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): 768 Minimal number of generators: 129 Number of equivalence classes of cusps: 56 Genus: 37 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -3/7 -5/13 -1/3 -3/11 -1/5 -1/7 0/1 1/7 3/13 1/3 7/17 5/11 1/2 7/13 3/5 5/7 7/9 1/1 9/7 7/5 3/2 49/31 5/3 9/5 31/17 13/7 2/1 11/5 7/3 45/19 17/7 5/2 23/9 3/1 49/15 10/3 7/2 25/7 11/3 4/1 13/3 9/2 14/3 53/11 5/1 27/5 11/2 6/1 13/2 7/1 8/1 9/1 29/3 10/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/2 -6/13 0/1 1/2 -5/11 0/1 1/3 -9/20 2/5 1/2 -4/9 1/2 1/1 -7/16 0/1 1/6 -10/23 1/4 2/7 -3/7 1/2 -14/33 3/4 4/5 -11/26 1/1 5/4 -8/19 0/1 1/0 -5/12 0/1 1/4 -12/29 1/3 1/2 -7/17 1/3 2/5 -9/22 4/9 1/2 -2/5 1/2 1/1 -7/18 0/1 1/2 -5/13 1/2 -13/34 1/2 4/7 -8/21 5/8 2/3 -3/8 3/4 1/1 -13/35 1/1 4/3 -10/27 1/1 3/2 -7/19 1/0 -11/30 -1/2 0/1 -15/41 0/1 1/3 -4/11 0/1 1/2 -5/14 1/2 2/3 -11/31 1/2 -6/17 2/3 3/4 -1/3 0/1 1/1 -7/22 1/1 13/12 -6/19 5/4 4/3 -5/16 3/2 2/1 -9/29 1/0 -4/13 2/1 1/0 -3/10 0/1 1/0 -8/27 0/1 1/4 -5/17 1/2 -7/24 3/4 1/1 -2/7 1/1 3/2 -5/18 2/1 5/2 -3/11 1/0 -7/26 -2/1 -3/2 -11/41 -1/1 -2/3 -4/15 0/1 1/0 -1/4 1/1 1/0 -4/17 -4/1 1/0 -7/30 -2/1 -3/2 -3/13 -2/1 -1/1 -11/48 -9/8 -1/1 -8/35 -1/1 -5/6 -5/22 -1/1 1/0 -2/9 -1/2 0/1 -3/14 0/1 1/2 -7/33 2/3 1/1 -4/19 5/6 1/1 -1/5 1/0 -4/21 -7/6 -1/1 -11/58 -1/1 -19/20 -7/37 -1/1 -6/7 -3/16 -2/3 -1/2 -5/27 -1/3 0/1 -2/11 -1/4 0/1 -1/6 1/1 1/0 -2/13 -2/1 1/0 -3/20 -1/1 -3/4 -1/7 -1/1 0/1 -3/22 -1/6 0/1 -2/15 0/1 1/4 -1/8 0/1 1/0 0/1 0/1 1/0 1/9 0/1 1/1 1/8 0/1 1/2 1/7 1/0 2/13 -1/1 -1/2 1/6 0/1 1/0 1/5 0/1 1/1 2/9 2/1 1/0 3/13 1/0 4/17 -4/1 1/0 1/4 -1/1 1/0 4/15 -1/4 0/1 3/11 -1/3 0/1 2/7 0/1 1/4 3/10 1/2 2/3 1/3 1/0 5/14 -3/2 -4/3 4/11 -1/1 -5/6 3/8 -1/2 0/1 8/21 -1/2 -1/3 5/13 -1/3 0/1 2/5 0/1 1/2 9/22 0/1 1/2 7/17 1/2 12/29 2/3 3/4 17/41 2/3 1/1 5/12 1/1 1/0 3/7 1/1 2/1 4/9 4/1 1/0 5/11 1/0 6/13 -6/1 1/0 1/2 -1/1 1/0 9/17 -1/1 0/1 8/15 0/1 1/0 7/13 1/0 6/11 -1/1 -1/2 5/9 -1/1 0/1 9/16 0/1 1/4 4/7 0/1 1/0 3/5 1/0 8/13 -4/1 1/0 5/8 -5/2 -2/1 17/27 1/0 29/46 -5/2 -2/1 41/65 -2/1 -1/1 12/19 -2/1 1/0 7/11 -3/1 -2/1 9/14 -2/1 -11/6 2/3 -3/2 -1/1 9/13 -1/1 -4/5 7/10 -1/1 -3/4 19/27 -1/1 -2/3 12/17 -1/1 -1/2 5/7 -1/2 13/18 -1/2 0/1 21/29 -1/3 0/1 8/11 -1/4 0/1 11/15 0/1 1/1 3/4 0/1 1/0 7/9 1/0 11/14 -4/1 1/0 4/5 -2/1 1/0 9/11 1/0 5/6 -2/1 -3/2 1/1 -1/1 0/1 8/7 -1/1 -13/14 7/6 -5/6 -4/5 6/5 -3/4 -2/3 17/14 -2/3 -1/2 11/9 -1/2 16/13 -1/1 -1/2 5/4 -2/3 -1/2 9/7 -1/2 13/10 -1/2 -2/5 4/3 -1/2 0/1 11/8 0/1 1/2 18/13 0/1 1/0 25/18 0/1 1/2 7/5 1/0 24/17 -2/1 1/0 17/12 -1/1 1/0 10/7 -3/2 -1/1 3/2 -1/1 -3/4 14/9 -11/16 -2/3 25/16 -2/3 -17/26 11/7 -2/3 -3/5 30/19 -3/5 -1/2 49/31 -1/2 19/12 -2/3 -1/2 46/29 -2/3 -5/8 27/17 -1/2 8/5 -2/3 -5/8 5/3 -1/2 12/7 -2/5 -3/8 19/11 -1/3 -2/7 7/4 -1/2 0/1 16/9 -1/6 0/1 25/14 0/1 1/4 9/5 -1/1 0/1 20/11 0/1 1/0 31/17 1/0 11/6 -1/1 1/0 13/7 -1/2 15/8 -1/2 0/1 2/1 -1/1 -1/2 11/5 -1/2 20/9 -1/2 -9/19 9/4 -1/2 -4/9 16/7 -2/5 -3/8 39/17 -1/2 23/10 -1/2 -2/5 30/13 -5/12 -2/5 7/3 -2/5 -1/3 26/11 -9/26 -1/3 45/19 -1/3 64/27 -1/3 -19/58 19/8 -1/3 -5/16 12/5 -1/2 -1/3 41/17 -1/3 -2/7 111/46 -1/3 -1/4 70/29 -1/3 -3/10 29/12 -3/10 -2/7 17/7 -1/4 22/9 -1/4 0/1 5/2 -1/4 0/1 23/9 0/1 41/16 0/1 1/16 18/7 0/1 1/6 13/5 0/1 1/1 47/18 0/1 1/2 81/31 1/2 34/13 1/2 1/1 21/8 1/1 1/0 8/3 0/1 1/0 19/7 -2/1 -1/1 49/18 -1/1 1/0 30/11 -2/1 1/0 11/4 -5/4 -1/1 3/1 -1/2 13/4 -7/20 -1/3 49/15 -1/3 85/26 -1/3 -45/136 36/11 -1/3 -19/58 23/7 -1/3 -6/19 10/3 -2/7 -1/4 17/5 -1/5 0/1 58/17 -1/8 0/1 99/29 0/1 41/12 -1/2 0/1 24/7 -1/4 0/1 31/9 -1/4 7/2 -1/6 0/1 25/7 0/1 43/12 0/1 1/22 18/5 0/1 1/8 11/3 0/1 1/1 37/10 1/1 1/0 63/17 1/0 26/7 0/1 1/0 15/4 0/1 1/2 4/1 -1/1 -1/2 13/3 -1/2 22/5 -1/2 -5/11 9/2 -1/2 -2/5 14/3 -1/3 -3/10 33/7 -1/4 19/4 -1/2 0/1 24/5 -1/2 -1/3 53/11 -1/3 82/17 -1/3 -3/10 29/6 -1/3 -1/4 5/1 -1/3 0/1 16/3 -1/8 0/1 27/5 0/1 38/7 0/1 1/12 11/2 0/1 1/2 6/1 -1/2 0/1 13/2 -1/1 1/0 7/1 -1/2 15/2 -3/8 -1/3 8/1 -1/4 0/1 9/1 -1/3 0/1 19/2 -1/6 0/1 29/3 0/1 10/1 0/1 1/0 1/0 -1/2 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(121,56,296,137) (-1/2,-6/13) -> (2/5,9/22) Hyperbolic Matrix(239,110,-654,-301) (-6/13,-5/11) -> (-15/41,-4/11) Hyperbolic Matrix(173,78,-814,-367) (-5/11,-9/20) -> (-3/14,-7/33) Hyperbolic Matrix(521,234,118,53) (-9/20,-4/9) -> (22/5,9/2) Hyperbolic Matrix(113,50,174,77) (-4/9,-7/16) -> (9/14,2/3) Hyperbolic Matrix(225,98,-962,-419) (-7/16,-10/23) -> (-4/17,-7/30) Hyperbolic Matrix(167,72,-392,-169) (-10/23,-3/7) -> (-3/7,-14/33) Parabolic Matrix(387,164,-1220,-517) (-14/33,-11/26) -> (-7/22,-6/19) Hyperbolic Matrix(109,46,-718,-303) (-11/26,-8/19) -> (-2/13,-3/20) Hyperbolic Matrix(219,92,388,163) (-8/19,-5/12) -> (9/16,4/7) Hyperbolic Matrix(53,22,330,137) (-5/12,-12/29) -> (2/13,1/6) Hyperbolic Matrix(547,226,-1474,-609) (-12/29,-7/17) -> (-13/35,-10/27) Hyperbolic Matrix(161,66,-866,-355) (-7/17,-9/22) -> (-3/16,-5/27) Hyperbolic Matrix(485,198,218,89) (-9/22,-2/5) -> (20/9,9/4) Hyperbolic Matrix(159,62,418,163) (-2/5,-7/18) -> (3/8,8/21) Hyperbolic Matrix(259,100,-676,-261) (-7/18,-5/13) -> (-5/13,-13/34) Parabolic Matrix(309,118,-982,-375) (-13/34,-8/21) -> (-6/19,-5/16) Hyperbolic Matrix(53,20,204,77) (-8/21,-3/8) -> (1/4,4/15) Hyperbolic Matrix(409,152,-1784,-663) (-3/8,-13/35) -> (-3/13,-11/48) Hyperbolic Matrix(303,112,560,207) (-10/27,-7/19) -> (7/13,6/11) Hyperbolic Matrix(605,222,962,353) (-7/19,-11/30) -> (5/8,17/27) Hyperbolic Matrix(153,56,-1112,-407) (-11/30,-15/41) -> (-1/7,-3/22) Hyperbolic Matrix(199,72,152,55) (-4/11,-5/14) -> (13/10,4/3) Hyperbolic Matrix(647,230,346,123) (-5/14,-11/31) -> (13/7,15/8) Hyperbolic Matrix(1885,668,1188,421) (-11/31,-6/17) -> (46/29,27/17) Hyperbolic Matrix(199,70,-742,-261) (-6/17,-1/3) -> (-11/41,-4/15) Hyperbolic Matrix(187,60,-988,-317) (-1/3,-7/22) -> (-11/58,-7/37) Hyperbolic Matrix(1113,346,1766,549) (-5/16,-9/29) -> (17/27,29/46) Hyperbolic Matrix(323,100,604,187) (-9/29,-4/13) -> (8/15,7/13) Hyperbolic Matrix(183,56,232,71) (-4/13,-3/10) -> (11/14,4/5) Hyperbolic Matrix(47,14,-366,-109) (-3/10,-8/27) -> (-2/15,-1/8) Hyperbolic Matrix(865,256,544,161) (-8/27,-5/17) -> (27/17,8/5) Hyperbolic Matrix(499,146,270,79) (-5/17,-7/24) -> (11/6,13/7) Hyperbolic Matrix(227,66,-994,-289) (-7/24,-2/7) -> (-8/35,-5/22) Hyperbolic Matrix(177,50,46,13) (-2/7,-5/18) -> (15/4,4/1) Hyperbolic Matrix(131,36,-484,-133) (-5/18,-3/11) -> (-3/11,-7/26) Parabolic Matrix(305,82,-1618,-435) (-7/26,-11/41) -> (-7/37,-3/16) Hyperbolic Matrix(347,92,132,35) (-4/15,-1/4) -> (21/8,8/3) Hyperbolic Matrix(41,10,86,21) (-1/4,-4/17) -> (6/13,1/2) Hyperbolic Matrix(43,10,374,87) (-7/30,-3/13) -> (1/9,1/8) Hyperbolic Matrix(3737,856,1576,361) (-11/48,-8/35) -> (64/27,19/8) Hyperbolic Matrix(249,56,40,9) (-5/22,-2/9) -> (6/1,13/2) Hyperbolic Matrix(287,62,162,35) (-2/9,-3/14) -> (7/4,16/9) Hyperbolic Matrix(1175,248,488,103) (-7/33,-4/19) -> (12/5,41/17) Hyperbolic Matrix(39,8,-200,-41) (-4/19,-1/5) -> (-1/5,-4/21) Parabolic Matrix(3873,736,1184,225) (-4/21,-11/58) -> (85/26,36/11) Hyperbolic Matrix(315,58,38,7) (-5/27,-2/11) -> (8/1,9/1) Hyperbolic Matrix(117,20,76,13) (-2/11,-1/6) -> (3/2,14/9) Hyperbolic Matrix(37,6,154,25) (-1/6,-2/13) -> (4/17,1/4) Hyperbolic Matrix(375,56,904,135) (-3/20,-1/7) -> (17/41,5/12) Hyperbolic Matrix(1001,136,184,25) (-3/22,-2/15) -> (38/7,11/2) Hyperbolic Matrix(181,22,74,9) (-1/8,0/1) -> (22/9,5/2) Hyperbolic Matrix(149,-12,236,-19) (0/1,1/9) -> (41/65,12/19) Hyperbolic Matrix(397,-52,84,-11) (1/8,1/7) -> (33/7,19/4) Hyperbolic Matrix(527,-80,112,-17) (1/7,2/13) -> (14/3,33/7) Hyperbolic Matrix(81,-14,110,-19) (1/6,1/5) -> (11/15,3/4) Hyperbolic Matrix(211,-46,78,-17) (1/5,2/9) -> (8/3,19/7) Hyperbolic Matrix(79,-18,338,-77) (2/9,3/13) -> (3/13,4/17) Parabolic Matrix(403,-108,556,-149) (4/15,3/11) -> (21/29,8/11) Hyperbolic Matrix(275,-76,76,-21) (3/11,2/7) -> (18/5,11/3) Hyperbolic Matrix(223,-66,98,-29) (2/7,3/10) -> (9/4,16/7) Hyperbolic Matrix(25,-8,72,-23) (3/10,1/3) -> (1/3,5/14) Parabolic Matrix(189,-68,164,-59) (5/14,4/11) -> (8/7,7/6) Hyperbolic Matrix(211,-78,46,-17) (4/11,3/8) -> (9/2,14/3) Hyperbolic Matrix(555,-212,788,-301) (8/21,5/13) -> (19/27,12/17) Hyperbolic Matrix(299,-116,116,-45) (5/13,2/5) -> (18/7,13/5) Hyperbolic Matrix(1249,-512,544,-223) (9/22,7/17) -> (39/17,23/10) Hyperbolic Matrix(1403,-580,612,-253) (7/17,12/29) -> (16/7,39/17) Hyperbolic Matrix(589,-244,1108,-459) (12/29,17/41) -> (9/17,8/15) Hyperbolic Matrix(157,-66,226,-95) (5/12,3/7) -> (9/13,7/10) Hyperbolic Matrix(223,-98,66,-29) (3/7,4/9) -> (10/3,17/5) Hyperbolic Matrix(111,-50,242,-109) (4/9,5/11) -> (5/11,6/13) Parabolic Matrix(1969,-1040,816,-431) (1/2,9/17) -> (41/17,111/46) Hyperbolic Matrix(391,-216,248,-137) (6/11,5/9) -> (11/7,30/19) Hyperbolic Matrix(369,-206,206,-115) (5/9,9/16) -> (25/14,9/5) Hyperbolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(139,-86,118,-73) (8/13,5/8) -> (7/6,6/5) Hyperbolic Matrix(1649,-1040,176,-111) (29/46,41/65) -> (9/1,19/2) Hyperbolic Matrix(391,-248,216,-137) (12/19,7/11) -> (9/5,20/11) Hyperbolic Matrix(429,-274,274,-175) (7/11,9/14) -> (25/16,11/7) Hyperbolic Matrix(359,-248,152,-105) (2/3,9/13) -> (7/3,26/11) Hyperbolic Matrix(1109,-780,300,-211) (7/10,19/27) -> (11/3,37/10) Hyperbolic Matrix(299,-212,244,-173) (12/17,5/7) -> (11/9,16/13) Hyperbolic Matrix(317,-228,260,-187) (5/7,13/18) -> (17/14,11/9) Hyperbolic Matrix(1597,-1156,612,-443) (13/18,21/29) -> (13/5,47/18) Hyperbolic Matrix(295,-216,56,-41) (8/11,11/15) -> (5/1,16/3) Hyperbolic Matrix(127,-98,162,-125) (3/4,7/9) -> (7/9,11/14) Parabolic Matrix(299,-244,212,-173) (4/5,9/11) -> (7/5,24/17) Hyperbolic Matrix(317,-260,228,-187) (9/11,5/6) -> (25/18,7/5) Hyperbolic Matrix(121,-102,70,-59) (5/6,1/1) -> (19/11,7/4) Hyperbolic Matrix(197,-220,60,-67) (1/1,8/7) -> (36/11,23/7) Hyperbolic Matrix(535,-648,232,-281) (6/5,17/14) -> (23/10,30/13) Hyperbolic Matrix(343,-424,72,-89) (16/13,5/4) -> (19/4,24/5) Hyperbolic Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(81,-110,14,-19) (4/3,11/8) -> (11/2,6/1) Hyperbolic Matrix(403,-556,108,-149) (11/8,18/13) -> (26/7,15/4) Hyperbolic Matrix(965,-1338,282,-391) (18/13,25/18) -> (41/12,24/7) Hyperbolic Matrix(1193,-1686,438,-619) (24/17,17/12) -> (49/18,30/11) Hyperbolic Matrix(555,-788,212,-301) (17/12,10/7) -> (34/13,21/8) Hyperbolic Matrix(157,-226,66,-95) (10/7,3/2) -> (19/8,12/5) Hyperbolic Matrix(675,-1052,188,-293) (14/9,25/16) -> (43/12,18/5) Hyperbolic Matrix(2593,-4096,992,-1567) (30/19,49/31) -> (81/31,34/13) Hyperbolic Matrix(2429,-3842,930,-1471) (49/31,19/12) -> (47/18,81/31) Hyperbolic Matrix(149,-236,12,-19) (19/12,46/29) -> (10/1,1/0) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(271,-466,82,-141) (12/7,19/11) -> (23/7,10/3) Hyperbolic Matrix(621,-1106,242,-431) (16/9,25/14) -> (41/16,18/7) Hyperbolic Matrix(1135,-2066,306,-557) (20/11,31/17) -> (63/17,26/7) Hyperbolic Matrix(1007,-1840,272,-497) (31/17,11/6) -> (37/10,63/17) Hyperbolic Matrix(589,-1108,244,-459) (15/8,2/1) -> (70/29,29/12) Hyperbolic Matrix(111,-242,50,-109) (2/1,11/5) -> (11/5,20/9) Parabolic Matrix(395,-916,116,-269) (30/13,7/3) -> (17/5,58/17) Hyperbolic Matrix(1711,-4050,722,-1709) (26/11,45/19) -> (45/19,64/27) Parabolic Matrix(4613,-11132,956,-2307) (111/46,70/29) -> (82/17,29/6) Hyperbolic Matrix(421,-1018,122,-295) (29/12,17/7) -> (31/9,7/2) Hyperbolic Matrix(447,-1090,130,-317) (17/7,22/9) -> (24/7,31/9) Hyperbolic Matrix(415,-1058,162,-413) (5/2,23/9) -> (23/9,41/16) Parabolic Matrix(293,-796,60,-163) (19/7,49/18) -> (29/6,5/1) Hyperbolic Matrix(197,-538,26,-71) (30/11,11/4) -> (15/2,8/1) Hyperbolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(1471,-4802,450,-1469) (13/4,49/15) -> (49/15,85/26) Parabolic Matrix(783,-2672,80,-273) (58/17,99/29) -> (29/3,10/1) Hyperbolic Matrix(899,-3070,94,-321) (99/29,41/12) -> (19/2,29/3) Hyperbolic Matrix(351,-1250,98,-349) (7/2,25/7) -> (25/7,43/12) Parabolic Matrix(79,-338,18,-77) (4/1,13/3) -> (13/3,22/5) Parabolic Matrix(1167,-5618,242,-1165) (24/5,53/11) -> (53/11,82/17) Parabolic Matrix(271,-1458,50,-269) (16/3,27/5) -> (27/5,38/7) Parabolic Matrix(29,-196,4,-27) (13/2,7/1) -> (7/1,15/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(121,56,296,137) -> Matrix(1,0,0,1) Matrix(239,110,-654,-301) -> Matrix(1,0,0,1) Matrix(173,78,-814,-367) -> Matrix(5,-2,8,-3) Matrix(521,234,118,53) -> Matrix(9,-4,-20,9) Matrix(113,50,174,77) -> Matrix(1,-2,0,1) Matrix(225,98,-962,-419) -> Matrix(9,-2,-4,1) Matrix(167,72,-392,-169) -> Matrix(5,-2,8,-3) Matrix(387,164,-1220,-517) -> Matrix(9,-8,8,-7) Matrix(109,46,-718,-303) -> Matrix(1,-2,0,1) Matrix(219,92,388,163) -> Matrix(1,0,0,1) Matrix(53,22,330,137) -> Matrix(1,0,-4,1) Matrix(547,226,-1474,-609) -> Matrix(7,-2,4,-1) Matrix(161,66,-866,-355) -> Matrix(5,-2,-12,5) Matrix(485,198,218,89) -> Matrix(17,-8,-36,17) Matrix(159,62,418,163) -> Matrix(1,0,-4,1) Matrix(259,100,-676,-261) -> Matrix(9,-4,16,-7) Matrix(309,118,-982,-375) -> Matrix(17,-10,12,-7) Matrix(53,20,204,77) -> Matrix(3,-2,-4,3) Matrix(409,152,-1784,-663) -> Matrix(5,-6,-4,5) Matrix(303,112,560,207) -> Matrix(1,-2,0,1) Matrix(605,222,962,353) -> Matrix(1,-2,0,1) Matrix(153,56,-1112,-407) -> Matrix(1,0,-4,1) Matrix(199,72,152,55) -> Matrix(1,0,-4,1) Matrix(647,230,346,123) -> Matrix(3,-2,-4,3) Matrix(1885,668,1188,421) -> Matrix(7,-4,-12,7) Matrix(199,70,-742,-261) -> Matrix(3,-2,-4,3) Matrix(187,60,-988,-317) -> Matrix(7,-6,-8,7) Matrix(1113,346,1766,549) -> Matrix(1,-4,0,1) Matrix(323,100,604,187) -> Matrix(1,-2,0,1) Matrix(183,56,232,71) -> Matrix(1,-4,0,1) Matrix(47,14,-366,-109) -> Matrix(1,0,0,1) Matrix(865,256,544,161) -> Matrix(3,-2,-4,3) Matrix(499,146,270,79) -> Matrix(3,-2,-4,3) Matrix(227,66,-994,-289) -> Matrix(3,-2,-4,3) Matrix(177,50,46,13) -> Matrix(1,-2,0,1) Matrix(131,36,-484,-133) -> Matrix(1,-4,0,1) Matrix(305,82,-1618,-435) -> Matrix(3,4,-4,-5) Matrix(347,92,132,35) -> Matrix(1,0,0,1) Matrix(41,10,86,21) -> Matrix(1,-2,0,1) Matrix(43,10,374,87) -> Matrix(1,2,0,1) Matrix(3737,856,1576,361) -> Matrix(13,14,-40,-43) Matrix(249,56,40,9) -> Matrix(1,0,0,1) Matrix(287,62,162,35) -> Matrix(1,0,-4,1) Matrix(1175,248,488,103) -> Matrix(5,-4,-16,13) Matrix(39,8,-200,-41) -> Matrix(1,-2,0,1) Matrix(3873,736,1184,225) -> Matrix(25,26,-76,-79) Matrix(315,58,38,7) -> Matrix(1,0,0,1) Matrix(117,20,76,13) -> Matrix(3,-2,-4,3) Matrix(37,6,154,25) -> Matrix(1,-2,0,1) Matrix(375,56,904,135) -> Matrix(3,2,4,3) Matrix(1001,136,184,25) -> Matrix(1,0,8,1) Matrix(181,22,74,9) -> Matrix(1,0,-4,1) Matrix(149,-12,236,-19) -> Matrix(1,-2,0,1) Matrix(397,-52,84,-11) -> Matrix(1,0,-4,1) Matrix(527,-80,112,-17) -> Matrix(1,2,-4,-7) Matrix(81,-14,110,-19) -> Matrix(1,0,0,1) Matrix(211,-46,78,-17) -> Matrix(1,-2,0,1) Matrix(79,-18,338,-77) -> Matrix(1,-6,0,1) Matrix(403,-108,556,-149) -> Matrix(1,0,0,1) Matrix(275,-76,76,-21) -> Matrix(1,0,4,1) Matrix(223,-66,98,-29) -> Matrix(5,-2,-12,5) Matrix(25,-8,72,-23) -> Matrix(1,-2,0,1) Matrix(189,-68,164,-59) -> Matrix(7,8,-8,-9) Matrix(211,-78,46,-17) -> Matrix(3,2,-8,-5) Matrix(555,-212,788,-301) -> Matrix(5,2,-8,-3) Matrix(299,-116,116,-45) -> Matrix(1,0,4,1) Matrix(1249,-512,544,-223) -> Matrix(5,-2,-12,5) Matrix(1403,-580,612,-253) -> Matrix(1,0,-4,1) Matrix(589,-244,1108,-459) -> Matrix(3,-2,-4,3) Matrix(157,-66,226,-95) -> Matrix(3,-2,-4,3) Matrix(223,-98,66,-29) -> Matrix(1,-2,-4,9) Matrix(111,-50,242,-109) -> Matrix(1,-10,0,1) Matrix(1969,-1040,816,-431) -> Matrix(1,2,-4,-7) Matrix(391,-216,248,-137) -> Matrix(5,2,-8,-3) Matrix(369,-206,206,-115) -> Matrix(1,0,0,1) Matrix(61,-36,100,-59) -> Matrix(1,-4,0,1) Matrix(139,-86,118,-73) -> Matrix(3,10,-4,-13) Matrix(1649,-1040,176,-111) -> Matrix(1,2,-4,-7) Matrix(391,-248,216,-137) -> Matrix(1,2,0,1) Matrix(429,-274,274,-175) -> Matrix(5,12,-8,-19) Matrix(359,-248,152,-105) -> Matrix(7,6,-20,-17) Matrix(1109,-780,300,-211) -> Matrix(3,2,4,3) Matrix(299,-212,244,-173) -> Matrix(1,0,0,1) Matrix(317,-228,260,-187) -> Matrix(5,2,-8,-3) Matrix(1597,-1156,612,-443) -> Matrix(1,0,4,1) Matrix(295,-216,56,-41) -> Matrix(1,0,-4,1) Matrix(127,-98,162,-125) -> Matrix(1,-4,0,1) Matrix(299,-244,212,-173) -> Matrix(1,0,0,1) Matrix(317,-260,228,-187) -> Matrix(1,2,0,1) Matrix(121,-102,70,-59) -> Matrix(1,2,-4,-7) Matrix(197,-220,60,-67) -> Matrix(5,6,-16,-19) Matrix(535,-648,232,-281) -> Matrix(7,4,-16,-9) Matrix(343,-424,72,-89) -> Matrix(3,2,-8,-5) Matrix(127,-162,98,-125) -> Matrix(7,4,-16,-9) Matrix(81,-110,14,-19) -> Matrix(1,0,0,1) Matrix(403,-556,108,-149) -> Matrix(1,0,0,1) Matrix(965,-1338,282,-391) -> Matrix(1,0,-4,1) Matrix(1193,-1686,438,-619) -> Matrix(1,0,0,1) Matrix(555,-788,212,-301) -> Matrix(1,2,0,1) Matrix(157,-226,66,-95) -> Matrix(1,2,-4,-7) Matrix(675,-1052,188,-293) -> Matrix(3,2,40,27) Matrix(2593,-4096,992,-1567) -> Matrix(7,4,12,7) Matrix(2429,-3842,930,-1471) -> Matrix(3,2,4,3) Matrix(149,-236,12,-19) -> Matrix(3,2,-8,-5) Matrix(61,-100,36,-59) -> Matrix(7,4,-16,-9) Matrix(271,-466,82,-141) -> Matrix(11,4,-36,-13) Matrix(621,-1106,242,-431) -> Matrix(1,0,12,1) Matrix(1135,-2066,306,-557) -> Matrix(1,0,0,1) Matrix(1007,-1840,272,-497) -> Matrix(1,2,0,1) Matrix(589,-1108,244,-459) -> Matrix(1,2,-4,-7) Matrix(111,-242,50,-109) -> Matrix(19,10,-40,-21) Matrix(395,-916,116,-269) -> Matrix(5,2,-28,-11) Matrix(1711,-4050,722,-1709) -> Matrix(83,28,-252,-85) Matrix(4613,-11132,956,-2307) -> Matrix(1,0,0,1) Matrix(421,-1018,122,-295) -> Matrix(7,2,-32,-9) Matrix(447,-1090,130,-317) -> Matrix(1,0,0,1) Matrix(415,-1058,162,-413) -> Matrix(1,0,20,1) Matrix(293,-796,60,-163) -> Matrix(1,2,-4,-7) Matrix(197,-538,26,-71) -> Matrix(1,2,-4,-7) Matrix(25,-72,8,-23) -> Matrix(3,2,-8,-5) Matrix(1471,-4802,450,-1469) -> Matrix(155,52,-468,-157) Matrix(783,-2672,80,-273) -> Matrix(1,0,8,1) Matrix(899,-3070,94,-321) -> Matrix(1,0,-4,1) Matrix(351,-1250,98,-349) -> Matrix(1,0,28,1) Matrix(79,-338,18,-77) -> Matrix(11,6,-24,-13) Matrix(1167,-5618,242,-1165) -> Matrix(11,4,-36,-13) Matrix(271,-1458,50,-269) -> Matrix(1,0,20,1) Matrix(29,-196,4,-27) -> Matrix(3,2,-8,-5) 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): 21 Degree of the the map X: 42 Degree of the the map Y: 128 ----------------------------------------------------------------------- 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): 384 Minimal number of generators: 65 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: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/7 1/3 3/5 5/7 1/1 9/7 7/5 5/3 31/17 2/1 11/5 7/3 45/19 23/9 3/1 49/15 99/29 25/7 11/3 4/1 13/3 9/2 14/3 53/11 5/1 27/5 11/2 6/1 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 0/1 1/0 1/8 0/1 1/2 1/7 1/0 1/6 0/1 1/0 1/5 0/1 1/1 2/9 2/1 1/0 3/13 1/0 1/4 -1/1 1/0 3/11 -1/3 0/1 2/7 0/1 1/4 3/10 1/2 2/3 1/3 1/0 5/14 -3/2 -4/3 4/11 -1/1 -5/6 3/8 -1/2 0/1 5/13 -1/3 0/1 2/5 0/1 1/2 7/17 1/2 12/29 2/3 3/4 5/12 1/1 1/0 3/7 1/1 2/1 4/9 4/1 1/0 5/11 1/0 1/2 -1/1 1/0 6/11 -1/1 -1/2 5/9 -1/1 0/1 4/7 0/1 1/0 3/5 1/0 8/13 -4/1 1/0 5/8 -5/2 -2/1 12/19 -2/1 1/0 7/11 -3/1 -2/1 2/3 -3/2 -1/1 9/13 -1/1 -4/5 7/10 -1/1 -3/4 12/17 -1/1 -1/2 5/7 -1/2 13/18 -1/2 0/1 8/11 -1/4 0/1 11/15 0/1 1/1 3/4 0/1 1/0 7/9 1/0 4/5 -2/1 1/0 9/11 1/0 5/6 -2/1 -3/2 1/1 -1/1 0/1 8/7 -1/1 -13/14 7/6 -5/6 -4/5 6/5 -3/4 -2/3 17/14 -2/3 -1/2 11/9 -1/2 16/13 -1/1 -1/2 5/4 -2/3 -1/2 9/7 -1/2 4/3 -1/2 0/1 11/8 0/1 1/2 18/13 0/1 1/0 25/18 0/1 1/2 7/5 1/0 24/17 -2/1 1/0 17/12 -1/1 1/0 10/7 -3/2 -1/1 3/2 -1/1 -3/4 11/7 -2/3 -3/5 30/19 -3/5 -1/2 49/31 -1/2 19/12 -2/3 -1/2 8/5 -2/3 -5/8 5/3 -1/2 12/7 -2/5 -3/8 19/11 -1/3 -2/7 7/4 -1/2 0/1 9/5 -1/1 0/1 20/11 0/1 1/0 31/17 1/0 11/6 -1/1 1/0 2/1 -1/1 -1/2 11/5 -1/2 9/4 -1/2 -4/9 16/7 -2/5 -3/8 23/10 -1/2 -2/5 30/13 -5/12 -2/5 7/3 -2/5 -1/3 26/11 -9/26 -1/3 45/19 -1/3 19/8 -1/3 -5/16 12/5 -1/2 -1/3 29/12 -3/10 -2/7 17/7 -1/4 5/2 -1/4 0/1 23/9 0/1 18/7 0/1 1/6 13/5 0/1 1/1 8/3 0/1 1/0 19/7 -2/1 -1/1 49/18 -1/1 1/0 30/11 -2/1 1/0 11/4 -5/4 -1/1 3/1 -1/2 13/4 -7/20 -1/3 49/15 -1/3 36/11 -1/3 -19/58 23/7 -1/3 -6/19 10/3 -2/7 -1/4 17/5 -1/5 0/1 58/17 -1/8 0/1 99/29 0/1 41/12 -1/2 0/1 24/7 -1/4 0/1 7/2 -1/6 0/1 25/7 0/1 18/5 0/1 1/8 11/3 0/1 1/1 4/1 -1/1 -1/2 13/3 -1/2 9/2 -1/2 -2/5 14/3 -1/3 -3/10 19/4 -1/2 0/1 24/5 -1/2 -1/3 53/11 -1/3 29/6 -1/3 -1/4 5/1 -1/3 0/1 16/3 -1/8 0/1 27/5 0/1 11/2 0/1 1/2 6/1 -1/2 0/1 7/1 -1/2 8/1 -1/4 0/1 1/0 -1/2 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(200,-23,87,-10) (0/1,1/8) -> (16/7,23/10) Hyperbolic Matrix(140,-19,339,-46) (1/8,1/7) -> (7/17,12/29) Hyperbolic Matrix(56,-9,137,-22) (1/7,1/6) -> (2/5,7/17) Hyperbolic Matrix(81,-14,110,-19) (1/6,1/5) -> (11/15,3/4) Hyperbolic Matrix(211,-46,78,-17) (1/5,2/9) -> (8/3,19/7) Hyperbolic Matrix(234,-53,53,-12) (2/9,3/13) -> (13/3,9/2) Hyperbolic Matrix(104,-25,25,-6) (3/13,1/4) -> (4/1,13/3) Hyperbolic Matrix(50,-13,77,-20) (1/4,3/11) -> (7/11,2/3) Hyperbolic Matrix(275,-76,76,-21) (3/11,2/7) -> (18/5,11/3) Hyperbolic Matrix(223,-66,98,-29) (2/7,3/10) -> (9/4,16/7) Hyperbolic Matrix(25,-8,72,-23) (3/10,1/3) -> (1/3,5/14) Parabolic Matrix(189,-68,164,-59) (5/14,4/11) -> (8/7,7/6) Hyperbolic Matrix(211,-78,46,-17) (4/11,3/8) -> (9/2,14/3) Hyperbolic Matrix(92,-35,163,-62) (3/8,5/13) -> (5/9,4/7) Hyperbolic Matrix(299,-116,116,-45) (5/13,2/5) -> (18/7,13/5) Hyperbolic Matrix(634,-263,135,-56) (12/29,5/12) -> (14/3,19/4) Hyperbolic Matrix(157,-66,226,-95) (5/12,3/7) -> (9/13,7/10) Hyperbolic Matrix(223,-98,66,-29) (3/7,4/9) -> (10/3,17/5) Hyperbolic Matrix(198,-89,89,-40) (4/9,5/11) -> (11/5,9/4) Hyperbolic Matrix(44,-21,21,-10) (5/11,1/2) -> (2/1,11/5) Hyperbolic Matrix(146,-79,207,-112) (1/2,6/11) -> (7/10,12/17) Hyperbolic Matrix(391,-216,248,-137) (6/11,5/9) -> (11/7,30/19) Hyperbolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(139,-86,118,-73) (8/13,5/8) -> (7/6,6/5) Hyperbolic Matrix(256,-161,353,-222) (5/8,12/19) -> (13/18,8/11) Hyperbolic Matrix(391,-248,216,-137) (12/19,7/11) -> (9/5,20/11) Hyperbolic Matrix(359,-248,152,-105) (2/3,9/13) -> (7/3,26/11) Hyperbolic Matrix(299,-212,244,-173) (12/17,5/7) -> (11/9,16/13) Hyperbolic Matrix(317,-228,260,-187) (5/7,13/18) -> (17/14,11/9) Hyperbolic Matrix(295,-216,56,-41) (8/11,11/15) -> (5/1,16/3) Hyperbolic Matrix(72,-55,55,-42) (3/4,7/9) -> (9/7,4/3) Hyperbolic Matrix(90,-71,71,-56) (7/9,4/5) -> (5/4,9/7) Hyperbolic Matrix(299,-244,212,-173) (4/5,9/11) -> (7/5,24/17) Hyperbolic Matrix(317,-260,228,-187) (9/11,5/6) -> (25/18,7/5) Hyperbolic Matrix(121,-102,70,-59) (5/6,1/1) -> (19/11,7/4) Hyperbolic Matrix(197,-220,60,-67) (1/1,8/7) -> (36/11,23/7) Hyperbolic Matrix(535,-648,232,-281) (6/5,17/14) -> (23/10,30/13) Hyperbolic Matrix(343,-424,72,-89) (16/13,5/4) -> (19/4,24/5) Hyperbolic Matrix(81,-110,14,-19) (4/3,11/8) -> (11/2,6/1) Hyperbolic Matrix(256,-353,161,-222) (11/8,18/13) -> (19/12,8/5) Hyperbolic Matrix(965,-1338,282,-391) (18/13,25/18) -> (41/12,24/7) Hyperbolic Matrix(1193,-1686,438,-619) (24/17,17/12) -> (49/18,30/11) Hyperbolic Matrix(146,-207,79,-112) (17/12,10/7) -> (11/6,2/1) Hyperbolic Matrix(157,-226,66,-95) (10/7,3/2) -> (19/8,12/5) Hyperbolic Matrix(50,-77,13,-20) (3/2,11/7) -> (11/3,4/1) Hyperbolic Matrix(930,-1469,509,-804) (30/19,49/31) -> (31/17,11/6) Hyperbolic Matrix(992,-1569,545,-862) (49/31,19/12) -> (20/11,31/17) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(271,-466,82,-141) (12/7,19/11) -> (23/7,10/3) Hyperbolic Matrix(92,-163,35,-62) (7/4,9/5) -> (13/5,8/3) Hyperbolic Matrix(395,-916,116,-269) (30/13,7/3) -> (17/5,58/17) Hyperbolic Matrix(856,-2025,361,-854) (26/11,45/19) -> (45/19,19/8) Parabolic Matrix(282,-679,103,-248) (12/5,29/12) -> (30/11,11/4) Hyperbolic Matrix(140,-339,19,-46) (29/12,17/7) -> (7/1,8/1) Hyperbolic Matrix(56,-137,9,-22) (17/7,5/2) -> (6/1,7/1) Hyperbolic Matrix(208,-529,81,-206) (5/2,23/9) -> (23/9,18/7) Parabolic Matrix(293,-796,60,-163) (19/7,49/18) -> (29/6,5/1) Hyperbolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(736,-2401,225,-734) (13/4,49/15) -> (49/15,36/11) Parabolic Matrix(2872,-9801,841,-2870) (58/17,99/29) -> (99/29,41/12) Parabolic Matrix(58,-199,7,-24) (24/7,7/2) -> (8/1,1/0) Hyperbolic Matrix(176,-625,49,-174) (7/2,25/7) -> (25/7,18/5) Parabolic Matrix(584,-2809,121,-582) (24/5,53/11) -> (53/11,29/6) Parabolic Matrix(136,-729,25,-134) (16/3,27/5) -> (27/5,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,2,1) Matrix(200,-23,87,-10) -> Matrix(1,-2,-2,5) Matrix(140,-19,339,-46) -> Matrix(1,-2,2,-3) Matrix(56,-9,137,-22) -> Matrix(1,0,2,1) Matrix(81,-14,110,-19) -> Matrix(1,0,0,1) Matrix(211,-46,78,-17) -> Matrix(1,-2,0,1) Matrix(234,-53,53,-12) -> Matrix(1,-4,-2,9) Matrix(104,-25,25,-6) -> Matrix(1,2,-2,-3) Matrix(50,-13,77,-20) -> Matrix(3,2,-2,-1) Matrix(275,-76,76,-21) -> Matrix(1,0,4,1) Matrix(223,-66,98,-29) -> Matrix(5,-2,-12,5) Matrix(25,-8,72,-23) -> Matrix(1,-2,0,1) Matrix(189,-68,164,-59) -> Matrix(7,8,-8,-9) Matrix(211,-78,46,-17) -> Matrix(3,2,-8,-5) Matrix(92,-35,163,-62) -> Matrix(1,0,2,1) Matrix(299,-116,116,-45) -> Matrix(1,0,4,1) Matrix(634,-263,135,-56) -> Matrix(3,-2,-10,7) Matrix(157,-66,226,-95) -> Matrix(3,-2,-4,3) Matrix(223,-98,66,-29) -> Matrix(1,-2,-4,9) Matrix(198,-89,89,-40) -> Matrix(1,-8,-2,17) Matrix(44,-21,21,-10) -> Matrix(1,2,-2,-3) Matrix(146,-79,207,-112) -> Matrix(1,2,-2,-3) Matrix(391,-216,248,-137) -> Matrix(5,2,-8,-3) Matrix(61,-36,100,-59) -> Matrix(1,-4,0,1) Matrix(139,-86,118,-73) -> Matrix(3,10,-4,-13) Matrix(256,-161,353,-222) -> Matrix(1,2,-2,-3) Matrix(391,-248,216,-137) -> Matrix(1,2,0,1) Matrix(359,-248,152,-105) -> Matrix(7,6,-20,-17) Matrix(299,-212,244,-173) -> Matrix(1,0,0,1) Matrix(317,-228,260,-187) -> Matrix(5,2,-8,-3) Matrix(295,-216,56,-41) -> Matrix(1,0,-4,1) Matrix(72,-55,55,-42) -> Matrix(1,0,-2,1) Matrix(90,-71,71,-56) -> Matrix(1,4,-2,-7) Matrix(299,-244,212,-173) -> Matrix(1,0,0,1) Matrix(317,-260,228,-187) -> Matrix(1,2,0,1) Matrix(121,-102,70,-59) -> Matrix(1,2,-4,-7) Matrix(197,-220,60,-67) -> Matrix(5,6,-16,-19) Matrix(535,-648,232,-281) -> Matrix(7,4,-16,-9) Matrix(343,-424,72,-89) -> Matrix(3,2,-8,-5) Matrix(81,-110,14,-19) -> Matrix(1,0,0,1) Matrix(256,-353,161,-222) -> Matrix(1,2,-2,-3) Matrix(965,-1338,282,-391) -> Matrix(1,0,-4,1) Matrix(1193,-1686,438,-619) -> Matrix(1,0,0,1) Matrix(146,-207,79,-112) -> Matrix(1,2,-2,-3) Matrix(157,-226,66,-95) -> Matrix(1,2,-4,-7) Matrix(50,-77,13,-20) -> Matrix(3,2,-2,-1) Matrix(930,-1469,509,-804) -> Matrix(7,4,-2,-1) Matrix(992,-1569,545,-862) -> Matrix(3,2,-2,-1) Matrix(61,-100,36,-59) -> Matrix(7,4,-16,-9) Matrix(271,-466,82,-141) -> Matrix(11,4,-36,-13) Matrix(92,-163,35,-62) -> Matrix(1,0,2,1) Matrix(395,-916,116,-269) -> Matrix(5,2,-28,-11) Matrix(856,-2025,361,-854) -> Matrix(41,14,-126,-43) Matrix(282,-679,103,-248) -> Matrix(13,4,-10,-3) Matrix(140,-339,19,-46) -> Matrix(7,2,-18,-5) Matrix(56,-137,9,-22) -> Matrix(1,0,2,1) Matrix(208,-529,81,-206) -> Matrix(1,0,10,1) Matrix(293,-796,60,-163) -> Matrix(1,2,-4,-7) Matrix(25,-72,8,-23) -> Matrix(3,2,-8,-5) Matrix(736,-2401,225,-734) -> Matrix(77,26,-234,-79) Matrix(2872,-9801,841,-2870) -> Matrix(1,0,6,1) Matrix(58,-199,7,-24) -> Matrix(1,0,2,1) Matrix(176,-625,49,-174) -> Matrix(1,0,14,1) Matrix(584,-2809,121,-582) -> Matrix(5,2,-18,-7) Matrix(136,-729,25,-134) -> Matrix(1,0,10,1) 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): 21 ----------------------------------------------------------------------- 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 -1/1 0/1 2 1 1/1 (-1/1,0/1) 0 16 6/5 (-3/4,-2/3) 0 32 11/9 -1/2 2 8 5/4 (-2/3,-1/2) 0 32 9/7 -1/2 4 2 4/3 (-1/2,0/1) 0 32 11/8 (0/1,1/2) 0 32 18/13 (0/1,1/0) 0 32 25/18 (0/1,1/2) 0 32 7/5 1/0 2 8 24/17 (-2/1,1/0) 0 32 17/12 (-1/1,1/0) 0 32 10/7 (-3/2,-1/1) 0 32 3/2 (-1/1,-3/4) 0 32 11/7 (-2/3,-3/5) 0 16 19/12 (-2/3,-1/2) 0 32 8/5 (-2/3,-5/8) 0 32 5/3 -1/2 4 4 12/7 (-2/5,-3/8) 0 32 19/11 (-1/3,-2/7) 0 16 7/4 (-1/2,0/1) 0 32 9/5 (-1/1,0/1) 0 16 20/11 (0/1,1/0) 0 32 31/17 1/0 2 2 11/6 (-1/1,1/0) 0 32 2/1 (-1/1,-1/2) 0 32 11/5 -1/2 10 2 9/4 (-1/2,-4/9) 0 32 7/3 (-2/5,-1/3) 0 16 45/19 -1/3 14 1 19/8 (-1/3,-5/16) 0 32 12/5 (-1/2,-1/3) 0 32 29/12 (-3/10,-2/7) 0 32 17/7 -1/4 2 4 5/2 (-1/4,0/1) 0 32 23/9 0/1 10 1 13/5 (0/1,1/1) 0 16 8/3 (0/1,1/0) 0 32 11/4 (-5/4,-1/1) 0 32 3/1 -1/2 2 8 13/4 (-7/20,-1/3) 0 32 49/15 -1/3 26 1 23/7 (-1/3,-6/19) 0 16 10/3 (-2/7,-1/4) 0 32 17/5 (-1/5,0/1) 0 16 99/29 0/1 6 1 41/12 (-1/2,0/1) 0 32 24/7 (-1/4,0/1) 0 32 7/2 (-1/6,0/1) 0 32 25/7 0/1 14 1 11/3 (0/1,1/1) 0 16 4/1 (-1/1,-1/2) 0 32 13/3 -1/2 6 2 9/2 (-1/2,-2/5) 0 32 14/3 (-1/3,-3/10) 0 32 19/4 (-1/2,0/1) 0 32 24/5 (-1/2,-1/3) 0 32 53/11 -1/3 2 1 5/1 (-1/3,0/1) 0 16 27/5 0/1 10 1 11/2 (0/1,1/2) 0 32 6/1 (-1/2,0/1) 0 32 7/1 -1/2 2 4 8/1 (-1/4,0/1) 0 32 1/0 (-1/2,0/1) 0 32 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(102,-121,59,-70) (1/1,6/5) -> (19/11,7/4) Glide Reflection Matrix(260,-317,187,-228) (6/5,11/9) -> (25/18,7/5) Glide Reflection Matrix(244,-299,173,-212) (11/9,5/4) -> (7/5,24/17) Glide Reflection Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(81,-110,14,-19) (4/3,11/8) -> (11/2,6/1) Hyperbolic Matrix(256,-353,161,-222) (11/8,18/13) -> (19/12,8/5) Hyperbolic Matrix(965,-1338,282,-391) (18/13,25/18) -> (41/12,24/7) Hyperbolic Matrix(636,-899,133,-188) (24/17,17/12) -> (19/4,24/5) Glide Reflection Matrix(146,-207,79,-112) (17/12,10/7) -> (11/6,2/1) Hyperbolic Matrix(157,-226,66,-95) (10/7,3/2) -> (19/8,12/5) Hyperbolic Matrix(50,-77,13,-20) (3/2,11/7) -> (11/3,4/1) Hyperbolic Matrix(248,-391,137,-216) (11/7,19/12) -> (9/5,20/11) Glide Reflection Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(271,-466,82,-141) (12/7,19/11) -> (23/7,10/3) Hyperbolic Matrix(92,-163,35,-62) (7/4,9/5) -> (13/5,8/3) Hyperbolic Matrix(681,-1240,374,-681) (20/11,31/17) -> (20/11,31/17) Reflection Matrix(373,-682,204,-373) (31/17,11/6) -> (31/17,11/6) Reflection Matrix(21,-44,10,-21) (2/1,11/5) -> (2/1,11/5) Reflection Matrix(89,-198,40,-89) (11/5,9/4) -> (11/5,9/4) Reflection Matrix(98,-223,29,-66) (9/4,7/3) -> (10/3,17/5) Glide Reflection Matrix(134,-315,57,-134) (7/3,45/19) -> (7/3,45/19) Reflection Matrix(721,-1710,304,-721) (45/19,19/8) -> (45/19,19/8) Reflection Matrix(263,-634,56,-135) (12/5,29/12) -> (14/3,19/4) Glide Reflection Matrix(140,-339,19,-46) (29/12,17/7) -> (7/1,8/1) Hyperbolic Matrix(56,-137,9,-22) (17/7,5/2) -> (6/1,7/1) Hyperbolic Matrix(91,-230,36,-91) (5/2,23/9) -> (5/2,23/9) Reflection Matrix(116,-299,45,-116) (23/9,13/5) -> (23/9,13/5) Reflection Matrix(78,-211,17,-46) (8/3,11/4) -> (9/2,14/3) Glide Reflection Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(391,-1274,120,-391) (13/4,49/15) -> (13/4,49/15) Reflection Matrix(344,-1127,105,-344) (49/15,23/7) -> (49/15,23/7) Reflection Matrix(494,-1683,145,-494) (17/5,99/29) -> (17/5,99/29) Reflection Matrix(2377,-8118,696,-2377) (99/29,41/12) -> (99/29,41/12) Reflection Matrix(58,-199,7,-24) (24/7,7/2) -> (8/1,1/0) Hyperbolic Matrix(99,-350,28,-99) (7/2,25/7) -> (7/2,25/7) Reflection Matrix(76,-275,21,-76) (25/7,11/3) -> (25/7,11/3) Reflection Matrix(25,-104,6,-25) (4/1,13/3) -> (4/1,13/3) Reflection Matrix(53,-234,12,-53) (13/3,9/2) -> (13/3,9/2) Reflection Matrix(529,-2544,110,-529) (24/5,53/11) -> (24/5,53/11) Reflection Matrix(54,-265,11,-54) (53/11,5/1) -> (53/11,5/1) Reflection Matrix(26,-135,5,-26) (5/1,27/5) -> (5/1,27/5) 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(1,2,0,-1) -> Matrix(-1,0,4,1) (-1/1,1/0) -> (-1/2,0/1) Matrix(0,1,1,0) -> Matrix(-1,0,2,1) (-1/1,1/1) -> (-1/1,0/1) Matrix(102,-121,59,-70) -> Matrix(3,2,-10,-7) Matrix(260,-317,187,-228) -> Matrix(3,2,2,1) Matrix(244,-299,173,-212) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(71,-90,56,-71) -> Matrix(7,4,-12,-7) (5/4,9/7) -> (-2/3,-1/2) Matrix(55,-72,42,-55) -> Matrix(-1,0,4,1) (9/7,4/3) -> (-1/2,0/1) Matrix(81,-110,14,-19) -> Matrix(1,0,0,1) Matrix(256,-353,161,-222) -> Matrix(1,2,-2,-3) -1/1 Matrix(965,-1338,282,-391) -> Matrix(1,0,-4,1) 0/1 Matrix(636,-899,133,-188) -> Matrix(1,2,-2,-5) Matrix(146,-207,79,-112) -> Matrix(1,2,-2,-3) -1/1 Matrix(157,-226,66,-95) -> Matrix(1,2,-4,-7) Matrix(50,-77,13,-20) -> Matrix(3,2,-2,-1) -1/1 Matrix(248,-391,137,-216) -> Matrix(3,2,2,1) Matrix(61,-100,36,-59) -> Matrix(7,4,-16,-9) -1/2 Matrix(271,-466,82,-141) -> Matrix(11,4,-36,-13) -1/3 Matrix(92,-163,35,-62) -> Matrix(1,0,2,1) 0/1 Matrix(681,-1240,374,-681) -> Matrix(1,0,0,-1) (20/11,31/17) -> (0/1,1/0) Matrix(373,-682,204,-373) -> Matrix(1,2,0,-1) (31/17,11/6) -> (-1/1,1/0) Matrix(21,-44,10,-21) -> Matrix(3,2,-4,-3) (2/1,11/5) -> (-1/1,-1/2) Matrix(89,-198,40,-89) -> Matrix(17,8,-36,-17) (11/5,9/4) -> (-1/2,-4/9) Matrix(98,-223,29,-66) -> Matrix(5,2,-22,-9) Matrix(134,-315,57,-134) -> Matrix(11,4,-30,-11) (7/3,45/19) -> (-2/5,-1/3) Matrix(721,-1710,304,-721) -> Matrix(31,10,-96,-31) (45/19,19/8) -> (-1/3,-5/16) Matrix(263,-634,56,-135) -> Matrix(7,2,-24,-7) *** -> (-1/3,-1/4) Matrix(140,-339,19,-46) -> Matrix(7,2,-18,-5) -1/3 Matrix(56,-137,9,-22) -> Matrix(1,0,2,1) 0/1 Matrix(91,-230,36,-91) -> Matrix(-1,0,8,1) (5/2,23/9) -> (-1/4,0/1) Matrix(116,-299,45,-116) -> Matrix(1,0,2,-1) (23/9,13/5) -> (0/1,1/1) Matrix(78,-211,17,-46) -> Matrix(1,2,-2,-5) Matrix(25,-72,8,-23) -> Matrix(3,2,-8,-5) -1/2 Matrix(391,-1274,120,-391) -> Matrix(41,14,-120,-41) (13/4,49/15) -> (-7/20,-1/3) Matrix(344,-1127,105,-344) -> Matrix(37,12,-114,-37) (49/15,23/7) -> (-1/3,-6/19) Matrix(494,-1683,145,-494) -> Matrix(-1,0,10,1) (17/5,99/29) -> (-1/5,0/1) Matrix(2377,-8118,696,-2377) -> Matrix(-1,0,4,1) (99/29,41/12) -> (-1/2,0/1) Matrix(58,-199,7,-24) -> Matrix(1,0,2,1) 0/1 Matrix(99,-350,28,-99) -> Matrix(-1,0,12,1) (7/2,25/7) -> (-1/6,0/1) Matrix(76,-275,21,-76) -> Matrix(1,0,2,-1) (25/7,11/3) -> (0/1,1/1) Matrix(25,-104,6,-25) -> Matrix(3,2,-4,-3) (4/1,13/3) -> (-1/1,-1/2) Matrix(53,-234,12,-53) -> Matrix(9,4,-20,-9) (13/3,9/2) -> (-1/2,-2/5) Matrix(529,-2544,110,-529) -> Matrix(5,2,-12,-5) (24/5,53/11) -> (-1/2,-1/3) Matrix(54,-265,11,-54) -> Matrix(-1,0,6,1) (53/11,5/1) -> (-1/3,0/1) Matrix(26,-135,5,-26) -> Matrix(-1,0,6,1) (5/1,27/5) -> (-1/3,0/1) Matrix(109,-594,20,-109) -> Matrix(1,0,4,-1) (27/5,11/2) -> (0/1,1/2) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.