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 1/31 -6/13 1/30 -5/11 1/30 -9/20 2/59 1/29 -4/9 1/28 -7/16 0/1 1/31 -10/23 3/86 -3/7 1/28 -14/33 1/26 -11/26 2/55 1/27 -8/19 3/80 -5/12 0/1 1/29 -12/29 1/28 -7/17 1/28 -9/22 4/109 1/27 -2/5 1/26 -7/18 1/27 2/53 -5/13 1/26 -13/34 5/129 4/103 -8/21 5/128 -3/8 0/1 1/25 -13/35 1/26 -10/27 1/26 -7/19 1/26 -11/30 7/179 2/51 -15/41 3/76 -4/11 3/76 -5/14 1/25 4/99 -11/31 3/74 -6/17 5/122 -1/3 1/24 -7/22 2/47 1/23 -6/19 1/22 -5/16 3/71 2/47 -9/29 3/70 -4/13 5/116 -3/10 1/23 4/91 -8/27 5/112 -5/17 1/22 -7/24 1/23 4/91 -2/7 1/22 -5/18 4/89 5/111 -3/11 1/22 -7/26 6/131 5/109 -11/41 1/22 -4/15 7/152 -1/4 2/43 1/21 -4/17 9/188 -7/30 6/125 11/229 -3/13 3/62 -11/48 5/103 2/41 -8/35 1/20 -5/22 1/21 4/83 -2/9 3/62 -3/14 2/41 3/61 -7/33 1/20 -4/19 1/20 -1/5 1/20 -4/21 1/20 -11/58 4/79 3/59 -7/37 1/20 -3/16 4/79 3/59 -5/27 5/98 -2/11 5/98 -1/6 4/77 1/19 -2/13 9/170 -3/20 1/19 4/75 -1/7 3/56 -3/22 13/241 2/37 -2/15 9/166 -1/8 3/55 4/73 0/1 1/16 1/9 5/72 1/8 4/57 5/71 1/7 1/14 2/13 1/14 1/6 4/55 3/41 1/5 3/40 2/9 9/118 3/13 1/13 4/17 17/220 1/4 1/13 4/51 4/15 7/88 3/11 3/38 2/7 5/62 3/10 3/37 4/49 1/3 1/12 5/14 3/35 2/23 4/11 1/12 3/8 3/35 2/23 8/21 1/12 5/13 5/58 2/5 3/34 9/22 2/23 5/57 7/17 3/34 12/29 5/56 17/41 5/56 5/12 4/45 1/11 3/7 3/34 4/9 9/100 5/11 1/11 6/13 13/142 1/2 1/11 2/21 9/17 5/52 8/15 5/52 7/13 1/10 6/11 1/10 5/9 1/10 9/16 2/21 7/73 4/7 7/72 3/5 1/10 8/13 13/128 5/8 5/49 4/39 17/27 7/68 29/46 37/359 10/97 41/65 33/320 12/19 19/184 7/11 5/48 9/14 2/19 5/47 2/3 1/10 9/13 3/28 7/10 4/37 1/9 19/27 3/26 12/17 1/8 5/7 1/10 13/18 2/19 1/9 21/29 1/10 8/11 5/48 11/15 7/66 3/4 4/37 1/9 7/9 1/9 11/14 1/9 14/125 4/5 5/44 9/11 3/26 5/6 2/17 3/25 1/1 1/8 8/7 1/8 7/6 0/1 1/7 6/5 5/38 17/14 2/15 5/37 11/9 3/22 16/13 1/8 5/4 4/29 1/7 9/7 1/7 13/10 1/7 10/69 4/3 3/20 11/8 2/13 7/45 18/13 13/82 25/18 4/25 13/81 7/5 1/6 24/17 1/4 17/12 0/1 1/7 10/7 1/6 3/2 0/1 1/7 14/9 7/46 25/16 2/13 11/71 11/7 1/6 30/19 1/6 49/31 1/7 19/12 1/7 2/13 46/29 9/58 27/17 1/6 8/5 5/32 5/3 1/6 12/7 7/40 19/11 1/6 7/4 2/11 1/5 16/9 5/28 25/14 2/11 9/49 9/5 3/16 20/11 11/56 31/17 1/5 11/6 1/5 4/19 13/7 1/4 15/8 1/5 2/9 2/1 1/6 11/5 1/5 20/9 9/44 9/4 1/5 4/19 16/7 1/4 39/17 3/14 23/10 1/5 2/9 30/13 11/50 7/3 1/4 26/11 3/14 45/19 2/9 64/27 9/40 19/8 2/9 3/13 12/5 1/4 41/17 1/4 111/46 11/47 4/17 70/29 9/38 29/12 5/21 6/25 17/7 1/4 22/9 9/34 5/2 0/1 1/3 23/9 0/1 41/16 0/1 1/17 18/7 1/10 13/5 1/6 47/18 0/1 1/7 81/31 1/7 34/13 1/6 21/8 0/1 1/7 8/3 3/16 19/7 3/14 49/18 2/9 3/13 30/11 3/14 11/4 1/5 2/9 3/1 1/4 13/4 0/1 1/3 49/15 0/1 85/26 0/1 1/5 36/11 1/4 23/7 1/4 10/3 3/10 17/5 3/8 58/17 15/38 99/29 2/5 41/12 2/5 11/27 24/7 7/16 31/9 1/2 7/2 0/1 1/1 25/7 0/1 43/12 0/1 1/19 18/5 1/10 11/3 1/6 37/10 8/41 1/5 63/17 1/5 26/7 7/34 15/4 2/9 3/13 4/1 1/4 13/3 1/3 22/5 5/14 9/2 1/3 2/5 14/3 1/2 33/7 1/2 19/4 2/3 1/1 24/5 1/0 53/11 0/1 82/17 1/10 29/6 0/1 1/5 5/1 1/2 16/3 -1/4 27/5 0/1 38/7 1/18 11/2 0/1 1/7 6/1 1/2 13/2 1/3 2/5 7/1 1/2 15/2 0/1 1/1 8/1 1/0 9/1 1/0 19/2 -1/7 0/1 29/3 0/1 10/1 1/6 1/0 0/1 1/1 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(1,0,30,1) Matrix(121,56,296,137) -> Matrix(57,-2,656,-23) Matrix(239,110,-654,-301) -> Matrix(57,-2,1454,-51) Matrix(173,78,-814,-367) -> Matrix(119,-4,2410,-81) Matrix(521,234,118,53) -> Matrix(117,-4,322,-11) Matrix(113,50,174,77) -> Matrix(57,-2,542,-19) Matrix(225,98,-962,-419) -> Matrix(175,-6,3646,-125) Matrix(167,72,-392,-169) -> Matrix(57,-2,1568,-55) Matrix(387,164,-1220,-517) -> Matrix(1,0,-4,1) Matrix(109,46,-718,-303) -> Matrix(163,-6,3070,-113) Matrix(219,92,388,163) -> Matrix(51,-2,536,-21) Matrix(53,22,330,137) -> Matrix(113,-4,1554,-55) Matrix(547,226,-1474,-609) -> Matrix(55,-2,1458,-53) Matrix(161,66,-866,-355) -> Matrix(219,-8,4298,-157) Matrix(485,198,218,89) -> Matrix(217,-8,1058,-39) Matrix(159,62,418,163) -> Matrix(105,-4,1234,-47) Matrix(259,100,-676,-261) -> Matrix(157,-6,4056,-155) Matrix(309,118,-982,-375) -> Matrix(51,-2,1250,-49) Matrix(53,20,204,77) -> Matrix(101,-4,1288,-51) Matrix(409,152,-1784,-663) -> Matrix(55,-2,1128,-41) Matrix(303,112,560,207) -> Matrix(51,-2,536,-21) Matrix(605,222,962,353) -> Matrix(461,-18,4482,-175) Matrix(153,56,-1112,-407) -> Matrix(305,-12,5668,-223) Matrix(199,72,152,55) -> Matrix(151,-6,1032,-41) Matrix(647,230,346,123) -> Matrix(49,-2,270,-11) Matrix(1885,668,1188,421) -> Matrix(197,-8,1256,-51) Matrix(199,70,-742,-261) -> Matrix(145,-6,3166,-131) Matrix(187,60,-988,-317) -> Matrix(49,-2,956,-39) Matrix(1113,346,1766,549) -> Matrix(1029,-44,9986,-427) Matrix(323,100,604,187) -> Matrix(233,-10,2400,-103) Matrix(183,56,232,71) -> Matrix(231,-10,2056,-89) Matrix(47,14,-366,-109) -> Matrix(181,-8,3326,-147) Matrix(865,256,544,161) -> Matrix(1,0,-16,1) Matrix(499,146,270,79) -> Matrix(1,0,-18,1) Matrix(227,66,-994,-289) -> Matrix(1,0,-2,1) Matrix(177,50,46,13) -> Matrix(45,-2,158,-7) Matrix(131,36,-484,-133) -> Matrix(221,-10,4840,-219) Matrix(305,82,-1618,-435) -> Matrix(43,-2,882,-41) Matrix(347,92,132,35) -> Matrix(43,-2,280,-13) Matrix(41,10,86,21) -> Matrix(85,-4,914,-43) Matrix(43,10,374,87) -> Matrix(291,-14,4178,-201) Matrix(3737,856,1576,361) -> Matrix(329,-16,1460,-71) Matrix(249,56,40,9) -> Matrix(41,-2,144,-7) Matrix(287,62,162,35) -> Matrix(81,-4,466,-23) Matrix(1175,248,488,103) -> Matrix(1,0,-16,1) Matrix(39,8,-200,-41) -> Matrix(121,-6,2400,-119) Matrix(3873,736,1184,225) -> Matrix(79,-4,336,-17) Matrix(315,58,38,7) -> Matrix(39,-2,98,-5) Matrix(117,20,76,13) -> Matrix(77,-4,520,-27) Matrix(37,6,154,25) -> Matrix(153,-8,1970,-103) Matrix(375,56,904,135) -> Matrix(151,-8,1680,-89) Matrix(1001,136,184,25) -> Matrix(37,-2,500,-27) Matrix(181,22,74,9) -> Matrix(73,-4,274,-15) Matrix(149,-12,236,-19) -> Matrix(237,-16,2296,-155) Matrix(397,-52,84,-11) -> Matrix(85,-6,156,-11) Matrix(527,-80,112,-17) -> Matrix(111,-8,236,-17) Matrix(81,-14,110,-19) -> Matrix(109,-8,1022,-75) Matrix(211,-46,78,-17) -> Matrix(79,-6,382,-29) Matrix(79,-18,338,-77) -> Matrix(339,-26,4394,-337) Matrix(403,-108,556,-149) -> Matrix(51,-4,472,-37) Matrix(275,-76,76,-21) -> Matrix(25,-2,188,-15) Matrix(223,-66,98,-29) -> Matrix(99,-8,458,-37) Matrix(25,-8,72,-23) -> Matrix(73,-6,864,-71) Matrix(189,-68,164,-59) -> Matrix(23,-2,196,-17) Matrix(211,-78,46,-17) -> Matrix(47,-4,106,-9) Matrix(555,-212,788,-301) -> Matrix(47,-4,388,-33) Matrix(299,-116,116,-45) -> Matrix(23,-2,196,-17) Matrix(1249,-512,544,-223) -> Matrix(137,-12,628,-55) Matrix(1403,-580,612,-253) -> Matrix(67,-6,324,-29) Matrix(589,-244,1108,-459) -> Matrix(113,-10,1164,-103) Matrix(157,-66,226,-95) -> Matrix(1,0,-2,1) Matrix(223,-98,66,-29) -> Matrix(67,-6,190,-17) Matrix(111,-50,242,-109) -> Matrix(243,-22,2662,-241) Matrix(1969,-1040,816,-431) -> Matrix(187,-18,800,-77) Matrix(391,-216,248,-137) -> Matrix(1,0,-4,1) Matrix(369,-206,206,-115) -> Matrix(43,-4,226,-21) Matrix(61,-36,100,-59) -> Matrix(101,-10,1000,-99) Matrix(139,-86,118,-73) -> Matrix(39,-4,322,-33) Matrix(1649,-1040,176,-111) -> Matrix(97,-10,-320,33) Matrix(391,-248,216,-137) -> Matrix(135,-14,704,-73) Matrix(429,-274,274,-175) -> Matrix(115,-12,738,-77) Matrix(359,-248,152,-105) -> Matrix(37,-4,176,-19) Matrix(1109,-780,300,-211) -> Matrix(35,-4,184,-21) Matrix(299,-212,244,-173) -> Matrix(17,-2,128,-15) Matrix(317,-228,260,-187) -> Matrix(77,-8,568,-59) Matrix(1597,-1156,612,-443) -> Matrix(19,-2,124,-13) Matrix(295,-216,56,-41) -> Matrix(19,-2,-28,3) Matrix(127,-98,162,-125) -> Matrix(163,-18,1458,-161) Matrix(299,-244,212,-173) -> Matrix(35,-4,184,-21) Matrix(317,-260,228,-187) -> Matrix(121,-14,752,-87) Matrix(121,-102,70,-59) -> Matrix(33,-4,190,-23) Matrix(197,-220,60,-67) -> Matrix(1,0,-4,1) Matrix(535,-648,232,-281) -> Matrix(89,-12,408,-55) Matrix(343,-424,72,-89) -> Matrix(15,-2,8,-1) Matrix(127,-162,98,-125) -> Matrix(99,-14,686,-97) Matrix(81,-110,14,-19) -> Matrix(13,-2,46,-7) Matrix(403,-556,108,-149) -> Matrix(51,-8,236,-37) Matrix(965,-1338,282,-391) -> Matrix(113,-18,270,-43) Matrix(1193,-1686,438,-619) -> Matrix(11,-2,50,-9) Matrix(555,-788,212,-301) -> Matrix(1,0,0,1) Matrix(157,-226,66,-95) -> Matrix(11,-2,50,-9) Matrix(675,-1052,188,-293) -> Matrix(13,-2,176,-27) Matrix(2593,-4096,992,-1567) -> Matrix(1,0,0,1) Matrix(2429,-3842,930,-1471) -> Matrix(13,-2,98,-15) Matrix(149,-236,12,-19) -> Matrix(13,-2,20,-3) Matrix(61,-100,36,-59) -> Matrix(37,-6,216,-35) Matrix(271,-466,82,-141) -> Matrix(11,-2,50,-9) Matrix(621,-1106,242,-431) -> Matrix(11,-2,138,-25) Matrix(1135,-2066,306,-557) -> Matrix(91,-18,450,-89) Matrix(1007,-1840,272,-497) -> Matrix(59,-12,300,-61) Matrix(589,-1108,244,-459) -> Matrix(15,-4,64,-17) Matrix(111,-242,50,-109) -> Matrix(51,-10,250,-49) Matrix(395,-916,116,-269) -> Matrix(35,-8,92,-21) Matrix(1711,-4050,722,-1709) -> Matrix(109,-24,486,-107) Matrix(4613,-11132,956,-2307) -> Matrix(17,-4,132,-31) Matrix(421,-1018,122,-295) -> Matrix(25,-6,46,-11) Matrix(447,-1090,130,-317) -> Matrix(31,-8,66,-17) Matrix(415,-1058,162,-413) -> Matrix(1,0,14,1) Matrix(293,-796,60,-163) -> Matrix(9,-2,32,-7) Matrix(197,-538,26,-71) -> Matrix(9,-2,14,-3) Matrix(25,-72,8,-23) -> Matrix(9,-2,32,-7) Matrix(1471,-4802,450,-1469) -> Matrix(1,0,2,1) Matrix(783,-2672,80,-273) -> Matrix(5,-2,68,-27) Matrix(899,-3070,94,-321) -> Matrix(5,-2,-62,25) Matrix(351,-1250,98,-349) -> Matrix(1,0,18,1) Matrix(79,-338,18,-77) -> Matrix(19,-6,54,-17) Matrix(1167,-5618,242,-1165) -> Matrix(1,0,10,1) Matrix(271,-1458,50,-269) -> Matrix(1,0,22,1) Matrix(29,-196,4,-27) -> Matrix(5,-2,8,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 64 Degree of the the map X: 64 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 -3/7 -1/3 -1/5 0/1 1/7 3/13 1/3 5/11 1/2 3/5 7/9 1/1 9/7 3/2 5/3 2/1 11/5 5/2 8/3 3/1 49/15 10/3 4/1 13/3 14/3 5/1 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 -1/1 0/1 -1/2 0/1 1/31 -5/11 1/30 -9/20 2/59 1/29 -4/9 1/28 -3/7 1/28 -11/26 2/55 1/27 -8/19 3/80 -5/12 0/1 1/29 -12/29 1/28 -7/17 1/28 -9/22 4/109 1/27 -2/5 1/26 -3/8 0/1 1/25 -4/11 3/76 -5/14 1/25 4/99 -1/3 1/24 -4/13 5/116 -3/10 1/23 4/91 -8/27 5/112 -5/17 1/22 -2/7 1/22 -1/4 2/43 1/21 -4/17 9/188 -3/13 3/62 -5/22 1/21 4/83 -2/9 3/62 -3/14 2/41 3/61 -7/33 1/20 -4/19 1/20 -1/5 1/20 -3/16 4/79 3/59 -5/27 5/98 -2/11 5/98 -1/6 4/77 1/19 -2/13 9/170 -3/20 1/19 4/75 -1/7 3/56 -2/15 9/166 -1/8 3/55 4/73 0/1 1/16 1/7 1/14 1/6 4/55 3/41 1/5 3/40 2/9 9/118 3/13 1/13 4/17 17/220 1/4 1/13 4/51 2/7 5/62 3/10 3/37 4/49 1/3 1/12 2/5 3/34 3/7 3/34 4/9 9/100 5/11 1/11 6/13 13/142 1/2 1/11 2/21 7/13 1/10 6/11 1/10 5/9 1/10 9/16 2/21 7/73 4/7 7/72 3/5 1/10 8/13 13/128 5/8 5/49 4/39 2/3 1/10 3/4 4/37 1/9 7/9 1/9 11/14 1/9 14/125 4/5 5/44 5/6 2/17 3/25 1/1 1/8 5/4 4/29 1/7 9/7 1/7 13/10 1/7 10/69 4/3 3/20 7/5 1/6 17/12 0/1 1/7 10/7 1/6 3/2 0/1 1/7 14/9 7/46 11/7 1/6 30/19 1/6 49/31 1/7 19/12 1/7 2/13 27/17 1/6 8/5 5/32 5/3 1/6 12/7 7/40 19/11 1/6 7/4 2/11 1/5 2/1 1/6 11/5 1/5 20/9 9/44 9/4 1/5 4/19 16/7 1/4 7/3 1/4 12/5 1/4 29/12 5/21 6/25 17/7 1/4 22/9 9/34 5/2 0/1 1/3 13/5 1/6 34/13 1/6 21/8 0/1 1/7 8/3 3/16 19/7 3/14 30/11 3/14 11/4 1/5 2/9 3/1 1/4 13/4 0/1 1/3 49/15 0/1 36/11 1/4 23/7 1/4 10/3 3/10 17/5 3/8 7/2 0/1 1/1 4/1 1/4 13/3 1/3 22/5 5/14 9/2 1/3 2/5 14/3 1/2 5/1 1/2 6/1 1/2 13/2 1/3 2/5 7/1 1/2 15/2 0/1 1/1 8/1 1/0 9/1 1/0 1/0 0/1 1/1 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(61,28,-268,-123) (-1/2,-5/11) -> (-3/13,-5/22) 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(59,26,-202,-89) (-4/9,-3/7) -> (-5/17,-2/7) Hyperbolic Matrix(835,354,526,223) (-3/7,-11/26) -> (19/12,27/17) 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(985,408,408,169) (-5/12,-12/29) -> (12/5,29/12) Hyperbolic Matrix(271,112,-1280,-529) (-12/29,-7/17) -> (-7/33,-4/19) 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(51,20,28,11) (-2/5,-3/8) -> (7/4,2/1) Hyperbolic Matrix(49,18,-226,-83) (-3/8,-4/11) -> (-2/9,-3/14) Hyperbolic Matrix(199,72,152,55) (-4/11,-5/14) -> (13/10,4/3) Hyperbolic Matrix(97,34,174,61) (-5/14,-1/3) -> (5/9,9/16) Hyperbolic Matrix(185,58,118,37) (-1/3,-4/13) -> (14/9,11/7) 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(43,12,68,19) (-2/7,-1/4) -> (5/8,2/3) Hyperbolic Matrix(41,10,86,21) (-1/4,-4/17) -> (6/13,1/2) Hyperbolic Matrix(43,10,-314,-73) (-4/17,-3/13) -> (-1/7,-2/15) Hyperbolic Matrix(249,56,40,9) (-5/22,-2/9) -> (6/1,13/2) Hyperbolic Matrix(163,34,302,63) (-4/19,-1/5) -> (7/13,6/11) Hyperbolic Matrix(117,22,218,41) (-1/5,-3/16) -> (1/2,7/13) 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(187,28,20,3) (-3/20,-1/7) -> (9/1,1/0) Hyperbolic Matrix(181,22,74,9) (-1/8,0/1) -> (22/9,5/2) Hyperbolic Matrix(171,-22,70,-9) (0/1,1/7) -> (17/7,22/9) Hyperbolic Matrix(305,-46,126,-19) (1/7,1/6) -> (29/12,17/7) Hyperbolic Matrix(137,-24,40,-7) (1/6,1/5) -> (17/5,7/2) 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(51,-14,62,-17) (1/4,2/7) -> (4/5,5/6) Hyperbolic Matrix(223,-66,98,-29) (2/7,3/10) -> (9/4,16/7) Hyperbolic Matrix(121,-38,86,-27) (3/10,1/3) -> (7/5,17/12) Hyperbolic Matrix(47,-18,34,-13) (1/3,2/5) -> (4/3,7/5) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) 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(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(337,-208,128,-79) (8/13,5/8) -> (21/8,8/3) Hyperbolic Matrix(37,-26,10,-7) (2/3,3/4) -> (7/2,4/1) Hyperbolic Matrix(127,-98,162,-125) (3/4,7/9) -> (7/9,11/14) Parabolic Matrix(121,-102,70,-59) (5/6,1/1) -> (19/11,7/4) Hyperbolic Matrix(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(555,-788,212,-301) (17/12,10/7) -> (34/13,21/8) Hyperbolic Matrix(91,-132,20,-29) (10/7,3/2) -> (9/2,14/3) Hyperbolic Matrix(2047,-3234,626,-989) (30/19,49/31) -> (49/15,36/11) Hyperbolic Matrix(991,-1568,304,-481) (49/31,19/12) -> (13/4,49/15) 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(111,-242,50,-109) (2/1,11/5) -> (11/5,20/9) Parabolic Matrix(223,-514,82,-189) (16/7,7/3) -> (19/7,30/11) Hyperbolic Matrix(67,-158,14,-33) (7/3,12/5) -> (14/3,5/1) Hyperbolic Matrix(479,-1250,146,-381) (13/5,34/13) -> (36/11,23/7) 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(79,-338,18,-77) (4/1,13/3) -> (13/3,22/5) 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(1,0,30,1) Matrix(61,28,-268,-123) -> Matrix(27,-1,568,-21) Matrix(173,78,-814,-367) -> Matrix(119,-4,2410,-81) Matrix(521,234,118,53) -> Matrix(117,-4,322,-11) Matrix(59,26,-202,-89) -> Matrix(27,-1,622,-23) Matrix(835,354,526,223) -> Matrix(83,-3,526,-19) Matrix(109,46,-718,-303) -> Matrix(163,-6,3070,-113) Matrix(219,92,388,163) -> Matrix(51,-2,536,-21) Matrix(985,408,408,169) -> Matrix(139,-5,584,-21) Matrix(271,112,-1280,-529) -> Matrix(139,-5,2808,-101) Matrix(161,66,-866,-355) -> Matrix(219,-8,4298,-157) Matrix(485,198,218,89) -> Matrix(217,-8,1058,-39) Matrix(51,20,28,11) -> Matrix(27,-1,136,-5) Matrix(49,18,-226,-83) -> Matrix(77,-3,1566,-61) Matrix(199,72,152,55) -> Matrix(151,-6,1032,-41) Matrix(97,34,174,61) -> Matrix(23,-1,254,-11) Matrix(185,58,118,37) -> Matrix(71,-3,450,-19) Matrix(183,56,232,71) -> Matrix(231,-10,2056,-89) Matrix(47,14,-366,-109) -> Matrix(181,-8,3326,-147) Matrix(865,256,544,161) -> Matrix(1,0,-16,1) Matrix(43,12,68,19) -> Matrix(67,-3,648,-29) Matrix(41,10,86,21) -> Matrix(85,-4,914,-43) Matrix(43,10,-314,-73) -> Matrix(187,-9,3470,-167) Matrix(249,56,40,9) -> Matrix(41,-2,144,-7) Matrix(163,34,302,63) -> Matrix(21,-1,190,-9) Matrix(117,22,218,41) -> Matrix(99,-5,1010,-51) Matrix(315,58,38,7) -> Matrix(39,-2,98,-5) Matrix(117,20,76,13) -> Matrix(77,-4,520,-27) Matrix(37,6,154,25) -> Matrix(153,-8,1970,-103) Matrix(187,28,20,3) -> Matrix(19,-1,-56,3) Matrix(181,22,74,9) -> Matrix(73,-4,274,-15) Matrix(171,-22,70,-9) -> Matrix(71,-5,270,-19) Matrix(305,-46,126,-19) -> Matrix(125,-9,514,-37) Matrix(137,-24,40,-7) -> Matrix(41,-3,96,-7) Matrix(211,-46,78,-17) -> Matrix(79,-6,382,-29) Matrix(79,-18,338,-77) -> Matrix(339,-26,4394,-337) Matrix(51,-14,62,-17) -> Matrix(63,-5,542,-43) Matrix(223,-66,98,-29) -> Matrix(99,-8,458,-37) Matrix(121,-38,86,-27) -> Matrix(37,-3,210,-17) Matrix(47,-18,34,-13) -> Matrix(35,-3,222,-19) Matrix(67,-28,12,-5) -> Matrix(11,-1,56,-5) Matrix(223,-98,66,-29) -> Matrix(67,-6,190,-17) Matrix(111,-50,242,-109) -> Matrix(243,-22,2662,-241) Matrix(391,-216,248,-137) -> Matrix(1,0,-4,1) Matrix(61,-36,100,-59) -> Matrix(101,-10,1000,-99) Matrix(337,-208,128,-79) -> Matrix(49,-5,304,-31) Matrix(37,-26,10,-7) -> Matrix(9,-1,46,-5) Matrix(127,-98,162,-125) -> Matrix(163,-18,1458,-161) Matrix(121,-102,70,-59) -> Matrix(33,-4,190,-23) Matrix(57,-70,22,-27) -> Matrix(7,-1,50,-7) Matrix(127,-162,98,-125) -> Matrix(99,-14,686,-97) Matrix(555,-788,212,-301) -> Matrix(1,0,0,1) Matrix(91,-132,20,-29) -> Matrix(5,-1,16,-3) Matrix(2047,-3234,626,-989) -> Matrix(7,-1,22,-3) Matrix(991,-1568,304,-481) -> Matrix(7,-1,8,-1) Matrix(61,-100,36,-59) -> Matrix(37,-6,216,-35) Matrix(271,-466,82,-141) -> Matrix(11,-2,50,-9) Matrix(111,-242,50,-109) -> Matrix(51,-10,250,-49) Matrix(223,-514,82,-189) -> Matrix(23,-5,106,-23) Matrix(67,-158,14,-33) -> Matrix(5,-1,6,-1) Matrix(479,-1250,146,-381) -> Matrix(7,-1,22,-3) Matrix(197,-538,26,-71) -> Matrix(9,-2,14,-3) Matrix(25,-72,8,-23) -> Matrix(9,-2,32,-7) Matrix(79,-338,18,-77) -> Matrix(19,-6,54,-17) Matrix(29,-196,4,-27) -> Matrix(5,-2,8,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 64 ----------------------------------------------------------------------- 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 15 1 0/1 1/16 1 16 1/7 1/14 7 2 1/6 (4/55,3/41) 0 16 1/5 3/40 1 8 2/9 9/118 1 16 3/13 1/13 13 1 1/4 (1/13,4/51) 0 16 2/7 5/62 1 16 3/10 (3/37,4/49) 0 16 1/3 1/12 3 4 2/5 3/34 1 16 3/7 3/34 1 8 4/9 9/100 1 16 5/11 1/11 11 1 1/2 (1/11,2/21) 0 16 6/11 1/10 1 16 5/9 1/10 1 8 4/7 7/72 1 16 3/5 1/10 5 2 2/3 1/10 1 16 3/4 (4/37,1/9) 0 16 7/9 1/9 9 1 4/5 5/44 1 16 5/6 (2/17,3/25) 0 16 1/1 1/8 1 8 5/4 (4/29,1/7) 0 16 9/7 1/7 7 1 4/3 3/20 1 16 7/5 1/6 3 4 17/12 (0/1,1/7) 0 16 10/7 1/6 1 16 3/2 (0/1,1/7) 0 16 11/7 1/6 1 8 30/19 1/6 1 16 49/31 1/7 1 1 19/12 (1/7,2/13) 0 16 8/5 5/32 1 16 5/3 1/6 3 2 2/1 1/6 1 16 11/5 1/5 5 1 9/4 (1/5,4/19) 0 16 16/7 1/4 1 16 7/3 1/4 1 8 12/5 1/4 1 16 17/7 1/4 7 2 5/2 (0/1,1/3) 0 16 13/5 1/6 1 8 8/3 3/16 1 16 19/7 3/14 1 8 30/11 3/14 1 16 11/4 (1/5,2/9) 0 16 3/1 1/4 1 4 13/4 (0/1,1/3) 0 16 10/3 3/10 1 16 17/5 3/8 1 8 7/2 (0/1,1/1) 0 16 4/1 1/4 1 16 13/3 1/3 3 1 9/2 (1/3,2/5) 0 16 14/3 1/2 1 16 5/1 1/2 1 8 6/1 1/2 1 16 7/1 1/2 1 2 8/1 1/0 1 16 1/0 (0/1,1/1) 0 16 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(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,14,-1) (0/1,1/7) -> (0/1,1/7) Reflection Matrix(137,-22,56,-9) (1/7,1/6) -> (17/7,5/2) Glide Reflection Matrix(137,-24,40,-7) (1/6,1/5) -> (17/5,7/2) Hyperbolic Matrix(211,-46,78,-17) (1/5,2/9) -> (8/3,19/7) Hyperbolic Matrix(53,-12,234,-53) (2/9,3/13) -> (2/9,3/13) Reflection Matrix(25,-6,104,-25) (3/13,1/4) -> (3/13,1/4) Reflection Matrix(51,-14,62,-17) (1/4,2/7) -> (4/5,5/6) Hyperbolic Matrix(223,-66,98,-29) (2/7,3/10) -> (9/4,16/7) Hyperbolic Matrix(121,-38,86,-27) (3/10,1/3) -> (7/5,17/12) Hyperbolic Matrix(47,-18,34,-13) (1/3,2/5) -> (4/3,7/5) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(223,-98,66,-29) (3/7,4/9) -> (10/3,17/5) Hyperbolic Matrix(89,-40,198,-89) (4/9,5/11) -> (4/9,5/11) Reflection Matrix(21,-10,44,-21) (5/11,1/2) -> (5/11,1/2) Reflection Matrix(207,-112,146,-79) (1/2,6/11) -> (17/12,10/7) Glide Reflection Matrix(391,-216,248,-137) (6/11,5/9) -> (11/7,30/19) Hyperbolic Matrix(163,-92,62,-35) (5/9,4/7) -> (13/5,8/3) Glide Reflection Matrix(41,-24,70,-41) (4/7,3/5) -> (4/7,3/5) Reflection Matrix(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(37,-26,10,-7) (2/3,3/4) -> (7/2,4/1) Hyperbolic Matrix(55,-42,72,-55) (3/4,7/9) -> (3/4,7/9) Reflection Matrix(71,-56,90,-71) (7/9,4/5) -> (7/9,4/5) Reflection Matrix(69,-58,44,-37) (5/6,1/1) -> (3/2,11/7) Glide Reflection Matrix(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic 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(91,-132,20,-29) (10/7,3/2) -> (9/2,14/3) Hyperbolic Matrix(1861,-2940,1178,-1861) (30/19,49/31) -> (30/19,49/31) Reflection Matrix(1177,-1862,744,-1177) (49/31,19/12) -> (49/31,19/12) Reflection Matrix(185,-294,56,-89) (19/12,8/5) -> (13/4,10/3) Glide Reflection Matrix(49,-80,30,-49) (8/5,5/3) -> (8/5,5/3) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) 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(223,-514,82,-189) (16/7,7/3) -> (19/7,30/11) Hyperbolic Matrix(67,-158,14,-33) (7/3,12/5) -> (14/3,5/1) Hyperbolic Matrix(169,-408,70,-169) (12/5,17/7) -> (12/5,17/7) Reflection Matrix(43,-118,4,-11) (30/11,11/4) -> (8/1,1/0) Glide Reflection Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic 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(13,-84,2,-13) (6/1,7/1) -> (6/1,7/1) Reflection Matrix(15,-112,2,-15) (7/1,8/1) -> (7/1,8/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,2,-1) (-1/1,1/0) -> (0/1,1/1) Matrix(-1,0,2,1) -> Matrix(1,0,32,-1) (-1/1,0/1) -> (0/1,1/16) Matrix(1,0,14,-1) -> Matrix(15,-1,224,-15) (0/1,1/7) -> (1/16,1/14) Matrix(137,-22,56,-9) -> Matrix(55,-4,206,-15) Matrix(137,-24,40,-7) -> Matrix(41,-3,96,-7) Matrix(211,-46,78,-17) -> Matrix(79,-6,382,-29) Matrix(53,-12,234,-53) -> Matrix(235,-18,3068,-235) (2/9,3/13) -> (9/118,1/13) Matrix(25,-6,104,-25) -> Matrix(103,-8,1326,-103) (3/13,1/4) -> (1/13,4/51) Matrix(51,-14,62,-17) -> Matrix(63,-5,542,-43) Matrix(223,-66,98,-29) -> Matrix(99,-8,458,-37) Matrix(121,-38,86,-27) -> Matrix(37,-3,210,-17) Matrix(47,-18,34,-13) -> Matrix(35,-3,222,-19) Matrix(67,-28,12,-5) -> Matrix(11,-1,56,-5) Matrix(223,-98,66,-29) -> Matrix(67,-6,190,-17) Matrix(89,-40,198,-89) -> Matrix(199,-18,2200,-199) (4/9,5/11) -> (9/100,1/11) Matrix(21,-10,44,-21) -> Matrix(43,-4,462,-43) (5/11,1/2) -> (1/11,2/21) Matrix(207,-112,146,-79) -> Matrix(21,-2,136,-13) Matrix(391,-216,248,-137) -> Matrix(1,0,-4,1) 0/1 Matrix(163,-92,62,-35) -> Matrix(21,-2,136,-13) Matrix(41,-24,70,-41) -> Matrix(71,-7,720,-71) (4/7,3/5) -> (7/72,1/10) Matrix(19,-12,30,-19) -> Matrix(29,-3,280,-29) (3/5,2/3) -> (1/10,3/28) Matrix(37,-26,10,-7) -> Matrix(9,-1,46,-5) Matrix(55,-42,72,-55) -> Matrix(73,-8,666,-73) (3/4,7/9) -> (4/37,1/9) Matrix(71,-56,90,-71) -> Matrix(89,-10,792,-89) (7/9,4/5) -> (1/9,5/44) Matrix(69,-58,44,-37) -> Matrix(25,-3,158,-19) Matrix(57,-70,22,-27) -> Matrix(7,-1,50,-7) (0/1,1/7).(1/8,1/6) Matrix(71,-90,56,-71) -> Matrix(57,-8,406,-57) (5/4,9/7) -> (4/29,1/7) Matrix(55,-72,42,-55) -> Matrix(41,-6,280,-41) (9/7,4/3) -> (1/7,3/20) Matrix(91,-132,20,-29) -> Matrix(5,-1,16,-3) 1/4 Matrix(1861,-2940,1178,-1861) -> Matrix(13,-2,84,-13) (30/19,49/31) -> (1/7,1/6) Matrix(1177,-1862,744,-1177) -> Matrix(27,-4,182,-27) (49/31,19/12) -> (1/7,2/13) Matrix(185,-294,56,-89) -> Matrix(7,-1,34,-5) Matrix(49,-80,30,-49) -> Matrix(31,-5,192,-31) (8/5,5/3) -> (5/32,1/6) Matrix(11,-20,6,-11) -> Matrix(5,-1,24,-5) (5/3,2/1) -> (1/6,1/4) Matrix(21,-44,10,-21) -> Matrix(11,-2,60,-11) (2/1,11/5) -> (1/6,1/5) Matrix(89,-198,40,-89) -> Matrix(39,-8,190,-39) (11/5,9/4) -> (1/5,4/19) Matrix(223,-514,82,-189) -> Matrix(23,-5,106,-23) (1/5,2/9).(3/14,1/4) Matrix(67,-158,14,-33) -> Matrix(5,-1,6,-1) Matrix(169,-408,70,-169) -> Matrix(21,-5,88,-21) (12/5,17/7) -> (5/22,1/4) Matrix(43,-118,4,-11) -> Matrix(5,-1,14,-3) Matrix(25,-72,8,-23) -> Matrix(9,-2,32,-7) 1/4 Matrix(25,-104,6,-25) -> Matrix(7,-2,24,-7) (4/1,13/3) -> (1/4,1/3) Matrix(53,-234,12,-53) -> Matrix(11,-4,30,-11) (13/3,9/2) -> (1/3,2/5) Matrix(13,-84,2,-13) -> Matrix(3,-1,8,-3) (6/1,7/1) -> (1/4,1/2) Matrix(15,-112,2,-15) -> Matrix(-1,1,0,1) (7/1,8/1) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.