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): 648 Minimal number of generators: 109 Number of equivalence classes of cusps: 54 Genus: 28 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -9/1 -6/1 -9/2 -21/5 -15/4 -18/5 -3/1 -21/8 -18/7 -12/5 -9/4 -15/7 -9/5 -3/2 -15/11 -9/7 -9/8 0/1 1/1 9/8 27/22 9/7 3/2 27/17 27/16 9/5 2/1 9/4 12/5 27/11 5/2 18/7 144/55 21/8 108/41 3/1 27/8 24/7 7/2 18/5 108/29 15/4 72/19 4/1 9/2 33/7 24/5 5/1 27/5 6/1 7/1 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -9/1 -1/2 1/0 -8/1 -1/2 -1/3 -7/1 -1/4 -6/1 -1/1 0/1 -11/2 -1/5 0/1 -27/5 0/1 -16/3 0/1 1/7 -5/1 1/0 -9/2 0/1 -13/3 1/2 -17/4 0/1 1/1 -21/5 1/2 -4/1 1/1 1/0 -19/5 1/0 -34/9 0/1 1/1 -15/4 0/1 1/0 -41/11 1/0 -26/7 1/1 1/0 -11/3 1/2 -18/5 1/1 -7/2 2/1 3/1 -3/1 1/0 -11/4 0/1 1/1 -30/11 3/1 1/0 -19/7 1/0 -27/10 1/0 -8/3 -7/1 1/0 -29/11 -11/2 -50/19 -5/1 -9/2 -21/8 -5/1 -4/1 -55/21 -9/2 -34/13 -21/5 -4/1 -13/5 -7/2 -18/7 -3/1 -5/2 -2/1 1/0 -27/11 1/0 -22/9 -5/1 1/0 -17/7 1/0 -12/5 -3/1 1/0 -19/8 -3/1 -2/1 -45/19 -5/2 1/0 -26/11 -1/1 1/0 -7/3 -5/2 -9/4 -2/1 -11/5 1/0 -13/6 -2/1 1/0 -15/7 1/0 -2/1 -2/1 -1/1 -9/5 -3/2 1/0 -16/9 -2/1 -1/1 -39/22 -2/1 -3/2 -23/13 -3/2 -30/17 -5/3 -3/2 -7/4 -4/3 -1/1 -26/15 -1/1 1/0 -45/26 -2/1 -19/11 -3/2 -12/7 -3/2 -1/1 -17/10 -4/3 -1/1 -22/13 -7/6 -1/1 -27/16 -1/1 -5/3 1/0 -18/11 -1/1 -13/8 -2/1 1/0 -34/21 -2/1 -9/5 -89/55 -7/4 -144/89 -5/3 -55/34 -2/1 -5/3 -21/13 -3/2 -50/31 -5/4 -1/1 -79/49 -9/8 -108/67 -1/1 -29/18 -1/1 0/1 -8/5 -3/2 -1/1 -3/2 -2/1 -1/1 -10/7 -3/2 -1/1 -27/19 -1/1 -17/12 -1/1 0/1 -41/29 1/0 -24/17 -1/1 1/0 -31/22 -2/1 1/0 -7/5 1/0 -18/13 -1/1 -11/8 -3/1 -2/1 -26/19 -5/3 -3/2 -67/49 -3/2 -108/79 -5/3 -1/1 -41/30 -2/1 -3/2 -15/11 -3/2 -34/25 -7/5 -4/3 -53/39 -5/4 -72/53 -1/1 -19/14 -4/3 -1/1 -4/3 -1/1 1/0 -9/7 -3/2 1/0 -14/11 -1/1 1/0 -33/26 -1/1 0/1 -19/15 1/0 -24/19 -3/1 1/0 -5/4 -2/1 1/0 -16/13 -11/5 -2/1 -27/22 -2/1 -11/9 -7/4 -6/5 -2/1 -1/1 -7/6 -2/1 -9/5 -8/7 -3/2 -1/1 -9/8 -2/1 -1/1 -3/2 0/1 -1/1 1/1 -3/4 9/8 -2/3 8/7 -1/1 -3/4 7/6 -9/13 -2/3 6/5 -1/1 -2/3 11/9 -7/10 27/22 -2/3 16/13 -2/3 -11/17 5/4 -2/3 -1/2 9/7 -3/4 -1/2 13/10 -2/3 -1/2 17/13 -1/2 21/16 -1/1 0/1 4/3 -1/1 -1/2 19/14 -1/1 -4/5 34/25 -4/5 -7/9 15/11 -3/4 41/30 -3/4 -2/3 26/19 -3/4 -5/7 11/8 -2/3 -3/5 18/13 -1/1 7/5 -1/2 3/2 -1/1 -2/3 11/7 -1/2 30/19 -1/1 -1/2 19/12 -1/1 0/1 27/17 -1/1 8/5 -1/1 -3/4 29/18 -1/1 0/1 50/31 -1/1 -5/6 21/13 -3/4 55/34 -5/7 -2/3 34/21 -9/13 -2/3 13/8 -2/3 -1/2 18/11 -1/1 5/3 -1/2 27/16 -1/1 22/13 -1/1 -7/8 17/10 -1/1 -4/5 12/7 -1/1 -3/4 19/11 -3/4 45/26 -2/3 26/15 -1/1 -1/2 7/4 -1/1 -4/5 9/5 -3/4 -1/2 11/6 -1/1 -4/5 13/7 -3/4 15/8 -3/4 -2/3 2/1 -1/1 -2/3 9/4 -2/3 16/7 -2/3 -3/5 39/17 -5/8 23/10 -2/3 -5/8 30/13 -5/8 -3/5 7/3 -5/8 26/11 -1/1 -1/2 45/19 -5/8 -1/2 19/8 -2/3 -3/5 12/5 -3/5 -1/2 17/7 -1/2 22/9 -5/9 -1/2 27/11 -1/2 5/2 -2/3 -1/2 18/7 -3/5 13/5 -7/12 34/13 -4/7 -21/37 89/34 -4/7 -13/23 144/55 -13/23 55/21 -9/16 21/8 -4/7 -5/9 50/19 -9/16 -5/9 79/30 -34/61 -5/9 108/41 -5/9 29/11 -11/20 8/3 -7/13 -1/2 3/1 -1/2 10/3 -5/9 -1/2 27/8 -1/2 17/5 -1/2 41/12 -1/2 -6/13 24/7 -1/2 -5/11 31/9 -1/2 7/2 -3/7 -2/5 18/5 -1/3 11/3 -1/4 26/7 -1/2 -1/3 67/18 -1/2 0/1 108/29 -1/1 -1/3 41/11 -1/2 15/4 -1/2 0/1 34/9 -1/3 0/1 53/14 -1/1 0/1 72/19 -1/1 19/5 -1/2 4/1 -1/2 -1/3 9/2 0/1 14/3 -1/1 1/0 33/7 -1/2 19/4 -1/1 0/1 24/5 -1/1 -1/2 5/1 -1/2 16/3 -1/9 0/1 27/5 0/1 11/2 0/1 1/3 6/1 -1/1 0/1 7/1 1/2 8/1 1/1 1/0 9/1 -1/2 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(19,216,8,91) (-9/1,1/0) -> (45/19,19/8) Hyperbolic Matrix(71,594,30,251) (-9/1,-8/1) -> (26/11,45/19) Hyperbolic Matrix(37,270,10,73) (-8/1,-7/1) -> (11/3,26/7) 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(109,594,20,109) (-11/2,-27/5) -> (27/5,11/2) Hyperbolic Matrix(161,864,30,161) (-27/5,-16/3) -> (16/3,27/5) Hyperbolic Matrix(73,378,28,145) (-16/3,-5/1) -> (13/5,34/13) Hyperbolic Matrix(35,162,-8,-37) (-5/1,-9/2) -> (-9/2,-13/3) Parabolic Matrix(215,918,-152,-649) (-13/3,-17/4) -> (-17/12,-41/29) Hyperbolic Matrix(359,1512,222,935) (-17/4,-21/5) -> (21/13,55/34) Hyperbolic Matrix(271,1134,168,703) (-21/5,-4/1) -> (50/31,21/13) Hyperbolic Matrix(127,486,52,199) (-4/1,-19/5) -> (17/7,22/9) Hyperbolic Matrix(971,3672,-600,-2269) (-19/5,-34/9) -> (-34/21,-89/55) Hyperbolic Matrix(143,540,76,287) (-34/9,-15/4) -> (15/8,2/1) Hyperbolic Matrix(217,810,116,433) (-15/4,-41/11) -> (13/7,15/8) Hyperbolic Matrix(755,2808,-552,-2053) (-41/11,-26/7) -> (-26/19,-67/49) Hyperbolic Matrix(73,270,10,37) (-26/7,-11/3) -> (7/1,8/1) Hyperbolic Matrix(89,324,64,233) (-11/3,-18/5) -> (18/13,7/5) Hyperbolic Matrix(91,324,66,235) (-18/5,-7/2) -> (11/8,18/13) Hyperbolic Matrix(17,54,-6,-19) (-7/2,-3/1) -> (-3/1,-11/4) Parabolic Matrix(197,540,-112,-307) (-11/4,-30/11) -> (-30/17,-7/4) Hyperbolic Matrix(377,1026,-298,-811) (-30/11,-19/7) -> (-19/15,-24/19) Hyperbolic Matrix(359,972,106,287) (-19/7,-27/10) -> (27/8,17/5) Hyperbolic Matrix(181,486,54,145) (-27/10,-8/3) -> (10/3,27/8) Hyperbolic Matrix(307,810,130,343) (-8/3,-29/11) -> (7/3,26/11) Hyperbolic Matrix(2051,5400,-1272,-3349) (-29/11,-50/19) -> (-50/31,-79/49) Hyperbolic Matrix(431,1134,328,863) (-50/19,-21/8) -> (21/16,4/3) Hyperbolic Matrix(577,1512,440,1153) (-21/8,-55/21) -> (17/13,21/16) 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(125,324,76,197) (-13/5,-18/7) -> (18/11,5/3) Hyperbolic Matrix(127,324,78,199) (-18/7,-5/2) -> (13/8,18/11) Hyperbolic Matrix(109,270,44,109) (-5/2,-27/11) -> (27/11,5/2) Hyperbolic Matrix(485,1188,198,485) (-27/11,-22/9) -> (22/9,27/11) Hyperbolic Matrix(199,486,52,127) (-22/9,-17/7) -> (19/5,4/1) 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(91,216,8,19) (-19/8,-45/19) -> (9/1,1/0) Hyperbolic Matrix(251,594,30,71) (-45/19,-26/11) -> (8/1,9/1) Hyperbolic Matrix(343,810,130,307) (-26/11,-7/3) -> (29/11,8/3) Hyperbolic Matrix(71,162,-32,-73) (-7/3,-9/4) -> (-9/4,-11/5) Parabolic Matrix(197,432,-140,-307) (-11/5,-13/6) -> (-31/22,-7/5) Hyperbolic Matrix(377,810,276,593) (-13/6,-15/7) -> (15/11,41/30) Hyperbolic Matrix(253,540,186,397) (-15/7,-2/1) -> (34/25,15/11) Hyperbolic Matrix(89,162,-50,-91) (-2/1,-9/5) -> (-9/5,-16/9) Parabolic Matrix(883,1566,234,415) (-16/9,-39/22) -> (15/4,34/9) Hyperbolic Matrix(1097,1944,294,521) (-39/22,-23/13) -> (41/11,15/4) Hyperbolic Matrix(397,702,82,145) (-23/13,-30/17) -> (24/5,5/1) Hyperbolic Matrix(467,810,290,503) (-7/4,-26/15) -> (8/5,29/18) Hyperbolic Matrix(343,594,302,523) (-26/15,-45/26) -> (9/8,8/7) Hyperbolic Matrix(125,216,114,197) (-45/26,-19/11) -> (1/1,9/8) 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(287,486,212,359) (-17/10,-22/13) -> (4/3,19/14) Hyperbolic Matrix(703,1188,416,703) (-22/13,-27/16) -> (27/16,22/13) Hyperbolic Matrix(161,270,96,161) (-27/16,-5/3) -> (5/3,27/16) 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(233,378,188,305) (-13/8,-34/21) -> (16/13,5/4) Hyperbolic Matrix(12815,20736,4894,7919) (-89/55,-144/89) -> (144/55,55/21) Hyperbolic Matrix(12817,20736,4896,7921) (-144/89,-55/34) -> (89/34,144/55) Hyperbolic Matrix(1369,2214,290,469) (-55/34,-21/13) -> (33/7,19/4) Hyperbolic Matrix(1205,1944,256,413) (-21/13,-50/31) -> (14/3,33/7) 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(503,810,290,467) (-29/18,-8/5) -> (26/15,7/4) Hyperbolic Matrix(35,54,-24,-37) (-8/5,-3/2) -> (-3/2,-10/7) Parabolic Matrix(341,486,214,305) (-10/7,-27/19) -> (27/17,8/5) Hyperbolic Matrix(685,972,432,613) (-27/19,-17/12) -> (19/12,27/17) Hyperbolic Matrix(1223,1728,356,503) (-41/29,-24/17) -> (24/7,31/9) Hyperbolic Matrix(1225,1728,358,505) (-24/17,-31/22) -> (41/12,24/7) 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(197,270,170,233) (-11/8,-26/19) -> (8/7,7/6) 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(1423,1944,620,847) (-41/30,-15/11) -> (39/17,23/10) Hyperbolic Matrix(1151,1566,502,683) (-15/11,-34/25) -> (16/7,39/17) 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(359,486,212,287) (-19/14,-4/3) -> (22/13,17/10) Hyperbolic Matrix(125,162,-98,-127) (-4/3,-9/7) -> (-9/7,-14/11) Parabolic Matrix(1531,1944,582,739) (-14/11,-33/26) -> (21/8,50/19) Hyperbolic Matrix(1745,2214,666,845) (-33/26,-19/15) -> (55/21,21/8) Hyperbolic Matrix(557,702,242,305) (-24/19,-5/4) -> (23/10,30/13) Hyperbolic Matrix(305,378,188,233) (-5/4,-16/13) -> (34/21,13/8) Hyperbolic Matrix(703,864,572,703) (-16/13,-27/22) -> (27/22,16/13) Hyperbolic Matrix(485,594,396,485) (-27/22,-11/9) -> (11/9,27/22) 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(233,270,170,197) (-7/6,-8/7) -> (26/19,11/8) Hyperbolic Matrix(523,594,302,343) (-8/7,-9/8) -> (45/26,26/15) Hyperbolic Matrix(197,216,114,125) (-9/8,-1/1) -> (19/11,45/26) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(703,-918,206,-269) (13/10,17/13) -> (17/5,41/12) Hyperbolic Matrix(2701,-3672,1032,-1403) (19/14,34/25) -> (34/13,89/34) Hyperbolic Matrix(2053,-2808,552,-755) (41/30,26/19) -> (26/7,67/18) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(343,-540,148,-233) (11/7,30/19) -> (30/13,7/3) Hyperbolic Matrix(649,-1026,136,-215) (30/19,19/12) -> (19/4,24/5) Hyperbolic Matrix(3349,-5400,1272,-2051) (29/18,50/31) -> (50/19,79/30) Hyperbolic Matrix(2269,-3672,600,-971) (55/34,34/21) -> (34/9,53/14) Hyperbolic Matrix(91,-162,50,-89) (7/4,9/5) -> (9/5,11/6) Parabolic Matrix(235,-432,68,-125) (11/6,13/7) -> (31/9,7/2) Hyperbolic Matrix(73,-162,32,-71) (2/1,9/4) -> (9/4,16/7) Parabolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(37,-162,8,-35) (4/1,9/2) -> (9/2,14/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(19,216,8,91) -> Matrix(5,2,-8,-3) Matrix(71,594,30,251) -> Matrix(5,2,-8,-3) Matrix(37,270,10,73) -> Matrix(1,0,0,1) Matrix(17,108,14,89) -> Matrix(1,2,-2,-3) Matrix(19,108,16,91) -> Matrix(1,2,-2,-3) Matrix(109,594,20,109) -> Matrix(1,0,8,1) Matrix(161,864,30,161) -> Matrix(1,0,-16,1) Matrix(73,378,28,145) -> Matrix(7,-4,-12,7) Matrix(35,162,-8,-37) -> Matrix(1,0,2,1) Matrix(215,918,-152,-649) -> Matrix(1,0,-2,1) Matrix(359,1512,222,935) -> Matrix(7,-2,-10,3) Matrix(271,1134,168,703) -> Matrix(5,-4,-6,5) Matrix(127,486,52,199) -> Matrix(1,4,-2,-7) Matrix(971,3672,-600,-2269) -> Matrix(7,2,-4,-1) Matrix(143,540,76,287) -> Matrix(3,-2,-4,3) Matrix(217,810,116,433) -> Matrix(3,-2,-4,3) Matrix(755,2808,-552,-2053) -> Matrix(3,2,-2,-1) Matrix(73,270,10,37) -> Matrix(1,0,0,1) Matrix(89,324,64,233) -> Matrix(3,-2,-4,3) Matrix(91,324,66,235) -> Matrix(1,0,-2,1) Matrix(17,54,-6,-19) -> Matrix(1,-2,0,1) Matrix(197,540,-112,-307) -> Matrix(3,-4,-2,3) Matrix(377,1026,-298,-811) -> Matrix(1,-6,0,1) Matrix(359,972,106,287) -> Matrix(1,-18,-2,37) Matrix(181,486,54,145) -> Matrix(1,12,-2,-23) Matrix(307,810,130,343) -> Matrix(1,8,-2,-15) Matrix(2051,5400,-1272,-3349) -> Matrix(7,34,-6,-29) Matrix(431,1134,328,863) -> Matrix(1,4,0,1) Matrix(577,1512,440,1153) -> Matrix(1,4,0,1) Matrix(1403,3672,-1032,-2701) -> Matrix(13,56,-10,-43) Matrix(145,378,28,73) -> Matrix(1,4,-4,-15) Matrix(125,324,76,197) -> Matrix(3,10,-4,-13) Matrix(127,324,78,199) -> Matrix(1,4,-2,-7) Matrix(109,270,44,109) -> Matrix(1,4,-2,-7) Matrix(485,1188,198,485) -> Matrix(1,10,-2,-19) Matrix(199,486,52,127) -> Matrix(1,4,-2,-7) Matrix(179,432,104,251) -> Matrix(3,10,-4,-13) Matrix(181,432,106,253) -> Matrix(3,10,-4,-13) Matrix(91,216,8,19) -> Matrix(1,2,0,1) Matrix(251,594,30,71) -> Matrix(1,2,0,1) Matrix(343,810,130,307) -> Matrix(1,8,-2,-15) Matrix(71,162,-32,-73) -> Matrix(3,8,-2,-5) Matrix(197,432,-140,-307) -> Matrix(1,0,0,1) Matrix(377,810,276,593) -> Matrix(3,4,-4,-5) Matrix(253,540,186,397) -> Matrix(3,10,-4,-13) Matrix(89,162,-50,-91) -> Matrix(1,0,0,1) Matrix(883,1566,234,415) -> Matrix(1,2,-4,-7) Matrix(1097,1944,294,521) -> Matrix(1,2,-4,-7) Matrix(397,702,82,145) -> Matrix(1,2,-4,-7) Matrix(467,810,290,503) -> Matrix(3,4,-4,-5) Matrix(343,594,302,523) -> Matrix(3,4,-4,-5) Matrix(125,216,114,197) -> Matrix(7,12,-10,-17) Matrix(251,432,104,179) -> Matrix(7,10,-12,-17) Matrix(253,432,106,181) -> Matrix(7,10,-12,-17) Matrix(287,486,212,359) -> Matrix(7,8,-8,-9) Matrix(703,1188,416,703) -> Matrix(13,14,-14,-15) Matrix(161,270,96,161) -> Matrix(1,2,-2,-3) Matrix(197,324,76,125) -> Matrix(7,10,-12,-17) Matrix(199,324,78,127) -> Matrix(1,4,-2,-7) Matrix(233,378,188,305) -> Matrix(1,4,-2,-7) Matrix(12815,20736,4894,7919) -> Matrix(79,136,-140,-241) Matrix(12817,20736,4896,7921) -> Matrix(1,6,-2,-11) Matrix(1369,2214,290,469) -> Matrix(1,2,-4,-7) Matrix(1205,1944,256,413) -> Matrix(3,4,-4,-5) Matrix(7235,11664,2746,4427) -> Matrix(51,56,-92,-101) Matrix(7237,11664,2748,4429) -> Matrix(39,34,-70,-61) Matrix(503,810,290,467) -> Matrix(3,4,-4,-5) Matrix(35,54,-24,-37) -> Matrix(1,0,0,1) Matrix(341,486,214,305) -> Matrix(5,6,-6,-7) Matrix(685,972,432,613) -> Matrix(1,0,0,1) Matrix(1223,1728,356,503) -> Matrix(1,-4,-2,9) Matrix(1225,1728,358,505) -> Matrix(1,-4,-2,9) Matrix(233,324,64,89) -> Matrix(1,2,-4,-7) Matrix(235,324,66,91) -> Matrix(1,0,-2,1) Matrix(197,270,170,233) -> Matrix(7,12,-10,-17) Matrix(8531,11664,2290,3131) -> Matrix(1,2,-4,-7) Matrix(8533,11664,2292,3133) -> Matrix(1,2,-4,-7) Matrix(1423,1944,620,847) -> Matrix(1,4,-2,-7) Matrix(1151,1566,502,683) -> Matrix(19,26,-30,-41) Matrix(3815,5184,1006,1367) -> Matrix(5,6,-6,-7) Matrix(3817,5184,1008,1369) -> Matrix(3,4,-4,-5) Matrix(359,486,212,287) -> Matrix(7,8,-8,-9) Matrix(125,162,-98,-127) -> Matrix(1,0,0,1) Matrix(1531,1944,582,739) -> Matrix(9,4,-16,-7) Matrix(1745,2214,666,845) -> Matrix(9,4,-16,-7) Matrix(557,702,242,305) -> Matrix(5,12,-8,-19) Matrix(305,378,188,233) -> Matrix(1,4,-2,-7) Matrix(703,864,572,703) -> Matrix(21,44,-32,-67) Matrix(485,594,396,485) -> Matrix(15,28,-22,-41) Matrix(89,108,14,17) -> Matrix(1,2,-2,-3) Matrix(91,108,16,19) -> Matrix(1,2,-2,-3) Matrix(233,270,170,197) -> Matrix(7,12,-10,-17) Matrix(523,594,302,343) -> Matrix(3,4,-4,-5) Matrix(197,216,114,125) -> Matrix(7,12,-10,-17) Matrix(1,0,2,1) -> Matrix(5,6,-6,-7) Matrix(127,-162,98,-125) -> Matrix(1,0,0,1) Matrix(703,-918,206,-269) -> Matrix(15,8,-32,-17) Matrix(2701,-3672,1032,-1403) -> Matrix(69,56,-122,-99) Matrix(2053,-2808,552,-755) -> Matrix(3,2,-2,-1) Matrix(37,-54,24,-35) -> Matrix(1,0,0,1) Matrix(343,-540,148,-233) -> Matrix(11,8,-18,-13) Matrix(649,-1026,136,-215) -> Matrix(1,0,0,1) Matrix(3349,-5400,1272,-2051) -> Matrix(39,34,-70,-61) Matrix(2269,-3672,600,-971) -> Matrix(3,2,4,3) Matrix(91,-162,50,-89) -> Matrix(1,0,0,1) Matrix(235,-432,68,-125) -> Matrix(13,10,-30,-23) Matrix(73,-162,32,-71) -> Matrix(11,8,-18,-13) Matrix(19,-54,6,-17) -> Matrix(3,2,-8,-5) Matrix(37,-162,8,-35) -> Matrix(1,0,2,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 27 Degree of the the map X: 27 Degree of the the map Y: 108 ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 3 1 1/1 -3/4 1 27 9/8 -2/3 3 3 8/7 (-1/1,-3/4) 0 27 7/6 (-9/13,-2/3) 0 27 6/5 0 9 11/9 -7/10 1 27 27/22 -2/3 9 1 16/13 (-2/3,-11/17) 0 27 5/4 (-2/3,-1/2) 0 27 9/7 0 3 13/10 (-2/3,-1/2) 0 27 17/13 -1/2 1 27 21/16 0 9 4/3 (-1/1,-1/2) 0 27 19/14 (-1/1,-4/5) 0 27 34/25 (-4/5,-7/9) 0 27 15/11 -3/4 3 9 41/30 (-3/4,-2/3) 0 27 26/19 (-3/4,-5/7) 0 27 11/8 (-2/3,-3/5) 0 27 18/13 -1/1 3 3 7/5 -1/2 1 27 3/2 0 9 11/7 -1/2 1 27 30/19 0 9 19/12 (-1/1,0/1) 0 27 27/17 -1/1 3 1 8/5 (-1/1,-3/4) 0 27 29/18 (-1/1,0/1) 0 27 50/31 (-1/1,-5/6) 0 27 21/13 -3/4 3 9 55/34 (-5/7,-2/3) 0 27 34/21 (-9/13,-2/3) 0 27 13/8 (-2/3,-1/2) 0 27 18/11 -1/1 3 3 5/3 -1/2 1 27 27/16 -1/1 6 1 22/13 (-1/1,-7/8) 0 27 17/10 (-1/1,-4/5) 0 27 12/7 0 9 19/11 -3/4 1 27 45/26 -2/3 3 3 26/15 (-1/1,-1/2) 0 27 7/4 (-1/1,-4/5) 0 27 9/5 0 3 11/6 (-1/1,-4/5) 0 27 13/7 -3/4 1 27 15/8 0 9 2/1 (-1/1,-2/3) 0 27 9/4 -2/3 3 3 16/7 (-2/3,-3/5) 0 27 39/17 -5/8 3 9 23/10 (-2/3,-5/8) 0 27 30/13 0 9 7/3 -5/8 1 27 26/11 (-1/1,-1/2) 0 27 45/19 0 3 19/8 (-2/3,-3/5) 0 27 12/5 0 9 17/7 -1/2 1 27 22/9 (-5/9,-1/2) 0 27 27/11 -1/2 3 1 5/2 (-2/3,-1/2) 0 27 18/7 -3/5 3 3 13/5 -7/12 1 27 34/13 (-4/7,-21/37) 0 27 89/34 (-4/7,-13/23) 0 27 144/55 -13/23 3 3 55/21 -9/16 1 27 21/8 0 9 50/19 (-9/16,-5/9) 0 27 79/30 (-34/61,-5/9) 0 27 108/41 -5/9 9 1 29/11 -11/20 1 27 8/3 (-7/13,-1/2) 0 27 3/1 -1/2 3 9 10/3 (-5/9,-1/2) 0 27 27/8 -1/2 15 1 17/5 -1/2 1 27 41/12 (-1/2,-6/13) 0 27 24/7 0 9 31/9 -1/2 1 27 7/2 (-3/7,-2/5) 0 27 18/5 -1/3 3 3 11/3 -1/4 1 27 26/7 (-1/2,-1/3) 0 27 67/18 (-1/2,0/1) 0 27 108/29 (-1/2,0/1) 0 1 41/11 -1/2 1 27 15/4 0 9 34/9 (-1/3,0/1) 0 27 53/14 (-1/1,0/1) 0 27 72/19 -1/1 3 3 19/5 -1/2 1 27 4/1 (-1/2,-1/3) 0 27 9/2 0/1 3 3 14/3 (-1/1,1/0) 0 27 33/7 -1/2 3 9 19/4 (-1/1,0/1) 0 27 24/5 0 9 5/1 -1/2 1 27 16/3 (-1/9,0/1) 0 27 27/5 0/1 12 1 11/2 (0/1,1/3) 0 27 6/1 0 9 7/1 1/2 1 27 8/1 (1/1,1/0) 0 27 9/1 0 3 1/0 (-1/1,0/1) 0 27 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(197,-216,114,-125) (1/1,9/8) -> (19/11,45/26) Glide Reflection Matrix(523,-594,302,-343) (9/8,8/7) -> (45/26,26/15) Glide Reflection Matrix(233,-270,170,-197) (8/7,7/6) -> (26/19,11/8) 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(485,-594,396,-485) (11/9,27/22) -> (11/9,27/22) Reflection Matrix(703,-864,572,-703) (27/22,16/13) -> (27/22,16/13) Reflection Matrix(305,-378,188,-233) (16/13,5/4) -> (34/21,13/8) Glide Reflection Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(703,-918,206,-269) (13/10,17/13) -> (17/5,41/12) Hyperbolic Matrix(1153,-1512,440,-577) (17/13,21/16) -> (55/21,21/8) Glide Reflection Matrix(863,-1134,328,-431) (21/16,4/3) -> (21/8,50/19) Glide Reflection Matrix(359,-486,212,-287) (4/3,19/14) -> (22/13,17/10) Glide Reflection Matrix(2701,-3672,1032,-1403) (19/14,34/25) -> (34/13,89/34) Hyperbolic Matrix(1151,-1566,502,-683) (34/25,15/11) -> (16/7,39/17) Glide Reflection Matrix(1423,-1944,620,-847) (15/11,41/30) -> (39/17,23/10) Glide Reflection Matrix(2053,-2808,552,-755) (41/30,26/19) -> (26/7,67/18) Hyperbolic Matrix(235,-324,66,-91) (11/8,18/13) -> (7/2,18/5) Glide Reflection Matrix(233,-324,64,-89) (18/13,7/5) -> (18/5,11/3) Glide Reflection Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(343,-540,148,-233) (11/7,30/19) -> (30/13,7/3) Hyperbolic Matrix(649,-1026,136,-215) (30/19,19/12) -> (19/4,24/5) Hyperbolic Matrix(647,-1026,408,-647) (19/12,27/17) -> (19/12,27/17) Reflection Matrix(271,-432,170,-271) (27/17,8/5) -> (27/17,8/5) Reflection Matrix(503,-810,290,-467) (8/5,29/18) -> (26/15,7/4) Glide Reflection Matrix(3349,-5400,1272,-2051) (29/18,50/31) -> (50/19,79/30) Hyperbolic Matrix(1205,-1944,256,-413) (50/31,21/13) -> (14/3,33/7) Glide Reflection Matrix(1369,-2214,290,-469) (21/13,55/34) -> (33/7,19/4) Glide Reflection Matrix(2269,-3672,600,-971) (55/34,34/21) -> (34/9,53/14) Hyperbolic Matrix(199,-324,78,-127) (13/8,18/11) -> (5/2,18/7) Glide Reflection Matrix(197,-324,76,-125) (18/11,5/3) -> (18/7,13/5) Glide Reflection Matrix(161,-270,96,-161) (5/3,27/16) -> (5/3,27/16) Reflection Matrix(703,-1188,416,-703) (27/16,22/13) -> (27/16,22/13) Reflection 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(91,-162,50,-89) (7/4,9/5) -> (9/5,11/6) Parabolic Matrix(235,-432,68,-125) (11/6,13/7) -> (31/9,7/2) Hyperbolic Matrix(433,-810,116,-217) (13/7,15/8) -> (41/11,15/4) Glide Reflection Matrix(287,-540,76,-143) (15/8,2/1) -> (15/4,34/9) Glide Reflection Matrix(73,-162,32,-71) (2/1,9/4) -> (9/4,16/7) Parabolic Matrix(773,-1782,226,-521) (23/10,30/13) -> (41/12,24/7) Glide Reflection Matrix(343,-810,130,-307) (7/3,26/11) -> (29/11,8/3) Glide Reflection Matrix(251,-594,30,-71) (26/11,45/19) -> (8/1,9/1) Glide Reflection Matrix(91,-216,8,-19) (45/19,19/8) -> (9/1,1/0) Glide Reflection Matrix(199,-486,52,-127) (17/7,22/9) -> (19/5,4/1) Glide Reflection Matrix(485,-1188,198,-485) (22/9,27/11) -> (22/9,27/11) Reflection Matrix(109,-270,44,-109) (27/11,5/2) -> (27/11,5/2) Reflection Matrix(145,-378,28,-73) (13/5,34/13) -> (5/1,16/3) Glide Reflection Matrix(5363,-14040,1416,-3707) (89/34,144/55) -> (53/14,72/19) Glide Reflection Matrix(2557,-6696,674,-1765) (144/55,55/21) -> (72/19,19/5) Glide Reflection Matrix(6479,-17064,2460,-6479) (79/30,108/41) -> (79/30,108/41) Reflection Matrix(2377,-6264,902,-2377) (108/41,29/11) -> (108/41,29/11) Reflection Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(161,-540,48,-161) (10/3,27/8) -> (10/3,27/8) Reflection Matrix(271,-918,80,-271) (27/8,17/5) -> (27/8,17/5) Reflection Matrix(251,-864,52,-179) (24/7,31/9) -> (24/5,5/1) Glide Reflection Matrix(73,-270,10,-37) (11/3,26/7) -> (7/1,8/1) Glide Reflection Matrix(3887,-14472,1044,-3887) (67/18,108/29) -> (67/18,108/29) Reflection Matrix(2377,-8856,638,-2377) (108/29,41/11) -> (108/29,41/11) Reflection Matrix(37,-162,8,-35) (4/1,9/2) -> (9/2,14/3) Parabolic Matrix(161,-864,30,-161) (16/3,27/5) -> (16/3,27/5) Reflection Matrix(109,-594,20,-109) (27/5,11/2) -> (27/5,11/2) 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(7,6,-8,-7) (0/1,1/1) -> (-1/1,-3/4) Matrix(197,-216,114,-125) -> Matrix(17,12,-24,-17) *** -> (-3/4,-2/3) Matrix(523,-594,302,-343) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(233,-270,170,-197) -> Matrix(17,12,-24,-17) *** -> (-3/4,-2/3) Matrix(91,-108,16,-19) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(89,-108,14,-17) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(485,-594,396,-485) -> Matrix(41,28,-60,-41) (11/9,27/22) -> (-7/10,-2/3) Matrix(703,-864,572,-703) -> Matrix(67,44,-102,-67) (27/22,16/13) -> (-2/3,-11/17) Matrix(305,-378,188,-233) -> Matrix(7,4,-12,-7) *** -> (-2/3,-1/2) Matrix(127,-162,98,-125) -> Matrix(1,0,0,1) Matrix(703,-918,206,-269) -> Matrix(15,8,-32,-17) -1/2 Matrix(1153,-1512,440,-577) -> Matrix(1,-4,-2,7) Matrix(863,-1134,328,-431) -> Matrix(1,-4,-2,7) Matrix(359,-486,212,-287) -> Matrix(9,8,-10,-9) *** -> (-1/1,-4/5) Matrix(2701,-3672,1032,-1403) -> Matrix(69,56,-122,-99) Matrix(1151,-1566,502,-683) -> Matrix(33,26,-52,-41) Matrix(1423,-1944,620,-847) -> Matrix(7,4,-12,-7) *** -> (-2/3,-1/2) Matrix(2053,-2808,552,-755) -> Matrix(3,2,-2,-1) -1/1 Matrix(235,-324,66,-91) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(233,-324,64,-89) -> Matrix(3,2,-10,-7) Matrix(37,-54,24,-35) -> Matrix(1,0,0,1) Matrix(343,-540,148,-233) -> Matrix(11,8,-18,-13) -2/3 Matrix(649,-1026,136,-215) -> Matrix(1,0,0,1) Matrix(647,-1026,408,-647) -> Matrix(-1,0,2,1) (19/12,27/17) -> (-1/1,0/1) Matrix(271,-432,170,-271) -> Matrix(7,6,-8,-7) (27/17,8/5) -> (-1/1,-3/4) Matrix(503,-810,290,-467) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(3349,-5400,1272,-2051) -> Matrix(39,34,-70,-61) Matrix(1205,-1944,256,-413) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(1369,-2214,290,-469) -> Matrix(3,2,-10,-7) Matrix(2269,-3672,600,-971) -> Matrix(3,2,4,3) Matrix(199,-324,78,-127) -> Matrix(7,4,-12,-7) *** -> (-2/3,-1/2) Matrix(197,-324,76,-125) -> Matrix(13,10,-22,-17) Matrix(161,-270,96,-161) -> Matrix(3,2,-4,-3) (5/3,27/16) -> (-1/1,-1/2) Matrix(703,-1188,416,-703) -> Matrix(15,14,-16,-15) (27/16,22/13) -> (-1/1,-7/8) Matrix(253,-432,106,-181) -> Matrix(13,10,-22,-17) Matrix(251,-432,104,-179) -> Matrix(13,10,-22,-17) Matrix(91,-162,50,-89) -> Matrix(1,0,0,1) Matrix(235,-432,68,-125) -> Matrix(13,10,-30,-23) Matrix(433,-810,116,-217) -> Matrix(3,2,-10,-7) Matrix(287,-540,76,-143) -> Matrix(3,2,-10,-7) Matrix(73,-162,32,-71) -> Matrix(11,8,-18,-13) -2/3 Matrix(773,-1782,226,-521) -> Matrix(45,28,-98,-61) Matrix(343,-810,130,-307) -> Matrix(15,8,-28,-15) *** -> (-4/7,-1/2) Matrix(251,-594,30,-71) -> Matrix(3,2,2,1) Matrix(91,-216,8,-19) -> Matrix(3,2,2,1) Matrix(199,-486,52,-127) -> Matrix(7,4,-12,-7) *** -> (-2/3,-1/2) Matrix(485,-1188,198,-485) -> Matrix(19,10,-36,-19) (22/9,27/11) -> (-5/9,-1/2) Matrix(109,-270,44,-109) -> Matrix(7,4,-12,-7) (27/11,5/2) -> (-2/3,-1/2) Matrix(145,-378,28,-73) -> Matrix(7,4,-26,-15) Matrix(5363,-14040,1416,-3707) -> Matrix(7,4,16,9) Matrix(2557,-6696,674,-1765) -> Matrix(39,22,-62,-35) Matrix(6479,-17064,2460,-6479) -> Matrix(611,340,-1098,-611) (79/30,108/41) -> (-34/61,-5/9) Matrix(2377,-6264,902,-2377) -> Matrix(199,110,-360,-199) (108/41,29/11) -> (-5/9,-11/20) Matrix(19,-54,6,-17) -> Matrix(3,2,-8,-5) -1/2 Matrix(161,-540,48,-161) -> Matrix(19,10,-36,-19) (10/3,27/8) -> (-5/9,-1/2) Matrix(271,-918,80,-271) -> Matrix(41,20,-84,-41) (27/8,17/5) -> (-1/2,-10/21) Matrix(251,-864,52,-179) -> Matrix(9,4,-20,-9) *** -> (-1/2,-2/5) Matrix(73,-270,10,-37) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(3887,-14472,1044,-3887) -> Matrix(-1,0,4,1) (67/18,108/29) -> (-1/2,0/1) Matrix(2377,-8856,638,-2377) -> Matrix(-1,0,4,1) (108/29,41/11) -> (-1/2,0/1) Matrix(37,-162,8,-35) -> Matrix(1,0,2,1) 0/1 Matrix(161,-864,30,-161) -> Matrix(-1,0,18,1) (16/3,27/5) -> (-1/9,0/1) Matrix(109,-594,20,-109) -> Matrix(1,0,6,-1) (27/5,11/2) -> (0/1,1/3) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.