> (13/22,3/5) Hyperbolic Matrix(73,-16,324,-71) (3/14,2/9) -> (2/9,5/22) Parabolic Matrix(2053,-468,3312,-755) (5/22,8/35) -> (44/71,31/50) Hyperbolic Matrix(505,-116,936,-215) (8/35,3/13) -> (7/13,20/37) Hyperbolic Matrix(395,-92,468,-109) (3/13,4/17) -> (16/19,11/13) Hyperbolic Matrix(467,-124,1224,-325) (5/19,4/15) -> (8/21,21/55) Hyperbolic Matrix(2053,-552,2808,-755) (11/41,7/26) -> (19/26,49/67) Hyperbolic Matrix(361,-98,792,-215) (10/37,3/11) -> (5/11,16/35) Hyperbolic Matrix(251,-76,360,-109) (3/10,4/13) -> (16/23,7/10) Hyperbolic Matrix(37,-12,108,-35) (5/16,1/3) -> (1/3,7/20) Parabolic Matrix(323,-116,504,-181) (5/14,4/11) -> (16/25,9/14) Hyperbolic Matrix(827,-304,1080,-397) (11/30,7/19) -> (13/17,23/30) Hyperbolic Matrix(253,-94,288,-107) (10/27,3/8) -> (7/8,8/9) Hyperbolic Matrix(3349,-1272,5400,-2051) (11/29,19/50) -> (31/50,49/79) Hyperbolic Matrix(2231,-848,2844,-1081) (19/50,8/21) -> (40/51,11/14) Hyperbolic Matrix(2701,-1032,3672,-1403) (21/55,13/34) -> (25/34,39/53) Hyperbolic Matrix(143,-58,180,-73) (2/5,7/17) -> (15/19,4/5) Hyperbolic Matrix(145,-64,324,-143) (7/16,4/9) -> (4/9,9/20) Parabolic Matrix(1043,-480,1332,-613) (11/24,6/13) -> (18/23,47/60) Hyperbolic Matrix(577,-314,792,-431) (13/24,6/11) -> (8/11,35/48) Hyperbolic Matrix(181,-100,324,-179) (6/11,5/9) -> (5/9,14/25) Parabolic Matrix(899,-556,1224,-757) (21/34,13/21) -> (11/15,25/34) Hyperbolic Matrix(325,-206,396,-251) (12/19,7/11) -> (9/11,14/17) Hyperbolic Matrix(73,-48,108,-71) (11/17,2/3) -> (2/3,13/19) Parabolic Matrix(253,-196,324,-251) (10/13,7/9) -> (7/9,18/23) Parabolic Matrix(181,-150,216,-179) (14/17,5/6) -> (5/6,16/19) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(3,2,-2,-1) Matrix(107,94,-288,-253) -> Matrix(9,4,2,1) Matrix(179,156,288,251) -> Matrix(5,2,12,5) Matrix(143,122,252,215) -> Matrix(1,0,2,1) Matrix(109,92,-468,-395) -> Matrix(1,0,4,1) Matrix(251,206,-396,-325) -> Matrix(1,0,-2,1) Matrix(179,146,396,323) -> Matrix(1,0,-2,1) Matrix(791,642,1152,935) -> Matrix(1,2,-2,-3) Matrix(1405,1138,2268,1837) -> Matrix(3,2,10,7) Matrix(685,554,936,757) -> Matrix(1,0,2,1) Matrix(73,58,-180,-143) -> Matrix(3,2,-2,-1) Matrix(251,196,-324,-253) -> Matrix(19,10,-40,-21) Matrix(1259,972,-3312,-2557) -> Matrix(9,4,20,9) Matrix(431,332,-936,-721) -> Matrix(9,4,20,9) Matrix(73,56,-468,-359) -> Matrix(9,4,-16,-7) Matrix(37,28,144,109) -> Matrix(5,2,-8,-3) Matrix(35,26,144,107) -> Matrix(1,0,2,1) Matrix(757,556,-1224,-899) -> Matrix(1,0,-4,1) Matrix(757,554,936,685) -> Matrix(1,0,2,1) Matrix(755,552,-2808,-2053) -> Matrix(7,2,-4,-1) Matrix(397,290,1332,973) -> Matrix(7,2,-18,-5) Matrix(431,314,-792,-577) -> Matrix(1,0,2,1) Matrix(287,208,396,287) -> Matrix(1,0,8,1) Matrix(181,130,252,181) -> Matrix(1,0,-2,1) Matrix(107,76,252,179) -> Matrix(1,0,0,1) Matrix(613,432,972,685) -> Matrix(1,0,0,1) Matrix(1223,860,1944,1367) -> Matrix(1,2,0,1) Matrix(359,252,1332,935) -> Matrix(1,0,0,1) Matrix(109,76,-360,-251) -> Matrix(1,0,0,1) Matrix(325,224,576,397) -> Matrix(1,0,0,1) Matrix(71,48,-108,-73) -> Matrix(3,2,-8,-5) Matrix(1081,702,1332,865) -> Matrix(11,4,-14,-5) Matrix(2665,1728,3888,2521) -> Matrix(53,18,-56,-19) Matrix(1331,862,1944,1259) -> Matrix(37,12,-34,-11) Matrix(397,256,504,325) -> Matrix(15,4,-4,-1) Matrix(181,116,-504,-323) -> Matrix(1,0,4,1) Matrix(253,160,-1080,-683) -> Matrix(1,0,4,1) Matrix(685,432,972,613) -> Matrix(1,0,0,1) Matrix(35,22,-288,-181) -> Matrix(1,0,2,1) Matrix(251,156,288,179) -> Matrix(9,2,4,1) Matrix(2051,1272,-5400,-3349) -> Matrix(11,2,16,3) Matrix(613,380,-2844,-1763) -> Matrix(7,2,-4,-1) Matrix(971,600,-3672,-2269) -> Matrix(1,0,0,1) Matrix(217,134,468,289) -> Matrix(1,0,2,1) Matrix(287,176,468,287) -> Matrix(1,0,12,1) Matrix(109,66,180,109) -> Matrix(1,0,-2,1) Matrix(37,22,-180,-107) -> Matrix(3,2,-2,-1) Matrix(181,106,432,253) -> Matrix(1,0,2,1) Matrix(179,104,432,251) -> Matrix(1,0,-4,1) Matrix(73,42,252,145) -> Matrix(1,2,-2,-3) Matrix(215,122,252,143) -> Matrix(1,0,2,1) Matrix(397,224,576,325) -> Matrix(1,0,0,1) Matrix(179,100,-324,-181) -> Matrix(3,2,-8,-5) Matrix(73,40,396,217) -> Matrix(5,2,-8,-3) Matrix(2341,1270,2988,1621) -> Matrix(7,4,-2,-1) Matrix(289,156,-1332,-719) -> Matrix(15,4,-4,-1) Matrix(179,96,468,251) -> Matrix(1,0,0,1) Matrix(289,134,468,217) -> Matrix(1,0,2,1) Matrix(1943,892,2664,1223) -> Matrix(3,-2,8,-5) Matrix(215,98,-792,-361) -> Matrix(1,0,-2,1) Matrix(143,64,-324,-145) -> Matrix(1,-2,0,1) Matrix(469,206,576,253) -> Matrix(1,2,-2,-3) Matrix(179,78,576,251) -> Matrix(1,2,-2,-3) Matrix(37,16,252,109) -> Matrix(1,0,0,1) Matrix(179,76,252,107) -> Matrix(1,0,0,1) Matrix(253,106,432,181) -> Matrix(1,0,2,1) Matrix(251,104,432,179) -> Matrix(1,0,-4,1) Matrix(395,162,612,251) -> Matrix(1,-2,2,-3) Matrix(71,28,180,71) -> Matrix(1,0,0,1) Matrix(181,70,468,181) -> Matrix(1,0,-6,1) Matrix(251,96,468,179) -> Matrix(1,0,0,1) Matrix(325,124,-1224,-467) -> Matrix(1,0,-4,1) Matrix(431,164,2268,863) -> Matrix(3,-2,-4,3) Matrix(7237,2748,11664,4429) -> Matrix(7,-6,20,-17) Matrix(7235,2746,11664,4427) -> Matrix(3,-4,10,-13) Matrix(37,14,288,109) -> Matrix(1,0,-2,1) Matrix(577,214,1944,721) -> Matrix(1,-8,-2,17) Matrix(287,106,972,359) -> Matrix(1,6,-2,-11) Matrix(71,26,-396,-145) -> Matrix(3,2,-2,-1) Matrix(829,300,1296,469) -> Matrix(1,0,4,1) Matrix(827,298,1296,467) -> Matrix(1,0,-2,1) Matrix(361,128,612,217) -> Matrix(1,0,0,1) Matrix(35,12,-108,-37) -> Matrix(1,-2,0,1) Matrix(685,216,1944,613) -> Matrix(1,2,-4,-7) Matrix(1367,430,3888,1223) -> Matrix(1,2,-6,-11) Matrix(217,68,1152,361) -> Matrix(3,4,-4,-5) Matrix(251,78,576,179) -> Matrix(1,2,-2,-3) Matrix(901,276,1296,397) -> Matrix(3,4,-4,-5) Matrix(899,274,1296,395) -> Matrix(3,2,-2,-1) Matrix(973,290,1332,397) -> Matrix(1,-2,2,-3) Matrix(323,96,360,107) -> Matrix(1,-4,0,1) Matrix(359,106,972,287) -> Matrix(1,6,-2,-11) Matrix(145,42,252,73) -> Matrix(1,2,-2,-3) Matrix(71,20,252,71) -> Matrix(3,4,-4,-5) Matrix(109,30,396,109) -> Matrix(3,2,-2,-1) Matrix(1441,390,2664,721) -> Matrix(1,0,2,1) Matrix(935,252,1332,359) -> Matrix(1,0,0,1) Matrix(8533,2292,11664,3133) -> Matrix(5,6,4,5) Matrix(8531,2290,11664,3131) -> Matrix(9,8,10,9) Matrix(179,48,936,251) -> Matrix(1,0,0,1) Matrix(3817,1008,5184,1369) -> Matrix(1,0,16,1) Matrix(3815,1006,5184,1367) -> Matrix(1,0,-14,1) Matrix(109,28,144,37) -> Matrix(1,2,0,1) Matrix(107,26,144,35) -> Matrix(1,0,2,1) Matrix(71,16,-324,-73) -> Matrix(1,-4,0,1) Matrix(1367,296,2988,647) -> Matrix(1,4,-4,-15) Matrix(2159,466,2664,575) -> Matrix(3,8,-2,-5) Matrix(179,38,504,107) -> Matrix(1,2,-2,-3) Matrix(251,48,936,179) -> Matrix(1,0,0,1) Matrix(2089,396,2664,505) -> Matrix(1,-4,0,1) Matrix(467,88,1332,251) -> Matrix(1,2,-4,-7) Matrix(323,60,576,107) -> Matrix(1,0,0,1) Matrix(35,6,-216,-37) -> Matrix(5,6,-6,-7) Matrix(109,16,252,37) -> Matrix(1,0,0,1) Matrix(109,14,288,37) -> Matrix(1,0,-2,1) Matrix(253,26,360,37) -> Matrix(1,2,-2,-3) Matrix(181,-22,288,-35) -> Matrix(1,0,2,1) Matrix(359,-56,468,-73) -> Matrix(1,4,0,1) Matrix(145,-26,396,-71) -> Matrix(3,2,-2,-1) Matrix(107,-22,180,-37) -> Matrix(3,2,-2,-1) Matrix(73,-16,324,-71) -> Matrix(7,4,-16,-9) Matrix(2053,-468,3312,-755) -> Matrix(5,2,12,5) Matrix(505,-116,936,-215) -> Matrix(1,0,4,1) Matrix(395,-92,468,-109) -> Matrix(1,0,4,1) Matrix(467,-124,1224,-325) -> Matrix(1,0,-4,1) Matrix(2053,-552,2808,-755) -> Matrix(3,2,4,3) Matrix(361,-98,792,-215) -> Matrix(1,0,-2,1) Matrix(251,-76,360,-109) -> Matrix(1,0,0,1) Matrix(37,-12,108,-35) -> Matrix(3,2,-8,-5) Matrix(323,-116,504,-181) -> Matrix(1,0,4,1) Matrix(827,-304,1080,-397) -> Matrix(11,8,4,3) Matrix(253,-94,288,-107) -> Matrix(9,4,2,1) Matrix(3349,-1272,5400,-2051) -> Matrix(7,2,24,7) Matrix(2231,-848,2844,-1081) -> Matrix(23,6,-4,-1) Matrix(2701,-1032,3672,-1403) -> Matrix(1,0,0,1) Matrix(143,-58,180,-73) -> Matrix(3,2,-2,-1) Matrix(145,-64,324,-143) -> Matrix(3,2,-8,-5) Matrix(1043,-480,1332,-613) -> Matrix(23,6,-4,-1) Matrix(577,-314,792,-431) -> Matrix(1,0,2,1) Matrix(181,-100,324,-179) -> Matrix(1,-2,0,1) Matrix(899,-556,1224,-757) -> Matrix(1,0,-4,1) Matrix(325,-206,396,-251) -> Matrix(1,0,-2,1) Matrix(73,-48,108,-71) -> Matrix(1,-2,0,1) Matrix(253,-196,324,-251) -> Matrix(1,-10,0,1) Matrix(181,-150,216,-179) -> Matrix(1,0,2,1) 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 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: 40 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/6 -7/9 -11/15 -2/3 -17/27 -5/9 -1/2 -3/8 -1/3 -5/17 -19/72 -1/4 -1/6 -1/7 0/1 1/6 1/5 2/9 3/13 1/4 3/11 5/18 2/7 1/3 3/8 7/18 2/5 4/9 1/2 5/9 11/18 17/27 2/3 13/18 11/15 7/9 5/6 8/9 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/2 -7/8 -3/7 -6/7 -1/3 0/1 -11/13 -1/4 -5/6 0/1 -9/11 1/0 -4/5 -1/1 -2/3 -7/9 -1/2 -10/13 -4/9 -3/7 -3/4 -1/3 -14/19 -1/7 0/1 -11/15 -1/2 -8/11 -1/5 0/1 -13/18 0/1 -5/7 1/0 -12/17 -1/1 0/1 -19/27 1/0 -7/10 0/1 -2/3 -1/2 -7/11 -1/2 -19/30 0/1 -12/19 -1/1 0/1 -17/27 -1/2 -5/8 -1/3 -18/29 -2/9 -1/5 -13/21 -1/6 -34/55 -1/11 0/1 -21/34 0/1 -8/13 -1/7 0/1 -11/18 0/1 -3/5 1/0 -7/12 0/1 -11/19 3/2 -4/7 -1/1 0/1 -5/9 -1/2 -6/11 -1/3 0/1 -7/13 -1/4 -1/2 0/1 -6/13 0/1 1/3 -5/11 1/0 -4/9 1/0 -3/7 -1/2 -5/12 0/1 -7/17 3/4 -9/22 2/1 -2/5 -1/1 0/1 -7/18 0/1 -5/13 1/2 -13/34 0/1 -8/21 1/2 -3/8 1/1 -1/3 1/0 -3/10 0/1 -11/37 1/0 -8/27 1/0 -5/17 -7/2 -2/7 -2/1 -1/1 -5/18 -1/1 -3/11 -1/2 -4/15 -1/2 -9/34 0/1 -14/53 -1/11 0/1 -19/72 0/1 -5/19 1/4 -1/4 -1/1 -4/17 0/1 1/3 -7/30 0/1 -3/13 1/0 -2/9 1/0 -1/5 -3/2 -1/6 -1/1 -1/7 1/0 0/1 -1/1 0/1 1/6 -1/1 1/5 -3/4 2/9 -1/2 3/13 -1/2 4/17 -1/5 0/1 1/4 -1/1 3/11 1/0 5/18 -1/1 2/7 -1/1 -2/3 1/3 -1/2 4/11 -1/1 0/1 7/19 -5/8 10/27 -1/2 3/8 -1/3 5/13 -1/4 7/18 0/1 2/5 -1/1 0/1 7/17 -3/10 5/12 0/1 8/19 -1/1 0/1 3/7 1/0 4/9 -1/2 5/11 -1/2 1/2 0/1 5/9 1/0 9/16 -1/1 13/23 -1/2 4/7 -1/1 0/1 11/19 -3/8 7/12 0/1 3/5 -1/2 11/18 0/1 8/13 0/1 1/5 21/34 0/1 13/21 1/4 5/8 1/1 17/27 1/0 12/19 -1/1 0/1 7/11 1/0 9/14 0/1 11/17 5/6 2/3 1/0 13/19 -7/6 11/16 -1/1 9/13 1/0 7/10 0/1 19/27 -1/2 12/17 -1/1 0/1 5/7 -1/2 13/18 0/1 8/11 0/1 1/3 19/26 2/3 11/15 1/0 25/34 0/1 39/53 -1/10 53/72 0/1 14/19 0/1 1/5 3/4 1/1 7/9 1/0 11/14 -4/1 15/19 -11/4 4/5 -2/1 -1/1 13/16 -1/1 9/11 -1/2 14/17 -1/3 0/1 5/6 0/1 16/19 -1/1 0/1 11/13 1/2 6/7 0/1 1/1 7/8 3/1 8/9 1/0 1/1 1/0 1/0 -1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(91,80,-306,-269) (-1/1,-7/8) -> (-3/10,-11/37) Hyperbolic Matrix(197,170,270,233) (-7/8,-6/7) -> (8/11,19/26) Hyperbolic Matrix(143,122,252,215) (-6/7,-11/13) -> (13/23,4/7) Hyperbolic Matrix(109,92,-468,-395) (-11/13,-5/6) -> (-7/30,-3/13) Hyperbolic Matrix(251,206,-396,-325) (-5/6,-9/11) -> (-7/11,-19/30) Hyperbolic Matrix(91,74,-198,-161) (-9/11,-4/5) -> (-6/13,-5/11) Hyperbolic Matrix(125,98,-162,-127) (-4/5,-7/9) -> (-7/9,-10/13) Parabolic Matrix(125,96,-306,-235) (-10/13,-3/4) -> (-9/22,-2/5) Hyperbolic Matrix(35,26,144,107) (-3/4,-14/19) -> (4/17,1/4) Hyperbolic Matrix(757,556,-1224,-899) (-14/19,-11/15) -> (-13/21,-34/55) Hyperbolic Matrix(413,302,-666,-487) (-11/15,-8/11) -> (-18/29,-13/21) Hyperbolic Matrix(287,208,396,287) (-8/11,-13/18) -> (13/18,8/11) Hyperbolic Matrix(181,130,252,181) (-13/18,-5/7) -> (5/7,13/18) Hyperbolic Matrix(107,76,252,179) (-5/7,-12/17) -> (8/19,3/7) Hyperbolic Matrix(613,432,972,685) (-12/17,-19/27) -> (17/27,12/19) Hyperbolic Matrix(305,214,486,341) (-19/27,-7/10) -> (5/8,17/27) Hyperbolic Matrix(89,62,-234,-163) (-7/10,-2/3) -> (-8/21,-3/8) Hyperbolic Matrix(53,34,-198,-127) (-2/3,-7/11) -> (-3/11,-4/15) Hyperbolic Matrix(253,160,-1080,-683) (-19/30,-12/19) -> (-4/17,-7/30) Hyperbolic Matrix(685,432,972,613) (-12/19,-17/27) -> (19/27,12/17) Hyperbolic Matrix(341,214,486,305) (-17/27,-5/8) -> (7/10,19/27) Hyperbolic Matrix(251,156,288,179) (-5/8,-18/29) -> (6/7,7/8) Hyperbolic Matrix(971,600,-3672,-2269) (-34/55,-21/34) -> (-9/34,-14/53) Hyperbolic Matrix(305,188,378,233) (-21/34,-8/13) -> (4/5,13/16) Hyperbolic Matrix(287,176,468,287) (-8/13,-11/18) -> (11/18,8/13) Hyperbolic Matrix(109,66,180,109) (-11/18,-3/5) -> (3/5,11/18) Hyperbolic Matrix(17,10,90,53) (-3/5,-7/12) -> (1/6,1/5) Hyperbolic Matrix(179,104,432,251) (-7/12,-11/19) -> (7/17,5/12) Hyperbolic Matrix(125,72,342,197) (-11/19,-4/7) -> (4/11,7/19) Hyperbolic Matrix(89,50,-162,-91) (-4/7,-5/9) -> (-5/9,-6/11) Parabolic Matrix(199,108,234,127) (-6/11,-7/13) -> (11/13,6/7) Hyperbolic Matrix(161,86,234,125) (-7/13,-1/2) -> (11/16,9/13) Hyperbolic Matrix(289,134,468,217) (-1/2,-6/13) -> (8/13,21/34) Hyperbolic Matrix(89,40,198,89) (-5/11,-4/9) -> (4/9,5/11) Hyperbolic Matrix(55,24,126,55) (-4/9,-3/7) -> (3/7,4/9) Hyperbolic Matrix(19,8,-126,-53) (-3/7,-5/12) -> (-1/6,-1/7) Hyperbolic Matrix(251,104,432,179) (-5/12,-7/17) -> (11/19,7/12) Hyperbolic Matrix(395,162,612,251) (-7/17,-9/22) -> (9/14,11/17) Hyperbolic Matrix(71,28,180,71) (-2/5,-7/18) -> (7/18,2/5) Hyperbolic Matrix(181,70,468,181) (-7/18,-5/13) -> (5/13,7/18) Hyperbolic Matrix(559,214,990,379) (-5/13,-13/34) -> (9/16,13/23) Hyperbolic Matrix(325,124,-1224,-467) (-13/34,-8/21) -> (-4/15,-9/34) Hyperbolic Matrix(17,6,-54,-19) (-3/8,-1/3) -> (-1/3,-3/10) Parabolic Matrix(323,96,360,107) (-11/37,-8/27) -> (8/9,1/1) Hyperbolic Matrix(359,106,972,287) (-8/27,-5/17) -> (7/19,10/27) Hyperbolic Matrix(145,42,252,73) (-5/17,-2/7) -> (4/7,11/19) Hyperbolic Matrix(71,20,252,71) (-2/7,-5/18) -> (5/18,2/7) Hyperbolic Matrix(109,30,396,109) (-5/18,-3/11) -> (3/11,5/18) Hyperbolic Matrix(3817,1008,5184,1369) (-14/53,-19/72) -> (53/72,14/19) Hyperbolic Matrix(3815,1006,5184,1367) (-19/72,-5/19) -> (39/53,53/72) Hyperbolic Matrix(269,70,342,89) (-5/19,-1/4) -> (11/14,15/19) Hyperbolic Matrix(107,26,144,35) (-1/4,-4/17) -> (14/19,3/4) Hyperbolic Matrix(53,12,234,53) (-3/13,-2/9) -> (2/9,3/13) Hyperbolic Matrix(19,4,90,19) (-2/9,-1/5) -> (1/5,2/9) Hyperbolic Matrix(53,10,90,17) (-1/5,-1/6) -> (7/12,3/5) Hyperbolic Matrix(89,12,126,17) (-1/7,0/1) -> (12/17,5/7) Hyperbolic Matrix(53,-8,126,-19) (0/1,1/6) -> (5/12,8/19) Hyperbolic Matrix(395,-92,468,-109) (3/13,4/17) -> (16/19,11/13) Hyperbolic Matrix(127,-34,198,-53) (1/4,3/11) -> (7/11,9/14) Hyperbolic Matrix(19,-6,54,-17) (2/7,1/3) -> (1/3,4/11) Parabolic Matrix(253,-94,288,-107) (10/27,3/8) -> (7/8,8/9) Hyperbolic Matrix(163,-62,234,-89) (3/8,5/13) -> (9/13,7/10) Hyperbolic Matrix(143,-58,180,-73) (2/5,7/17) -> (15/19,4/5) Hyperbolic Matrix(161,-74,198,-91) (5/11,1/2) -> (13/16,9/11) Hyperbolic Matrix(91,-50,162,-89) (1/2,5/9) -> (5/9,9/16) Parabolic Matrix(899,-556,1224,-757) (21/34,13/21) -> (11/15,25/34) Hyperbolic Matrix(487,-302,666,-413) (13/21,5/8) -> (19/26,11/15) Hyperbolic Matrix(325,-206,396,-251) (12/19,7/11) -> (9/11,14/17) Hyperbolic Matrix(73,-48,108,-71) (11/17,2/3) -> (2/3,13/19) Parabolic Matrix(1099,-754,1494,-1025) (13/19,11/16) -> (25/34,39/53) Hyperbolic Matrix(127,-98,162,-125) (3/4,7/9) -> (7/9,11/14) Parabolic Matrix(181,-150,216,-179) (14/17,5/6) -> (5/6,16/19) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(3,2,-2,-1) Matrix(91,80,-306,-269) -> Matrix(7,3,2,1) Matrix(197,170,270,233) -> Matrix(3,1,8,3) Matrix(143,122,252,215) -> Matrix(1,0,2,1) Matrix(109,92,-468,-395) -> Matrix(1,0,4,1) Matrix(251,206,-396,-325) -> Matrix(1,0,-2,1) Matrix(91,74,-198,-161) -> Matrix(1,1,0,1) Matrix(125,98,-162,-127) -> Matrix(9,5,-20,-11) Matrix(125,96,-306,-235) -> Matrix(7,3,2,1) Matrix(35,26,144,107) -> Matrix(1,0,2,1) Matrix(757,556,-1224,-899) -> Matrix(1,0,-4,1) Matrix(413,302,-666,-487) -> Matrix(3,1,-16,-5) Matrix(287,208,396,287) -> Matrix(1,0,8,1) Matrix(181,130,252,181) -> Matrix(1,0,-2,1) Matrix(107,76,252,179) -> Matrix(1,0,0,1) Matrix(613,432,972,685) -> Matrix(1,0,0,1) Matrix(305,214,486,341) -> Matrix(1,1,0,1) Matrix(89,62,-234,-163) -> Matrix(1,1,0,1) Matrix(53,34,-198,-127) -> Matrix(3,1,-4,-1) Matrix(253,160,-1080,-683) -> Matrix(1,0,4,1) Matrix(685,432,972,613) -> Matrix(1,0,0,1) Matrix(341,214,486,305) -> Matrix(3,1,-4,-1) Matrix(251,156,288,179) -> Matrix(9,2,4,1) Matrix(971,600,-3672,-2269) -> Matrix(1,0,0,1) Matrix(305,188,378,233) -> Matrix(5,1,-6,-1) Matrix(287,176,468,287) -> Matrix(1,0,12,1) Matrix(109,66,180,109) -> Matrix(1,0,-2,1) Matrix(17,10,90,53) -> Matrix(3,1,-4,-1) Matrix(179,104,432,251) -> Matrix(1,0,-4,1) Matrix(125,72,342,197) -> Matrix(1,1,-2,-1) Matrix(89,50,-162,-91) -> Matrix(1,1,-4,-3) Matrix(199,108,234,127) -> Matrix(3,1,2,1) Matrix(161,86,234,125) -> Matrix(3,1,-4,-1) Matrix(289,134,468,217) -> Matrix(1,0,2,1) Matrix(89,40,198,89) -> Matrix(1,-1,-2,3) Matrix(55,24,126,55) -> Matrix(1,1,-2,-1) Matrix(19,8,-126,-53) -> Matrix(1,1,-2,-1) Matrix(251,104,432,179) -> Matrix(1,0,-4,1) Matrix(395,162,612,251) -> Matrix(1,-2,2,-3) Matrix(71,28,180,71) -> Matrix(1,0,0,1) Matrix(181,70,468,181) -> Matrix(1,0,-6,1) Matrix(559,214,990,379) -> Matrix(1,-1,0,1) Matrix(325,124,-1224,-467) -> Matrix(1,0,-4,1) Matrix(17,6,-54,-19) -> Matrix(1,-1,0,1) Matrix(323,96,360,107) -> Matrix(1,-4,0,1) Matrix(359,106,972,287) -> Matrix(1,6,-2,-11) Matrix(145,42,252,73) -> Matrix(1,2,-2,-3) Matrix(71,20,252,71) -> Matrix(3,4,-4,-5) Matrix(109,30,396,109) -> Matrix(3,2,-2,-1) Matrix(3817,1008,5184,1369) -> Matrix(1,0,16,1) Matrix(3815,1006,5184,1367) -> Matrix(1,0,-14,1) Matrix(269,70,342,89) -> Matrix(1,-3,0,1) Matrix(107,26,144,35) -> Matrix(1,0,2,1) Matrix(53,12,234,53) -> Matrix(1,-1,-2,3) Matrix(19,4,90,19) -> Matrix(1,3,-2,-5) Matrix(53,10,90,17) -> Matrix(1,1,0,1) Matrix(89,12,126,17) -> Matrix(1,1,-2,-1) Matrix(53,-8,126,-19) -> Matrix(1,1,-2,-1) Matrix(395,-92,468,-109) -> Matrix(1,0,4,1) Matrix(127,-34,198,-53) -> Matrix(1,1,0,1) Matrix(19,-6,54,-17) -> Matrix(1,1,-4,-3) Matrix(253,-94,288,-107) -> Matrix(9,4,2,1) Matrix(163,-62,234,-89) -> Matrix(3,1,-4,-1) Matrix(143,-58,180,-73) -> Matrix(3,2,-2,-1) Matrix(161,-74,198,-91) -> Matrix(3,1,-4,-1) Matrix(91,-50,162,-89) -> Matrix(1,-1,0,1) Matrix(899,-556,1224,-757) -> Matrix(1,0,-4,1) Matrix(487,-302,666,-413) -> Matrix(3,-1,4,-1) Matrix(325,-206,396,-251) -> Matrix(1,0,-2,1) Matrix(73,-48,108,-71) -> Matrix(1,-2,0,1) Matrix(1099,-754,1494,-1025) -> Matrix(1,1,-4,-3) Matrix(127,-98,162,-125) -> Matrix(1,-5,0,1) Matrix(181,-150,216,-179) -> Matrix(1,0,2,1) 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 0/1 (-1/1,0/1) 0 18 1/6 -1/1 3 3 1/5 -3/4 1 18 2/9 -1/2 2 2 3/13 -1/2 1 18 4/17 (-1/5,0/1) 0 18 1/4 -1/1 1 9 3/11 1/0 1 18 5/18 -1/1 3 1 2/7 (-1/1,-2/3) 0 18 1/3 -1/2 1 6 4/11 (-1/1,0/1) 0 18 7/19 -5/8 1 18 10/27 -1/2 7 2 3/8 -1/3 1 9 5/13 -1/4 1 18 7/18 0/1 3 1 2/5 (-1/1,0/1) 0 18 7/17 -3/10 1 18 5/12 0/1 3 3 8/19 (-1/1,0/1) 0 18 3/7 1/0 1 18 4/9 -1/2 1 2 5/11 -1/2 1 18 1/2 0/1 1 9 5/9 1/0 1 2 9/16 -1/1 1 9 13/23 -1/2 1 18 4/7 (-1/1,0/1) 0 18 11/19 -3/8 1 18 7/12 0/1 3 3 3/5 -1/2 1 18 11/18 0/1 7 1 8/13 (0/1,1/5) 0 18 21/34 0/1 1 9 13/21 1/4 1 6 5/8 1/1 1 9 17/27 1/0 1 2 12/19 (-1/1,0/1) 0 18 7/11 1/0 1 18 9/14 0/1 1 9 11/17 5/6 1 18 2/3 1/0 1 6 13/19 -7/6 1 18 11/16 -1/1 1 9 9/13 1/0 1 18 7/10 0/1 1 9 19/27 -1/2 1 2 12/17 (-1/1,0/1) 0 18 5/7 -1/2 1 18 13/18 0/1 5 1 8/11 (0/1,1/3) 0 18 19/26 2/3 1 9 11/15 1/0 1 6 25/34 0/1 1 9 39/53 -1/10 1 18 53/72 0/1 15 1 14/19 (0/1,1/5) 0 18 3/4 1/1 1 9 7/9 1/0 5 2 11/14 -4/1 1 9 15/19 -11/4 1 18 4/5 (-2/1,-1/1) 0 18 13/16 -1/1 1 9 9/11 -1/2 1 18 14/17 (-1/3,0/1) 0 18 5/6 0/1 1 3 16/19 (-1/1,0/1) 0 18 11/13 1/2 1 18 6/7 (0/1,1/1) 0 18 7/8 3/1 1 9 8/9 1/0 7 2 1/1 1/0 1 18 1/0 -1/1 1 1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(53,-8,126,-19) (0/1,1/6) -> (5/12,8/19) Hyperbolic Matrix(53,-10,90,-17) (1/6,1/5) -> (7/12,3/5) Glide Reflection Matrix(19,-4,90,-19) (1/5,2/9) -> (1/5,2/9) Reflection Matrix(53,-12,234,-53) (2/9,3/13) -> (2/9,3/13) Reflection Matrix(395,-92,468,-109) (3/13,4/17) -> (16/19,11/13) Hyperbolic Matrix(107,-26,144,-35) (4/17,1/4) -> (14/19,3/4) Glide Reflection Matrix(127,-34,198,-53) (1/4,3/11) -> (7/11,9/14) Hyperbolic Matrix(109,-30,396,-109) (3/11,5/18) -> (3/11,5/18) Reflection Matrix(71,-20,252,-71) (5/18,2/7) -> (5/18,2/7) Reflection Matrix(19,-6,54,-17) (2/7,1/3) -> (1/3,4/11) Parabolic Matrix(197,-72,342,-125) (4/11,7/19) -> (4/7,11/19) Glide Reflection Matrix(379,-140,1026,-379) (7/19,10/27) -> (7/19,10/27) Reflection Matrix(253,-94,288,-107) (10/27,3/8) -> (7/8,8/9) Hyperbolic Matrix(163,-62,234,-89) (3/8,5/13) -> (9/13,7/10) Hyperbolic Matrix(181,-70,468,-181) (5/13,7/18) -> (5/13,7/18) Reflection Matrix(71,-28,180,-71) (7/18,2/5) -> (7/18,2/5) Reflection Matrix(143,-58,180,-73) (2/5,7/17) -> (15/19,4/5) Hyperbolic Matrix(251,-104,432,-179) (7/17,5/12) -> (11/19,7/12) Glide Reflection Matrix(179,-76,252,-107) (8/19,3/7) -> (12/17,5/7) Glide Reflection Matrix(55,-24,126,-55) (3/7,4/9) -> (3/7,4/9) Reflection Matrix(89,-40,198,-89) (4/9,5/11) -> (4/9,5/11) Reflection Matrix(161,-74,198,-91) (5/11,1/2) -> (13/16,9/11) Hyperbolic Matrix(91,-50,162,-89) (1/2,5/9) -> (5/9,9/16) Parabolic Matrix(397,-224,576,-325) (9/16,13/23) -> (11/16,9/13) Glide Reflection Matrix(215,-122,252,-143) (13/23,4/7) -> (11/13,6/7) Glide Reflection Matrix(109,-66,180,-109) (3/5,11/18) -> (3/5,11/18) Reflection Matrix(287,-176,468,-287) (11/18,8/13) -> (11/18,8/13) Reflection Matrix(305,-188,378,-233) (8/13,21/34) -> (4/5,13/16) Glide Reflection Matrix(899,-556,1224,-757) (21/34,13/21) -> (11/15,25/34) Hyperbolic Matrix(487,-302,666,-413) (13/21,5/8) -> (19/26,11/15) Hyperbolic Matrix(341,-214,486,-305) (5/8,17/27) -> (7/10,19/27) Glide Reflection Matrix(685,-432,972,-613) (17/27,12/19) -> (19/27,12/17) Glide Reflection Matrix(325,-206,396,-251) (12/19,7/11) -> (9/11,14/17) Hyperbolic Matrix(397,-256,504,-325) (9/14,11/17) -> (11/14,15/19) Glide Reflection Matrix(73,-48,108,-71) (11/17,2/3) -> (2/3,13/19) Parabolic Matrix(1099,-754,1494,-1025) (13/19,11/16) -> (25/34,39/53) Hyperbolic Matrix(181,-130,252,-181) (5/7,13/18) -> (5/7,13/18) Reflection Matrix(287,-208,396,-287) (13/18,8/11) -> (13/18,8/11) Reflection Matrix(233,-170,270,-197) (8/11,19/26) -> (6/7,7/8) Glide Reflection Matrix(5617,-4134,7632,-5617) (39/53,53/72) -> (39/53,53/72) Reflection Matrix(2015,-1484,2736,-2015) (53/72,14/19) -> (53/72,14/19) Reflection Matrix(127,-98,162,-125) (3/4,7/9) -> (7/9,11/14) Parabolic Matrix(181,-150,216,-179) (14/17,5/6) -> (5/6,16/19) Parabolic Matrix(17,-16,18,-17) (8/9,1/1) -> (8/9,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(53,-8,126,-19) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(53,-10,90,-17) -> Matrix(1,1,2,1) Matrix(19,-4,90,-19) -> Matrix(5,3,-8,-5) (1/5,2/9) -> (-3/4,-1/2) Matrix(53,-12,234,-53) -> Matrix(3,1,-8,-3) (2/9,3/13) -> (-1/2,-1/4) Matrix(395,-92,468,-109) -> Matrix(1,0,4,1) 0/1 Matrix(107,-26,144,-35) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(127,-34,198,-53) -> Matrix(1,1,0,1) 1/0 Matrix(109,-30,396,-109) -> Matrix(1,2,0,-1) (3/11,5/18) -> (-1/1,1/0) Matrix(71,-20,252,-71) -> Matrix(5,4,-6,-5) (5/18,2/7) -> (-1/1,-2/3) Matrix(19,-6,54,-17) -> Matrix(1,1,-4,-3) -1/2 Matrix(197,-72,342,-125) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(379,-140,1026,-379) -> Matrix(9,5,-16,-9) (7/19,10/27) -> (-5/8,-1/2) Matrix(253,-94,288,-107) -> Matrix(9,4,2,1) Matrix(163,-62,234,-89) -> Matrix(3,1,-4,-1) -1/2 Matrix(181,-70,468,-181) -> Matrix(-1,0,8,1) (5/13,7/18) -> (-1/4,0/1) Matrix(71,-28,180,-71) -> Matrix(-1,0,2,1) (7/18,2/5) -> (-1/1,0/1) Matrix(143,-58,180,-73) -> Matrix(3,2,-2,-1) -1/1 Matrix(251,-104,432,-179) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(179,-76,252,-107) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(55,-24,126,-55) -> Matrix(1,1,0,-1) (3/7,4/9) -> (-1/2,1/0) Matrix(89,-40,198,-89) -> Matrix(3,1,-8,-3) (4/9,5/11) -> (-1/2,-1/4) Matrix(161,-74,198,-91) -> Matrix(3,1,-4,-1) -1/2 Matrix(91,-50,162,-89) -> Matrix(1,-1,0,1) 1/0 Matrix(397,-224,576,-325) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(215,-122,252,-143) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(109,-66,180,-109) -> Matrix(-1,0,4,1) (3/5,11/18) -> (-1/2,0/1) Matrix(287,-176,468,-287) -> Matrix(1,0,10,-1) (11/18,8/13) -> (0/1,1/5) Matrix(305,-188,378,-233) -> Matrix(3,-1,-4,1) Matrix(899,-556,1224,-757) -> Matrix(1,0,-4,1) 0/1 Matrix(487,-302,666,-413) -> Matrix(3,-1,4,-1) 1/2 Matrix(341,-214,486,-305) -> Matrix(1,-1,-2,1) Matrix(685,-432,972,-613) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(325,-206,396,-251) -> Matrix(1,0,-2,1) 0/1 Matrix(397,-256,504,-325) -> Matrix(7,-4,-2,1) Matrix(73,-48,108,-71) -> Matrix(1,-2,0,1) 1/0 Matrix(1099,-754,1494,-1025) -> Matrix(1,1,-4,-3) -1/2 Matrix(181,-130,252,-181) -> Matrix(-1,0,4,1) (5/7,13/18) -> (-1/2,0/1) Matrix(287,-208,396,-287) -> Matrix(1,0,6,-1) (13/18,8/11) -> (0/1,1/3) Matrix(233,-170,270,-197) -> Matrix(3,-1,2,-1) Matrix(5617,-4134,7632,-5617) -> Matrix(-1,0,20,1) (39/53,53/72) -> (-1/10,0/1) Matrix(2015,-1484,2736,-2015) -> Matrix(1,0,10,-1) (53/72,14/19) -> (0/1,1/5) Matrix(127,-98,162,-125) -> Matrix(1,-5,0,1) 1/0 Matrix(181,-150,216,-179) -> Matrix(1,0,2,1) 0/1 Matrix(17,-16,18,-17) -> Matrix(1,1,0,-1) (8/9,1/1) -> (-1/2,1/0) Matrix(-1,2,0,1) -> Matrix(1,2,0,-1) (1/1,1/0) -> (-1/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.