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 -8/1 -6/1 -16/3 -14/3 -9/2 -4/1 -15/4 -27/8 -10/3 -3/1 -12/5 -9/4 -24/11 -2/1 -9/5 -27/17 -3/2 -15/11 -9/7 -6/5 -9/8 0/1 1/1 6/5 9/7 15/11 18/13 3/2 54/35 18/11 9/5 2/1 24/11 9/4 12/5 5/2 18/7 108/41 36/13 3/1 54/17 36/11 10/3 7/2 18/5 11/3 108/29 15/4 72/19 4/1 17/4 13/3 9/2 14/3 5/1 16/3 11/2 17/3 6/1 13/2 7/1 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 0/1 -7/1 0/1 1/0 -13/2 -2/1 1/0 -6/1 -1/1 -11/2 0/1 1/0 -16/3 0/1 -37/7 -5/1 1/0 -21/4 -2/1 -5/1 -1/1 0/1 -14/3 -2/1 0/1 -9/2 -2/1 0/1 -22/5 -2/1 0/1 -35/8 -1/1 0/1 -13/3 0/1 1/0 -17/4 0/1 1/0 -4/1 -2/1 -19/5 -5/3 -3/2 -15/4 -4/3 -41/11 -6/5 -1/1 -26/7 -4/3 -6/5 -37/10 -6/5 -1/1 -11/3 -6/5 -1/1 -18/5 -1/1 -7/2 -1/1 -2/3 -17/5 -1/3 -1/4 -27/8 0/1 -37/11 1/5 1/4 -10/3 0/1 2/1 -13/4 -2/1 1/0 -16/5 0/1 -3/1 -2/1 -1/1 1/0 -20/7 0/1 -37/13 5/1 1/0 -54/19 1/0 -17/6 -4/1 1/0 -14/5 -4/1 -2/1 -11/4 -2/1 1/0 -30/11 -3/1 -19/7 -5/2 -7/3 -27/10 -2/1 -8/3 -2/1 -29/11 -2/1 1/0 -50/19 -2/1 0/1 -21/8 -2/1 -55/21 1/1 1/0 -34/13 -4/1 -2/1 -13/5 -5/2 -2/1 -18/7 -2/1 -5/2 -2/1 -5/3 -17/7 -5/3 -13/8 -12/5 -3/2 -19/8 -3/2 -4/3 -7/3 -3/2 -4/3 -23/10 -2/1 -3/2 -16/7 -4/3 -9/4 -2/1 -4/3 -20/9 -4/3 -11/5 -4/3 -1/1 -35/16 -4/3 -1/1 -24/11 -1/1 -13/6 -2/1 1/0 -2/1 -2/1 -4/3 -13/7 -4/3 -5/4 -37/20 -4/3 -1/1 -24/13 -1/1 -11/6 -3/2 -4/3 -9/5 -3/2 -4/3 -1/1 -25/14 -3/2 -4/3 -16/9 -4/3 -23/13 -3/2 -4/3 -7/4 -4/3 -1/1 -19/11 -7/4 -5/3 -12/7 -3/2 -17/10 -3/2 -4/3 -22/13 -10/7 -4/3 -5/3 -7/5 -4/3 -18/11 -4/3 -13/8 -4/3 -13/10 -34/21 -4/3 -14/11 -21/13 -4/3 -5/4 -1/1 -71/44 -2/1 -1/1 -50/31 -2/1 -4/3 -79/49 -3/2 -4/3 -108/67 -4/3 -29/18 -4/3 -1/1 -8/5 -4/3 -35/22 -4/3 -9/7 -27/17 -4/3 -9/7 -5/4 -19/12 -4/3 -5/4 -11/7 -4/3 -1/1 -36/23 -4/3 -25/16 -4/3 -5/4 -14/9 -4/3 -6/5 -17/11 -7/6 -1/1 -3/2 -4/3 -19/13 -31/24 -9/7 -54/37 -9/7 -35/24 -9/7 -50/39 -16/11 -14/11 -13/9 -14/11 -5/4 -36/25 -5/4 -23/16 -5/4 -6/5 -10/7 -4/3 -14/11 -37/26 -9/7 -14/11 -27/19 -9/7 -14/11 -5/4 -17/12 -14/11 -5/4 -7/5 -14/11 -5/4 -18/13 -5/4 -11/8 -5/4 -26/21 -48/35 -21/17 -37/27 -37/30 -53/43 -26/19 -16/13 -38/31 -67/49 -38/31 -11/9 -108/79 -11/9 -41/30 -11/9 -6/5 -15/11 -5/4 -11/9 -6/5 -34/25 -4/3 -6/5 -53/39 -19/15 -5/4 -72/53 -5/4 -19/14 -5/4 -16/13 -4/3 -6/5 -17/13 -5/4 -11/9 -30/23 -1/1 -13/10 -5/4 -6/5 -9/7 -5/4 -6/5 -1/1 -23/18 -5/4 -6/5 -60/47 -1/1 -37/29 -9/7 -5/4 -51/40 -6/5 -14/11 -4/3 -6/5 -19/15 -3/2 -1/1 -5/4 -6/5 -1/1 -21/17 -5/4 -6/5 -1/1 -37/30 -8/7 -1/1 -16/13 -4/3 -11/9 -4/3 -1/1 -17/14 -4/3 -5/4 -6/5 -5/4 -19/16 -5/4 -6/5 -13/11 -5/4 -6/5 -7/6 -11/9 -6/5 -8/7 -6/5 -9/8 -6/5 -1/1 -7/6 -1/1 0/1 -1/1 1/1 -1/1 -7/8 8/7 -6/7 7/6 -6/7 -11/13 13/11 -6/7 -5/6 6/5 -5/6 11/9 -1/1 -4/5 16/13 -4/5 37/30 -1/1 -8/9 21/17 -1/1 -6/7 -5/6 5/4 -1/1 -6/7 14/11 -6/7 -4/5 9/7 -1/1 -6/7 -5/6 22/17 -6/7 -4/5 35/27 -5/6 -9/11 13/10 -6/7 -5/6 17/13 -11/13 -5/6 4/3 -6/7 19/14 -16/19 -5/6 15/11 -6/7 -11/13 -5/6 41/30 -6/7 -11/13 26/19 -38/45 -16/19 37/27 -53/63 -37/44 11/8 -26/31 -5/6 18/13 -5/6 7/5 -5/6 -14/17 17/12 -5/6 -14/17 27/19 -5/6 -14/17 -9/11 37/26 -14/17 -9/11 10/7 -14/17 -4/5 13/9 -5/6 -14/17 16/11 -14/17 3/2 -4/5 20/13 -2/3 37/24 -2/1 -1/1 54/35 -1/1 17/11 -1/1 -7/8 14/9 -6/7 -4/5 11/7 -1/1 -4/5 30/19 -5/6 19/12 -5/6 -4/5 27/17 -5/6 -9/11 -4/5 8/5 -4/5 29/18 -1/1 -4/5 50/31 -4/5 -2/3 21/13 -1/1 -5/6 -4/5 55/34 -6/7 -5/6 34/21 -14/17 -4/5 13/8 -13/16 -4/5 18/11 -4/5 5/3 -4/5 -7/9 17/10 -4/5 -3/4 12/7 -3/4 19/11 -5/7 -7/10 7/4 -1/1 -4/5 23/13 -4/5 -3/4 16/9 -4/5 9/5 -1/1 -4/5 -3/4 20/11 -4/5 11/6 -4/5 -3/4 35/19 -3/5 -1/2 24/13 -1/1 13/7 -5/6 -4/5 2/1 -4/5 -2/3 13/6 -2/3 -1/2 37/17 -1/2 -1/3 24/11 -1/1 11/5 -1/1 -4/5 9/4 -4/5 -2/3 25/11 -1/1 -4/5 16/7 -4/5 23/10 -3/4 -2/3 7/3 -4/5 -3/4 19/8 -4/5 -3/4 12/5 -3/4 17/7 -13/18 -5/7 22/9 -8/11 -2/3 5/2 -5/7 -2/3 18/7 -2/3 13/5 -2/3 -5/8 34/13 -2/3 -4/7 21/8 -2/3 71/27 -1/2 -1/3 50/19 -2/3 0/1 79/30 -1/1 -2/3 108/41 -2/3 29/11 -2/3 -1/2 8/3 -2/3 35/13 -5/7 -7/10 27/10 -2/3 19/7 -7/11 -5/8 11/4 -2/3 -1/2 36/13 -2/3 25/9 -2/3 -3/5 14/5 -2/3 -4/7 17/6 -4/7 -1/2 3/1 -1/1 -2/3 -1/2 19/6 -4/7 -1/2 54/17 -1/2 35/11 -1/2 -5/11 16/5 0/1 13/4 -2/3 -1/2 36/11 -1/2 23/7 -1/2 -2/5 10/3 -2/5 0/1 37/11 -1/6 -1/7 27/8 0/1 17/5 1/2 1/1 7/2 -2/1 -1/1 18/5 -1/1 11/3 -1/1 -6/7 48/13 -1/1 37/10 -1/1 -6/7 26/7 -6/7 -4/5 67/18 -1/1 -2/3 108/29 -1/1 41/11 -1/1 -6/7 15/4 -4/5 34/9 -4/5 -10/13 53/14 -16/21 -3/4 72/19 -3/4 19/5 -3/4 -5/7 4/1 -2/3 17/4 -1/2 0/1 30/7 -1/2 13/3 -1/2 0/1 9/2 -2/3 0/1 23/5 -1/2 0/1 60/13 -1/1 37/8 -1/1 0/1 51/11 -1/1 -3/4 -2/3 14/3 -2/3 0/1 19/4 -1/2 0/1 5/1 -1/1 0/1 21/4 -2/3 37/7 -5/9 -1/2 16/3 0/1 11/2 -1/2 0/1 17/3 1/1 1/0 6/1 -1/1 19/3 -3/5 -1/2 13/2 -2/3 -1/2 7/1 -1/2 0/1 8/1 0/1 9/1 -1/1 0/1 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(35,288,-22,-181) (-8/1,1/0) -> (-8/5,-35/22) Hyperbolic Matrix(37,288,14,109) (-8/1,-7/1) -> (29/11,8/3) Hyperbolic Matrix(37,252,16,109) (-7/1,-13/2) -> (23/10,7/3) Hyperbolic Matrix(73,468,-56,-359) (-13/2,-6/1) -> (-30/23,-13/10) Hyperbolic Matrix(71,396,-26,-145) (-6/1,-11/2) -> (-11/4,-30/11) Hyperbolic Matrix(73,396,40,217) (-11/2,-16/3) -> (20/11,11/6) Hyperbolic Matrix(217,1152,68,361) (-16/3,-37/7) -> (35/11,16/5) Hyperbolic Matrix(431,2268,164,863) (-37/7,-21/4) -> (21/8,71/27) Hyperbolic Matrix(179,936,48,251) (-21/4,-5/1) -> (41/11,15/4) Hyperbolic Matrix(37,180,-22,-107) (-5/1,-14/3) -> (-22/13,-5/3) Hyperbolic Matrix(71,324,-16,-73) (-14/3,-9/2) -> (-9/2,-22/5) Parabolic Matrix(755,3312,-468,-2053) (-22/5,-35/8) -> (-71/44,-50/31) Hyperbolic Matrix(215,936,-116,-505) (-35/8,-13/3) -> (-13/7,-37/20) Hyperbolic Matrix(109,468,-92,-395) (-13/3,-17/4) -> (-19/16,-13/11) Hyperbolic Matrix(35,144,26,107) (-17/4,-4/1) -> (4/3,19/14) Hyperbolic Matrix(37,144,28,109) (-4/1,-19/5) -> (17/13,4/3) Hyperbolic Matrix(325,1224,-124,-467) (-19/5,-15/4) -> (-21/8,-55/21) Hyperbolic Matrix(251,936,48,179) (-15/4,-41/11) -> (5/1,21/4) Hyperbolic Matrix(755,2808,-552,-2053) (-41/11,-26/7) -> (-26/19,-67/49) Hyperbolic Matrix(359,1332,252,935) (-26/7,-37/10) -> (37/26,10/7) Hyperbolic Matrix(215,792,-98,-361) (-37/10,-11/3) -> (-11/5,-35/16) Hyperbolic Matrix(109,396,30,109) (-11/3,-18/5) -> (18/5,11/3) Hyperbolic Matrix(71,252,20,71) (-18/5,-7/2) -> (7/2,18/5) Hyperbolic Matrix(73,252,42,145) (-7/2,-17/5) -> (19/11,7/4) Hyperbolic Matrix(287,972,106,359) (-17/5,-27/8) -> (27/10,19/7) Hyperbolic Matrix(577,1944,214,721) (-27/8,-37/11) -> (35/13,27/10) Hyperbolic Matrix(397,1332,290,973) (-37/11,-10/3) -> (26/19,37/27) Hyperbolic Matrix(109,360,-76,-251) (-10/3,-13/4) -> (-23/16,-10/7) Hyperbolic Matrix(179,576,78,251) (-13/4,-16/5) -> (16/7,23/10) Hyperbolic Matrix(35,108,-12,-37) (-16/5,-3/1) -> (-3/1,-20/7) Parabolic Matrix(467,1332,88,251) (-20/7,-37/13) -> (37/7,16/3) Hyperbolic Matrix(1367,3888,430,1223) (-37/13,-54/19) -> (54/17,35/11) Hyperbolic Matrix(685,1944,216,613) (-54/19,-17/6) -> (19/6,54/17) Hyperbolic Matrix(179,504,38,107) (-17/6,-14/5) -> (14/3,19/4) Hyperbolic Matrix(181,504,-116,-323) (-14/5,-11/4) -> (-25/16,-14/9) Hyperbolic Matrix(397,1080,-304,-827) (-30/11,-19/7) -> (-17/13,-30/23) Hyperbolic Matrix(359,972,106,287) (-19/7,-27/10) -> (27/8,17/5) Hyperbolic Matrix(107,288,-94,-253) (-27/10,-8/3) -> (-8/7,-9/8) Hyperbolic Matrix(109,288,14,37) (-8/3,-29/11) -> (7/1,8/1) Hyperbolic Matrix(2051,5400,-1272,-3349) (-29/11,-50/19) -> (-50/31,-79/49) Hyperbolic Matrix(1081,2844,-848,-2231) (-50/19,-21/8) -> (-51/40,-14/11) Hyperbolic Matrix(1403,3672,-1032,-2701) (-55/21,-34/13) -> (-34/25,-53/39) Hyperbolic Matrix(179,468,96,251) (-34/13,-13/5) -> (13/7,2/1) Hyperbolic Matrix(181,468,70,181) (-13/5,-18/7) -> (18/7,13/5) Hyperbolic Matrix(71,180,28,71) (-18/7,-5/2) -> (5/2,18/7) Hyperbolic Matrix(73,180,-58,-143) (-5/2,-17/7) -> (-19/15,-5/4) Hyperbolic Matrix(179,432,104,251) (-17/7,-12/5) -> (12/7,19/11) Hyperbolic Matrix(181,432,106,253) (-12/5,-19/8) -> (17/10,12/7) Hyperbolic Matrix(107,252,76,179) (-19/8,-7/3) -> (7/5,17/12) Hyperbolic Matrix(109,252,16,37) (-7/3,-23/10) -> (13/2,7/1) Hyperbolic Matrix(251,576,78,179) (-23/10,-16/7) -> (16/5,13/4) Hyperbolic Matrix(143,324,-64,-145) (-16/7,-9/4) -> (-9/4,-20/9) Parabolic Matrix(179,396,146,323) (-20/9,-11/5) -> (11/9,16/13) Hyperbolic Matrix(1367,2988,296,647) (-35/16,-24/11) -> (60/13,37/8) Hyperbolic Matrix(613,1332,-480,-1043) (-24/11,-13/6) -> (-23/18,-60/47) Hyperbolic Matrix(217,468,134,289) (-13/6,-2/1) -> (34/21,13/8) Hyperbolic Matrix(251,468,96,179) (-2/1,-13/7) -> (13/5,34/13) Hyperbolic Matrix(1441,2664,390,721) (-37/20,-24/13) -> (48/13,37/10) Hyperbolic Matrix(431,792,-314,-577) (-24/13,-11/6) -> (-11/8,-48/35) Hyperbolic Matrix(179,324,-100,-181) (-11/6,-9/5) -> (-9/5,-25/14) Parabolic Matrix(323,576,60,107) (-25/14,-16/9) -> (16/3,11/2) Hyperbolic Matrix(325,576,224,397) (-16/9,-23/13) -> (13/9,16/11) Hyperbolic Matrix(143,252,122,215) (-23/13,-7/4) -> (7/6,13/11) Hyperbolic Matrix(145,252,42,73) (-7/4,-19/11) -> (17/5,7/2) Hyperbolic Matrix(251,432,104,179) (-19/11,-12/7) -> (12/5,17/7) Hyperbolic Matrix(253,432,106,181) (-12/7,-17/10) -> (19/8,12/5) Hyperbolic Matrix(361,612,128,217) (-17/10,-22/13) -> (14/5,17/6) Hyperbolic Matrix(109,180,66,109) (-5/3,-18/11) -> (18/11,5/3) Hyperbolic Matrix(287,468,176,287) (-18/11,-13/8) -> (13/8,18/11) Hyperbolic Matrix(289,468,134,217) (-13/8,-34/21) -> (2/1,13/6) Hyperbolic Matrix(757,1224,-556,-899) (-34/21,-21/13) -> (-15/11,-34/25) Hyperbolic Matrix(1405,2268,1138,1837) (-21/13,-71/44) -> (37/30,21/17) Hyperbolic Matrix(7235,11664,2746,4427) (-79/49,-108/67) -> (108/41,29/11) Hyperbolic Matrix(7237,11664,2748,4429) (-108/67,-29/18) -> (79/30,108/41) Hyperbolic Matrix(179,288,156,251) (-29/18,-8/5) -> (8/7,7/6) Hyperbolic Matrix(1223,1944,860,1367) (-35/22,-27/17) -> (27/19,37/26) Hyperbolic Matrix(613,972,432,685) (-27/17,-19/12) -> (17/12,27/19) Hyperbolic Matrix(251,396,-206,-325) (-19/12,-11/7) -> (-11/9,-17/14) Hyperbolic Matrix(827,1296,298,467) (-11/7,-36/23) -> (36/13,25/9) Hyperbolic Matrix(829,1296,300,469) (-36/23,-25/16) -> (11/4,36/13) Hyperbolic Matrix(395,612,162,251) (-14/9,-17/11) -> (17/7,22/9) Hyperbolic Matrix(71,108,-48,-73) (-17/11,-3/2) -> (-3/2,-19/13) Parabolic Matrix(1331,1944,862,1259) (-19/13,-54/37) -> (54/35,17/11) Hyperbolic Matrix(2665,3888,1728,2521) (-54/37,-35/24) -> (37/24,54/35) Hyperbolic Matrix(791,1152,642,935) (-35/24,-16/11) -> (16/13,37/30) Hyperbolic Matrix(397,576,224,325) (-16/11,-13/9) -> (23/13,16/9) Hyperbolic Matrix(899,1296,274,395) (-13/9,-36/25) -> (36/11,23/7) Hyperbolic Matrix(901,1296,276,397) (-36/25,-23/16) -> (13/4,36/11) Hyperbolic Matrix(935,1332,252,359) (-10/7,-37/26) -> (37/10,26/7) Hyperbolic Matrix(253,360,26,37) (-37/26,-27/19) -> (9/1,1/0) Hyperbolic Matrix(685,972,432,613) (-27/19,-17/12) -> (19/12,27/17) Hyperbolic Matrix(179,252,76,107) (-17/12,-7/5) -> (7/3,19/8) Hyperbolic Matrix(181,252,130,181) (-7/5,-18/13) -> (18/13,7/5) Hyperbolic Matrix(287,396,208,287) (-18/13,-11/8) -> (11/8,18/13) Hyperbolic Matrix(1943,2664,892,1223) (-48/35,-37/27) -> (37/17,24/11) Hyperbolic Matrix(973,1332,290,397) (-37/27,-26/19) -> (10/3,37/11) Hyperbolic Matrix(8531,11664,2290,3131) (-67/49,-108/79) -> (108/29,41/11) Hyperbolic Matrix(8533,11664,2292,3133) (-108/79,-41/30) -> (67/18,108/29) Hyperbolic Matrix(685,936,554,757) (-41/30,-15/11) -> (21/17,5/4) Hyperbolic Matrix(3815,5184,1006,1367) (-53/39,-72/53) -> (72/19,19/5) Hyperbolic Matrix(3817,5184,1008,1369) (-72/53,-19/14) -> (53/14,72/19) Hyperbolic Matrix(107,144,26,35) (-19/14,-4/3) -> (4/1,17/4) Hyperbolic Matrix(109,144,28,37) (-4/3,-17/13) -> (19/5,4/1) Hyperbolic Matrix(251,324,-196,-253) (-13/10,-9/7) -> (-9/7,-23/18) Parabolic Matrix(2341,2988,1270,1621) (-60/47,-37/29) -> (35/19,24/13) Hyperbolic Matrix(2089,2664,396,505) (-37/29,-51/40) -> (21/4,37/7) Hyperbolic Matrix(397,504,256,325) (-14/11,-19/15) -> (17/11,14/9) Hyperbolic Matrix(757,936,554,685) (-5/4,-21/17) -> (15/11,41/30) Hyperbolic Matrix(2159,2664,466,575) (-21/17,-37/30) -> (37/8,51/11) Hyperbolic Matrix(1081,1332,702,865) (-37/30,-16/13) -> (20/13,37/24) Hyperbolic Matrix(469,576,206,253) (-16/13,-11/9) -> (25/11,16/7) Hyperbolic Matrix(179,216,-150,-181) (-17/14,-6/5) -> (-6/5,-19/16) Parabolic Matrix(215,252,122,143) (-13/11,-7/6) -> (7/4,23/13) Hyperbolic Matrix(251,288,156,179) (-7/6,-8/7) -> (8/5,29/18) Hyperbolic Matrix(323,360,96,107) (-9/8,-1/1) -> (37/11,27/8) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(253,-288,94,-107) (1/1,8/7) -> (8/3,35/13) Hyperbolic Matrix(395,-468,92,-109) (13/11,6/5) -> (30/7,13/3) Hyperbolic Matrix(325,-396,206,-251) (6/5,11/9) -> (11/7,30/19) Hyperbolic Matrix(143,-180,58,-73) (5/4,14/11) -> (22/9,5/2) Hyperbolic Matrix(253,-324,196,-251) (14/11,9/7) -> (9/7,22/17) Parabolic Matrix(2557,-3312,972,-1259) (22/17,35/27) -> (71/27,50/19) Hyperbolic Matrix(721,-936,332,-431) (35/27,13/10) -> (13/6,37/17) Hyperbolic Matrix(359,-468,56,-73) (13/10,17/13) -> (19/3,13/2) Hyperbolic Matrix(899,-1224,556,-757) (19/14,15/11) -> (21/13,55/34) Hyperbolic Matrix(2053,-2808,552,-755) (41/30,26/19) -> (26/7,67/18) Hyperbolic Matrix(577,-792,314,-431) (37/27,11/8) -> (11/6,35/19) Hyperbolic Matrix(251,-360,76,-109) (10/7,13/9) -> (23/7,10/3) Hyperbolic Matrix(73,-108,48,-71) (16/11,3/2) -> (3/2,20/13) Parabolic Matrix(323,-504,116,-181) (14/9,11/7) -> (25/9,14/5) Hyperbolic Matrix(683,-1080,160,-253) (30/19,19/12) -> (17/4,30/7) Hyperbolic Matrix(181,-288,22,-35) (27/17,8/5) -> (8/1,9/1) Hyperbolic Matrix(3349,-5400,1272,-2051) (29/18,50/31) -> (50/19,79/30) Hyperbolic Matrix(1763,-2844,380,-613) (50/31,21/13) -> (51/11,14/3) Hyperbolic Matrix(2269,-3672,600,-971) (55/34,34/21) -> (34/9,53/14) Hyperbolic Matrix(107,-180,22,-37) (5/3,17/10) -> (19/4,5/1) Hyperbolic Matrix(181,-324,100,-179) (16/9,9/5) -> (9/5,20/11) Parabolic Matrix(719,-1332,156,-289) (24/13,13/7) -> (23/5,60/13) Hyperbolic Matrix(361,-792,98,-215) (24/11,11/5) -> (11/3,48/13) Hyperbolic Matrix(145,-324,64,-143) (11/5,9/4) -> (9/4,25/11) Parabolic Matrix(467,-1224,124,-325) (34/13,21/8) -> (15/4,34/9) Hyperbolic Matrix(145,-396,26,-71) (19/7,11/4) -> (11/2,17/3) Hyperbolic Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic Matrix(73,-324,16,-71) (13/3,9/2) -> (9/2,23/5) Parabolic Matrix(37,-216,6,-35) (17/3,6/1) -> (6/1,19/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(35,288,-22,-181) -> Matrix(13,4,-10,-3) Matrix(37,288,14,109) -> Matrix(1,2,-2,-3) Matrix(37,252,16,109) -> Matrix(3,4,-4,-5) Matrix(73,468,-56,-359) -> Matrix(5,4,-4,-3) Matrix(71,396,-26,-145) -> Matrix(1,-2,0,1) Matrix(73,396,40,217) -> Matrix(3,4,-4,-5) Matrix(217,1152,68,361) -> Matrix(1,0,-2,1) Matrix(431,2268,164,863) -> Matrix(1,4,-2,-7) Matrix(179,936,48,251) -> Matrix(5,6,-6,-7) Matrix(37,180,-22,-107) -> Matrix(3,-4,-2,3) Matrix(71,324,-16,-73) -> Matrix(1,0,0,1) Matrix(755,3312,-468,-2053) -> Matrix(3,2,-2,-1) Matrix(215,936,-116,-505) -> Matrix(5,4,-4,-3) Matrix(109,468,-92,-395) -> Matrix(5,-6,-4,5) Matrix(35,144,26,107) -> Matrix(5,16,-6,-19) Matrix(37,144,28,109) -> Matrix(7,8,-8,-9) Matrix(325,1224,-124,-467) -> Matrix(1,2,-2,-3) Matrix(251,936,48,179) -> Matrix(5,6,-6,-7) Matrix(755,2808,-552,-2053) -> Matrix(17,28,-14,-23) Matrix(359,1332,252,935) -> Matrix(31,40,-38,-49) Matrix(215,792,-98,-361) -> Matrix(1,2,-2,-3) Matrix(109,396,30,109) -> Matrix(11,12,-12,-13) Matrix(71,252,20,71) -> Matrix(5,4,-4,-3) Matrix(73,252,42,145) -> Matrix(1,2,-2,-3) Matrix(287,972,106,359) -> Matrix(13,2,-20,-3) Matrix(577,1944,214,721) -> Matrix(15,-2,-22,3) Matrix(397,1332,290,973) -> Matrix(27,-16,-32,19) Matrix(109,360,-76,-251) -> Matrix(5,4,-4,-3) Matrix(179,576,78,251) -> Matrix(3,4,-4,-5) Matrix(35,108,-12,-37) -> Matrix(1,0,0,1) Matrix(467,1332,88,251) -> Matrix(1,0,-2,1) Matrix(1367,3888,430,1223) -> Matrix(1,-10,-2,21) Matrix(685,1944,216,613) -> Matrix(1,8,-2,-15) Matrix(179,504,38,107) -> Matrix(1,4,-2,-7) Matrix(181,504,-116,-323) -> Matrix(5,14,-4,-11) Matrix(397,1080,-304,-827) -> Matrix(7,20,-6,-17) Matrix(359,972,106,287) -> Matrix(1,2,4,9) Matrix(107,288,-94,-253) -> Matrix(17,28,-14,-23) Matrix(109,288,14,37) -> Matrix(1,2,-2,-3) Matrix(2051,5400,-1272,-3349) -> Matrix(3,2,-2,-1) Matrix(1081,2844,-848,-2231) -> Matrix(5,4,-4,-3) Matrix(1403,3672,-1032,-2701) -> Matrix(5,14,-4,-11) Matrix(179,468,96,251) -> Matrix(3,10,-4,-13) Matrix(181,468,70,181) -> Matrix(9,20,-14,-31) Matrix(71,180,28,71) -> Matrix(11,20,-16,-29) Matrix(73,180,-58,-143) -> Matrix(17,28,-14,-23) Matrix(179,432,104,251) -> Matrix(19,30,-26,-41) Matrix(181,432,106,253) -> Matrix(17,24,-22,-31) Matrix(107,252,76,179) -> Matrix(13,22,-16,-27) Matrix(109,252,16,37) -> Matrix(3,4,-4,-5) Matrix(251,576,78,179) -> Matrix(3,4,-4,-5) Matrix(143,324,-64,-145) -> Matrix(1,0,0,1) Matrix(179,396,146,323) -> Matrix(7,8,-8,-9) Matrix(1367,2988,296,647) -> Matrix(3,4,-4,-5) Matrix(613,1332,-480,-1043) -> Matrix(5,4,-4,-3) Matrix(217,468,134,289) -> Matrix(13,22,-16,-27) Matrix(251,468,96,179) -> Matrix(7,10,-12,-17) Matrix(1441,2664,390,721) -> Matrix(9,10,-10,-11) Matrix(431,792,-314,-577) -> Matrix(37,58,-30,-47) Matrix(179,324,-100,-181) -> Matrix(1,0,0,1) Matrix(323,576,60,107) -> Matrix(3,4,-4,-5) Matrix(325,576,224,397) -> Matrix(13,22,-16,-27) Matrix(143,252,122,215) -> Matrix(27,38,-32,-45) Matrix(145,252,42,73) -> Matrix(1,2,-2,-3) Matrix(251,432,104,179) -> Matrix(19,30,-26,-41) Matrix(253,432,106,181) -> Matrix(17,24,-22,-31) Matrix(361,612,128,217) -> Matrix(11,16,-20,-29) Matrix(109,180,66,109) -> Matrix(41,56,-52,-71) Matrix(287,468,176,287) -> Matrix(79,104,-98,-129) Matrix(289,468,134,217) -> Matrix(17,22,-24,-31) Matrix(757,1224,-556,-899) -> Matrix(39,50,-32,-41) Matrix(1405,2268,1138,1837) -> Matrix(9,10,-10,-11) Matrix(7235,11664,2746,4427) -> Matrix(7,10,-12,-17) Matrix(7237,11664,2748,4429) -> Matrix(5,6,-6,-7) Matrix(179,288,156,251) -> Matrix(27,38,-32,-45) Matrix(1223,1944,860,1367) -> Matrix(71,90,-86,-109) Matrix(613,972,432,685) -> Matrix(71,90,-86,-109) Matrix(251,396,-206,-325) -> Matrix(1,0,0,1) Matrix(827,1296,298,467) -> Matrix(7,10,-12,-17) Matrix(829,1296,300,469) -> Matrix(5,6,-6,-7) Matrix(395,612,162,251) -> Matrix(17,22,-24,-31) Matrix(71,108,-48,-73) -> Matrix(23,32,-18,-25) Matrix(1331,1944,862,1259) -> Matrix(73,94,-80,-103) Matrix(2665,3888,1728,2521) -> Matrix(53,68,-46,-59) Matrix(791,1152,642,935) -> Matrix(17,22,-24,-31) Matrix(397,576,224,325) -> Matrix(17,22,-24,-31) Matrix(899,1296,274,395) -> Matrix(19,24,-42,-53) Matrix(901,1296,276,397) -> Matrix(13,16,-22,-27) Matrix(935,1332,252,359) -> Matrix(31,40,-38,-49) Matrix(253,360,26,37) -> Matrix(11,14,-4,-5) Matrix(685,972,432,613) -> Matrix(71,90,-86,-109) Matrix(179,252,76,107) -> Matrix(17,22,-24,-31) Matrix(181,252,130,181) -> Matrix(111,140,-134,-169) Matrix(287,396,208,287) -> Matrix(209,260,-250,-311) Matrix(1943,2664,892,1223) -> Matrix(13,16,4,5) Matrix(973,1332,290,397) -> Matrix(13,16,-48,-59) Matrix(8531,11664,2290,3131) -> Matrix(85,104,-94,-115) Matrix(8533,11664,2292,3133) -> Matrix(13,16,-22,-27) Matrix(685,936,554,757) -> Matrix(49,60,-58,-71) Matrix(3815,5184,1006,1367) -> Matrix(65,82,-88,-111) Matrix(3817,5184,1008,1369) -> Matrix(103,128,-136,-169) Matrix(107,144,26,35) -> Matrix(13,16,-22,-27) Matrix(109,144,28,37) -> Matrix(7,8,-8,-9) Matrix(251,324,-196,-253) -> Matrix(1,0,0,1) Matrix(2341,2988,1270,1621) -> Matrix(5,6,-6,-7) Matrix(2089,2664,396,505) -> Matrix(13,16,-22,-27) Matrix(397,504,256,325) -> Matrix(9,10,-10,-11) Matrix(757,936,554,685) -> Matrix(49,60,-58,-71) Matrix(2159,2664,466,575) -> Matrix(7,8,-8,-9) Matrix(1081,1332,702,865) -> Matrix(5,6,-6,-7) Matrix(469,576,206,253) -> Matrix(7,8,-8,-9) Matrix(179,216,-150,-181) -> Matrix(39,50,-32,-41) Matrix(215,252,122,143) -> Matrix(31,38,-40,-49) Matrix(251,288,156,179) -> Matrix(31,38,-40,-49) Matrix(323,360,96,107) -> Matrix(5,6,-36,-43) Matrix(1,0,2,1) -> Matrix(13,14,-14,-15) Matrix(253,-288,94,-107) -> Matrix(33,28,-46,-39) Matrix(395,-468,92,-109) -> Matrix(7,6,-20,-17) Matrix(325,-396,206,-251) -> Matrix(1,0,0,1) Matrix(143,-180,58,-73) -> Matrix(33,28,-46,-39) Matrix(253,-324,196,-251) -> Matrix(1,0,0,1) Matrix(2557,-3312,972,-1259) -> Matrix(5,4,-4,-3) Matrix(721,-936,332,-431) -> Matrix(5,4,-4,-3) Matrix(359,-468,56,-73) -> Matrix(5,4,-4,-3) Matrix(899,-1224,556,-757) -> Matrix(59,50,-72,-61) Matrix(2053,-2808,552,-755) -> Matrix(33,28,-46,-39) Matrix(577,-792,314,-431) -> Matrix(69,58,-94,-79) Matrix(251,-360,76,-109) -> Matrix(5,4,-4,-3) Matrix(73,-108,48,-71) -> Matrix(39,32,-50,-41) Matrix(323,-504,116,-181) -> Matrix(17,14,-28,-23) Matrix(683,-1080,160,-253) -> Matrix(5,4,-4,-3) Matrix(181,-288,22,-35) -> Matrix(5,4,6,5) Matrix(3349,-5400,1272,-2051) -> Matrix(3,2,-2,-1) Matrix(1763,-2844,380,-613) -> Matrix(3,2,-2,-1) Matrix(2269,-3672,600,-971) -> Matrix(75,62,-98,-81) Matrix(107,-180,22,-37) -> Matrix(5,4,-14,-11) Matrix(181,-324,100,-179) -> Matrix(1,0,0,1) Matrix(719,-1332,156,-289) -> Matrix(5,4,-4,-3) Matrix(361,-792,98,-215) -> Matrix(1,2,-2,-3) Matrix(145,-324,64,-143) -> Matrix(1,0,0,1) Matrix(467,-1224,124,-325) -> Matrix(1,2,-2,-3) Matrix(145,-396,26,-71) -> Matrix(3,2,-8,-5) Matrix(37,-108,12,-35) -> Matrix(1,0,0,1) Matrix(73,-324,16,-71) -> Matrix(1,0,0,1) Matrix(37,-216,6,-35) -> Matrix(1,2,-2,-3) 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): 36 Degree of the the map X: 36 Degree of the the map Y: 144 ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 432 Minimal number of generators: 73 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 32 Genus: 21 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -3/1 -12/5 -3/2 0/1 1/1 9/7 3/2 27/17 9/5 2/1 9/4 12/5 5/2 18/7 8/3 36/13 3/1 54/17 36/11 10/3 27/8 7/2 18/5 4/1 9/2 14/3 5/1 16/3 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 0/1 1/0 -6/1 -1/1 -11/2 0/1 1/0 -5/1 -1/1 0/1 -14/3 -2/1 0/1 -9/2 -2/1 0/1 -4/1 -2/1 -7/2 -1/1 -2/3 -17/5 -1/3 -1/4 -27/8 0/1 -10/3 0/1 2/1 -13/4 -2/1 1/0 -16/5 0/1 -3/1 -2/1 -1/1 1/0 -20/7 0/1 -17/6 -4/1 1/0 -14/5 -4/1 -2/1 -11/4 -2/1 1/0 -19/7 -5/2 -7/3 -27/10 -2/1 -8/3 -2/1 -5/2 -2/1 -5/3 -17/7 -5/3 -13/8 -12/5 -3/2 -19/8 -3/2 -4/3 -7/3 -3/2 -4/3 -16/7 -4/3 -9/4 -2/1 -4/3 -2/1 -2/1 -4/3 -9/5 -3/2 -4/3 -1/1 -16/9 -4/3 -23/13 -3/2 -4/3 -7/4 -4/3 -1/1 -19/11 -7/4 -5/3 -12/7 -3/2 -17/10 -3/2 -4/3 -22/13 -10/7 -4/3 -5/3 -7/5 -4/3 -18/11 -4/3 -13/8 -4/3 -13/10 -8/5 -4/3 -27/17 -4/3 -9/7 -5/4 -19/12 -4/3 -5/4 -11/7 -4/3 -1/1 -36/23 -4/3 -25/16 -4/3 -5/4 -14/9 -4/3 -6/5 -17/11 -7/6 -1/1 -3/2 -4/3 -19/13 -31/24 -9/7 -54/37 -9/7 -35/24 -9/7 -50/39 -16/11 -14/11 -13/9 -14/11 -5/4 -36/25 -5/4 -23/16 -5/4 -6/5 -10/7 -4/3 -14/11 -27/19 -9/7 -14/11 -5/4 -17/12 -14/11 -5/4 -7/5 -14/11 -5/4 -18/13 -5/4 -11/8 -5/4 -26/21 -4/3 -6/5 -9/7 -5/4 -6/5 -1/1 -14/11 -4/3 -6/5 -19/15 -3/2 -1/1 -5/4 -6/5 -1/1 -16/13 -4/3 -11/9 -4/3 -1/1 -6/5 -5/4 -7/6 -11/9 -6/5 -1/1 -7/6 -1/1 0/1 -1/1 1/1 -1/1 -7/8 7/6 -6/7 -11/13 6/5 -5/6 11/9 -1/1 -4/5 5/4 -1/1 -6/7 14/11 -6/7 -4/5 9/7 -1/1 -6/7 -5/6 4/3 -6/7 7/5 -5/6 -14/17 17/12 -5/6 -14/17 27/19 -5/6 -14/17 -9/11 10/7 -14/17 -4/5 13/9 -5/6 -14/17 16/11 -14/17 3/2 -4/5 20/13 -2/3 17/11 -1/1 -7/8 14/9 -6/7 -4/5 11/7 -1/1 -4/5 19/12 -5/6 -4/5 27/17 -5/6 -9/11 -4/5 8/5 -4/5 5/3 -4/5 -7/9 17/10 -4/5 -3/4 12/7 -3/4 19/11 -5/7 -7/10 7/4 -1/1 -4/5 16/9 -4/5 9/5 -1/1 -4/5 -3/4 2/1 -4/5 -2/3 9/4 -4/5 -2/3 16/7 -4/5 23/10 -3/4 -2/3 7/3 -4/5 -3/4 19/8 -4/5 -3/4 12/5 -3/4 17/7 -13/18 -5/7 22/9 -8/11 -2/3 5/2 -5/7 -2/3 18/7 -2/3 13/5 -2/3 -5/8 8/3 -2/3 27/10 -2/3 19/7 -7/11 -5/8 11/4 -2/3 -1/2 36/13 -2/3 25/9 -2/3 -3/5 14/5 -2/3 -4/7 17/6 -4/7 -1/2 3/1 -1/1 -2/3 -1/2 19/6 -4/7 -1/2 54/17 -1/2 35/11 -1/2 -5/11 16/5 0/1 13/4 -2/3 -1/2 36/11 -1/2 23/7 -1/2 -2/5 10/3 -2/5 0/1 27/8 0/1 17/5 1/2 1/1 7/2 -2/1 -1/1 18/5 -1/1 11/3 -1/1 -6/7 4/1 -2/3 9/2 -2/3 0/1 14/3 -2/3 0/1 19/4 -1/2 0/1 5/1 -1/1 0/1 16/3 0/1 11/2 -1/2 0/1 6/1 -1/1 7/1 -1/2 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(19,144,12,91) (-7/1,1/0) -> (11/7,19/12) Hyperbolic Matrix(17,108,14,89) (-7/1,-6/1) -> (6/5,11/9) Hyperbolic Matrix(19,108,16,91) (-6/1,-11/2) -> (7/6,6/5) Hyperbolic Matrix(53,288,-30,-163) (-11/2,-5/1) -> (-23/13,-7/4) Hyperbolic Matrix(37,180,-22,-107) (-5/1,-14/3) -> (-22/13,-5/3) Hyperbolic Matrix(55,252,12,55) (-14/3,-9/2) -> (9/2,14/3) Hyperbolic Matrix(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(19,72,-14,-53) (-4/1,-7/2) -> (-11/8,-4/3) Hyperbolic Matrix(73,252,42,145) (-7/2,-17/5) -> (19/11,7/4) Hyperbolic Matrix(287,972,106,359) (-17/5,-27/8) -> (27/10,19/7) Hyperbolic Matrix(161,540,48,161) (-27/8,-10/3) -> (10/3,27/8) Hyperbolic Matrix(109,360,-76,-251) (-10/3,-13/4) -> (-23/16,-10/7) Hyperbolic Matrix(179,576,78,251) (-13/4,-16/5) -> (16/7,23/10) Hyperbolic Matrix(35,108,-12,-37) (-16/5,-3/1) -> (-3/1,-20/7) Parabolic Matrix(341,972,-234,-667) (-20/7,-17/6) -> (-35/24,-16/11) Hyperbolic Matrix(179,504,38,107) (-17/6,-14/5) -> (14/3,19/4) Hyperbolic Matrix(181,504,-116,-323) (-14/5,-11/4) -> (-25/16,-14/9) Hyperbolic Matrix(53,144,46,125) (-11/4,-19/7) -> (1/1,7/6) Hyperbolic Matrix(359,972,106,287) (-19/7,-27/10) -> (27/8,17/5) Hyperbolic Matrix(161,432,60,161) (-27/10,-8/3) -> (8/3,27/10) Hyperbolic Matrix(55,144,-34,-89) (-8/3,-5/2) -> (-13/8,-8/5) Hyperbolic Matrix(73,180,-58,-143) (-5/2,-17/7) -> (-19/15,-5/4) Hyperbolic Matrix(179,432,104,251) (-17/7,-12/5) -> (12/7,19/11) Hyperbolic Matrix(181,432,106,253) (-12/5,-19/8) -> (17/10,12/7) Hyperbolic Matrix(107,252,76,179) (-19/8,-7/3) -> (7/5,17/12) Hyperbolic Matrix(125,288,-102,-235) (-7/3,-16/7) -> (-16/13,-11/9) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(17,36,8,17) (-9/4,-2/1) -> (2/1,9/4) Hyperbolic Matrix(19,36,10,19) (-2/1,-9/5) -> (9/5,2/1) Hyperbolic Matrix(161,288,90,161) (-9/5,-16/9) -> (16/9,9/5) Hyperbolic Matrix(325,576,224,397) (-16/9,-23/13) -> (13/9,16/11) Hyperbolic Matrix(145,252,42,73) (-7/4,-19/11) -> (17/5,7/2) Hyperbolic Matrix(251,432,104,179) (-19/11,-12/7) -> (12/5,17/7) Hyperbolic Matrix(253,432,106,181) (-12/7,-17/10) -> (19/8,12/5) Hyperbolic Matrix(361,612,128,217) (-17/10,-22/13) -> (14/5,17/6) Hyperbolic Matrix(197,324,76,125) (-5/3,-18/11) -> (18/7,13/5) Hyperbolic Matrix(199,324,78,127) (-18/11,-13/8) -> (5/2,18/7) Hyperbolic Matrix(271,432,170,271) (-8/5,-27/17) -> (27/17,8/5) Hyperbolic Matrix(613,972,432,685) (-27/17,-19/12) -> (17/12,27/19) Hyperbolic Matrix(91,144,12,19) (-19/12,-11/7) -> (7/1,1/0) Hyperbolic Matrix(827,1296,298,467) (-11/7,-36/23) -> (36/13,25/9) Hyperbolic Matrix(829,1296,300,469) (-36/23,-25/16) -> (11/4,36/13) Hyperbolic Matrix(395,612,162,251) (-14/9,-17/11) -> (17/7,22/9) Hyperbolic Matrix(71,108,-48,-73) (-17/11,-3/2) -> (-3/2,-19/13) Parabolic Matrix(1997,2916,628,917) (-19/13,-54/37) -> (54/17,35/11) Hyperbolic Matrix(1999,2916,630,919) (-54/37,-35/24) -> (19/6,54/17) Hyperbolic Matrix(199,288,38,55) (-16/11,-13/9) -> (5/1,16/3) Hyperbolic Matrix(899,1296,274,395) (-13/9,-36/25) -> (36/11,23/7) Hyperbolic Matrix(901,1296,276,397) (-36/25,-23/16) -> (13/4,36/11) Hyperbolic Matrix(379,540,266,379) (-10/7,-27/19) -> (27/19,10/7) Hyperbolic Matrix(685,972,432,613) (-27/19,-17/12) -> (19/12,27/17) Hyperbolic Matrix(179,252,76,107) (-17/12,-7/5) -> (7/3,19/8) Hyperbolic Matrix(233,324,64,89) (-7/5,-18/13) -> (18/5,11/3) Hyperbolic Matrix(235,324,66,91) (-18/13,-11/8) -> (7/2,18/5) Hyperbolic Matrix(55,72,42,55) (-4/3,-9/7) -> (9/7,4/3) Hyperbolic Matrix(197,252,154,197) (-9/7,-14/11) -> (14/11,9/7) Hyperbolic Matrix(397,504,256,325) (-14/11,-19/15) -> (17/11,14/9) Hyperbolic Matrix(233,288,72,89) (-5/4,-16/13) -> (16/5,13/4) Hyperbolic Matrix(89,108,14,17) (-11/9,-6/5) -> (6/1,7/1) Hyperbolic Matrix(91,108,16,19) (-6/5,-7/6) -> (11/2,6/1) Hyperbolic Matrix(125,144,46,53) (-7/6,-1/1) -> (19/7,11/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(235,-288,102,-125) (11/9,5/4) -> (23/10,7/3) Hyperbolic Matrix(143,-180,58,-73) (5/4,14/11) -> (22/9,5/2) Hyperbolic Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(251,-360,76,-109) (10/7,13/9) -> (23/7,10/3) Hyperbolic Matrix(73,-108,48,-71) (16/11,3/2) -> (3/2,20/13) Parabolic Matrix(631,-972,198,-305) (20/13,17/11) -> (35/11,16/5) Hyperbolic Matrix(323,-504,116,-181) (14/9,11/7) -> (25/9,14/5) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(107,-180,22,-37) (5/3,17/10) -> (19/4,5/1) Hyperbolic Matrix(163,-288,30,-53) (7/4,16/9) -> (16/3,11/2) Hyperbolic Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(19,144,12,91) -> Matrix(1,-4,-1,5) Matrix(17,108,14,89) -> Matrix(1,-4,-1,5) Matrix(19,108,16,91) -> Matrix(11,6,-13,-7) Matrix(53,288,-30,-163) -> Matrix(1,4,-1,-3) Matrix(37,180,-22,-107) -> Matrix(3,-4,-2,3) Matrix(55,252,12,55) -> Matrix(1,0,-1,1) Matrix(17,72,4,17) -> Matrix(1,0,-1,1) Matrix(19,72,-14,-53) -> Matrix(11,16,-9,-13) Matrix(73,252,42,145) -> Matrix(1,2,-2,-3) Matrix(287,972,106,359) -> Matrix(13,2,-20,-3) Matrix(161,540,48,161) -> Matrix(1,0,-3,1) Matrix(109,360,-76,-251) -> Matrix(5,4,-4,-3) Matrix(179,576,78,251) -> Matrix(3,4,-4,-5) Matrix(35,108,-12,-37) -> Matrix(1,0,0,1) Matrix(341,972,-234,-667) -> Matrix(9,-14,-7,11) Matrix(179,504,38,107) -> Matrix(1,4,-2,-7) Matrix(181,504,-116,-323) -> Matrix(5,14,-4,-11) Matrix(53,144,46,125) -> Matrix(11,28,-13,-33) Matrix(359,972,106,287) -> Matrix(1,2,4,9) Matrix(161,432,60,161) -> Matrix(1,0,-1,1) Matrix(55,144,-34,-89) -> Matrix(1,6,-1,-5) Matrix(73,180,-58,-143) -> Matrix(17,28,-14,-23) Matrix(179,432,104,251) -> Matrix(19,30,-26,-41) Matrix(181,432,106,253) -> Matrix(17,24,-22,-31) Matrix(107,252,76,179) -> Matrix(13,22,-16,-27) Matrix(125,288,-102,-235) -> Matrix(11,16,-9,-13) Matrix(127,288,56,127) -> Matrix(5,8,-7,-11) Matrix(17,36,8,17) -> Matrix(5,8,-7,-11) Matrix(19,36,10,19) -> Matrix(5,8,-7,-11) Matrix(161,288,90,161) -> Matrix(5,8,-7,-11) Matrix(325,576,224,397) -> Matrix(13,22,-16,-27) Matrix(145,252,42,73) -> Matrix(1,2,-2,-3) Matrix(251,432,104,179) -> Matrix(19,30,-26,-41) Matrix(253,432,106,181) -> Matrix(17,24,-22,-31) Matrix(361,612,128,217) -> Matrix(11,16,-20,-29) Matrix(197,324,76,125) -> Matrix(25,34,-39,-53) Matrix(199,324,78,127) -> Matrix(35,46,-51,-67) Matrix(271,432,170,271) -> Matrix(43,56,-53,-69) Matrix(613,972,432,685) -> Matrix(71,90,-86,-109) Matrix(91,144,12,19) -> Matrix(3,4,-7,-9) Matrix(827,1296,298,467) -> Matrix(7,10,-12,-17) Matrix(829,1296,300,469) -> Matrix(5,6,-6,-7) Matrix(395,612,162,251) -> Matrix(17,22,-24,-31) Matrix(71,108,-48,-73) -> Matrix(23,32,-18,-25) Matrix(1997,2916,628,917) -> Matrix(59,76,-125,-161) Matrix(1999,2916,630,919) -> Matrix(67,86,-127,-163) Matrix(199,288,38,55) -> Matrix(11,14,-15,-19) Matrix(899,1296,274,395) -> Matrix(19,24,-42,-53) Matrix(901,1296,276,397) -> Matrix(13,16,-22,-27) Matrix(379,540,266,379) -> Matrix(43,56,-53,-69) Matrix(685,972,432,613) -> Matrix(71,90,-86,-109) Matrix(179,252,76,107) -> Matrix(17,22,-24,-31) Matrix(233,324,64,89) -> Matrix(35,44,-39,-49) Matrix(235,324,66,91) -> Matrix(29,36,-25,-31) Matrix(55,72,42,55) -> Matrix(19,24,-23,-29) Matrix(197,252,154,197) -> Matrix(19,24,-23,-29) Matrix(397,504,256,325) -> Matrix(9,10,-10,-11) Matrix(233,288,72,89) -> Matrix(3,4,-7,-9) Matrix(89,108,14,17) -> Matrix(3,4,-7,-9) Matrix(91,108,16,19) -> Matrix(5,6,-1,-1) Matrix(125,144,46,53) -> Matrix(23,28,-37,-45) Matrix(1,0,2,1) -> Matrix(13,14,-14,-15) Matrix(235,-288,102,-125) -> Matrix(19,16,-25,-21) Matrix(143,-180,58,-73) -> Matrix(33,28,-46,-39) Matrix(53,-72,14,-19) -> Matrix(19,16,-25,-21) Matrix(251,-360,76,-109) -> Matrix(5,4,-4,-3) Matrix(73,-108,48,-71) -> Matrix(39,32,-50,-41) Matrix(631,-972,198,-305) -> Matrix(3,2,-5,-3) Matrix(323,-504,116,-181) -> Matrix(17,14,-28,-23) Matrix(89,-144,34,-55) -> Matrix(7,6,-13,-11) Matrix(107,-180,22,-37) -> Matrix(5,4,-14,-11) Matrix(163,-288,30,-53) -> Matrix(5,4,-9,-7) Matrix(37,-108,12,-35) -> Matrix(1,0,0,1) 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): 36 ----------------------------------------------------------------------- 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 7 1 1/1 (-1/1,-7/8) 0 18 7/6 (-6/7,-11/13) 0 18 6/5 -5/6 1 3 11/9 (-1/1,-4/5) 0 18 5/4 (-1/1,-6/7) 0 18 14/11 (-6/7,-4/5) 0 9 9/7 (-6/7,-4/5) 0 2 4/3 -6/7 1 9 7/5 (-5/6,-14/17) 0 18 17/12 (-5/6,-14/17) 0 18 27/19 (-14/17,-4/5) 0 2 10/7 (-14/17,-4/5) 0 9 13/9 (-5/6,-14/17) 0 18 16/11 -14/17 1 9 3/2 -4/5 1 6 20/13 -2/3 1 9 17/11 (-1/1,-7/8) 0 18 14/9 (-6/7,-4/5) 0 9 11/7 (-1/1,-4/5) 0 18 19/12 (-5/6,-4/5) 0 18 27/17 (-14/17,-4/5) 0 2 8/5 -4/5 1 9 5/3 (-4/5,-7/9) 0 18 17/10 (-4/5,-3/4) 0 18 12/7 -3/4 3 3 19/11 (-5/7,-7/10) 0 18 7/4 (-1/1,-4/5) 0 18 16/9 -4/5 1 9 9/5 (-4/5,-2/3) 0 2 2/1 (-4/5,-2/3) 0 9 9/4 (-4/5,-2/3) 0 2 16/7 -4/5 1 9 23/10 (-3/4,-2/3) 0 18 7/3 (-4/5,-3/4) 0 18 19/8 (-4/5,-3/4) 0 18 12/5 -3/4 3 3 17/7 (-13/18,-5/7) 0 18 22/9 (-8/11,-2/3) 0 9 5/2 (-5/7,-2/3) 0 18 18/7 -2/3 5 1 13/5 (-2/3,-5/8) 0 18 8/3 -2/3 1 9 27/10 -2/3 7 2 19/7 (-7/11,-5/8) 0 18 11/4 (-2/3,-1/2) 0 18 36/13 -2/3 1 1 25/9 (-2/3,-3/5) 0 18 14/5 (-2/3,-4/7) 0 9 17/6 (-4/7,-1/2) 0 18 3/1 0 6 19/6 (-4/7,-1/2) 0 18 54/17 -1/2 9 1 35/11 (-1/2,-5/11) 0 18 16/5 0/1 1 9 13/4 (-2/3,-1/2) 0 18 36/11 -1/2 4 1 23/7 (-1/2,-2/5) 0 18 10/3 (-2/5,0/1) 0 9 27/8 0/1 7 2 17/5 (1/2,1/1) 0 18 7/2 (-2/1,-1/1) 0 18 18/5 -1/1 8 1 11/3 (-1/1,-6/7) 0 18 4/1 -2/3 1 9 9/2 (-2/3,0/1) 0 2 14/3 (-2/3,0/1) 0 9 19/4 (-1/2,0/1) 0 18 5/1 (-1/1,0/1) 0 18 16/3 0/1 1 9 11/2 (-1/2,0/1) 0 18 6/1 -1/1 1 3 7/1 (-1/2,0/1) 0 18 1/0 (-1/1,0/1) 0 18 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(125,-144,46,-53) (1/1,7/6) -> (19/7,11/4) Glide Reflection Matrix(91,-108,16,-19) (7/6,6/5) -> (11/2,6/1) Glide Reflection Matrix(89,-108,14,-17) (6/5,11/9) -> (6/1,7/1) Glide Reflection Matrix(235,-288,102,-125) (11/9,5/4) -> (23/10,7/3) Hyperbolic Matrix(143,-180,58,-73) (5/4,14/11) -> (22/9,5/2) Hyperbolic Matrix(197,-252,154,-197) (14/11,9/7) -> (14/11,9/7) Reflection Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(179,-252,76,-107) (7/5,17/12) -> (7/3,19/8) Glide Reflection Matrix(685,-972,432,-613) (17/12,27/19) -> (19/12,27/17) Glide Reflection Matrix(379,-540,266,-379) (27/19,10/7) -> (27/19,10/7) Reflection Matrix(251,-360,76,-109) (10/7,13/9) -> (23/7,10/3) Hyperbolic Matrix(199,-288,38,-55) (13/9,16/11) -> (5/1,16/3) Glide Reflection Matrix(73,-108,48,-71) (16/11,3/2) -> (3/2,20/13) Parabolic Matrix(631,-972,198,-305) (20/13,17/11) -> (35/11,16/5) Hyperbolic Matrix(395,-612,162,-251) (17/11,14/9) -> (17/7,22/9) Glide Reflection Matrix(323,-504,116,-181) (14/9,11/7) -> (25/9,14/5) Hyperbolic Matrix(91,-144,12,-19) (11/7,19/12) -> (7/1,1/0) Glide Reflection Matrix(271,-432,170,-271) (27/17,8/5) -> (27/17,8/5) Reflection Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(107,-180,22,-37) (5/3,17/10) -> (19/4,5/1) Hyperbolic Matrix(253,-432,106,-181) (17/10,12/7) -> (19/8,12/5) Glide Reflection Matrix(251,-432,104,-179) (12/7,19/11) -> (12/5,17/7) Glide Reflection Matrix(145,-252,42,-73) (19/11,7/4) -> (17/5,7/2) Glide Reflection Matrix(163,-288,30,-53) (7/4,16/9) -> (16/3,11/2) Hyperbolic Matrix(161,-288,90,-161) (16/9,9/5) -> (16/9,9/5) Reflection Matrix(19,-36,10,-19) (9/5,2/1) -> (9/5,2/1) Reflection Matrix(17,-36,8,-17) (2/1,9/4) -> (2/1,9/4) Reflection Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(251,-576,78,-179) (16/7,23/10) -> (16/5,13/4) Glide Reflection Matrix(71,-180,28,-71) (5/2,18/7) -> (5/2,18/7) Reflection Matrix(181,-468,70,-181) (18/7,13/5) -> (18/7,13/5) Reflection Matrix(161,-432,60,-161) (8/3,27/10) -> (8/3,27/10) Reflection Matrix(359,-972,106,-287) (27/10,19/7) -> (27/8,17/5) Glide Reflection Matrix(287,-792,104,-287) (11/4,36/13) -> (11/4,36/13) Reflection Matrix(649,-1800,234,-649) (36/13,25/9) -> (36/13,25/9) Reflection Matrix(179,-504,38,-107) (14/5,17/6) -> (14/3,19/4) Glide Reflection Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic Matrix(647,-2052,204,-647) (19/6,54/17) -> (19/6,54/17) Reflection Matrix(1189,-3780,374,-1189) (54/17,35/11) -> (54/17,35/11) Reflection Matrix(287,-936,88,-287) (13/4,36/11) -> (13/4,36/11) Reflection Matrix(505,-1656,154,-505) (36/11,23/7) -> (36/11,23/7) Reflection Matrix(161,-540,48,-161) (10/3,27/8) -> (10/3,27/8) Reflection Matrix(71,-252,20,-71) (7/2,18/5) -> (7/2,18/5) Reflection Matrix(109,-396,30,-109) (18/5,11/3) -> (18/5,11/3) Reflection Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) 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(1,0,2,-1) -> Matrix(15,14,-16,-15) (0/1,1/1) -> (-1/1,-7/8) Matrix(125,-144,46,-53) -> Matrix(33,28,-53,-45) Matrix(91,-108,16,-19) -> Matrix(7,6,-1,-1) Matrix(89,-108,14,-17) -> Matrix(5,4,-11,-9) Matrix(235,-288,102,-125) -> Matrix(19,16,-25,-21) -4/5 Matrix(143,-180,58,-73) -> Matrix(33,28,-46,-39) Matrix(197,-252,154,-197) -> Matrix(29,24,-35,-29) (14/11,9/7) -> (-6/7,-4/5) Matrix(55,-72,42,-55) -> Matrix(29,24,-35,-29) (9/7,4/3) -> (-6/7,-4/5) Matrix(53,-72,14,-19) -> Matrix(19,16,-25,-21) -4/5 Matrix(179,-252,76,-107) -> Matrix(27,22,-38,-31) Matrix(685,-972,432,-613) -> Matrix(109,90,-132,-109) *** -> (-5/6,-9/11) Matrix(379,-540,266,-379) -> Matrix(69,56,-85,-69) (27/19,10/7) -> (-14/17,-4/5) Matrix(251,-360,76,-109) -> Matrix(5,4,-4,-3) -1/1 Matrix(199,-288,38,-55) -> Matrix(17,14,-23,-19) Matrix(73,-108,48,-71) -> Matrix(39,32,-50,-41) -4/5 Matrix(631,-972,198,-305) -> Matrix(3,2,-5,-3) (-1/1,-1/2).(-2/3,0/1) Matrix(395,-612,162,-251) -> Matrix(27,22,-38,-31) Matrix(323,-504,116,-181) -> Matrix(17,14,-28,-23) Matrix(91,-144,12,-19) -> Matrix(5,4,-11,-9) Matrix(271,-432,170,-271) -> Matrix(69,56,-85,-69) (27/17,8/5) -> (-14/17,-4/5) Matrix(89,-144,34,-55) -> Matrix(7,6,-13,-11) Matrix(107,-180,22,-37) -> Matrix(5,4,-14,-11) Matrix(253,-432,106,-181) -> Matrix(31,24,-40,-31) *** -> (-4/5,-3/4) Matrix(251,-432,104,-179) -> Matrix(41,30,-56,-41) *** -> (-3/4,-5/7) Matrix(145,-252,42,-73) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(163,-288,30,-53) -> Matrix(5,4,-9,-7) -2/3 Matrix(161,-288,90,-161) -> Matrix(11,8,-15,-11) (16/9,9/5) -> (-4/5,-2/3) Matrix(19,-36,10,-19) -> Matrix(11,8,-15,-11) (9/5,2/1) -> (-4/5,-2/3) Matrix(17,-36,8,-17) -> Matrix(11,8,-15,-11) (2/1,9/4) -> (-4/5,-2/3) Matrix(127,-288,56,-127) -> Matrix(11,8,-15,-11) (9/4,16/7) -> (-4/5,-2/3) Matrix(251,-576,78,-179) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(71,-180,28,-71) -> Matrix(29,20,-42,-29) (5/2,18/7) -> (-5/7,-2/3) Matrix(181,-468,70,-181) -> Matrix(31,20,-48,-31) (18/7,13/5) -> (-2/3,-5/8) Matrix(161,-432,60,-161) -> Matrix(-1,0,3,1) (8/3,27/10) -> (-2/3,0/1) Matrix(359,-972,106,-287) -> Matrix(3,2,14,9) Matrix(287,-792,104,-287) -> Matrix(7,4,-12,-7) (11/4,36/13) -> (-2/3,-1/2) Matrix(649,-1800,234,-649) -> Matrix(19,12,-30,-19) (36/13,25/9) -> (-2/3,-3/5) Matrix(179,-504,38,-107) -> Matrix(7,4,-12,-7) *** -> (-2/3,-1/2) Matrix(37,-108,12,-35) -> Matrix(1,0,0,1) Matrix(647,-2052,204,-647) -> Matrix(15,8,-28,-15) (19/6,54/17) -> (-4/7,-1/2) Matrix(1189,-3780,374,-1189) -> Matrix(21,10,-44,-21) (54/17,35/11) -> (-1/2,-5/11) Matrix(287,-936,88,-287) -> Matrix(7,4,-12,-7) (13/4,36/11) -> (-2/3,-1/2) Matrix(505,-1656,154,-505) -> Matrix(9,4,-20,-9) (36/11,23/7) -> (-1/2,-2/5) Matrix(161,-540,48,-161) -> Matrix(-1,0,5,1) (10/3,27/8) -> (-2/5,0/1) Matrix(71,-252,20,-71) -> Matrix(3,4,-2,-3) (7/2,18/5) -> (-2/1,-1/1) Matrix(109,-396,30,-109) -> Matrix(13,12,-14,-13) (18/5,11/3) -> (-1/1,-6/7) Matrix(17,-72,4,-17) -> Matrix(-1,0,3,1) (4/1,9/2) -> (-2/3,0/1) Matrix(55,-252,12,-55) -> Matrix(-1,0,3,1) (9/2,14/3) -> (-2/3,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.