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 0/1 1/0 -7/8 0/1 -6/7 0/1 1/2 -11/13 0/1 1/2 -5/6 1/0 -9/11 -1/1 -1/2 -13/16 0/1 -30/37 -1/2 0/1 -17/21 -1/2 -4/5 0/1 1/2 -11/14 1/1 2/1 -7/9 1/0 -17/22 -3/1 -2/1 -27/35 -2/1 1/0 -10/13 -2/1 1/0 -13/17 -7/6 -1/1 -3/4 0/1 -14/19 7/8 1/1 -11/15 3/2 -30/41 11/6 2/1 -19/26 2/1 3/1 -27/37 2/1 1/0 -8/11 3/1 1/0 -13/18 1/0 -5/7 -2/1 1/0 -12/17 -1/1 1/0 -19/27 -1/2 -26/37 -1/2 0/1 -7/10 -1/1 0/1 -9/13 0/1 1/2 -11/16 0/1 -2/3 1/0 -13/20 -2/1 -24/37 -2/1 -3/2 -35/54 -3/2 -11/17 -3/2 -1/1 -9/14 -2/1 -1/1 -7/11 -1/1 -1/2 -19/30 1/0 -12/19 -1/1 -3/4 -17/27 -1/2 -5/8 0/1 -18/29 -1/2 0/1 -31/50 0/1 1/3 -13/21 1/2 -34/55 11/12 1/1 -21/34 1/1 4/3 -8/13 2/1 1/0 -11/18 1/0 -3/5 -2/1 1/0 -10/17 -5/4 -1/1 -7/12 -1/1 -11/19 -1/1 -1/2 -4/7 -1/2 0/1 -13/23 -1/2 0/1 -9/16 0/1 -5/9 1/0 -11/20 -2/1 -6/11 -1/1 1/0 -19/35 -2/1 1/0 -13/24 -2/1 -7/13 -2/1 -3/2 -1/2 -1/1 0/1 -6/13 -3/4 -2/3 -17/37 -3/4 -2/3 -11/24 -2/3 -5/11 -1/1 -1/2 -4/9 -1/2 -11/25 -1/2 -1/3 -7/16 0/1 -10/23 0/1 1/0 -3/7 0/1 1/0 -8/19 -1/1 1/0 -5/12 -1/1 -7/17 -1/1 -5/6 -9/22 -1/1 -4/5 -2/5 -2/3 -1/2 -7/18 -1/2 -5/13 -1/2 -2/5 -13/34 -4/11 -1/3 -8/21 -1/4 -27/71 -1/4 0/1 -19/50 -1/5 0/1 -30/79 -1/10 0/1 -41/108 0/1 -11/29 0/1 1/0 -3/8 0/1 -13/35 0/1 1/0 -10/27 1/0 -7/19 -3/2 -1/1 -4/11 -1/1 1/0 -13/36 -1/1 -9/25 -1/1 -5/6 -5/14 -1/1 -2/3 -6/17 -1/1 -3/4 -1/3 -1/2 -6/19 -1/3 -1/4 -17/54 -1/4 -11/35 -1/4 0/1 -5/16 0/1 -4/13 -1/4 0/1 -11/36 0/1 -7/23 0/1 1/2 -3/10 -1/1 0/1 -11/37 0/1 1/0 -8/27 1/0 -5/17 -1/1 -1/2 -2/7 -2/3 -1/2 -5/18 -1/2 -3/11 -1/2 -3/7 -13/48 -2/5 -10/37 -1/2 -2/5 -7/26 -3/7 -2/5 -18/67 -17/42 -2/5 -29/108 -2/5 -11/41 -2/5 -11/28 -4/15 -3/8 -9/34 -8/23 -1/3 -14/53 -23/68 -1/3 -19/72 -1/3 -5/19 -1/3 -7/22 -1/4 0/1 -4/17 -1/1 -7/8 -7/30 -3/4 -3/13 -2/3 -1/2 -2/9 -1/2 -5/23 -1/2 -2/5 -13/60 -2/5 -8/37 -1/2 -2/5 -11/51 -1/2 -3/14 -2/5 -1/3 -4/19 -3/8 -1/3 -1/5 -1/4 0/1 -4/21 1/0 -7/37 0/1 1/0 -3/16 0/1 -2/11 -1/1 1/0 -3/17 -3/2 -1/1 -1/6 -1/2 -3/19 -5/14 -1/3 -2/13 -1/4 0/1 -1/7 -1/4 0/1 -1/8 0/1 -1/9 -1/2 0/1 -1/2 0/1 1/8 0/1 1/7 0/1 1/0 2/13 0/1 1/0 1/6 -1/2 2/11 -1/3 -1/4 3/16 0/1 7/37 -1/4 0/1 4/21 -1/4 1/5 0/1 1/0 3/14 -1/1 -2/3 2/9 -1/2 5/22 -3/7 -2/5 8/35 -1/2 -2/5 3/13 -1/2 -2/5 4/17 -7/20 -1/3 1/4 0/1 5/19 -7/6 -1/1 4/15 -3/4 11/41 -11/16 -2/3 7/26 -2/3 -3/5 10/37 -2/3 -1/2 3/11 -3/5 -1/2 5/18 -1/2 2/7 -1/2 -2/5 5/17 -1/2 -1/3 8/27 -1/4 11/37 -1/4 0/1 3/10 -1/3 0/1 4/13 0/1 1/0 5/16 0/1 1/3 -1/2 7/20 -2/5 13/37 -2/5 -3/8 19/54 -3/8 6/17 -3/8 -1/3 5/14 -2/5 -1/3 4/11 -1/3 -1/4 11/30 -1/2 7/19 -1/3 -3/10 10/27 -1/4 3/8 0/1 11/29 -1/4 0/1 19/50 0/1 1/1 8/21 1/0 21/55 -11/10 -1/1 13/34 -1/1 -4/5 5/13 -2/3 -1/2 7/18 -1/2 2/5 -1/2 -2/5 7/17 -5/14 -1/3 5/12 -1/3 8/19 -1/3 -1/4 3/7 -1/4 0/1 10/23 -1/4 0/1 7/16 0/1 4/9 -1/2 9/20 -2/5 5/11 -1/2 -1/3 16/35 -1/2 -2/5 11/24 -2/5 6/13 -2/5 -3/8 1/2 -1/3 0/1 7/13 -3/10 -2/7 20/37 -3/10 -2/7 13/24 -2/7 6/11 -1/3 -1/4 5/9 -1/4 14/25 -1/4 -1/5 9/16 0/1 13/23 -1/2 0/1 4/7 -1/2 0/1 11/19 -1/2 -1/3 7/12 -1/3 10/17 -1/3 -5/16 13/22 -1/3 -4/13 3/5 -2/7 -1/4 11/18 -1/4 8/13 -1/4 -2/9 21/34 -4/19 -1/5 13/21 -1/6 44/71 -1/6 0/1 31/50 -1/7 0/1 49/79 -1/12 0/1 67/108 0/1 18/29 -1/2 0/1 5/8 0/1 22/35 -1/2 0/1 17/27 -1/2 12/19 -3/8 -1/3 7/11 -1/2 -1/3 23/36 -1/3 16/25 -1/3 -5/16 9/14 -1/3 -2/7 11/17 -1/3 -3/10 2/3 -1/4 13/19 -1/5 -1/6 37/54 -1/6 24/35 -1/6 0/1 11/16 0/1 9/13 -1/6 0/1 25/36 0/1 16/23 0/1 1/0 7/10 -1/3 0/1 26/37 -1/2 0/1 19/27 -1/2 12/17 -1/3 -1/4 5/7 -2/7 -1/4 13/18 -1/4 8/11 -1/4 -3/13 35/48 -2/9 27/37 -1/4 -2/9 19/26 -3/13 -2/9 49/67 -17/76 -2/9 79/108 -2/9 30/41 -2/9 -11/50 11/15 -3/14 25/34 -8/39 -1/5 39/53 -23/114 -1/5 53/72 -1/5 14/19 -1/5 -7/36 3/4 0/1 13/17 -1/3 -7/22 23/30 -3/10 10/13 -2/7 -1/4 7/9 -1/4 18/23 -1/4 -2/9 47/60 -2/9 29/37 -1/4 -2/9 40/51 -1/4 11/14 -2/9 -1/5 15/19 -3/14 -1/5 4/5 -1/6 0/1 17/21 -1/2 30/37 -1/2 0/1 13/16 0/1 9/11 -1/2 -1/3 14/17 -3/8 -1/3 5/6 -1/4 16/19 -5/24 -1/5 11/13 -1/6 0/1 6/7 -1/6 0/1 7/8 0/1 8/9 -1/4 1/1 -1/4 0/1 1/0 0/1 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(1,0,-4,1) Matrix(107,94,-288,-253) -> Matrix(1,0,0,1) Matrix(179,156,288,251) -> Matrix(1,0,-4,1) Matrix(143,122,252,215) -> Matrix(1,0,-4,1) Matrix(109,92,-468,-395) -> Matrix(3,-2,-4,3) Matrix(251,206,-396,-325) -> Matrix(1,0,0,1) Matrix(179,146,396,323) -> Matrix(3,2,-8,-5) Matrix(791,642,1152,935) -> Matrix(1,0,-4,1) Matrix(1405,1138,2268,1837) -> Matrix(1,0,-4,1) Matrix(685,554,936,757) -> Matrix(7,2,-32,-9) Matrix(73,58,-180,-143) -> Matrix(3,-2,-4,3) Matrix(251,196,-324,-253) -> Matrix(1,-4,0,1) Matrix(1259,972,-3312,-2557) -> Matrix(1,2,-4,-7) Matrix(431,332,-936,-721) -> Matrix(3,4,-4,-5) Matrix(73,56,-468,-359) -> Matrix(1,2,-4,-7) Matrix(37,28,144,109) -> Matrix(1,0,0,1) Matrix(35,26,144,107) -> Matrix(1,0,-4,1) Matrix(757,556,-1224,-899) -> Matrix(3,-4,4,-5) Matrix(757,554,936,685) -> Matrix(1,-2,0,1) Matrix(755,552,-2808,-2053) -> Matrix(5,-12,-12,29) Matrix(397,290,1332,973) -> Matrix(1,-2,-4,9) Matrix(431,314,-792,-577) -> Matrix(1,-4,0,1) Matrix(287,208,396,287) -> Matrix(1,-6,-4,25) Matrix(181,130,252,181) -> Matrix(1,4,-4,-15) Matrix(107,76,252,179) -> Matrix(1,2,-4,-7) Matrix(613,432,972,685) -> Matrix(3,2,-8,-5) Matrix(1223,860,1944,1367) -> Matrix(1,0,0,1) Matrix(359,252,1332,935) -> Matrix(5,2,-8,-3) Matrix(109,76,-360,-251) -> Matrix(1,0,0,1) Matrix(325,224,576,397) -> Matrix(1,0,-4,1) Matrix(71,48,-108,-73) -> Matrix(1,-2,0,1) Matrix(1081,702,1332,865) -> Matrix(1,2,-4,-7) Matrix(2665,1728,3888,2521) -> Matrix(1,2,-8,-15) Matrix(1331,862,1944,1259) -> Matrix(3,4,-16,-21) Matrix(397,256,504,325) -> Matrix(1,0,-4,1) Matrix(181,116,-504,-323) -> Matrix(3,4,-4,-5) Matrix(253,160,-1080,-683) -> Matrix(3,4,-4,-5) Matrix(685,432,972,613) -> Matrix(3,2,-8,-5) Matrix(35,22,-288,-181) -> Matrix(1,0,0,1) Matrix(251,156,288,179) -> Matrix(1,0,-4,1) Matrix(2051,1272,-5400,-3349) -> Matrix(1,0,-8,1) Matrix(613,380,-2844,-1763) -> Matrix(5,-2,-12,5) Matrix(971,600,-3672,-2269) -> Matrix(11,-12,-32,35) Matrix(217,134,468,289) -> Matrix(3,-4,-8,11) Matrix(287,176,468,287) -> Matrix(1,-4,-4,17) Matrix(109,66,180,109) -> Matrix(1,4,-4,-15) Matrix(37,22,-180,-107) -> Matrix(1,2,-4,-7) Matrix(181,106,432,253) -> Matrix(5,6,-16,-19) Matrix(179,104,432,251) -> Matrix(7,6,-20,-17) Matrix(73,42,252,145) -> Matrix(3,2,-8,-5) Matrix(215,122,252,143) -> Matrix(1,0,-4,1) Matrix(397,224,576,325) -> Matrix(1,0,-4,1) Matrix(179,100,-324,-181) -> Matrix(1,-2,0,1) Matrix(73,40,396,217) -> Matrix(1,2,-4,-7) Matrix(2341,1270,2988,1621) -> Matrix(1,0,-4,1) Matrix(289,156,-1332,-719) -> Matrix(5,8,-12,-19) Matrix(179,96,468,251) -> Matrix(3,4,-4,-5) Matrix(289,134,468,217) -> Matrix(5,4,-24,-19) Matrix(1943,892,2664,1223) -> Matrix(5,4,-24,-19) Matrix(215,98,-792,-361) -> Matrix(7,4,-16,-9) Matrix(143,64,-324,-145) -> Matrix(3,2,-8,-5) Matrix(469,206,576,253) -> Matrix(1,0,0,1) Matrix(179,78,576,251) -> Matrix(1,0,0,1) Matrix(37,16,252,109) -> Matrix(1,0,0,1) Matrix(179,76,252,107) -> Matrix(1,2,-4,-7) Matrix(253,106,432,181) -> Matrix(5,6,-16,-19) Matrix(251,104,432,179) -> Matrix(7,6,-20,-17) Matrix(395,162,612,251) -> Matrix(3,2,-8,-5) Matrix(71,28,180,71) -> Matrix(7,4,-16,-9) Matrix(181,70,468,181) -> Matrix(9,4,-16,-7) Matrix(251,96,468,179) -> Matrix(11,4,-36,-13) Matrix(325,124,-1224,-467) -> Matrix(13,4,-36,-11) Matrix(431,164,2268,863) -> Matrix(1,0,0,1) Matrix(7237,2748,11664,4429) -> Matrix(1,0,8,1) Matrix(7235,2746,11664,4427) -> Matrix(1,0,-12,1) Matrix(37,14,288,109) -> Matrix(1,0,0,1) Matrix(577,214,1944,721) -> Matrix(1,0,-4,1) Matrix(287,106,972,359) -> Matrix(1,2,-4,-7) Matrix(71,26,-396,-145) -> Matrix(1,0,0,1) Matrix(829,300,1296,469) -> Matrix(5,6,-16,-19) Matrix(827,298,1296,467) -> Matrix(7,6,-20,-17) Matrix(361,128,612,217) -> Matrix(1,2,-4,-7) Matrix(35,12,-108,-37) -> Matrix(3,2,-8,-5) Matrix(685,216,1944,613) -> Matrix(13,4,-36,-11) Matrix(1367,430,3888,1223) -> Matrix(11,2,-28,-5) Matrix(217,68,1152,361) -> Matrix(1,0,0,1) Matrix(251,78,576,179) -> Matrix(1,0,0,1) Matrix(901,276,1296,397) -> Matrix(1,0,4,1) Matrix(899,274,1296,395) -> Matrix(1,0,-8,1) Matrix(973,290,1332,397) -> Matrix(1,-2,-4,9) Matrix(323,96,360,107) -> Matrix(1,0,-4,1) Matrix(359,106,972,287) -> Matrix(1,2,-4,-7) Matrix(145,42,252,73) -> Matrix(3,2,-8,-5) Matrix(71,20,252,71) -> Matrix(7,4,-16,-9) Matrix(109,30,396,109) -> Matrix(13,6,-24,-11) Matrix(1441,390,2664,721) -> Matrix(11,4,-36,-13) Matrix(935,252,1332,359) -> Matrix(5,2,-8,-3) Matrix(8533,2292,11664,3133) -> Matrix(139,56,-628,-253) Matrix(8531,2290,11664,3131) -> Matrix(141,56,-632,-251) Matrix(179,48,936,251) -> Matrix(5,2,-28,-11) Matrix(3817,1008,5184,1369) -> Matrix(89,30,-448,-151) Matrix(3815,1006,5184,1367) -> Matrix(91,30,-452,-149) Matrix(109,28,144,37) -> Matrix(1,0,0,1) Matrix(107,26,144,35) -> Matrix(1,0,-4,1) Matrix(71,16,-324,-73) -> Matrix(7,4,-16,-9) Matrix(1367,296,2988,647) -> Matrix(1,0,0,1) Matrix(2159,466,2664,575) -> Matrix(5,2,-8,-3) Matrix(179,38,504,107) -> Matrix(1,0,0,1) Matrix(251,48,936,179) -> Matrix(3,-2,-4,3) Matrix(2089,396,2664,505) -> Matrix(1,-2,-4,9) Matrix(467,88,1332,251) -> Matrix(3,2,-8,-5) Matrix(323,60,576,107) -> Matrix(1,0,-4,1) Matrix(35,6,-216,-37) -> Matrix(3,2,-8,-5) Matrix(109,16,252,37) -> Matrix(1,0,0,1) Matrix(109,14,288,37) -> Matrix(1,0,0,1) Matrix(253,26,360,37) -> Matrix(1,0,0,1) Matrix(181,-22,288,-35) -> Matrix(1,0,0,1) Matrix(359,-56,468,-73) -> Matrix(1,2,-4,-7) Matrix(145,-26,396,-71) -> Matrix(1,0,0,1) Matrix(107,-22,180,-37) -> Matrix(1,2,-4,-7) Matrix(73,-16,324,-71) -> Matrix(7,4,-16,-9) Matrix(2053,-468,3312,-755) -> Matrix(5,2,-28,-11) Matrix(505,-116,936,-215) -> Matrix(11,4,-36,-13) Matrix(395,-92,468,-109) -> Matrix(5,2,-28,-11) Matrix(467,-124,1224,-325) -> Matrix(5,4,-4,-3) Matrix(2053,-552,2808,-755) -> Matrix(19,12,-84,-53) Matrix(361,-98,792,-215) -> Matrix(7,4,-16,-9) 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(11,4,-36,-13) Matrix(827,-304,1080,-397) -> Matrix(11,4,-36,-13) Matrix(253,-94,288,-107) -> Matrix(1,0,0,1) Matrix(3349,-1272,5400,-2051) -> Matrix(1,0,-8,1) Matrix(2231,-848,2844,-1081) -> Matrix(1,-2,-4,9) Matrix(2701,-1032,3672,-1403) -> Matrix(13,12,-64,-59) Matrix(143,-58,180,-73) -> Matrix(5,2,-28,-11) Matrix(145,-64,324,-143) -> Matrix(3,2,-8,-5) Matrix(1043,-480,1332,-613) -> Matrix(21,8,-92,-35) Matrix(577,-314,792,-431) -> Matrix(15,4,-64,-17) Matrix(181,-100,324,-179) -> Matrix(7,2,-32,-9) Matrix(899,-556,1224,-757) -> Matrix(21,4,-100,-19) Matrix(325,-206,396,-251) -> Matrix(1,0,0,1) Matrix(73,-48,108,-71) -> Matrix(7,2,-32,-9) Matrix(253,-196,324,-251) -> Matrix(15,4,-64,-17) Matrix(181,-150,216,-179) -> Matrix(7,2,-32,-9) 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: 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 0/1 1/8 1/7 2/13 1/6 2/11 3/16 1/5 3/14 2/9 4/17 1/4 4/15 5/18 8/27 3/10 5/16 1/3 19/54 3/8 7/18 5/12 4/9 11/24 1/2 13/24 5/9 7/12 67/108 23/36 2/3 25/36 79/108 11/15 53/72 3/4 7/9 5/6 8/9 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/2 0/1 1/8 0/1 1/7 0/1 1/0 2/13 0/1 1/0 1/6 -1/2 2/11 -1/3 -1/4 3/16 0/1 7/37 -1/4 0/1 4/21 -1/4 1/5 0/1 1/0 3/14 -1/1 -2/3 2/9 -1/2 5/22 -3/7 -2/5 8/35 -1/2 -2/5 3/13 -1/2 -2/5 4/17 -7/20 -1/3 1/4 0/1 5/19 -7/6 -1/1 4/15 -3/4 11/41 -11/16 -2/3 7/26 -2/3 -3/5 10/37 -2/3 -1/2 3/11 -3/5 -1/2 5/18 -1/2 2/7 -1/2 -2/5 5/17 -1/2 -1/3 8/27 -1/4 11/37 -1/4 0/1 3/10 -1/3 0/1 4/13 0/1 1/0 5/16 0/1 1/3 -1/2 7/20 -2/5 13/37 -2/5 -3/8 19/54 -3/8 6/17 -3/8 -1/3 5/14 -2/5 -1/3 4/11 -1/3 -1/4 11/30 -1/2 7/19 -1/3 -3/10 10/27 -1/4 3/8 0/1 11/29 -1/4 0/1 19/50 0/1 1/1 8/21 1/0 21/55 -11/10 -1/1 13/34 -1/1 -4/5 5/13 -2/3 -1/2 7/18 -1/2 2/5 -1/2 -2/5 7/17 -5/14 -1/3 5/12 -1/3 8/19 -1/3 -1/4 3/7 -1/4 0/1 10/23 -1/4 0/1 7/16 0/1 4/9 -1/2 9/20 -2/5 5/11 -1/2 -1/3 16/35 -1/2 -2/5 11/24 -2/5 6/13 -2/5 -3/8 1/2 -1/3 0/1 7/13 -3/10 -2/7 20/37 -3/10 -2/7 13/24 -2/7 6/11 -1/3 -1/4 5/9 -1/4 14/25 -1/4 -1/5 9/16 0/1 13/23 -1/2 0/1 4/7 -1/2 0/1 11/19 -1/2 -1/3 7/12 -1/3 10/17 -1/3 -5/16 13/22 -1/3 -4/13 3/5 -2/7 -1/4 11/18 -1/4 8/13 -1/4 -2/9 21/34 -4/19 -1/5 13/21 -1/6 44/71 -1/6 0/1 31/50 -1/7 0/1 49/79 -1/12 0/1 67/108 0/1 18/29 -1/2 0/1 5/8 0/1 22/35 -1/2 0/1 17/27 -1/2 12/19 -3/8 -1/3 7/11 -1/2 -1/3 23/36 -1/3 16/25 -1/3 -5/16 9/14 -1/3 -2/7 11/17 -1/3 -3/10 2/3 -1/4 13/19 -1/5 -1/6 37/54 -1/6 24/35 -1/6 0/1 11/16 0/1 9/13 -1/6 0/1 25/36 0/1 16/23 0/1 1/0 7/10 -1/3 0/1 26/37 -1/2 0/1 19/27 -1/2 12/17 -1/3 -1/4 5/7 -2/7 -1/4 13/18 -1/4 8/11 -1/4 -3/13 35/48 -2/9 27/37 -1/4 -2/9 19/26 -3/13 -2/9 49/67 -17/76 -2/9 79/108 -2/9 30/41 -2/9 -11/50 11/15 -3/14 25/34 -8/39 -1/5 39/53 -23/114 -1/5 53/72 -1/5 14/19 -1/5 -7/36 3/4 0/1 13/17 -1/3 -7/22 23/30 -3/10 10/13 -2/7 -1/4 7/9 -1/4 18/23 -1/4 -2/9 47/60 -2/9 29/37 -1/4 -2/9 40/51 -1/4 11/14 -2/9 -1/5 15/19 -3/14 -1/5 4/5 -1/6 0/1 17/21 -1/2 30/37 -1/2 0/1 13/16 0/1 9/11 -1/2 -1/3 14/17 -3/8 -1/3 5/6 -1/4 16/19 -5/24 -1/5 11/13 -1/6 0/1 6/7 -1/6 0/1 7/8 0/1 8/9 -1/4 1/1 -1/4 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(181,-22,288,-35) (0/1,1/8) -> (5/8,22/35) Hyperbolic Matrix(179,-23,288,-37) (1/8,1/7) -> (18/29,5/8) Hyperbolic Matrix(143,-21,252,-37) (1/7,2/13) -> (13/23,4/7) 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(179,-33,396,-73) (2/11,3/16) -> (9/20,5/11) Hyperbolic Matrix(791,-149,1152,-217) (3/16,7/37) -> (24/35,11/16) Hyperbolic Matrix(1405,-267,2268,-431) (7/37,4/21) -> (13/21,44/71) Hyperbolic Matrix(685,-131,936,-179) (4/21,1/5) -> (30/41,11/15) 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(37,-9,144,-35) (4/17,1/4) -> (1/4,5/19) Parabolic Matrix(467,-124,1224,-325) (5/19,4/15) -> (8/21,21/55) Hyperbolic Matrix(757,-203,936,-251) (4/15,11/41) -> (4/5,17/21) Hyperbolic Matrix(2053,-552,2808,-755) (11/41,7/26) -> (19/26,49/67) Hyperbolic Matrix(397,-107,1332,-359) (7/26,10/37) -> (11/37,3/10) Hyperbolic Matrix(361,-98,792,-215) (10/37,3/11) -> (5/11,16/35) Hyperbolic Matrix(287,-79,396,-109) (3/11,5/18) -> (13/18,8/11) Hyperbolic Matrix(181,-51,252,-71) (5/18,2/7) -> (5/7,13/18) Hyperbolic Matrix(107,-31,252,-73) (2/7,5/17) -> (8/19,3/7) Hyperbolic Matrix(613,-181,972,-287) (5/17,8/27) -> (17/27,12/19) Hyperbolic Matrix(1223,-363,1944,-577) (8/27,11/37) -> (22/35,17/27) Hyperbolic Matrix(251,-76,360,-109) (3/10,4/13) -> (16/23,7/10) Hyperbolic Matrix(325,-101,576,-179) (4/13,5/16) -> (9/16,13/23) Hyperbolic Matrix(37,-12,108,-35) (5/16,1/3) -> (1/3,7/20) Parabolic Matrix(1081,-379,1332,-467) (7/20,13/37) -> (30/37,13/16) Hyperbolic Matrix(2665,-937,3888,-1367) (13/37,19/54) -> (37/54,24/35) Hyperbolic Matrix(1331,-469,1944,-685) (19/54,6/17) -> (13/19,37/54) Hyperbolic Matrix(397,-141,504,-179) (6/17,5/14) -> (11/14,15/19) Hyperbolic 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(685,-253,972,-359) (7/19,10/27) -> (19/27,12/17) Hyperbolic Matrix(253,-94,288,-107) (10/27,3/8) -> (7/8,8/9) Hyperbolic Matrix(251,-95,288,-109) (3/8,11/29) -> (6/7,7/8) 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(217,-83,468,-179) (13/34,5/13) -> (6/13,1/2) Hyperbolic Matrix(287,-111,468,-181) (5/13,7/18) -> (11/18,8/13) Hyperbolic Matrix(109,-43,180,-71) (7/18,2/5) -> (3/5,11/18) Hyperbolic Matrix(143,-58,180,-73) (2/5,7/17) -> (15/19,4/5) Hyperbolic Matrix(181,-75,432,-179) (7/17,5/12) -> (5/12,8/19) Parabolic Matrix(215,-93,252,-109) (3/7,10/23) -> (11/13,6/7) Hyperbolic Matrix(397,-173,576,-251) (10/23,7/16) -> (11/16,9/13) Hyperbolic Matrix(145,-64,324,-143) (7/16,4/9) -> (4/9,9/20) Parabolic Matrix(2341,-1071,2988,-1367) (16/35,11/24) -> (47/60,29/37) Hyperbolic Matrix(1043,-480,1332,-613) (11/24,6/13) -> (18/23,47/60) Hyperbolic Matrix(289,-155,468,-251) (1/2,7/13) -> (8/13,21/34) Hyperbolic Matrix(1943,-1051,2664,-1441) (20/37,13/24) -> (35/48,27/37) 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(469,-263,576,-323) (14/25,9/16) -> (13/16,9/11) Hyperbolic Matrix(179,-103,252,-145) (4/7,11/19) -> (12/17,5/7) Hyperbolic Matrix(253,-147,432,-251) (11/19,7/12) -> (7/12,10/17) Parabolic Matrix(395,-233,612,-361) (10/17,13/22) -> (9/14,11/17) Hyperbolic Matrix(899,-556,1224,-757) (21/34,13/21) -> (11/15,25/34) Hyperbolic Matrix(7237,-4489,11664,-7235) (49/79,67/108) -> (67/108,18/29) Parabolic Matrix(325,-206,396,-251) (12/19,7/11) -> (9/11,14/17) Hyperbolic Matrix(829,-529,1296,-827) (7/11,23/36) -> (23/36,16/25) Parabolic Matrix(73,-48,108,-71) (11/17,2/3) -> (2/3,13/19) Parabolic Matrix(901,-625,1296,-899) (9/13,25/36) -> (25/36,16/23) Parabolic Matrix(973,-683,1332,-935) (7/10,26/37) -> (27/37,19/26) Hyperbolic Matrix(323,-227,360,-253) (26/37,19/27) -> (8/9,1/1) Hyperbolic Matrix(8533,-6241,11664,-8531) (49/67,79/108) -> (79/108,30/41) Parabolic Matrix(3817,-2809,5184,-3815) (39/53,53/72) -> (53/72,14/19) Parabolic Matrix(109,-81,144,-107) (14/19,3/4) -> (3/4,13/17) Parabolic Matrix(253,-196,324,-251) (10/13,7/9) -> (7/9,18/23) Parabolic Matrix(2159,-1693,2664,-2089) (29/37,40/51) -> (17/21,30/37) Hyperbolic Matrix(181,-150,216,-179) (14/17,5/6) -> (5/6,16/19) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-2,1) Matrix(181,-22,288,-35) -> Matrix(1,0,0,1) Matrix(179,-23,288,-37) -> Matrix(1,0,-2,1) Matrix(143,-21,252,-37) -> Matrix(1,0,-2,1) Matrix(359,-56,468,-73) -> Matrix(1,2,-4,-7) Matrix(145,-26,396,-71) -> Matrix(1,0,0,1) Matrix(179,-33,396,-73) -> Matrix(7,2,-18,-5) Matrix(791,-149,1152,-217) -> Matrix(1,0,-2,1) Matrix(1405,-267,2268,-431) -> Matrix(1,0,-2,1) Matrix(685,-131,936,-179) -> Matrix(11,2,-50,-9) Matrix(107,-22,180,-37) -> Matrix(1,2,-4,-7) Matrix(73,-16,324,-71) -> Matrix(7,4,-16,-9) Matrix(2053,-468,3312,-755) -> Matrix(5,2,-28,-11) Matrix(505,-116,936,-215) -> Matrix(11,4,-36,-13) Matrix(395,-92,468,-109) -> Matrix(5,2,-28,-11) Matrix(37,-9,144,-35) -> Matrix(1,0,2,1) Matrix(467,-124,1224,-325) -> Matrix(5,4,-4,-3) Matrix(757,-203,936,-251) -> Matrix(3,2,-2,-1) Matrix(2053,-552,2808,-755) -> Matrix(19,12,-84,-53) Matrix(397,-107,1332,-359) -> Matrix(3,2,-14,-9) Matrix(361,-98,792,-215) -> Matrix(7,4,-16,-9) Matrix(287,-79,396,-109) -> Matrix(11,6,-46,-25) Matrix(181,-51,252,-71) -> Matrix(9,4,-34,-15) Matrix(107,-31,252,-73) -> Matrix(5,2,-18,-7) Matrix(613,-181,972,-287) -> Matrix(7,2,-18,-5) Matrix(1223,-363,1944,-577) -> Matrix(1,0,2,1) Matrix(251,-76,360,-109) -> Matrix(1,0,0,1) Matrix(325,-101,576,-179) -> Matrix(1,0,-2,1) Matrix(37,-12,108,-35) -> Matrix(3,2,-8,-5) Matrix(1081,-379,1332,-467) -> Matrix(5,2,-18,-7) Matrix(2665,-937,3888,-1367) -> Matrix(5,2,-38,-15) Matrix(1331,-469,1944,-685) -> Matrix(11,4,-58,-21) Matrix(397,-141,504,-179) -> Matrix(1,0,-2,1) Matrix(323,-116,504,-181) -> Matrix(11,4,-36,-13) Matrix(827,-304,1080,-397) -> Matrix(11,4,-36,-13) Matrix(685,-253,972,-359) -> Matrix(7,2,-18,-5) Matrix(253,-94,288,-107) -> Matrix(1,0,0,1) Matrix(251,-95,288,-109) -> Matrix(1,0,-2,1) Matrix(3349,-1272,5400,-2051) -> Matrix(1,0,-8,1) Matrix(2231,-848,2844,-1081) -> Matrix(1,-2,-4,9) Matrix(2701,-1032,3672,-1403) -> Matrix(13,12,-64,-59) Matrix(217,-83,468,-179) -> Matrix(5,4,-14,-11) Matrix(287,-111,468,-181) -> Matrix(7,4,-30,-17) Matrix(109,-43,180,-71) -> Matrix(9,4,-34,-15) Matrix(143,-58,180,-73) -> Matrix(5,2,-28,-11) Matrix(181,-75,432,-179) -> Matrix(17,6,-54,-19) Matrix(215,-93,252,-109) -> Matrix(1,0,-2,1) Matrix(397,-173,576,-251) -> Matrix(1,0,-2,1) Matrix(145,-64,324,-143) -> Matrix(3,2,-8,-5) Matrix(2341,-1071,2988,-1367) -> Matrix(1,0,-2,1) Matrix(1043,-480,1332,-613) -> Matrix(21,8,-92,-35) Matrix(289,-155,468,-251) -> Matrix(13,4,-62,-19) Matrix(1943,-1051,2664,-1441) -> Matrix(13,4,-62,-19) Matrix(577,-314,792,-431) -> Matrix(15,4,-64,-17) Matrix(181,-100,324,-179) -> Matrix(7,2,-32,-9) Matrix(469,-263,576,-323) -> Matrix(1,0,2,1) Matrix(179,-103,252,-145) -> Matrix(5,2,-18,-7) Matrix(253,-147,432,-251) -> Matrix(17,6,-54,-19) Matrix(395,-233,612,-361) -> Matrix(7,2,-18,-5) Matrix(899,-556,1224,-757) -> Matrix(21,4,-100,-19) Matrix(7237,-4489,11664,-7235) -> Matrix(1,0,10,1) Matrix(325,-206,396,-251) -> Matrix(1,0,0,1) Matrix(829,-529,1296,-827) -> Matrix(17,6,-54,-19) Matrix(73,-48,108,-71) -> Matrix(7,2,-32,-9) Matrix(901,-625,1296,-899) -> Matrix(1,0,6,1) Matrix(973,-683,1332,-935) -> Matrix(3,2,-14,-9) Matrix(323,-227,360,-253) -> Matrix(1,0,-2,1) Matrix(8533,-6241,11664,-8531) -> Matrix(251,56,-1134,-253) Matrix(3817,-2809,5184,-3815) -> Matrix(149,30,-750,-151) Matrix(109,-81,144,-107) -> Matrix(1,0,2,1) Matrix(253,-196,324,-251) -> Matrix(15,4,-64,-17) Matrix(2159,-1693,2664,-2089) -> Matrix(9,2,-14,-3) Matrix(181,-150,216,-179) -> Matrix(7,2,-32,-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 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): 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 0/1 (-1/2,0/1) 0 36 1/8 0/1 2 9 1/7 (0/1,1/0) 0 36 2/13 (0/1,1/0) 0 36 1/6 -1/2 2 6 2/11 (-1/3,-1/4) 0 36 3/16 0/1 2 9 7/37 (-1/4,0/1) 0 36 4/21 -1/4 2 12 1/5 (0/1,1/0) 0 36 3/14 (-1/1,-2/3) 0 18 2/9 -1/2 4 4 5/22 (-3/7,-2/5) 0 18 8/35 (-1/2,-2/5) 0 36 3/13 (-1/2,-2/5) 0 36 4/17 (-7/20,-1/3) 0 36 1/4 0/1 2 9 5/19 (-7/6,-1/1) 0 36 4/15 -3/4 2 12 11/41 (-11/16,-2/3) 0 36 29/108 -2/3 14 1 7/26 (-2/3,-3/5) 0 18 10/37 (-2/3,-1/2) 0 36 3/11 (-3/5,-1/2) 0 36 5/18 -1/2 10 2 2/7 (-1/2,-2/5) 0 36 5/17 (-1/2,-1/3) 0 36 8/27 -1/4 2 4 11/37 (-1/4,0/1) 0 36 3/10 (-1/3,0/1) 0 18 11/36 0/1 6 1 4/13 (0/1,1/0) 0 36 5/16 0/1 2 9 1/3 -1/2 2 12 7/20 -2/5 2 9 13/37 (-2/5,-3/8) 0 36 19/54 -3/8 2 2 6/17 (-3/8,-1/3) 0 36 5/14 (-2/5,-1/3) 0 18 13/36 -1/3 6 1 4/11 (-1/3,-1/4) 0 36 11/30 -1/2 2 6 18/49 (-2/5,-3/8) 0 36 7/19 (-1/3,-3/10) 0 36 10/27 -1/4 2 4 13/35 (-1/4,0/1) 0 36 3/8 0/1 2 9 11/29 (-1/4,0/1) 0 36 41/108 0/1 10 1 19/50 (0/1,1/1) 0 18 27/71 (0/1,1/0) 0 36 8/21 1/0 2 12 21/55 (-11/10,-1/1) 0 36 55/144 -1/1 30 1 13/34 (-1/1,-4/5) 0 18 5/13 (-2/3,-1/2) 0 36 7/18 -1/2 8 2 2/5 (-1/2,-2/5) 0 36 9/22 (-4/11,-1/3) 0 18 7/17 (-5/14,-1/3) 0 36 5/12 -1/3 6 3 8/19 (-1/3,-1/4) 0 36 3/7 (-1/4,0/1) 0 36 10/23 (-1/4,0/1) 0 36 7/16 0/1 2 9 4/9 -1/2 2 4 9/20 -2/5 2 9 5/11 (-1/2,-1/3) 0 36 16/35 (-1/2,-2/5) 0 36 11/24 -2/5 2 3 17/37 (-2/5,-3/8) 0 36 6/13 (-2/5,-3/8) 0 36 1/2 (-1/3,0/1) 0 18 1/0 0/1 2 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(107,-13,288,-35) (0/1,1/8) -> (13/35,3/8) Glide Reflection Matrix(109,-14,288,-37) (1/8,1/7) -> (3/8,11/29) Glide Reflection Matrix(109,-16,252,-37) (1/7,2/13) -> (3/7,10/23) Glide Reflection Matrix(251,-40,684,-109) (2/13,1/6) -> (11/30,18/49) Hyperbolic Matrix(145,-26,396,-71) (1/6,2/11) -> (4/11,11/30) Hyperbolic Matrix(179,-33,396,-73) (2/11,3/16) -> (9/20,5/11) Hyperbolic Matrix(467,-88,1332,-251) (3/16,7/37) -> (7/20,13/37) Glide Reflection Matrix(863,-164,2268,-431) (7/37,4/21) -> (27/71,8/21) Glide Reflection Matrix(251,-48,936,-179) (4/21,1/5) -> (4/15,11/41) Glide Reflection Matrix(73,-15,180,-37) (1/5,3/14) -> (2/5,9/22) Glide Reflection Matrix(73,-16,324,-71) (3/14,2/9) -> (2/9,5/22) Parabolic Matrix(1259,-287,3312,-755) (5/22,8/35) -> (19/50,27/71) Glide Reflection Matrix(431,-99,936,-215) (8/35,3/13) -> (17/37,6/13) Glide Reflection Matrix(397,-93,1080,-253) (3/13,4/17) -> (18/49,7/19) Glide Reflection Matrix(37,-9,144,-35) (4/17,1/4) -> (1/4,5/19) Parabolic Matrix(467,-124,1224,-325) (5/19,4/15) -> (8/21,21/55) Hyperbolic Matrix(2377,-638,8856,-2377) (11/41,29/108) -> (11/41,29/108) Reflection Matrix(755,-203,2808,-755) (29/108,7/26) -> (29/108,7/26) Reflection Matrix(397,-107,1332,-359) (7/26,10/37) -> (11/37,3/10) Hyperbolic Matrix(361,-98,792,-215) (10/37,3/11) -> (5/11,16/35) 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(107,-31,252,-73) (2/7,5/17) -> (8/19,3/7) Hyperbolic Matrix(359,-106,972,-287) (5/17,8/27) -> (7/19,10/27) Glide Reflection Matrix(721,-214,1944,-577) (8/27,11/37) -> (10/27,13/35) Glide Reflection Matrix(109,-33,360,-109) (3/10,11/36) -> (3/10,11/36) Reflection Matrix(287,-88,936,-287) (11/36,4/13) -> (11/36,4/13) Reflection Matrix(251,-78,576,-179) (4/13,5/16) -> (10/23,7/16) Glide Reflection Matrix(37,-12,108,-35) (5/16,1/3) -> (1/3,7/20) Parabolic Matrix(1405,-494,3996,-1405) (13/37,19/54) -> (13/37,19/54) Reflection Matrix(647,-228,1836,-647) (19/54,6/17) -> (19/54,6/17) Reflection Matrix(251,-89,612,-217) (6/17,5/14) -> (9/22,7/17) Hyperbolic Matrix(181,-65,504,-181) (5/14,13/36) -> (5/14,13/36) Reflection Matrix(287,-104,792,-287) (13/36,4/11) -> (13/36,4/11) Reflection Matrix(2377,-902,6264,-2377) (11/29,41/108) -> (11/29,41/108) Reflection Matrix(2051,-779,5400,-2051) (41/108,19/50) -> (41/108,19/50) Reflection Matrix(6049,-2310,15840,-6049) (21/55,55/144) -> (21/55,55/144) Reflection Matrix(1871,-715,4896,-1871) (55/144,13/34) -> (55/144,13/34) Reflection Matrix(217,-83,468,-179) (13/34,5/13) -> (6/13,1/2) 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(181,-75,432,-179) (7/17,5/12) -> (5/12,8/19) Parabolic Matrix(145,-64,324,-143) (7/16,4/9) -> (4/9,9/20) Parabolic Matrix(793,-363,1728,-791) (16/35,11/24) -> (11/24,17/37) Parabolic Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,4,1) (0/1,1/0) -> (-1/2,0/1) Matrix(107,-13,288,-35) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(109,-14,288,-37) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(109,-16,252,-37) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(251,-40,684,-109) -> Matrix(3,2,-8,-5) -1/2 Matrix(145,-26,396,-71) -> Matrix(1,0,0,1) Matrix(179,-33,396,-73) -> Matrix(7,2,-18,-5) -1/3 Matrix(467,-88,1332,-251) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(863,-164,2268,-431) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(251,-48,936,-179) -> Matrix(11,2,-16,-3) Matrix(73,-15,180,-37) -> Matrix(1,2,-2,-5) Matrix(73,-16,324,-71) -> Matrix(7,4,-16,-9) -1/2 Matrix(1259,-287,3312,-755) -> Matrix(5,2,-2,-1) Matrix(431,-99,936,-215) -> Matrix(11,4,-30,-11) *** -> (-2/5,-1/3) Matrix(397,-93,1080,-253) -> Matrix(11,4,-30,-11) *** -> (-2/5,-1/3) Matrix(37,-9,144,-35) -> Matrix(1,0,2,1) 0/1 Matrix(467,-124,1224,-325) -> Matrix(5,4,-4,-3) -1/1 Matrix(2377,-638,8856,-2377) -> Matrix(65,44,-96,-65) (11/41,29/108) -> (-11/16,-2/3) Matrix(755,-203,2808,-755) -> Matrix(19,12,-30,-19) (29/108,7/26) -> (-2/3,-3/5) Matrix(397,-107,1332,-359) -> Matrix(3,2,-14,-9) Matrix(361,-98,792,-215) -> Matrix(7,4,-16,-9) -1/2 Matrix(109,-30,396,-109) -> Matrix(11,6,-20,-11) (3/11,5/18) -> (-3/5,-1/2) Matrix(71,-20,252,-71) -> Matrix(9,4,-20,-9) (5/18,2/7) -> (-1/2,-2/5) Matrix(107,-31,252,-73) -> Matrix(5,2,-18,-7) -1/3 Matrix(359,-106,972,-287) -> Matrix(7,2,-24,-7) *** -> (-1/3,-1/4) Matrix(721,-214,1944,-577) -> Matrix(-1,0,8,1) *** -> (-1/4,0/1) Matrix(109,-33,360,-109) -> Matrix(-1,0,6,1) (3/10,11/36) -> (-1/3,0/1) Matrix(287,-88,936,-287) -> Matrix(1,0,0,-1) (11/36,4/13) -> (0/1,1/0) Matrix(251,-78,576,-179) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(37,-12,108,-35) -> Matrix(3,2,-8,-5) -1/2 Matrix(1405,-494,3996,-1405) -> Matrix(31,12,-80,-31) (13/37,19/54) -> (-2/5,-3/8) Matrix(647,-228,1836,-647) -> Matrix(17,6,-48,-17) (19/54,6/17) -> (-3/8,-1/3) Matrix(251,-89,612,-217) -> Matrix(7,2,-18,-5) -1/3 Matrix(181,-65,504,-181) -> Matrix(11,4,-30,-11) (5/14,13/36) -> (-2/5,-1/3) Matrix(287,-104,792,-287) -> Matrix(7,2,-24,-7) (13/36,4/11) -> (-1/3,-1/4) Matrix(2377,-902,6264,-2377) -> Matrix(-1,0,8,1) (11/29,41/108) -> (-1/4,0/1) Matrix(2051,-779,5400,-2051) -> Matrix(1,0,2,-1) (41/108,19/50) -> (0/1,1/1) Matrix(6049,-2310,15840,-6049) -> Matrix(21,22,-20,-21) (21/55,55/144) -> (-11/10,-1/1) Matrix(1871,-715,4896,-1871) -> Matrix(9,8,-10,-9) (55/144,13/34) -> (-1/1,-4/5) Matrix(217,-83,468,-179) -> Matrix(5,4,-14,-11) Matrix(181,-70,468,-181) -> Matrix(7,4,-12,-7) (5/13,7/18) -> (-2/3,-1/2) Matrix(71,-28,180,-71) -> Matrix(9,4,-20,-9) (7/18,2/5) -> (-1/2,-2/5) Matrix(181,-75,432,-179) -> Matrix(17,6,-54,-19) -1/3 Matrix(145,-64,324,-143) -> Matrix(3,2,-8,-5) -1/2 Matrix(793,-363,1728,-791) -> Matrix(19,8,-50,-21) -2/5 Matrix(-1,1,0,1) -> Matrix(-1,0,6,1) (1/2,1/0) -> (-1/3,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.