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 -1/1 0/1 1/0 -8/1 0/1 1/0 -7/1 -2/1 -1/1 -6/1 -1/1 0/1 -11/2 -2/1 -1/1 -27/5 -1/1 -16/3 -1/1 -4/5 -5/1 -1/1 -1/2 -9/2 -1/1 -1/2 0/1 -13/3 -1/1 -1/2 -17/4 0/1 1/0 -21/5 -1/1 0/1 -4/1 -1/1 -1/2 -19/5 -1/2 -2/5 -34/9 -1/3 0/1 -15/4 -1/2 -1/3 -41/11 -1/1 -1/2 -26/7 -1/2 -2/5 -11/3 -1/3 0/1 -18/5 -1/2 -1/3 0/1 -7/2 -1/3 0/1 -3/1 -1/2 0/1 -11/4 -1/3 0/1 -30/11 -1/4 0/1 -19/7 -1/6 0/1 -27/10 0/1 -8/3 0/1 1/0 -29/11 -2/1 -1/1 -50/19 -1/1 -1/2 -21/8 -1/1 0/1 -55/21 0/1 1/0 -34/13 -2/1 -1/1 -13/5 -1/1 -1/2 -18/7 -1/1 -1/2 0/1 -5/2 -1/1 -1/2 -27/11 -1/2 -22/9 -1/2 -5/11 -17/7 -1/2 -2/5 -12/5 -1/2 -1/3 -19/8 -1/2 -2/5 -45/19 -1/2 -2/5 -1/3 -26/11 -1/2 -2/5 -7/3 -1/3 0/1 -9/4 -1/2 -1/3 0/1 -11/5 -1/3 0/1 -13/6 -1/1 -1/2 -15/7 -1/2 -1/3 -2/1 -1/3 0/1 -9/5 -1/3 -1/4 0/1 -16/9 -1/3 0/1 -39/22 -1/3 -1/4 -23/13 -1/4 -1/5 -30/17 -1/4 0/1 -7/4 -1/3 0/1 -26/15 -2/7 -1/4 -45/26 -1/3 -2/7 -1/4 -19/11 -2/7 -1/4 -12/7 -1/3 -1/4 -17/10 -2/7 -1/4 -22/13 -5/19 -1/4 -27/16 -1/4 -5/3 -1/4 -1/5 -18/11 -1/4 -1/5 0/1 -13/8 -1/4 -1/5 -34/21 -1/5 -2/11 -89/55 -1/6 0/1 -144/89 -1/5 -1/6 0/1 -55/34 -1/6 0/1 -21/13 -1/5 0/1 -50/31 -1/4 -1/5 -79/49 -4/19 -1/5 -108/67 -1/5 -29/18 -1/5 -2/11 -8/5 -1/6 0/1 -3/2 -1/4 0/1 -10/7 -1/6 0/1 -27/19 0/1 -17/12 0/1 1/0 -41/29 -1/1 -1/2 -24/17 -1/2 0/1 -31/22 -1/1 -1/2 -7/5 -1/3 0/1 -18/13 -1/3 -1/4 0/1 -11/8 -1/3 0/1 -26/19 -2/7 -1/4 -67/49 -5/19 -1/4 -108/79 -1/4 -41/30 -1/4 -1/5 -15/11 -1/3 -1/4 -34/25 -1/3 0/1 -53/39 -2/7 -1/4 -72/53 -1/3 -2/7 -1/4 -19/14 -2/7 -1/4 -4/3 -1/4 -1/5 -9/7 -1/4 -1/5 0/1 -14/11 -1/4 -1/5 -33/26 -1/5 0/1 -19/15 -1/6 0/1 -24/19 -1/4 0/1 -5/4 -1/4 -1/5 -16/13 -4/19 -1/5 -27/22 -1/5 -11/9 -1/5 -2/11 -6/5 -1/5 0/1 -7/6 -1/5 -2/11 -8/7 -1/6 0/1 -9/8 -1/5 -1/6 0/1 -1/1 -1/6 0/1 0/1 0/1 1/1 0/1 1/6 9/8 0/1 1/6 1/5 8/7 0/1 1/6 7/6 2/11 1/5 6/5 0/1 1/5 11/9 2/11 1/5 27/22 1/5 16/13 1/5 4/19 5/4 1/5 1/4 9/7 0/1 1/5 1/4 13/10 1/5 1/4 17/13 0/1 1/6 21/16 0/1 1/5 4/3 1/5 1/4 19/14 1/4 2/7 34/25 0/1 1/3 15/11 1/4 1/3 41/30 1/5 1/4 26/19 1/4 2/7 11/8 0/1 1/3 18/13 0/1 1/4 1/3 7/5 0/1 1/3 3/2 0/1 1/4 11/7 0/1 1/3 30/19 0/1 1/2 19/12 0/1 1/0 27/17 0/1 8/5 0/1 1/6 29/18 2/11 1/5 50/31 1/5 1/4 21/13 0/1 1/5 55/34 0/1 1/6 34/21 2/11 1/5 13/8 1/5 1/4 18/11 0/1 1/5 1/4 5/3 1/5 1/4 27/16 1/4 22/13 1/4 5/19 17/10 1/4 2/7 12/7 1/4 1/3 19/11 1/4 2/7 45/26 1/4 2/7 1/3 26/15 1/4 2/7 7/4 0/1 1/3 9/5 0/1 1/4 1/3 11/6 0/1 1/3 13/7 1/5 1/4 15/8 1/4 1/3 2/1 0/1 1/3 9/4 0/1 1/3 1/2 16/7 0/1 1/3 39/17 1/3 1/2 23/10 1/2 1/1 30/13 0/1 1/2 7/3 0/1 1/3 26/11 2/5 1/2 45/19 1/3 2/5 1/2 19/8 2/5 1/2 12/5 1/3 1/2 17/7 2/5 1/2 22/9 5/11 1/2 27/11 1/2 5/2 1/2 1/1 18/7 0/1 1/2 1/1 13/5 1/2 1/1 34/13 1/1 2/1 89/34 0/1 1/0 144/55 0/1 1/1 1/0 55/21 0/1 1/0 21/8 0/1 1/1 50/19 1/2 1/1 79/30 4/5 1/1 108/41 1/1 29/11 1/1 2/1 8/3 0/1 1/0 3/1 0/1 1/2 10/3 0/1 1/0 27/8 0/1 17/5 0/1 1/6 41/12 1/5 1/4 24/7 0/1 1/4 31/9 1/5 1/4 7/2 0/1 1/3 18/5 0/1 1/3 1/2 11/3 0/1 1/3 26/7 2/5 1/2 67/18 5/11 1/2 108/29 1/2 41/11 1/2 1/1 15/4 1/3 1/2 34/9 0/1 1/3 53/14 2/5 1/2 72/19 1/3 2/5 1/2 19/5 2/5 1/2 4/1 1/2 1/1 9/2 0/1 1/2 1/1 14/3 1/2 1/1 33/7 0/1 1/1 19/4 0/1 1/0 24/5 0/1 1/2 5/1 1/2 1/1 16/3 4/5 1/1 27/5 1/1 11/2 1/1 2/1 6/1 0/1 1/1 7/1 1/1 2/1 8/1 0/1 1/0 9/1 0/1 1/1 1/0 1/0 0/1 1/0 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(1,2,2,5) Matrix(71,594,30,251) -> Matrix(1,2,2,5) Matrix(37,270,10,73) -> Matrix(1,2,2,5) Matrix(17,108,14,89) -> Matrix(1,0,6,1) Matrix(19,108,16,91) -> Matrix(1,0,6,1) Matrix(109,594,20,109) -> Matrix(3,4,2,3) Matrix(161,864,30,161) -> Matrix(9,8,10,9) Matrix(73,378,28,145) -> Matrix(3,2,4,3) Matrix(35,162,-8,-37) -> Matrix(1,0,0,1) Matrix(215,918,-152,-649) -> Matrix(1,0,0,1) Matrix(359,1512,222,935) -> Matrix(1,0,6,1) Matrix(271,1134,168,703) -> Matrix(1,0,6,1) Matrix(127,486,52,199) -> Matrix(9,4,20,9) Matrix(971,3672,-600,-2269) -> Matrix(5,2,-28,-11) Matrix(143,540,76,287) -> Matrix(1,0,6,1) Matrix(217,810,116,433) -> Matrix(1,0,6,1) Matrix(755,2808,-552,-2053) -> Matrix(9,4,-34,-15) Matrix(73,270,10,37) -> Matrix(5,2,2,1) Matrix(89,324,64,233) -> Matrix(1,0,6,1) Matrix(91,324,66,235) -> Matrix(1,0,6,1) Matrix(17,54,-6,-19) -> Matrix(1,0,0,1) Matrix(197,540,-112,-307) -> Matrix(1,0,0,1) Matrix(377,1026,-298,-811) -> Matrix(1,0,0,1) Matrix(359,972,106,287) -> Matrix(1,0,12,1) Matrix(181,486,54,145) -> Matrix(1,0,0,1) Matrix(307,810,130,343) -> Matrix(1,2,2,5) Matrix(2051,5400,-1272,-3349) -> Matrix(3,2,-14,-9) Matrix(431,1134,328,863) -> Matrix(1,0,6,1) Matrix(577,1512,440,1153) -> Matrix(1,0,6,1) Matrix(1403,3672,-1032,-2701) -> Matrix(1,2,-4,-7) Matrix(145,378,28,73) -> Matrix(3,2,4,3) Matrix(125,324,76,197) -> Matrix(1,0,6,1) Matrix(127,324,78,199) -> Matrix(1,0,6,1) Matrix(109,270,44,109) -> Matrix(3,2,4,3) Matrix(485,1188,198,485) -> Matrix(21,10,44,21) Matrix(199,486,52,127) -> Matrix(9,4,20,9) Matrix(179,432,104,251) -> Matrix(1,0,6,1) Matrix(181,432,106,253) -> Matrix(1,0,6,1) Matrix(91,216,8,19) -> Matrix(5,2,2,1) Matrix(251,594,30,71) -> Matrix(5,2,2,1) Matrix(343,810,130,307) -> Matrix(5,2,2,1) Matrix(71,162,-32,-73) -> Matrix(1,0,0,1) Matrix(197,432,-140,-307) -> Matrix(1,0,0,1) Matrix(377,810,276,593) -> Matrix(1,0,6,1) Matrix(253,540,186,397) -> Matrix(1,0,6,1) Matrix(89,162,-50,-91) -> Matrix(1,0,0,1) Matrix(883,1566,234,415) -> Matrix(1,0,6,1) Matrix(1097,1944,294,521) -> Matrix(1,0,6,1) Matrix(397,702,82,145) -> Matrix(1,0,6,1) Matrix(467,810,290,503) -> Matrix(7,2,38,11) Matrix(343,594,302,523) -> Matrix(7,2,38,11) Matrix(125,216,114,197) -> Matrix(7,2,38,11) Matrix(251,432,104,179) -> Matrix(1,0,6,1) Matrix(253,432,106,181) -> Matrix(1,0,6,1) Matrix(287,486,212,359) -> Matrix(15,4,56,15) Matrix(703,1188,416,703) -> Matrix(39,10,152,39) Matrix(161,270,96,161) -> Matrix(9,2,40,9) Matrix(197,324,76,125) -> Matrix(1,0,6,1) Matrix(199,324,78,127) -> Matrix(1,0,6,1) Matrix(233,378,188,305) -> Matrix(9,2,40,9) Matrix(12815,20736,4894,7919) -> Matrix(1,0,6,1) Matrix(12817,20736,4896,7921) -> Matrix(1,0,6,1) Matrix(1369,2214,290,469) -> Matrix(1,0,6,1) Matrix(1205,1944,256,413) -> Matrix(1,0,6,1) Matrix(7235,11664,2746,4427) -> Matrix(29,6,24,5) Matrix(7237,11664,2748,4429) -> Matrix(31,6,36,7) Matrix(503,810,290,467) -> Matrix(11,2,38,7) Matrix(35,54,-24,-37) -> Matrix(1,0,0,1) Matrix(341,486,214,305) -> Matrix(1,0,12,1) Matrix(685,972,432,613) -> Matrix(1,0,0,1) Matrix(1223,1728,356,503) -> Matrix(1,0,6,1) Matrix(1225,1728,358,505) -> Matrix(1,0,6,1) Matrix(233,324,64,89) -> Matrix(1,0,6,1) Matrix(235,324,66,91) -> Matrix(1,0,6,1) Matrix(197,270,170,233) -> Matrix(7,2,38,11) Matrix(8531,11664,2290,3131) -> Matrix(23,6,42,11) Matrix(8533,11664,2292,3133) -> Matrix(25,6,54,13) Matrix(1423,1944,620,847) -> Matrix(1,0,6,1) Matrix(1151,1566,502,683) -> Matrix(1,0,6,1) Matrix(3815,5184,1006,1367) -> Matrix(1,0,6,1) Matrix(3817,5184,1008,1369) -> Matrix(1,0,6,1) Matrix(359,486,212,287) -> Matrix(15,4,56,15) Matrix(125,162,-98,-127) -> Matrix(1,0,0,1) Matrix(1531,1944,582,739) -> Matrix(1,0,6,1) Matrix(1745,2214,666,845) -> Matrix(1,0,6,1) Matrix(557,702,242,305) -> Matrix(1,0,6,1) Matrix(305,378,188,233) -> Matrix(9,2,40,9) Matrix(703,864,572,703) -> Matrix(39,8,190,39) Matrix(485,594,396,485) -> Matrix(21,4,110,21) Matrix(89,108,14,17) -> Matrix(1,0,6,1) Matrix(91,108,16,19) -> Matrix(1,0,6,1) Matrix(233,270,170,197) -> Matrix(11,2,38,7) Matrix(523,594,302,343) -> Matrix(11,2,38,7) Matrix(197,216,114,125) -> Matrix(11,2,38,7) Matrix(1,0,2,1) -> Matrix(1,0,12,1) Matrix(127,-162,98,-125) -> Matrix(1,0,0,1) Matrix(703,-918,206,-269) -> Matrix(1,0,0,1) Matrix(2701,-3672,1032,-1403) -> Matrix(7,-2,4,-1) Matrix(2053,-2808,552,-755) -> Matrix(15,-4,34,-9) Matrix(37,-54,24,-35) -> Matrix(1,0,0,1) Matrix(343,-540,148,-233) -> Matrix(1,0,0,1) Matrix(649,-1026,136,-215) -> Matrix(1,0,0,1) Matrix(3349,-5400,1272,-2051) -> Matrix(9,-2,14,-3) Matrix(2269,-3672,600,-971) -> Matrix(11,-2,28,-5) Matrix(91,-162,50,-89) -> Matrix(1,0,0,1) Matrix(235,-432,68,-125) -> Matrix(1,0,0,1) Matrix(73,-162,32,-71) -> Matrix(1,0,0,1) Matrix(19,-54,6,-17) -> Matrix(1,0,0,1) Matrix(37,-162,8,-35) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 18 Minimal number of generators: 4 Number of equivalence classes of cusps: 3 Genus: 1 Degree of H/liftables -> H/(image of liftables): 6 Degree of the the map X: 18 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. ----------------------------------------------------------------------- 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): 36 Minimal number of generators: 7 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: 6 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 3/2 9/4 3/1 9/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 0/1 1/6 4/3 1/5 1/4 3/2 0/1 1/4 5/3 1/5 1/4 2/1 0/1 1/3 9/4 0/1 1/3 1/2 7/3 0/1 1/3 5/2 1/2 1/1 3/1 0/1 1/2 4/1 1/2 1/1 9/2 0/1 1/2 1/1 5/1 1/2 1/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(22,-27,9,-11) (1/1,4/3) -> (7/3,5/2) Hyperbolic Matrix(19,-27,12,-17) (4/3,3/2) -> (3/2,5/3) Parabolic Matrix(16,-27,3,-5) (5/3,2/1) -> (5/1,1/0) Hyperbolic Matrix(37,-81,16,-35) (2/1,9/4) -> (9/4,7/3) Parabolic Matrix(10,-27,3,-8) (5/2,3/1) -> (3/1,4/1) Parabolic Matrix(19,-81,4,-17) (4/1,9/2) -> (9/2,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,6,1) Matrix(22,-27,9,-11) -> Matrix(5,-1,11,-2) Matrix(19,-27,12,-17) -> Matrix(1,0,0,1) Matrix(16,-27,3,-5) -> Matrix(4,-1,5,-1) Matrix(37,-81,16,-35) -> Matrix(1,0,0,1) Matrix(10,-27,3,-8) -> Matrix(1,0,0,1) Matrix(19,-81,4,-17) -> Matrix(1,0,0,1) 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: 1 Genus: 1 Degree of H/liftables -> H/(image of liftables): 1 ----------------------------------------------------------------------- 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 6 1 2/1 (0/1,1/3) 0 27 5/2 (1/2,1/1) 0 27 3/1 0 9 4/1 (1/2,1/1) 0 27 9/2 0 3 5/1 (1/2,1/1) 0 27 1/0 (0/1,1/0) 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(11,-27,2,-5) (2/1,5/2) -> (5/1,1/0) Glide Reflection Matrix(10,-27,3,-8) (5/2,3/1) -> (3/1,4/1) Parabolic Matrix(19,-81,4,-17) (4/1,9/2) -> (9/2,5/1) Parabolic 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(1,0,1,-1) -> Matrix(1,0,6,-1) (0/1,2/1) -> (0/1,1/3) Matrix(11,-27,2,-5) -> Matrix(2,-1,1,-1) Matrix(10,-27,3,-8) -> Matrix(1,0,0,1) Matrix(19,-81,4,-17) -> Matrix(1,0,0,1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.