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 -7/1 -6/1 -5/1 -9/2 -17/4 -4/1 -15/4 -27/8 -3/1 -27/10 -9/4 -2/1 -3/2 -6/5 -1/1 -3/4 -9/14 -3/5 -15/26 -9/16 0/1 1/2 9/16 3/5 9/14 9/13 3/4 9/11 1/1 6/5 5/4 9/7 18/13 3/2 27/17 18/11 7/4 9/5 2/1 9/4 5/2 18/7 27/10 11/4 17/6 3/1 13/4 27/8 7/2 18/5 15/4 72/19 4/1 9/2 5/1 11/2 6/1 13/2 7/1 15/2 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -9/1 0/1 -8/1 1/2 1/0 -7/1 1/2 1/0 -13/2 -1/2 -6/1 0/1 -23/4 1/2 1/1 -17/3 3/2 1/0 -11/2 -1/2 -5/1 1/2 1/0 -9/2 1/0 -13/3 -3/2 1/0 -17/4 -1/1 1/0 -4/1 -1/2 1/0 -19/5 -1/2 1/0 -15/4 -1/1 0/1 -11/3 -1/2 -1/4 -18/5 0/1 -7/2 1/2 -17/5 5/2 1/0 -27/8 1/0 -10/3 -3/2 1/0 -3/1 0/1 -11/4 1/1 1/0 -30/11 2/1 -19/7 5/2 1/0 -27/10 1/0 -8/3 -1/2 1/0 -29/11 -1/2 -1/4 -21/8 -1/1 0/1 -55/21 -1/2 1/0 -34/13 -1/2 -1/4 -13/5 -1/2 1/0 -18/7 0/1 -5/2 1/2 -12/5 1/1 -19/8 1/1 1/0 -7/3 3/2 1/0 -9/4 1/0 -11/5 -5/2 1/0 -13/6 -3/2 -2/1 -1/2 1/0 -13/7 -1/2 -1/4 -11/6 -1/6 -9/5 0/1 -7/4 0/1 1/4 -19/11 1/4 3/10 -12/7 1/3 -17/10 1/2 -22/13 9/20 1/2 -5/3 1/2 3/4 -18/11 1/1 -13/8 1/1 3/2 -34/21 3/2 7/4 -21/13 2/1 -8/5 1/2 1/0 -3/2 1/0 -10/7 -3/2 1/0 -37/26 -3/2 -27/19 -1/1 -17/12 -1/1 1/0 -7/5 -3/2 1/0 -18/13 -1/1 -11/8 -1/1 -3/4 -15/11 -2/3 -34/25 -3/4 -1/2 -53/39 -5/8 -1/2 -72/53 -3/5 -19/14 -1/2 -4/3 -1/2 -1/4 -17/13 -1/2 -1/4 -30/23 0/1 -13/10 -1/6 -9/7 0/1 -5/4 0/1 1/2 -6/5 1/1 -7/6 3/2 -1/1 -1/2 1/0 -6/7 -1/1 -5/6 -3/2 -9/11 -1/1 -13/16 -1/1 -5/6 -17/21 -3/4 -1/2 -4/5 -1/2 1/0 -11/14 -3/2 -18/23 -1/1 -7/9 -3/4 -1/2 -3/4 -1/1 0/1 -11/15 -3/4 -1/2 -19/26 -1/2 -27/37 0/1 -8/11 -1/2 1/0 -13/18 -3/2 -18/25 -1/1 -5/7 -1/2 1/0 -17/24 -1/1 1/0 -12/17 -1/1 -7/10 -3/2 -9/13 -1/1 -11/16 -1/1 -7/8 -2/3 -3/4 -1/2 -9/14 -1/2 -16/25 -1/2 -9/20 -23/36 -1/2 -3/7 -7/11 -1/2 -3/8 -19/30 -1/2 -12/19 -1/3 -5/8 -1/2 0/1 -18/29 0/1 -13/21 1/4 1/2 -34/55 3/2 1/0 -21/34 1/0 -8/13 1/2 1/0 -27/44 1/0 -19/31 -7/2 1/0 -11/18 -3/2 -14/23 -1/2 1/0 -17/28 -1/1 1/0 -3/5 -1/1 -19/32 -1/1 -3/4 -16/27 -5/6 -3/4 -13/22 -1/2 -10/17 -5/6 -3/4 -27/46 -3/4 -17/29 -3/4 -13/18 -7/12 -3/4 -2/3 -18/31 -2/3 -11/19 -9/14 -5/8 -26/45 -9/14 -5/8 -15/26 -5/8 -34/59 -5/8 -1/2 -53/92 -5/8 -3/5 -72/125 -3/5 -19/33 -5/8 -1/2 -4/7 -7/12 -1/2 -9/16 -1/2 -14/25 -1/2 -17/36 -5/9 -1/2 -5/12 -16/29 -3/8 -5/14 -11/20 -1/3 -1/4 -6/11 0/1 -13/24 -1/1 -1/2 -7/13 -1/2 -1/4 -8/15 -1/2 1/0 -1/2 -1/2 0/1 0/1 1/2 1/2 7/13 1/4 1/2 13/24 1/2 1/1 6/11 0/1 11/20 1/4 1/3 5/9 5/12 1/2 9/16 1/2 13/23 1/2 19/36 4/7 1/2 7/12 19/33 1/2 5/8 15/26 5/8 11/19 5/8 9/14 7/12 2/3 3/4 10/17 3/4 5/6 3/5 1/1 11/18 3/2 19/31 7/2 1/0 8/13 -1/2 1/0 21/34 1/0 55/89 -3/2 1/0 34/55 -3/2 1/0 13/21 -1/2 -1/4 5/8 0/1 1/2 7/11 3/8 1/2 9/14 1/2 11/17 1/2 9/16 2/3 1/2 3/4 13/19 13/16 5/6 11/16 7/8 1/1 9/13 1/1 7/10 3/2 12/17 1/1 5/7 1/2 1/0 13/18 3/2 21/29 1/1 8/11 1/2 1/0 3/4 0/1 1/1 10/13 1/2 1/0 27/35 0/1 17/22 1/2 7/9 1/2 3/4 11/14 3/2 15/19 1/1 4/5 1/2 1/0 17/21 1/2 3/4 13/16 5/6 1/1 9/11 1/1 5/6 3/2 1/1 1/2 1/0 7/6 -3/2 6/5 -1/1 11/9 -3/4 -1/2 5/4 -1/2 0/1 9/7 0/1 13/10 1/6 17/13 1/4 1/2 4/3 1/4 1/2 11/8 3/4 1/1 18/13 1/1 25/18 7/6 7/5 3/2 1/0 3/2 1/0 11/7 -1/2 1/0 19/12 -1/1 1/0 27/17 -1/1 35/22 -1/2 8/5 -1/2 1/0 13/8 -3/2 -1/1 18/11 -1/1 23/14 -5/6 5/3 -3/4 -1/2 22/13 -1/2 -9/20 17/10 -1/2 29/17 -1/2 -1/4 12/7 -1/3 19/11 -3/10 -1/4 26/15 -1/4 -3/14 7/4 -1/4 0/1 9/5 0/1 11/6 1/6 2/1 1/2 1/0 9/4 1/0 16/7 -7/2 1/0 39/17 -3/1 23/10 -5/2 7/3 -3/2 1/0 19/8 -1/1 1/0 31/13 -3/2 1/0 12/5 -1/1 17/7 -3/2 1/0 5/2 -1/2 18/7 0/1 31/12 0/1 1/6 13/5 1/2 1/0 34/13 1/4 1/2 21/8 0/1 1/1 8/3 1/2 1/0 27/10 1/0 46/17 -11/2 1/0 19/7 -5/2 1/0 30/11 -2/1 11/4 -1/1 1/0 14/5 -5/8 -1/2 17/6 -1/2 3/1 0/1 19/6 1/2 16/5 1/4 1/2 29/9 1/2 5/8 13/4 1/2 1/1 10/3 3/2 1/0 27/8 1/0 44/13 -13/2 1/0 17/5 -5/2 1/0 7/2 -1/2 18/5 0/1 29/8 0/1 1/8 11/3 1/4 1/2 37/10 1/2 26/7 1/2 1/0 15/4 0/1 1/1 34/9 1/4 1/2 53/14 1/2 72/19 1/1 19/5 1/2 1/0 4/1 1/2 1/0 9/2 1/0 14/3 -3/2 1/0 33/7 -1/1 52/11 -3/2 1/0 71/15 -3/2 1/0 90/19 -1/1 19/4 -1/1 1/0 5/1 -1/2 1/0 16/3 -1/2 -1/4 11/2 1/2 17/3 -3/2 1/0 23/4 -1/1 -1/2 6/1 0/1 19/3 -1/2 1/0 13/2 1/2 7/1 -1/2 1/0 15/2 1/0 8/1 -1/2 1/0 9/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,18,0,1) (-9/1,1/0) -> (9/1,1/0) Parabolic Matrix(37,306,48,397) (-9/1,-8/1) -> (10/13,27/35) Hyperbolic Matrix(37,270,-64,-467) (-8/1,-7/1) -> (-11/19,-26/45) 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(37,216,68,397) (-6/1,-23/4) -> (13/24,6/11) Hyperbolic Matrix(107,612,132,755) (-23/4,-17/3) -> (17/21,13/16) Hyperbolic Matrix(71,396,116,647) (-17/3,-11/2) -> (11/18,19/31) Hyperbolic Matrix(37,198,-20,-107) (-11/2,-5/1) -> (-13/7,-11/6) Hyperbolic Matrix(35,162,-8,-37) (-5/1,-9/2) -> (-9/2,-13/3) Parabolic Matrix(181,774,76,325) (-13/3,-17/4) -> (19/8,31/13) Hyperbolic Matrix(73,306,-120,-503) (-17/4,-4/1) -> (-14/23,-17/28) 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(179,666,-68,-253) (-15/4,-11/3) -> (-29/11,-21/8) Hyperbolic Matrix(109,396,-188,-683) (-11/3,-18/5) -> (-18/31,-11/19) Hyperbolic Matrix(71,252,20,71) (-18/5,-7/2) -> (7/2,18/5) Hyperbolic Matrix(37,126,32,109) (-7/2,-17/5) -> (1/1,7/6) Hyperbolic Matrix(287,972,-468,-1585) (-17/5,-27/8) -> (-27/44,-19/31) Hyperbolic Matrix(145,486,-236,-791) (-27/8,-10/3) -> (-8/13,-27/44) Hyperbolic Matrix(71,234,-44,-145) (-10/3,-3/1) -> (-21/13,-8/5) Hyperbolic Matrix(71,198,-52,-145) (-3/1,-11/4) -> (-11/8,-15/11) Hyperbolic Matrix(145,396,264,721) (-11/4,-30/11) -> (6/11,11/20) Hyperbolic Matrix(397,1080,-304,-827) (-30/11,-19/7) -> (-17/13,-30/23) Hyperbolic Matrix(359,972,-612,-1657) (-19/7,-27/10) -> (-27/46,-17/29) Hyperbolic Matrix(181,486,-308,-827) (-27/10,-8/3) -> (-10/17,-27/46) Hyperbolic Matrix(109,288,-204,-539) (-8/3,-29/11) -> (-7/13,-8/15) Hyperbolic Matrix(1403,3672,-1032,-2701) (-55/21,-34/13) -> (-34/25,-53/39) Hyperbolic Matrix(145,378,28,73) (-34/13,-13/5) -> (5/1,16/3) Hyperbolic Matrix(181,468,-292,-755) (-13/5,-18/7) -> (-18/29,-13/21) Hyperbolic Matrix(71,180,28,71) (-18/7,-5/2) -> (5/2,18/7) Hyperbolic Matrix(37,90,-44,-107) (-5/2,-12/5) -> (-6/7,-5/6) Hyperbolic Matrix(181,432,-256,-611) (-12/5,-19/8) -> (-17/24,-12/17) Hyperbolic Matrix(145,342,92,217) (-19/8,-7/3) -> (11/7,19/12) Hyperbolic Matrix(71,162,-32,-73) (-7/3,-9/4) -> (-9/4,-11/5) Parabolic Matrix(107,234,16,35) (-11/5,-13/6) -> (13/2,7/1) Hyperbolic Matrix(109,234,-184,-395) (-13/6,-2/1) -> (-16/27,-13/22) Hyperbolic Matrix(251,468,96,179) (-2/1,-13/7) -> (13/5,34/13) Hyperbolic Matrix(109,198,60,109) (-11/6,-9/5) -> (9/5,11/6) Hyperbolic Matrix(71,126,40,71) (-9/5,-7/4) -> (7/4,9/5) Hyperbolic Matrix(145,252,-248,-431) (-7/4,-19/11) -> (-17/29,-7/12) Hyperbolic Matrix(251,432,104,179) (-19/11,-12/7) -> (12/5,17/7) Hyperbolic Matrix(253,432,-400,-683) (-12/7,-17/10) -> (-19/30,-12/19) Hyperbolic Matrix(361,612,128,217) (-17/10,-22/13) -> (14/5,17/6) Hyperbolic Matrix(181,306,320,541) (-22/13,-5/3) -> (13/23,4/7) Hyperbolic Matrix(109,180,-152,-251) (-5/3,-18/11) -> (-18/25,-5/7) Hyperbolic Matrix(287,468,176,287) (-18/11,-13/8) -> (13/8,18/11) Hyperbolic Matrix(611,990,-956,-1549) (-13/8,-34/21) -> (-16/25,-23/36) Hyperbolic Matrix(757,1224,-556,-899) (-34/21,-21/13) -> (-15/11,-34/25) Hyperbolic Matrix(35,54,-24,-37) (-8/5,-3/2) -> (-3/2,-10/7) Parabolic Matrix(935,1332,252,359) (-10/7,-37/26) -> (37/10,26/7) Hyperbolic Matrix(1367,1944,860,1223) (-37/26,-27/19) -> (27/17,35/22) 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,-232,-323) (-7/5,-18/13) -> (-18/23,-7/9) Hyperbolic Matrix(287,396,208,287) (-18/13,-11/8) -> (11/8,18/13) Hyperbolic Matrix(3815,5184,-6624,-9001) (-53/39,-72/53) -> (-72/125,-19/33) Hyperbolic Matrix(3817,5184,1008,1369) (-72/53,-19/14) -> (53/14,72/19) Hyperbolic Matrix(359,486,212,287) (-19/14,-4/3) -> (22/13,17/10) Hyperbolic Matrix(109,144,28,37) (-4/3,-17/13) -> (19/5,4/1) Hyperbolic Matrix(181,234,140,181) (-13/10,-9/7) -> (9/7,13/10) Hyperbolic Matrix(71,90,56,71) (-9/7,-5/4) -> (5/4,9/7) Hyperbolic Matrix(73,90,-116,-143) (-5/4,-6/5) -> (-12/19,-5/8) Hyperbolic Matrix(107,126,152,179) (-6/5,-7/6) -> (7/10,12/17) Hyperbolic Matrix(109,126,32,37) (-7/6,-1/1) -> (17/5,7/2) Hyperbolic Matrix(145,126,84,73) (-1/1,-6/7) -> (12/7,19/11) Hyperbolic Matrix(109,90,132,109) (-5/6,-9/11) -> (9/11,5/6) Hyperbolic Matrix(287,234,352,287) (-9/11,-13/16) -> (13/16,9/11) Hyperbolic Matrix(755,612,132,107) (-13/16,-17/21) -> (17/3,23/4) Hyperbolic Matrix(179,144,312,251) (-17/21,-4/5) -> (4/7,19/33) Hyperbolic Matrix(251,198,-412,-325) (-4/5,-11/14) -> (-11/18,-14/23) Hyperbolic Matrix(827,648,596,467) (-11/14,-18/23) -> (18/13,25/18) Hyperbolic Matrix(71,54,-96,-73) (-7/9,-3/4) -> (-3/4,-11/15) Parabolic Matrix(1009,738,592,433) (-11/15,-19/26) -> (17/10,29/17) Hyperbolic Matrix(1331,972,1724,1259) (-19/26,-27/37) -> (27/35,17/22) Hyperbolic Matrix(395,288,48,35) (-27/37,-8/11) -> (8/1,9/1) Hyperbolic Matrix(323,234,-548,-397) (-8/11,-13/18) -> (-13/22,-10/17) Hyperbolic Matrix(899,648,548,395) (-13/18,-18/25) -> (18/11,23/14) Hyperbolic Matrix(253,180,52,37) (-5/7,-17/24) -> (19/4,5/1) Hyperbolic Matrix(179,126,152,107) (-12/17,-7/10) -> (7/6,6/5) Hyperbolic Matrix(181,126,260,181) (-7/10,-9/13) -> (9/13,7/10) Hyperbolic Matrix(287,198,416,287) (-9/13,-11/16) -> (11/16,9/13) Hyperbolic Matrix(289,198,-524,-359) (-11/16,-2/3) -> (-16/29,-11/20) Hyperbolic Matrix(251,162,-392,-253) (-2/3,-9/14) -> (-9/14,-16/25) Parabolic Matrix(395,252,732,467) (-23/36,-7/11) -> (7/13,13/24) Hyperbolic Matrix(397,252,512,325) (-7/11,-19/30) -> (17/22,7/9) Hyperbolic Matrix(1043,648,404,251) (-5/8,-18/29) -> (18/7,31/12) Hyperbolic Matrix(757,468,1108,685) (-13/21,-34/55) -> (2/3,13/19) Hyperbolic Matrix(1981,1224,-3436,-2123) (-34/55,-21/34) -> (-15/26,-34/59) Hyperbolic Matrix(1079,666,-1868,-1153) (-21/34,-8/13) -> (-26/45,-15/26) Hyperbolic Matrix(647,396,116,71) (-19/31,-11/18) -> (11/2,17/3) Hyperbolic Matrix(179,108,-300,-181) (-17/28,-3/5) -> (-3/5,-19/32) Parabolic Matrix(2519,1494,-4372,-2593) (-19/32,-16/27) -> (-34/59,-53/92) Hyperbolic Matrix(1115,648,308,179) (-7/12,-18/31) -> (18/5,29/8) Hyperbolic Matrix(10655,6138,2248,1295) (-53/92,-72/125) -> (90/19,19/4) Hyperbolic Matrix(251,144,312,179) (-19/33,-4/7) -> (4/5,17/21) Hyperbolic Matrix(287,162,-512,-289) (-4/7,-9/16) -> (-9/16,-14/25) Parabolic Matrix(323,180,192,107) (-14/25,-5/9) -> (5/3,22/13) Hyperbolic Matrix(683,378,1104,611) (-5/9,-16/29) -> (34/55,13/21) Hyperbolic Matrix(721,396,264,145) (-11/20,-6/11) -> (30/11,11/4) Hyperbolic Matrix(397,216,68,37) (-6/11,-13/24) -> (23/4,6/1) Hyperbolic Matrix(865,468,268,145) (-13/24,-7/13) -> (29/9,13/4) Hyperbolic Matrix(541,288,340,181) (-8/15,-1/2) -> (35/22,8/5) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(503,-270,136,-73) (1/2,7/13) -> (11/3,37/10) Hyperbolic Matrix(359,-198,524,-289) (11/20,5/9) -> (13/19,11/16) Hyperbolic Matrix(289,-162,512,-287) (5/9,9/16) -> (9/16,13/23) Parabolic Matrix(2123,-1224,3436,-1981) (19/33,15/26) -> (21/34,55/89) Hyperbolic Matrix(467,-270,64,-37) (15/26,11/19) -> (7/1,15/2) Hyperbolic Matrix(683,-396,188,-109) (11/19,7/12) -> (29/8,11/3) Hyperbolic Matrix(431,-252,248,-145) (7/12,10/17) -> (26/15,7/4) Hyperbolic Matrix(397,-234,548,-323) (10/17,3/5) -> (21/29,8/11) Hyperbolic Matrix(325,-198,412,-251) (3/5,11/18) -> (11/14,15/19) Hyperbolic Matrix(1585,-972,468,-287) (19/31,8/13) -> (44/13,17/5) Hyperbolic Matrix(467,-288,60,-37) (8/13,21/34) -> (15/2,8/1) Hyperbolic Matrix(8533,-5274,1804,-1115) (55/89,34/55) -> (52/11,71/15) Hyperbolic Matrix(755,-468,292,-181) (13/21,5/8) -> (31/12,13/5) Hyperbolic Matrix(143,-90,116,-73) (5/8,7/11) -> (11/9,5/4) Hyperbolic Matrix(253,-162,392,-251) (7/11,9/14) -> (9/14,11/17) Parabolic Matrix(359,-234,112,-73) (11/17,2/3) -> (16/5,29/9) Hyperbolic Matrix(611,-432,256,-181) (12/17,5/7) -> (31/13,12/5) Hyperbolic Matrix(251,-180,152,-109) (5/7,13/18) -> (23/14,5/3) Hyperbolic Matrix(1369,-990,596,-431) (13/18,21/29) -> (39/17,23/10) Hyperbolic Matrix(73,-54,96,-71) (8/11,3/4) -> (3/4,10/13) Parabolic Matrix(323,-252,232,-181) (7/9,11/14) -> (25/18,7/5) Hyperbolic Matrix(431,-342,92,-73) (15/19,4/5) -> (14/3,33/7) Hyperbolic Matrix(107,-90,44,-37) (5/6,1/1) -> (17/7,5/2) Hyperbolic Matrix(253,-306,148,-179) (6/5,11/9) -> (29/17,12/7) Hyperbolic Matrix(359,-468,56,-73) (13/10,17/13) -> (19/3,13/2) Hyperbolic Matrix(145,-198,52,-71) (4/3,11/8) -> (11/4,14/5) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(145,-234,44,-71) (8/5,13/8) -> (13/4,10/3) Hyperbolic Matrix(791,-1368,292,-505) (19/11,26/15) -> (46/17,19/7) Hyperbolic Matrix(107,-198,20,-37) (11/6,2/1) -> (16/3,11/2) Hyperbolic Matrix(73,-162,32,-71) (2/1,9/4) -> (9/4,16/7) Parabolic Matrix(1115,-2556,236,-541) (16/7,39/17) -> (33/7,52/11) Hyperbolic Matrix(467,-1224,124,-325) (34/13,21/8) -> (15/4,34/9) Hyperbolic Matrix(253,-666,68,-179) (21/8,8/3) -> (26/7,15/4) Hyperbolic Matrix(541,-1458,200,-539) (8/3,27/10) -> (27/10,46/17) Parabolic Matrix(251,-684,40,-109) (19/7,30/11) -> (6/1,19/3) Hyperbolic Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic Matrix(469,-1494,124,-395) (19/6,16/5) -> (34/9,53/14) Hyperbolic Matrix(433,-1458,128,-431) (10/3,27/8) -> (27/8,44/13) Parabolic Matrix(1799,-6822,380,-1441) (72/19,19/5) -> (71/15,90/19) Hyperbolic Matrix(37,-162,8,-35) (4/1,9/2) -> (9/2,14/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,18,0,1) -> Matrix(1,0,0,1) Matrix(37,306,48,397) -> Matrix(1,0,0,1) Matrix(37,270,-64,-467) -> Matrix(5,2,-8,-3) Matrix(37,252,16,109) -> Matrix(1,-2,0,1) Matrix(73,468,-56,-359) -> Matrix(1,0,-4,1) Matrix(37,216,68,397) -> Matrix(1,0,0,1) Matrix(107,612,132,755) -> Matrix(3,-4,4,-5) Matrix(71,396,116,647) -> Matrix(1,2,0,1) Matrix(37,198,-20,-107) -> Matrix(1,0,-4,1) Matrix(35,162,-8,-37) -> Matrix(1,-2,0,1) Matrix(181,774,76,325) -> Matrix(1,0,0,1) Matrix(73,306,-120,-503) -> Matrix(1,0,0,1) Matrix(37,144,28,109) -> Matrix(1,0,4,1) Matrix(325,1224,-124,-467) -> Matrix(1,0,0,1) Matrix(179,666,-68,-253) -> Matrix(1,0,0,1) Matrix(109,396,-188,-683) -> Matrix(13,2,-20,-3) Matrix(71,252,20,71) -> Matrix(1,0,-4,1) Matrix(37,126,32,109) -> Matrix(1,-2,0,1) Matrix(287,972,-468,-1585) -> Matrix(1,-6,0,1) Matrix(145,486,-236,-791) -> Matrix(1,2,0,1) Matrix(71,234,-44,-145) -> Matrix(1,2,0,1) Matrix(71,198,-52,-145) -> Matrix(3,-2,-4,3) Matrix(145,396,264,721) -> Matrix(1,-2,4,-7) Matrix(397,1080,-304,-827) -> Matrix(1,-2,-4,9) Matrix(359,972,-612,-1657) -> Matrix(3,-14,-4,19) Matrix(181,486,-308,-827) -> Matrix(3,4,-4,-5) Matrix(109,288,-204,-539) -> Matrix(1,0,0,1) Matrix(1403,3672,-1032,-2701) -> Matrix(5,2,-8,-3) Matrix(145,378,28,73) -> Matrix(1,0,0,1) Matrix(181,468,-292,-755) -> Matrix(1,0,4,1) Matrix(71,180,28,71) -> Matrix(1,0,-4,1) Matrix(37,90,-44,-107) -> Matrix(1,-2,0,1) Matrix(181,432,-256,-611) -> Matrix(1,-2,0,1) Matrix(145,342,92,217) -> Matrix(1,-2,0,1) Matrix(71,162,-32,-73) -> Matrix(1,-4,0,1) Matrix(107,234,16,35) -> Matrix(1,2,0,1) Matrix(109,234,-184,-395) -> Matrix(3,4,-4,-5) Matrix(251,468,96,179) -> Matrix(1,0,4,1) Matrix(109,198,60,109) -> Matrix(1,0,12,1) Matrix(71,126,40,71) -> Matrix(1,0,-8,1) Matrix(145,252,-248,-431) -> Matrix(11,-2,-16,3) Matrix(251,432,104,179) -> Matrix(1,0,-4,1) Matrix(253,432,-400,-683) -> Matrix(5,-2,-12,5) Matrix(361,612,128,217) -> Matrix(5,-2,-12,5) Matrix(181,306,320,541) -> Matrix(17,-8,32,-15) Matrix(109,180,-152,-251) -> Matrix(3,-2,-4,3) Matrix(287,468,176,287) -> Matrix(5,-6,-4,5) Matrix(611,990,-956,-1549) -> Matrix(7,-10,-16,23) Matrix(757,1224,-556,-899) -> Matrix(5,-8,-8,13) Matrix(35,54,-24,-37) -> Matrix(1,-2,0,1) Matrix(935,1332,252,359) -> Matrix(1,2,0,1) Matrix(1367,1944,860,1223) -> Matrix(3,4,-4,-5) Matrix(685,972,432,613) -> Matrix(1,0,0,1) Matrix(179,252,76,107) -> Matrix(1,0,0,1) Matrix(181,252,-232,-323) -> Matrix(3,4,-4,-5) Matrix(287,396,208,287) -> Matrix(7,6,8,7) Matrix(3815,5184,-6624,-9001) -> Matrix(1,0,0,1) Matrix(3817,5184,1008,1369) -> Matrix(7,4,12,7) Matrix(359,486,212,287) -> Matrix(7,4,-16,-9) Matrix(109,144,28,37) -> Matrix(1,0,4,1) Matrix(181,234,140,181) -> Matrix(1,0,12,1) Matrix(71,90,56,71) -> Matrix(1,0,-4,1) Matrix(73,90,-116,-143) -> Matrix(1,0,-4,1) Matrix(107,126,152,179) -> Matrix(1,0,0,1) Matrix(109,126,32,37) -> Matrix(1,-2,0,1) Matrix(145,126,84,73) -> Matrix(1,2,-4,-7) Matrix(109,90,132,109) -> Matrix(5,6,4,5) Matrix(287,234,352,287) -> Matrix(11,10,12,11) Matrix(755,612,132,107) -> Matrix(5,4,-4,-3) Matrix(179,144,312,251) -> Matrix(7,4,12,7) Matrix(251,198,-412,-325) -> Matrix(1,0,0,1) Matrix(827,648,596,467) -> Matrix(9,10,8,9) Matrix(71,54,-96,-73) -> Matrix(1,0,0,1) Matrix(1009,738,592,433) -> Matrix(3,2,-8,-5) Matrix(1331,972,1724,1259) -> Matrix(1,0,4,1) Matrix(395,288,48,35) -> Matrix(1,0,0,1) Matrix(323,234,-548,-397) -> Matrix(3,4,-4,-5) Matrix(899,648,548,395) -> Matrix(7,8,-8,-9) Matrix(253,180,52,37) -> Matrix(1,0,0,1) Matrix(179,126,152,107) -> Matrix(1,0,0,1) Matrix(181,126,260,181) -> Matrix(5,6,4,5) Matrix(287,198,416,287) -> Matrix(15,14,16,15) Matrix(289,198,-524,-359) -> Matrix(7,6,-20,-17) Matrix(251,162,-392,-253) -> Matrix(11,6,-24,-13) Matrix(395,252,732,467) -> Matrix(5,2,12,5) Matrix(397,252,512,325) -> Matrix(1,0,4,1) Matrix(1043,648,404,251) -> Matrix(1,0,8,1) Matrix(757,468,1108,685) -> Matrix(3,-4,4,-5) Matrix(1981,1224,-3436,-2123) -> Matrix(5,-8,-8,13) Matrix(1079,666,-1868,-1153) -> Matrix(5,2,-8,-3) Matrix(647,396,116,71) -> Matrix(1,2,0,1) Matrix(179,108,-300,-181) -> Matrix(3,4,-4,-5) Matrix(2519,1494,-4372,-2593) -> Matrix(17,14,-28,-23) Matrix(1115,648,308,179) -> Matrix(3,2,28,19) Matrix(10655,6138,2248,1295) -> Matrix(3,2,-8,-5) Matrix(251,144,312,179) -> Matrix(7,4,12,7) Matrix(287,162,-512,-289) -> Matrix(23,12,-48,-25) Matrix(323,180,192,107) -> Matrix(9,4,-16,-7) Matrix(683,378,1104,611) -> Matrix(5,2,-8,-3) Matrix(721,396,264,145) -> Matrix(7,2,-4,-1) Matrix(397,216,68,37) -> Matrix(1,0,0,1) Matrix(865,468,268,145) -> Matrix(3,2,4,3) Matrix(541,288,340,181) -> Matrix(1,0,0,1) Matrix(1,0,4,1) -> Matrix(1,0,4,1) Matrix(503,-270,136,-73) -> Matrix(1,0,0,1) Matrix(359,-198,524,-289) -> Matrix(17,-6,20,-7) Matrix(289,-162,512,-287) -> Matrix(25,-12,48,-23) Matrix(2123,-1224,3436,-1981) -> Matrix(13,-8,-8,5) Matrix(467,-270,64,-37) -> Matrix(3,-2,8,-5) Matrix(683,-396,188,-109) -> Matrix(3,-2,20,-13) Matrix(431,-252,248,-145) -> Matrix(3,-2,-16,11) Matrix(397,-234,548,-323) -> Matrix(5,-4,4,-3) Matrix(325,-198,412,-251) -> Matrix(1,0,0,1) Matrix(1585,-972,468,-287) -> Matrix(1,-6,0,1) Matrix(467,-288,60,-37) -> Matrix(1,0,0,1) Matrix(8533,-5274,1804,-1115) -> Matrix(1,0,0,1) Matrix(755,-468,292,-181) -> Matrix(1,0,4,1) Matrix(143,-90,116,-73) -> Matrix(1,0,-4,1) Matrix(253,-162,392,-251) -> Matrix(13,-6,24,-11) Matrix(359,-234,112,-73) -> Matrix(3,-2,8,-5) Matrix(611,-432,256,-181) -> Matrix(1,-2,0,1) Matrix(251,-180,152,-109) -> Matrix(3,-2,-4,3) Matrix(1369,-990,596,-431) -> Matrix(1,-4,0,1) Matrix(73,-54,96,-71) -> Matrix(1,0,0,1) Matrix(323,-252,232,-181) -> Matrix(5,-4,4,-3) Matrix(431,-342,92,-73) -> Matrix(1,-2,0,1) Matrix(107,-90,44,-37) -> Matrix(1,-2,0,1) Matrix(253,-306,148,-179) -> Matrix(3,2,-8,-5) Matrix(359,-468,56,-73) -> Matrix(1,0,-4,1) Matrix(145,-198,52,-71) -> Matrix(3,-2,-4,3) Matrix(37,-54,24,-35) -> Matrix(1,-2,0,1) Matrix(145,-234,44,-71) -> Matrix(1,2,0,1) Matrix(791,-1368,292,-505) -> Matrix(15,4,-4,-1) Matrix(107,-198,20,-37) -> Matrix(1,0,-4,1) Matrix(73,-162,32,-71) -> Matrix(1,-4,0,1) Matrix(1115,-2556,236,-541) -> Matrix(1,2,0,1) Matrix(467,-1224,124,-325) -> Matrix(1,0,0,1) Matrix(253,-666,68,-179) -> Matrix(1,0,0,1) Matrix(541,-1458,200,-539) -> Matrix(1,-6,0,1) Matrix(251,-684,40,-109) -> Matrix(1,2,0,1) Matrix(37,-108,12,-35) -> Matrix(1,0,4,1) Matrix(469,-1494,124,-395) -> Matrix(1,0,0,1) Matrix(433,-1458,128,-431) -> Matrix(1,-8,0,1) Matrix(1799,-6822,380,-1441) -> Matrix(1,-2,0,1) Matrix(37,-162,8,-35) -> Matrix(1,-2,0,1) 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): 216 Minimal number of generators: 37 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: 24 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/1 -3/2 -3/4 0/1 1/2 9/16 9/14 3/4 1/1 5/4 9/7 3/2 7/4 9/5 2/1 9/4 5/2 27/10 3/1 27/8 7/2 4/1 9/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/2 1/0 -7/2 1/2 -10/3 -3/2 1/0 -3/1 0/1 -11/4 1/1 1/0 -8/3 -1/2 1/0 -5/2 1/2 -2/1 -1/2 1/0 -9/5 0/1 -7/4 0/1 1/4 -5/3 1/2 3/4 -3/2 1/0 -4/3 -1/2 -1/4 -9/7 0/1 -5/4 0/1 1/2 -1/1 -1/2 1/0 -4/5 -1/2 1/0 -3/4 -1/1 0/1 -5/7 -1/2 1/0 -2/3 -3/4 -1/2 -5/8 -1/2 0/1 -3/5 -1/1 -4/7 -7/12 -1/2 -1/2 -1/2 0/1 0/1 1/2 1/2 5/9 5/12 1/2 9/16 1/2 4/7 1/2 7/12 7/12 2/3 3/4 3/5 1/1 8/13 -1/2 1/0 5/8 0/1 1/2 7/11 3/8 1/2 9/14 1/2 2/3 1/2 3/4 7/10 3/2 5/7 1/2 1/0 3/4 0/1 1/1 4/5 1/2 1/0 5/6 3/2 1/1 1/2 1/0 6/5 -1/1 5/4 -1/2 0/1 9/7 0/1 13/10 1/6 4/3 1/4 1/2 3/2 1/0 5/3 -3/4 -1/2 12/7 -1/3 7/4 -1/4 0/1 9/5 0/1 11/6 1/6 2/1 1/2 1/0 9/4 1/0 7/3 -3/2 1/0 5/2 -1/2 18/7 0/1 13/5 1/2 1/0 8/3 1/2 1/0 27/10 1/0 19/7 -5/2 1/0 11/4 -1/1 1/0 3/1 0/1 10/3 3/2 1/0 27/8 1/0 17/5 -5/2 1/0 7/2 -1/2 18/5 0/1 11/3 1/4 1/2 4/1 1/2 1/0 9/2 1/0 5/1 -1/2 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,9,0,1) (-4/1,1/0) -> (5/1,1/0) Parabolic Matrix(17,63,24,89) (-4/1,-7/2) -> (7/10,5/7) Hyperbolic Matrix(55,189,16,55) (-7/2,-10/3) -> (17/5,7/2) Hyperbolic Matrix(17,54,28,89) (-10/3,-3/1) -> (3/5,8/13) Hyperbolic Matrix(19,54,32,91) (-3/1,-11/4) -> (7/12,3/5) Hyperbolic Matrix(109,297,40,109) (-11/4,-8/3) -> (19/7,11/4) Hyperbolic Matrix(17,45,20,53) (-8/3,-5/2) -> (5/6,1/1) Hyperbolic Matrix(19,45,8,19) (-5/2,-2/1) -> (7/3,5/2) Hyperbolic Matrix(73,135,20,37) (-2/1,-9/5) -> (18/5,11/3) Hyperbolic Matrix(71,126,40,71) (-9/5,-7/4) -> (7/4,9/5) Hyperbolic Matrix(37,63,64,109) (-7/4,-5/3) -> (4/7,7/12) Hyperbolic Matrix(17,27,-12,-19) (-5/3,-3/2) -> (-3/2,-4/3) Parabolic Matrix(145,189,56,73) (-4/3,-9/7) -> (18/7,13/5) Hyperbolic Matrix(71,90,56,71) (-9/7,-5/4) -> (5/4,9/7) Hyperbolic Matrix(37,45,60,73) (-5/4,-1/1) -> (8/13,5/8) Hyperbolic Matrix(53,45,20,17) (-1/1,-4/5) -> (13/5,8/3) Hyperbolic Matrix(35,27,-48,-37) (-4/5,-3/4) -> (-3/4,-5/7) Parabolic Matrix(89,63,24,17) (-5/7,-2/3) -> (11/3,4/1) Hyperbolic Matrix(71,45,112,71) (-2/3,-5/8) -> (5/8,7/11) Hyperbolic Matrix(73,45,60,37) (-5/8,-3/5) -> (6/5,5/4) Hyperbolic Matrix(109,63,64,37) (-3/5,-4/7) -> (5/3,12/7) Hyperbolic Matrix(17,9,32,17) (-4/7,-1/2) -> (1/2,5/9) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(145,-81,256,-143) (5/9,9/16) -> (9/16,4/7) Parabolic Matrix(127,-81,196,-125) (7/11,9/14) -> (9/14,2/3) Parabolic Matrix(53,-36,28,-19) (2/3,7/10) -> (11/6,2/1) Hyperbolic Matrix(37,-27,48,-35) (5/7,3/4) -> (3/4,4/5) Parabolic Matrix(89,-72,68,-55) (4/5,5/6) -> (13/10,4/3) Hyperbolic Matrix(53,-63,16,-19) (1/1,6/5) -> (3/1,10/3) Hyperbolic Matrix(215,-279,84,-109) (9/7,13/10) -> (5/2,18/7) Hyperbolic Matrix(19,-27,12,-17) (4/3,3/2) -> (3/2,5/3) Parabolic Matrix(89,-153,32,-55) (12/7,7/4) -> (11/4,3/1) Hyperbolic Matrix(143,-261,40,-73) (9/5,11/6) -> (7/2,18/5) Hyperbolic Matrix(37,-81,16,-35) (2/1,9/4) -> (9/4,7/3) Parabolic Matrix(271,-729,100,-269) (8/3,27/10) -> (27/10,19/7) Parabolic Matrix(217,-729,64,-215) (10/3,27/8) -> (27/8,17/5) Parabolic Matrix(19,-81,4,-17) (4/1,9/2) -> (9/2,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,9,0,1) -> Matrix(1,0,0,1) Matrix(17,63,24,89) -> Matrix(1,1,0,1) Matrix(55,189,16,55) -> Matrix(1,-1,0,1) Matrix(17,54,28,89) -> Matrix(1,1,0,1) Matrix(19,54,32,91) -> Matrix(3,-1,4,-1) Matrix(109,297,40,109) -> Matrix(1,-2,0,1) Matrix(17,45,20,53) -> Matrix(1,1,0,1) Matrix(19,45,8,19) -> Matrix(1,-1,0,1) Matrix(73,135,20,37) -> Matrix(1,0,4,1) Matrix(71,126,40,71) -> Matrix(1,0,-8,1) Matrix(37,63,64,109) -> Matrix(5,-2,8,-3) Matrix(17,27,-12,-19) -> Matrix(1,-1,0,1) Matrix(145,189,56,73) -> Matrix(1,0,4,1) Matrix(71,90,56,71) -> Matrix(1,0,-4,1) Matrix(37,45,60,73) -> Matrix(1,0,0,1) Matrix(53,45,20,17) -> Matrix(1,1,0,1) Matrix(35,27,-48,-37) -> Matrix(1,0,0,1) Matrix(89,63,24,17) -> Matrix(1,1,0,1) Matrix(71,45,112,71) -> Matrix(1,0,4,1) Matrix(73,45,60,37) -> Matrix(1,0,0,1) Matrix(109,63,64,37) -> Matrix(3,2,-8,-5) Matrix(17,9,32,17) -> Matrix(1,1,0,1) Matrix(1,0,4,1) -> Matrix(1,0,4,1) Matrix(145,-81,256,-143) -> Matrix(13,-6,24,-11) Matrix(127,-81,196,-125) -> Matrix(7,-3,12,-5) Matrix(53,-36,28,-19) -> Matrix(1,-1,4,-3) Matrix(37,-27,48,-35) -> Matrix(1,0,0,1) Matrix(89,-72,68,-55) -> Matrix(1,-1,4,-3) Matrix(53,-63,16,-19) -> Matrix(1,1,0,1) Matrix(215,-279,84,-109) -> Matrix(1,0,-8,1) Matrix(19,-27,12,-17) -> Matrix(1,-1,0,1) Matrix(89,-153,32,-55) -> Matrix(3,1,-4,-1) Matrix(143,-261,40,-73) -> Matrix(1,0,-8,1) Matrix(37,-81,16,-35) -> Matrix(1,-2,0,1) Matrix(271,-729,100,-269) -> Matrix(1,-3,0,1) Matrix(217,-729,64,-215) -> Matrix(1,-4,0,1) Matrix(19,-81,4,-17) -> Matrix(1,-1,0,1) 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 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): 12 ----------------------------------------------------------------------- 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 0/1 2 2 1/2 1/2 2 18 9/16 1/2 12 2 4/7 (1/2,7/12) 0 18 3/5 1/1 2 6 5/8 (0/1,1/2) 0 18 9/14 1/2 6 2 2/3 (1/2,3/4) 0 18 5/7 (1/2,1/0) 0 18 3/4 0 6 4/5 (1/2,1/0) 0 18 1/1 (1/2,1/0) 0 18 6/5 -1/1 2 6 5/4 (-1/2,0/1) 0 18 9/7 0/1 8 2 4/3 (1/4,1/2) 0 18 3/2 1/0 2 6 5/3 (-3/4,-1/2) 0 18 12/7 -1/3 2 6 7/4 (-1/4,0/1) 0 18 9/5 0/1 10 2 2/1 (1/2,1/0) 0 18 9/4 1/0 4 2 5/2 -1/2 2 18 18/7 0/1 8 2 13/5 (1/2,1/0) 0 18 8/3 (1/2,1/0) 0 18 27/10 1/0 6 2 11/4 (-1/1,1/0) 0 18 3/1 0/1 2 6 10/3 (3/2,1/0) 0 18 27/8 1/0 8 2 7/2 -1/2 2 18 18/5 0/1 10 2 11/3 (1/4,1/2) 0 18 4/1 (1/2,1/0) 0 18 9/2 1/0 2 2 1/0 (0/1,1/0) 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(17,-9,32,-17) (1/2,9/16) -> (1/2,9/16) Reflection Matrix(127,-72,224,-127) (9/16,4/7) -> (9/16,4/7) Reflection Matrix(109,-63,64,-37) (4/7,3/5) -> (5/3,12/7) Glide Reflection Matrix(73,-45,60,-37) (3/5,5/8) -> (6/5,5/4) Glide Reflection Matrix(71,-45,112,-71) (5/8,9/14) -> (5/8,9/14) Reflection Matrix(55,-36,84,-55) (9/14,2/3) -> (9/14,2/3) Reflection Matrix(89,-63,24,-17) (2/3,5/7) -> (11/3,4/1) Glide Reflection Matrix(37,-27,48,-35) (5/7,3/4) -> (3/4,4/5) Parabolic Matrix(53,-45,20,-17) (4/5,1/1) -> (13/5,8/3) Glide Reflection Matrix(53,-63,16,-19) (1/1,6/5) -> (3/1,10/3) Hyperbolic Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(145,-189,56,-73) (9/7,4/3) -> (18/7,13/5) Glide Reflection Matrix(19,-27,12,-17) (4/3,3/2) -> (3/2,5/3) Parabolic Matrix(89,-153,32,-55) (12/7,7/4) -> (11/4,3/1) Hyperbolic Matrix(71,-126,40,-71) (7/4,9/5) -> (7/4,9/5) Reflection Matrix(73,-135,20,-37) (9/5,2/1) -> (18/5,11/3) Glide Reflection Matrix(17,-36,8,-17) (2/1,9/4) -> (2/1,9/4) Reflection Matrix(19,-45,8,-19) (9/4,5/2) -> (9/4,5/2) Reflection Matrix(71,-180,28,-71) (5/2,18/7) -> (5/2,18/7) Reflection Matrix(161,-432,60,-161) (8/3,27/10) -> (8/3,27/10) Reflection Matrix(109,-297,40,-109) (27/10,11/4) -> (27/10,11/4) Reflection Matrix(161,-540,48,-161) (10/3,27/8) -> (10/3,27/8) Reflection Matrix(55,-189,16,-55) (27/8,7/2) -> (27/8,7/2) Reflection Matrix(71,-252,20,-71) (7/2,18/5) -> (7/2,18/5) Reflection Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(-1,9,0,1) (9/2,1/0) -> (9/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,4,-1) -> Matrix(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Matrix(17,-9,32,-17) -> Matrix(-1,1,0,1) (1/2,9/16) -> (1/2,1/0) Matrix(127,-72,224,-127) -> Matrix(13,-7,24,-13) (9/16,4/7) -> (1/2,7/12) Matrix(109,-63,64,-37) -> Matrix(3,-2,-8,5) Matrix(73,-45,60,-37) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(71,-45,112,-71) -> Matrix(1,0,4,-1) (5/8,9/14) -> (0/1,1/2) Matrix(55,-36,84,-55) -> Matrix(5,-3,8,-5) (9/14,2/3) -> (1/2,3/4) Matrix(89,-63,24,-17) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(37,-27,48,-35) -> Matrix(1,0,0,1) Matrix(53,-45,20,-17) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(53,-63,16,-19) -> Matrix(1,1,0,1) 1/0 Matrix(71,-90,56,-71) -> Matrix(-1,0,4,1) (5/4,9/7) -> (-1/2,0/1) Matrix(145,-189,56,-73) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(19,-27,12,-17) -> Matrix(1,-1,0,1) 1/0 Matrix(89,-153,32,-55) -> Matrix(3,1,-4,-1) -1/2 Matrix(71,-126,40,-71) -> Matrix(-1,0,8,1) (7/4,9/5) -> (-1/4,0/1) Matrix(73,-135,20,-37) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(17,-36,8,-17) -> Matrix(-1,1,0,1) (2/1,9/4) -> (1/2,1/0) Matrix(19,-45,8,-19) -> Matrix(1,1,0,-1) (9/4,5/2) -> (-1/2,1/0) Matrix(71,-180,28,-71) -> Matrix(-1,0,4,1) (5/2,18/7) -> (-1/2,0/1) Matrix(161,-432,60,-161) -> Matrix(-1,1,0,1) (8/3,27/10) -> (1/2,1/0) Matrix(109,-297,40,-109) -> Matrix(1,2,0,-1) (27/10,11/4) -> (-1/1,1/0) Matrix(161,-540,48,-161) -> Matrix(-1,3,0,1) (10/3,27/8) -> (3/2,1/0) Matrix(55,-189,16,-55) -> Matrix(1,1,0,-1) (27/8,7/2) -> (-1/2,1/0) Matrix(71,-252,20,-71) -> Matrix(-1,0,4,1) (7/2,18/5) -> (-1/2,0/1) Matrix(17,-72,4,-17) -> Matrix(-1,1,0,1) (4/1,9/2) -> (1/2,1/0) Matrix(-1,9,0,1) -> Matrix(1,0,0,-1) (9/2,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.