INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF PURE MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 864 Minimal number of generators: 145 Number of equivalence classes of cusps: 64 Genus: 41 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/3 -13/9 -4/3 -11/9 -1/1 -8/9 -5/6 -7/9 -11/15 -2/3 -17/27 -5/9 -1/2 -7/15 -4/9 -7/17 -3/8 -1/3 -5/17 -3/11 -1/4 -17/72 -3/13 -2/9 -1/5 -1/6 -1/7 -1/8 -1/9 0/1 1/7 1/6 1/5 2/9 3/13 1/4 3/11 5/18 2/7 1/3 7/18 2/5 4/9 1/2 5/9 4/7 11/18 17/27 2/3 13/18 11/15 7/9 4/5 5/6 8/9 1/1 11/9 19/15 35/27 4/3 13/9 5/3 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 1/1 -11/6 -1/1 -9/5 -1/2 0/1 -7/4 1/3 1/1 -19/11 2/1 1/0 -12/7 -3/1 -1/1 -5/3 0/1 -18/11 3/5 1/1 -13/8 1/1 5/3 -21/13 3/1 1/0 -8/5 -3/1 -1/1 -27/17 -1/1 -19/12 -1/1 -11/7 -1/2 0/1 -3/2 -1/1 1/1 -13/9 0/1 -23/16 1/5 1/3 -10/7 1/3 1/1 -17/12 1/1 -7/5 -1/1 -18/13 1/3 1/1 -29/21 1/2 -11/8 1/1 3/1 -37/27 1/0 -26/19 -3/1 -1/1 -15/11 0/1 1/0 -19/14 -5/3 -1/1 -4/3 0/1 -21/16 1/1 3/1 -17/13 1/1 1/0 -13/10 -1/1 -1/3 -9/7 -1/1 -14/11 1/3 1/1 -33/26 1/1 7/5 -19/15 1/0 -5/4 -1/1 1/1 -11/9 0/1 -17/14 1/3 1/1 -6/5 -1/1 1/1 -13/11 1/1 -7/6 0/1 -8/7 -1/1 1/1 -9/8 -1/1 1/1 -1/1 0/1 1/0 -8/9 0/1 -7/8 -1/1 1/1 -6/7 -1/1 1/1 -17/20 1/3 1/1 -11/13 -1/1 -5/6 0/1 -19/23 1/1 1/0 -14/17 -1/1 -1/3 -9/11 -1/1 -4/5 -1/1 1/1 -7/9 0/1 -10/13 1/3 1/1 -13/17 0/1 1/1 -3/4 -1/1 1/1 -14/19 3/5 1/1 -11/15 1/0 -8/11 -1/1 -1/3 -13/18 0/1 -18/25 1/5 1/3 -5/7 1/1 -12/17 -1/1 1/1 -19/27 0/1 -7/10 1/3 1/1 -2/3 0/1 -7/11 0/1 1/0 -19/30 0/1 -12/19 1/1 3/1 -17/27 1/0 -5/8 -3/1 -1/1 -18/29 -1/1 -5/7 -13/21 -1/2 -34/55 -7/19 -1/3 -21/34 -1/3 -1/5 -8/13 -1/1 -1/3 -11/18 0/1 -14/23 1/5 1/3 -3/5 1/1 -7/12 -1/1 -11/19 -2/1 -1/1 -15/26 -1/1 -9/11 -4/7 -1/1 -1/3 -5/9 0/1 -6/11 1/3 1/1 -13/24 0/1 -7/13 1/3 -1/2 -1/1 1/1 -7/15 -1/2 -6/13 -1/1 -1/3 -5/11 -1/3 -4/9 0/1 -11/25 1/7 -18/41 1/7 1/5 -7/16 1/5 1/3 -3/7 0/1 1/2 -8/19 1/3 1/1 -5/12 1/1 -7/17 1/1 -23/56 1/1 3/1 -16/39 0/1 -9/22 5/7 1/1 -2/5 1/1 3/1 -7/18 1/0 -12/31 -9/1 -7/1 -5/13 -3/1 1/0 -13/34 -5/1 -3/1 -8/21 -2/1 -3/8 -5/3 -1/1 -1/3 0/1 -3/10 5/7 1/1 -11/37 1/1 -8/27 1/1 -13/44 1/1 21/19 -5/17 1/1 4/3 -7/24 1/1 -2/7 1/1 3/1 -5/18 1/0 -8/29 -7/1 -5/1 -3/11 -2/1 1/0 -4/15 -2/1 -9/34 -7/5 -1/1 -14/53 -15/13 -1/1 -19/72 -1/1 -5/19 -1/1 -1/4 -1/1 -1/3 -5/21 -1/2 -9/38 -5/13 -1/3 -13/55 -1/3 -17/72 -1/3 -4/17 -1/3 -3/11 -7/30 0/1 -3/13 -1/5 -2/9 0/1 -5/23 1/9 -8/37 3/23 1/7 -3/14 1/7 1/5 -1/5 0/1 1/2 -1/6 1/1 -1/7 -1/1 -1/8 -1/1 -1/3 -1/9 0/1 0/1 -1/1 1/1 1/7 0/1 1/0 1/6 1/1 1/5 3/1 2/9 1/0 3/13 -5/1 1/0 4/17 -11/3 -3/1 1/4 -1/1 1/1 3/11 3/1 5/18 1/0 2/7 -1/1 1/1 1/3 1/0 4/11 -5/1 -3/1 7/19 -3/1 10/27 -2/1 13/35 -2/1 1/0 3/8 -3/1 -1/1 5/13 1/1 7/18 1/0 2/5 -3/1 -1/1 9/22 -5/3 -1/1 7/17 -2/1 -1/1 5/12 -1/1 8/19 -1/1 1/1 3/7 1/1 4/9 1/0 5/11 -6/1 1/0 1/2 -3/1 -1/1 5/9 1/0 9/16 -5/1 -3/1 13/23 -3/1 1/0 4/7 -3/1 -1/1 11/19 -3/1 7/12 -3/1 3/5 -2/1 1/0 11/18 -2/1 8/13 -9/5 -5/3 21/34 -7/5 -1/1 13/21 -2/1 5/8 -5/3 -1/1 17/27 -2/1 12/19 -9/5 -5/3 19/30 -2/1 7/11 -5/3 9/14 -7/5 -1/1 11/17 -4/3 -1/1 2/3 -1/1 13/19 -2/1 -1/1 11/16 -1/1 -1/3 9/13 1/1 7/10 -3/1 -1/1 19/27 1/0 12/17 -3/1 -1/1 5/7 -2/1 1/0 13/18 -2/1 8/11 -5/3 -1/1 19/26 -5/3 -1/1 11/15 -2/1 25/34 -19/11 -5/3 39/53 -22/13 -5/3 53/72 -5/3 14/19 -5/3 -21/13 3/4 -7/5 -1/1 7/9 -1/1 11/14 -1/1 -9/11 15/19 -1/1 4/5 -1/1 -1/3 13/16 1/1 3/1 9/11 -2/1 1/0 14/17 -5/1 -3/1 19/23 -1/1 5/6 -2/1 16/19 -9/5 -5/3 11/13 -3/2 -1/1 6/7 -7/5 -1/1 7/8 -15/13 -1/1 8/9 -1/1 1/1 -1/1 9/8 -1/1 -15/17 8/7 -1/1 -7/9 15/13 -1/1 -3/4 7/6 -2/3 13/11 -2/3 -1/2 6/5 -1/1 1/1 11/9 -1/1 16/13 -1/1 -15/17 5/4 -1/1 -7/9 24/19 -21/29 -5/7 19/15 -2/3 14/11 -1/1 -5/7 23/18 -2/3 9/7 -2/3 -1/2 22/17 -1/1 -3/5 35/27 -1/2 13/10 -1/1 -3/5 4/3 -1/1 15/11 -5/7 26/19 -5/7 -9/13 37/27 -2/3 11/8 -1/1 -5/7 40/29 -1/1 -5/7 29/21 -2/3 76/55 -1/1 1/1 47/34 -1/1 -7/9 18/13 -5/7 -9/13 25/18 -2/3 7/5 -2/3 -1/2 17/12 -3/5 27/19 -3/5 10/7 -1/1 -3/5 13/9 -1/2 16/11 -1/1 -1/3 19/13 -1/1 -1/2 3/2 -1/1 -3/5 20/13 -13/23 -5/9 17/11 -6/11 -1/2 14/9 -1/2 11/7 -1/3 19/12 -1/1 27/17 -1/1 -2/3 8/5 -1/1 -3/5 29/18 -1/2 21/13 -1/3 55/34 -1/1 1/1 34/21 -1/1 13/8 -1/1 -3/5 5/3 -1/2 17/10 -3/7 -1/3 46/27 -2/5 29/17 -1/3 12/7 -1/1 -1/3 31/18 -1/2 19/11 -3/7 26/15 -1/3 59/34 -5/13 -1/3 92/53 -13/37 -1/3 125/72 -1/3 33/19 -1/3 0/1 7/4 -1/1 -1/3 30/17 -3/5 -11/19 23/13 -5/9 -1/2 16/9 -1/2 9/5 -3/7 11/6 -1/3 2/1 -1/1 -1/3 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(53,98,126,233) (-2/1,-11/6) -> (5/12,8/19) Hyperbolic Matrix(127,230,90,163) (-11/6,-9/5) -> (7/5,17/12) Hyperbolic Matrix(19,34,-90,-161) (-9/5,-7/4) -> (-3/14,-1/5) Hyperbolic Matrix(269,466,-198,-343) (-7/4,-19/11) -> (-15/11,-19/14) Hyperbolic Matrix(109,188,-396,-683) (-19/11,-12/7) -> (-8/29,-3/11) Hyperbolic Matrix(89,150,-54,-91) (-12/7,-5/3) -> (-5/3,-18/11) Parabolic Matrix(197,322,-342,-559) (-18/11,-13/8) -> (-15/26,-4/7) Hyperbolic Matrix(305,494,-234,-379) (-13/8,-21/13) -> (-17/13,-13/10) Hyperbolic Matrix(181,292,-468,-755) (-21/13,-8/5) -> (-12/31,-5/13) Hyperbolic Matrix(143,228,180,287) (-8/5,-27/17) -> (15/19,4/5) Hyperbolic Matrix(613,972,432,685) (-27/17,-19/12) -> (17/12,27/19) Hyperbolic Matrix(19,30,126,199) (-19/12,-11/7) -> (1/7,1/6) Hyperbolic Matrix(55,86,-126,-197) (-11/7,-3/2) -> (-7/16,-3/7) Hyperbolic Matrix(233,338,-162,-235) (-3/2,-13/9) -> (-13/9,-23/16) Parabolic Matrix(215,308,-252,-361) (-23/16,-10/7) -> (-6/7,-17/20) Hyperbolic Matrix(73,104,-252,-359) (-10/7,-17/12) -> (-7/24,-2/7) Hyperbolic Matrix(163,230,90,127) (-17/12,-7/5) -> (9/5,11/6) Hyperbolic Matrix(109,152,-180,-251) (-7/5,-18/13) -> (-14/23,-3/5) Hyperbolic Matrix(217,300,-468,-647) (-18/13,-29/21) -> (-7/15,-6/13) Hyperbolic Matrix(845,1166,-666,-919) (-29/21,-11/8) -> (-33/26,-19/15) Hyperbolic Matrix(631,866,486,667) (-11/8,-37/27) -> (35/27,13/10) Hyperbolic Matrix(1259,1724,972,1331) (-37/27,-26/19) -> (22/17,35/27) Hyperbolic Matrix(325,444,396,541) (-26/19,-15/11) -> (9/11,14/17) Hyperbolic Matrix(251,340,-612,-829) (-19/14,-4/3) -> (-16/39,-9/22) Hyperbolic Matrix(325,428,-792,-1043) (-4/3,-21/16) -> (-23/56,-16/39) Hyperbolic Matrix(343,450,234,307) (-21/16,-17/13) -> (19/13,3/2) Hyperbolic Matrix(17,22,-126,-163) (-13/10,-9/7) -> (-1/7,-1/8) Hyperbolic Matrix(181,232,-252,-323) (-9/7,-14/11) -> (-18/25,-5/7) Hyperbolic Matrix(307,390,270,343) (-14/11,-33/26) -> (9/8,8/7) Hyperbolic Matrix(35,44,-144,-181) (-19/15,-5/4) -> (-1/4,-5/21) Hyperbolic Matrix(197,242,-162,-199) (-5/4,-11/9) -> (-11/9,-17/14) Parabolic Matrix(73,88,180,217) (-17/14,-6/5) -> (2/5,9/22) Hyperbolic Matrix(253,300,-576,-683) (-6/5,-13/11) -> (-11/25,-18/41) Hyperbolic Matrix(251,296,396,467) (-13/11,-7/6) -> (19/30,7/11) Hyperbolic Matrix(127,146,-234,-269) (-7/6,-8/7) -> (-6/11,-13/24) Hyperbolic Matrix(397,448,288,325) (-8/7,-9/8) -> (11/8,40/29) Hyperbolic Matrix(107,120,288,323) (-9/8,-1/1) -> (13/35,3/8) Hyperbolic Matrix(127,114,342,307) (-1/1,-8/9) -> (10/27,13/35) Hyperbolic Matrix(521,458,306,269) (-8/9,-7/8) -> (17/10,46/27) Hyperbolic Matrix(197,170,270,233) (-7/8,-6/7) -> (8/11,19/26) Hyperbolic Matrix(1135,962,702,595) (-17/20,-11/13) -> (21/13,55/34) Hyperbolic Matrix(109,92,-468,-395) (-11/13,-5/6) -> (-7/30,-3/13) Hyperbolic Matrix(251,208,216,179) (-5/6,-19/23) -> (15/13,7/6) Hyperbolic Matrix(1081,892,612,505) (-19/23,-14/17) -> (30/17,23/13) Hyperbolic Matrix(541,444,396,325) (-14/17,-9/11) -> (15/11,26/19) Hyperbolic Matrix(91,74,-198,-161) (-9/11,-4/5) -> (-6/13,-5/11) Hyperbolic Matrix(125,98,-162,-127) (-4/5,-7/9) -> (-7/9,-10/13) Parabolic Matrix(487,374,306,235) (-10/13,-13/17) -> (27/17,8/5) Hyperbolic Matrix(197,150,306,233) (-13/17,-3/4) -> (9/14,11/17) Hyperbolic Matrix(89,66,-414,-307) (-3/4,-14/19) -> (-8/37,-3/14) Hyperbolic Matrix(757,556,-1224,-899) (-14/19,-11/15) -> (-13/21,-34/55) Hyperbolic Matrix(413,302,-666,-487) (-11/15,-8/11) -> (-18/29,-13/21) Hyperbolic Matrix(287,208,396,287) (-8/11,-13/18) -> (13/18,8/11) Hyperbolic Matrix(827,596,648,467) (-13/18,-18/25) -> (14/11,23/18) Hyperbolic Matrix(107,76,252,179) (-5/7,-12/17) -> (8/19,3/7) Hyperbolic Matrix(613,432,972,685) (-12/17,-19/27) -> (17/27,12/19) Hyperbolic Matrix(305,214,486,341) (-19/27,-7/10) -> (5/8,17/27) Hyperbolic Matrix(89,62,-234,-163) (-7/10,-2/3) -> (-8/21,-3/8) Hyperbolic Matrix(53,34,-198,-127) (-2/3,-7/11) -> (-3/11,-4/15) Hyperbolic Matrix(467,296,396,251) (-7/11,-19/30) -> (7/6,13/11) Hyperbolic Matrix(253,160,-1080,-683) (-19/30,-12/19) -> (-4/17,-7/30) Hyperbolic Matrix(685,432,972,613) (-12/19,-17/27) -> (19/27,12/17) Hyperbolic Matrix(341,214,486,305) (-17/27,-5/8) -> (7/10,19/27) Hyperbolic Matrix(251,156,288,179) (-5/8,-18/29) -> (6/7,7/8) Hyperbolic Matrix(971,600,-3672,-2269) (-34/55,-21/34) -> (-9/34,-14/53) Hyperbolic Matrix(703,434,-1602,-989) (-21/34,-8/13) -> (-18/41,-7/16) Hyperbolic Matrix(287,176,468,287) (-8/13,-11/18) -> (11/18,8/13) Hyperbolic Matrix(899,548,648,395) (-11/18,-14/23) -> (18/13,25/18) Hyperbolic Matrix(17,10,90,53) (-3/5,-7/12) -> (1/6,1/5) Hyperbolic Matrix(179,104,432,251) (-7/12,-11/19) -> (7/17,5/12) Hyperbolic Matrix(377,218,-1278,-739) (-11/19,-15/26) -> (-13/44,-5/17) Hyperbolic Matrix(89,50,-162,-91) (-4/7,-5/9) -> (-5/9,-6/11) Parabolic Matrix(521,282,630,341) (-13/24,-7/13) -> (19/23,5/6) Hyperbolic Matrix(161,86,234,125) (-7/13,-1/2) -> (11/16,9/13) Hyperbolic Matrix(145,68,-612,-287) (-1/2,-7/15) -> (-5/21,-9/38) Hyperbolic Matrix(143,64,-324,-145) (-5/11,-4/9) -> (-4/9,-11/25) Parabolic Matrix(179,76,252,107) (-3/7,-8/19) -> (12/17,5/7) Hyperbolic Matrix(233,98,126,53) (-8/19,-5/12) -> (11/6,2/1) Hyperbolic Matrix(251,104,432,179) (-5/12,-7/17) -> (11/19,7/12) Hyperbolic Matrix(881,362,-3726,-1531) (-7/17,-23/56) -> (-9/38,-13/55) Hyperbolic Matrix(377,154,306,125) (-9/22,-2/5) -> (16/13,5/4) Hyperbolic Matrix(71,28,180,71) (-2/5,-7/18) -> (7/18,2/5) Hyperbolic Matrix(1043,404,648,251) (-7/18,-12/31) -> (8/5,29/18) Hyperbolic Matrix(559,214,990,379) (-5/13,-13/34) -> (9/16,13/23) Hyperbolic Matrix(325,124,-1224,-467) (-13/34,-8/21) -> (-4/15,-9/34) Hyperbolic Matrix(17,6,-54,-19) (-3/8,-1/3) -> (-1/3,-3/10) Parabolic Matrix(343,102,306,91) (-3/10,-11/37) -> (1/1,9/8) Hyperbolic Matrix(323,96,360,107) (-11/37,-8/27) -> (8/9,1/1) Hyperbolic Matrix(541,160,612,181) (-8/27,-13/44) -> (7/8,8/9) Hyperbolic Matrix(971,284,612,179) (-5/17,-7/24) -> (19/12,27/17) Hyperbolic Matrix(71,20,252,71) (-2/7,-5/18) -> (5/18,2/7) Hyperbolic Matrix(1115,308,648,179) (-5/18,-8/29) -> (12/7,31/18) Hyperbolic Matrix(3817,1008,5184,1369) (-14/53,-19/72) -> (53/72,14/19) Hyperbolic Matrix(1259,332,-5328,-1405) (-19/72,-5/19) -> (-13/55,-17/72) Hyperbolic Matrix(269,70,342,89) (-5/19,-1/4) -> (11/14,15/19) Hyperbolic Matrix(8749,2064,5040,1189) (-17/72,-4/17) -> (92/53,125/72) Hyperbolic Matrix(71,16,-324,-73) (-3/13,-2/9) -> (-2/9,-5/23) Parabolic Matrix(1025,222,1242,269) (-5/23,-8/37) -> (14/17,19/23) Hyperbolic Matrix(53,10,90,17) (-1/5,-1/6) -> (7/12,3/5) Hyperbolic Matrix(199,30,126,19) (-1/6,-1/7) -> (11/7,19/12) Hyperbolic Matrix(395,48,288,35) (-1/8,-1/9) -> (37/27,11/8) Hyperbolic Matrix(271,26,198,19) (-1/9,0/1) -> (26/19,37/27) Hyperbolic Matrix(163,-22,126,-17) (0/1,1/7) -> (9/7,22/17) Hyperbolic Matrix(161,-34,90,-19) (1/5,2/9) -> (16/9,9/5) Hyperbolic Matrix(415,-94,234,-53) (2/9,3/13) -> (23/13,16/9) Hyperbolic Matrix(395,-92,468,-109) (3/13,4/17) -> (16/19,11/13) Hyperbolic Matrix(181,-44,144,-35) (4/17,1/4) -> (5/4,24/19) Hyperbolic Matrix(127,-34,198,-53) (1/4,3/11) -> (7/11,9/14) Hyperbolic Matrix(683,-188,396,-109) (3/11,5/18) -> (31/18,19/11) Hyperbolic Matrix(19,-6,54,-17) (2/7,1/3) -> (1/3,4/11) Parabolic Matrix(487,-178,342,-125) (4/11,7/19) -> (27/19,10/7) Hyperbolic Matrix(1657,-612,972,-359) (7/19,10/27) -> (46/27,29/17) Hyperbolic Matrix(163,-62,234,-89) (3/8,5/13) -> (9/13,7/10) Hyperbolic Matrix(755,-292,468,-181) (5/13,7/18) -> (29/18,21/13) Hyperbolic Matrix(845,-346,486,-199) (9/22,7/17) -> (33/19,7/4) Hyperbolic Matrix(197,-86,126,-55) (3/7,4/9) -> (14/9,11/7) Hyperbolic Matrix(307,-138,198,-89) (4/9,5/11) -> (17/11,14/9) Hyperbolic Matrix(161,-74,198,-91) (5/11,1/2) -> (13/16,9/11) Hyperbolic Matrix(91,-50,162,-89) (1/2,5/9) -> (5/9,9/16) Parabolic Matrix(289,-164,252,-143) (13/23,4/7) -> (8/7,15/13) Hyperbolic Matrix(431,-248,252,-145) (4/7,11/19) -> (29/17,12/7) Hyperbolic Matrix(251,-152,180,-109) (3/5,11/18) -> (25/18,7/5) Hyperbolic Matrix(719,-444,468,-289) (8/13,21/34) -> (3/2,20/13) Hyperbolic Matrix(899,-556,1224,-757) (21/34,13/21) -> (11/15,25/34) Hyperbolic Matrix(487,-302,666,-413) (13/21,5/8) -> (19/26,11/15) Hyperbolic Matrix(575,-364,684,-433) (12/19,19/30) -> (5/6,16/19) Hyperbolic Matrix(73,-48,108,-71) (11/17,2/3) -> (2/3,13/19) Parabolic Matrix(1099,-754,1494,-1025) (13/19,11/16) -> (25/34,39/53) Hyperbolic Matrix(323,-232,252,-181) (5/7,13/18) -> (23/18,9/7) Hyperbolic Matrix(9001,-6624,5184,-3815) (39/53,53/72) -> (125/72,33/19) Hyperbolic Matrix(253,-188,144,-107) (14/19,3/4) -> (7/4,30/17) Hyperbolic Matrix(127,-98,162,-125) (3/4,7/9) -> (7/9,11/14) Parabolic Matrix(523,-422,378,-305) (4/5,13/16) -> (47/34,18/13) Hyperbolic Matrix(341,-290,234,-199) (11/13,6/7) -> (16/11,19/13) Hyperbolic Matrix(305,-362,198,-235) (13/11,6/5) -> (20/13,17/11) Hyperbolic Matrix(199,-242,162,-197) (6/5,11/9) -> (11/9,16/13) Parabolic Matrix(1691,-2140,1224,-1549) (24/19,19/15) -> (29/21,76/55) Hyperbolic Matrix(919,-1166,666,-845) (19/15,14/11) -> (40/29,29/21) Hyperbolic Matrix(379,-494,234,-305) (13/10,4/3) -> (34/21,13/8) Hyperbolic Matrix(343,-466,198,-269) (4/3,15/11) -> (19/11,26/15) Hyperbolic Matrix(6373,-8808,3672,-5075) (76/55,47/34) -> (59/34,92/53) Hyperbolic Matrix(235,-338,162,-233) (10/7,13/9) -> (13/9,16/11) Parabolic Matrix(2123,-3436,1224,-1981) (55/34,34/21) -> (26/15,59/34) Hyperbolic Matrix(91,-150,54,-89) (13/8,5/3) -> (5/3,17/10) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-2,1) Matrix(53,98,126,233) -> Matrix(1,0,0,1) Matrix(127,230,90,163) -> Matrix(5,2,-8,-3) Matrix(19,34,-90,-161) -> Matrix(1,0,4,1) Matrix(269,466,-198,-343) -> Matrix(1,-2,0,1) Matrix(109,188,-396,-683) -> Matrix(1,-4,0,1) Matrix(89,150,-54,-91) -> Matrix(1,0,2,1) Matrix(197,322,-342,-559) -> Matrix(3,-2,-4,3) Matrix(305,494,-234,-379) -> Matrix(1,-2,0,1) Matrix(181,292,-468,-755) -> Matrix(1,-6,0,1) Matrix(143,228,180,287) -> Matrix(1,2,-2,-3) Matrix(613,972,432,685) -> Matrix(5,2,-8,-3) Matrix(19,30,126,199) -> Matrix(1,0,2,1) Matrix(55,86,-126,-197) -> Matrix(1,0,4,1) Matrix(233,338,-162,-235) -> Matrix(1,0,4,1) Matrix(215,308,-252,-361) -> Matrix(1,0,-2,1) Matrix(73,104,-252,-359) -> Matrix(3,-2,2,-1) Matrix(163,230,90,127) -> Matrix(1,-2,-2,5) Matrix(109,152,-180,-251) -> Matrix(1,0,2,1) Matrix(217,300,-468,-647) -> Matrix(1,0,-4,1) Matrix(845,1166,-666,-919) -> Matrix(3,-2,2,-1) Matrix(631,866,486,667) -> Matrix(1,0,-2,1) Matrix(1259,1724,972,1331) -> Matrix(1,4,-2,-7) Matrix(325,444,396,541) -> Matrix(1,-2,0,1) Matrix(251,340,-612,-829) -> Matrix(1,0,2,1) Matrix(325,428,-792,-1043) -> Matrix(1,0,0,1) Matrix(343,450,234,307) -> Matrix(1,0,-2,1) Matrix(17,22,-126,-163) -> Matrix(1,0,0,1) Matrix(181,232,-252,-323) -> Matrix(1,0,2,1) Matrix(307,390,270,343) -> Matrix(5,-4,-6,5) Matrix(35,44,-144,-181) -> Matrix(1,0,-2,1) Matrix(197,242,-162,-199) -> Matrix(1,0,2,1) Matrix(73,88,180,217) -> Matrix(1,-2,0,1) Matrix(253,300,-576,-683) -> Matrix(1,0,6,1) Matrix(251,296,396,467) -> Matrix(3,2,-2,-1) Matrix(127,146,-234,-269) -> Matrix(1,0,2,1) Matrix(397,448,288,325) -> Matrix(3,-2,-4,3) Matrix(107,120,288,323) -> Matrix(1,-2,0,1) Matrix(127,114,342,307) -> Matrix(1,-2,0,1) Matrix(521,458,306,269) -> Matrix(1,-2,-2,5) Matrix(197,170,270,233) -> Matrix(3,2,-2,-1) Matrix(1135,962,702,595) -> Matrix(1,0,-2,1) Matrix(109,92,-468,-395) -> Matrix(1,0,-4,1) Matrix(251,208,216,179) -> Matrix(3,-2,-4,3) Matrix(1081,892,612,505) -> Matrix(1,4,-2,-7) Matrix(541,444,396,325) -> Matrix(3,-2,-4,3) Matrix(91,74,-198,-161) -> Matrix(1,0,-2,1) Matrix(125,98,-162,-127) -> Matrix(1,0,2,1) Matrix(487,374,306,235) -> Matrix(3,-2,-4,3) Matrix(197,150,306,233) -> Matrix(3,-4,-2,3) Matrix(89,66,-414,-307) -> Matrix(1,0,6,1) Matrix(757,556,-1224,-899) -> Matrix(1,-2,-2,5) Matrix(413,302,-666,-487) -> Matrix(1,2,-2,-3) Matrix(287,208,396,287) -> Matrix(7,2,-4,-1) Matrix(827,596,648,467) -> Matrix(11,-2,-16,3) Matrix(107,76,252,179) -> Matrix(1,0,0,1) Matrix(613,432,972,685) -> Matrix(7,2,-4,-1) Matrix(305,214,486,341) -> Matrix(1,-2,0,1) Matrix(89,62,-234,-163) -> Matrix(1,-2,0,1) Matrix(53,34,-198,-127) -> Matrix(1,-2,0,1) Matrix(467,296,396,251) -> Matrix(1,2,-2,-3) Matrix(253,160,-1080,-683) -> Matrix(1,0,-4,1) Matrix(685,432,972,613) -> Matrix(1,-4,0,1) Matrix(341,214,486,305) -> Matrix(1,0,0,1) Matrix(251,156,288,179) -> Matrix(7,6,-6,-5) Matrix(971,600,-3672,-2269) -> Matrix(13,4,-10,-3) Matrix(703,434,-1602,-989) -> Matrix(1,0,8,1) Matrix(287,176,468,287) -> Matrix(11,2,-6,-1) Matrix(899,548,648,395) -> Matrix(15,-2,-22,3) Matrix(17,10,90,53) -> Matrix(1,2,0,1) Matrix(179,104,432,251) -> Matrix(1,0,0,1) Matrix(377,218,-1278,-739) -> Matrix(5,6,4,5) Matrix(89,50,-162,-91) -> Matrix(1,0,4,1) Matrix(521,282,630,341) -> Matrix(5,-2,-2,1) Matrix(161,86,234,125) -> Matrix(1,0,-2,1) Matrix(145,68,-612,-287) -> Matrix(3,2,-8,-5) Matrix(143,64,-324,-145) -> Matrix(1,0,10,1) Matrix(179,76,252,107) -> Matrix(5,-2,-2,1) Matrix(233,98,126,53) -> Matrix(1,0,-4,1) Matrix(251,104,432,179) -> Matrix(5,-2,-2,1) Matrix(881,362,-3726,-1531) -> Matrix(3,-4,-8,11) Matrix(377,154,306,125) -> Matrix(7,-6,-8,7) Matrix(71,28,180,71) -> Matrix(1,-4,0,1) Matrix(1043,404,648,251) -> Matrix(1,10,-2,-19) Matrix(559,214,990,379) -> Matrix(1,0,0,1) Matrix(325,124,-1224,-467) -> Matrix(3,8,-2,-5) Matrix(17,6,-54,-19) -> Matrix(1,0,2,1) Matrix(343,102,306,91) -> Matrix(11,-10,-12,11) Matrix(323,96,360,107) -> Matrix(9,-8,-10,9) Matrix(541,160,612,181) -> Matrix(17,-18,-16,17) Matrix(971,284,612,179) -> Matrix(1,-2,0,1) Matrix(71,20,252,71) -> Matrix(1,-2,0,1) Matrix(1115,308,648,179) -> Matrix(1,6,-2,-11) Matrix(3817,1008,5184,1369) -> Matrix(43,48,-26,-29) Matrix(1259,332,-5328,-1405) -> Matrix(5,4,-14,-11) Matrix(269,70,342,89) -> Matrix(3,4,-4,-5) Matrix(8749,2064,5040,1189) -> Matrix(25,8,-72,-23) Matrix(71,16,-324,-73) -> Matrix(1,0,14,1) Matrix(1025,222,1242,269) -> Matrix(17,-2,-8,1) Matrix(53,10,90,17) -> Matrix(5,-2,-2,1) Matrix(199,30,126,19) -> Matrix(1,0,-2,1) Matrix(395,48,288,35) -> Matrix(1,2,-2,-3) Matrix(271,26,198,19) -> Matrix(7,-2,-10,3) Matrix(163,-22,126,-17) -> Matrix(1,2,-2,-3) Matrix(161,-34,90,-19) -> Matrix(1,-6,-2,13) Matrix(415,-94,234,-53) -> Matrix(1,10,-2,-19) Matrix(395,-92,468,-109) -> Matrix(3,14,-2,-9) Matrix(181,-44,144,-35) -> Matrix(3,4,-4,-5) Matrix(127,-34,198,-53) -> Matrix(3,-4,-2,3) Matrix(683,-188,396,-109) -> Matrix(1,-6,-2,13) Matrix(19,-6,54,-17) -> Matrix(1,-4,0,1) Matrix(487,-178,342,-125) -> Matrix(1,6,-2,-11) Matrix(1657,-612,972,-359) -> Matrix(3,8,-8,-21) Matrix(163,-62,234,-89) -> Matrix(1,0,0,1) Matrix(755,-292,468,-181) -> Matrix(1,-2,-2,5) Matrix(845,-346,486,-199) -> Matrix(1,2,-4,-7) Matrix(197,-86,126,-55) -> Matrix(1,-2,-2,5) Matrix(307,-138,198,-89) -> Matrix(1,12,-2,-23) Matrix(161,-74,198,-91) -> Matrix(1,4,0,1) Matrix(91,-50,162,-89) -> Matrix(1,-2,0,1) Matrix(289,-164,252,-143) -> Matrix(3,10,-4,-13) Matrix(431,-248,252,-145) -> Matrix(1,2,-2,-3) Matrix(251,-152,180,-109) -> Matrix(1,4,-2,-7) Matrix(719,-444,468,-289) -> Matrix(7,10,-12,-17) Matrix(899,-556,1224,-757) -> Matrix(3,8,-2,-5) Matrix(487,-302,666,-413) -> Matrix(1,0,0,1) Matrix(575,-364,684,-433) -> Matrix(1,0,0,1) Matrix(73,-48,108,-71) -> Matrix(1,2,-2,-3) Matrix(1099,-754,1494,-1025) -> Matrix(17,12,-10,-7) Matrix(323,-232,252,-181) -> Matrix(1,4,-2,-7) Matrix(9001,-6624,5184,-3815) -> Matrix(13,22,-42,-71) Matrix(253,-188,144,-107) -> Matrix(3,4,-4,-5) Matrix(127,-98,162,-125) -> Matrix(7,8,-8,-9) Matrix(523,-422,378,-305) -> Matrix(3,-2,-4,3) Matrix(341,-290,234,-199) -> Matrix(3,4,-4,-5) Matrix(305,-362,198,-235) -> Matrix(9,4,-16,-7) Matrix(199,-242,162,-197) -> Matrix(7,8,-8,-9) Matrix(1691,-2140,1224,-1549) -> Matrix(11,8,-18,-13) Matrix(919,-1166,666,-845) -> Matrix(1,0,0,1) Matrix(379,-494,234,-305) -> Matrix(1,0,0,1) Matrix(343,-466,198,-269) -> Matrix(5,4,-14,-11) Matrix(6373,-8808,3672,-5075) -> Matrix(7,6,-20,-17) Matrix(235,-338,162,-233) -> Matrix(3,2,-8,-5) Matrix(2123,-3436,1224,-1981) -> Matrix(3,2,-8,-5) Matrix(91,-150,54,-89) -> Matrix(7,4,-16,-9) 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): 44 Degree of the the map X: 44 Degree of the the map Y: 144 ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 432 Minimal number of generators: 73 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: 40 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/3 -13/9 -4/3 -19/15 -1/1 -7/9 -11/15 -2/3 -17/27 -5/9 -1/2 -4/9 -3/7 -1/3 -5/17 -3/11 -1/4 -2/9 -1/5 0/1 1/5 2/9 3/13 1/4 1/3 3/7 4/9 1/2 5/9 17/27 2/3 7/9 8/9 1/1 11/9 4/3 37/27 5/3 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 1/1 -7/4 1/3 1/1 -19/11 2/1 1/0 -12/7 -3/1 -1/1 -5/3 0/1 -18/11 3/5 1/1 -13/8 1/1 5/3 -21/13 3/1 1/0 -8/5 -3/1 -1/1 -3/2 -1/1 1/1 -13/9 0/1 -10/7 1/3 1/1 -17/12 1/1 -7/5 -1/1 -18/13 1/3 1/1 -29/21 1/2 -11/8 1/1 3/1 -15/11 0/1 1/0 -4/3 0/1 -17/13 1/1 1/0 -13/10 -1/1 -1/3 -9/7 -1/1 -14/11 1/3 1/1 -33/26 1/1 7/5 -19/15 1/0 -5/4 -1/1 1/1 -1/1 0/1 1/0 -4/5 -1/1 1/1 -7/9 0/1 -10/13 1/3 1/1 -3/4 -1/1 1/1 -11/15 1/0 -8/11 -1/1 -1/3 -13/18 0/1 -5/7 1/1 -7/10 1/3 1/1 -2/3 0/1 -7/11 0/1 1/0 -12/19 1/1 3/1 -17/27 1/0 -5/8 -3/1 -1/1 -18/29 -1/1 -5/7 -13/21 -1/2 -8/13 -1/1 -1/3 -11/18 0/1 -3/5 1/1 -7/12 -1/1 -11/19 -2/1 -1/1 -15/26 -1/1 -9/11 -4/7 -1/1 -1/3 -5/9 0/1 -1/2 -1/1 1/1 -5/11 -1/3 -4/9 0/1 -3/7 0/1 1/2 -2/5 1/1 3/1 -5/13 -3/1 1/0 -8/21 -2/1 -3/8 -5/3 -1/1 -1/3 0/1 -3/10 5/7 1/1 -11/37 1/1 -8/27 1/1 -13/44 1/1 21/19 -5/17 1/1 4/3 -7/24 1/1 -2/7 1/1 3/1 -3/11 -2/1 1/0 -1/4 -1/1 -1/3 -3/13 -1/5 -2/9 0/1 -1/5 0/1 1/2 0/1 -1/1 1/1 1/6 1/1 1/5 3/1 2/9 1/0 3/13 -5/1 1/0 1/4 -1/1 1/1 1/3 1/0 2/5 -3/1 -1/1 9/22 -5/3 -1/1 7/17 -2/1 -1/1 5/12 -1/1 3/7 1/1 4/9 1/0 5/11 -6/1 1/0 1/2 -3/1 -1/1 5/9 1/0 9/16 -5/1 -3/1 13/23 -3/1 1/0 4/7 -3/1 -1/1 3/5 -2/1 1/0 11/18 -2/1 8/13 -9/5 -5/3 5/8 -5/3 -1/1 17/27 -2/1 12/19 -9/5 -5/3 7/11 -5/3 2/3 -1/1 7/10 -3/1 -1/1 19/27 1/0 12/17 -3/1 -1/1 5/7 -2/1 1/0 13/18 -2/1 8/11 -5/3 -1/1 3/4 -7/5 -1/1 7/9 -1/1 4/5 -1/1 -1/3 9/11 -2/1 1/0 5/6 -2/1 11/13 -3/2 -1/1 6/7 -7/5 -1/1 7/8 -15/13 -1/1 8/9 -1/1 1/1 -1/1 9/8 -1/1 -15/17 8/7 -1/1 -7/9 15/13 -1/1 -3/4 7/6 -2/3 13/11 -2/3 -1/2 6/5 -1/1 1/1 11/9 -1/1 5/4 -1/1 -7/9 9/7 -2/3 -1/2 13/10 -1/1 -3/5 4/3 -1/1 15/11 -5/7 26/19 -5/7 -9/13 37/27 -2/3 11/8 -1/1 -5/7 7/5 -2/3 -1/2 3/2 -1/1 -3/5 17/11 -6/11 -1/2 14/9 -1/2 11/7 -1/3 19/12 -1/1 27/17 -1/1 -2/3 8/5 -1/1 -3/5 5/3 -1/2 7/4 -1/1 -1/3 23/13 -5/9 -1/2 16/9 -1/2 9/5 -3/7 11/6 -1/3 2/1 -1/1 -1/3 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(28,51,45,82) (-2/1,-7/4) -> (8/13,5/8) Hyperbolic Matrix(118,205,99,172) (-7/4,-19/11) -> (13/11,6/5) Hyperbolic Matrix(152,261,99,170) (-19/11,-12/7) -> (3/2,17/11) Hyperbolic Matrix(89,150,-54,-91) (-12/7,-5/3) -> (-5/3,-18/11) Parabolic Matrix(197,322,-342,-559) (-18/11,-13/8) -> (-15/26,-4/7) Hyperbolic Matrix(305,494,-234,-379) (-13/8,-21/13) -> (-17/13,-13/10) Hyperbolic Matrix(206,331,117,188) (-21/13,-8/5) -> (7/4,23/13) Hyperbolic Matrix(46,73,63,100) (-8/5,-3/2) -> (8/11,3/4) Hyperbolic Matrix(116,169,-81,-118) (-3/2,-13/9) -> (-13/9,-10/7) Parabolic Matrix(73,104,-252,-359) (-10/7,-17/12) -> (-7/24,-2/7) Hyperbolic Matrix(163,230,90,127) (-17/12,-7/5) -> (9/5,11/6) Hyperbolic Matrix(44,61,-189,-262) (-7/5,-18/13) -> (-1/4,-3/13) Hyperbolic Matrix(352,487,-279,-386) (-18/13,-29/21) -> (-19/15,-5/4) Hyperbolic Matrix(845,1166,-666,-919) (-29/21,-11/8) -> (-33/26,-19/15) Hyperbolic Matrix(188,257,-297,-406) (-11/8,-15/11) -> (-7/11,-12/19) Hyperbolic Matrix(116,157,99,134) (-15/11,-4/3) -> (7/6,13/11) Hyperbolic Matrix(136,179,117,154) (-4/3,-17/13) -> (15/13,7/6) Hyperbolic Matrix(332,429,243,314) (-13/10,-9/7) -> (15/11,26/19) Hyperbolic Matrix(62,79,-135,-172) (-9/7,-14/11) -> (-1/2,-5/11) Hyperbolic Matrix(307,390,270,343) (-14/11,-33/26) -> (9/8,8/7) Hyperbolic Matrix(8,9,-9,-10) (-5/4,-1/1) -> (-1/1,-4/5) Parabolic Matrix(125,98,-162,-127) (-4/5,-7/9) -> (-7/9,-10/13) Parabolic Matrix(62,47,153,116) (-10/13,-3/4) -> (2/5,9/22) Hyperbolic Matrix(172,127,-279,-206) (-3/4,-11/15) -> (-13/21,-8/13) Hyperbolic Matrix(413,302,-666,-487) (-11/15,-8/11) -> (-18/29,-13/21) Hyperbolic Matrix(287,208,396,287) (-8/11,-13/18) -> (13/18,8/11) Hyperbolic Matrix(118,85,-261,-188) (-13/18,-5/7) -> (-5/11,-4/9) Hyperbolic Matrix(154,109,243,172) (-5/7,-7/10) -> (12/19,7/11) Hyperbolic Matrix(89,62,-234,-163) (-7/10,-2/3) -> (-8/21,-3/8) Hyperbolic Matrix(82,53,99,64) (-2/3,-7/11) -> (9/11,5/6) Hyperbolic Matrix(685,432,972,613) (-12/19,-17/27) -> (19/27,12/17) Hyperbolic Matrix(341,214,486,305) (-17/27,-5/8) -> (7/10,19/27) Hyperbolic Matrix(251,156,288,179) (-5/8,-18/29) -> (6/7,7/8) Hyperbolic Matrix(287,176,468,287) (-8/13,-11/18) -> (11/18,8/13) Hyperbolic Matrix(64,39,-279,-170) (-11/18,-3/5) -> (-3/13,-2/9) Hyperbolic Matrix(17,10,90,53) (-3/5,-7/12) -> (1/6,1/5) Hyperbolic Matrix(179,104,432,251) (-7/12,-11/19) -> (7/17,5/12) Hyperbolic Matrix(377,218,-1278,-739) (-11/19,-15/26) -> (-13/44,-5/17) Hyperbolic Matrix(44,25,-81,-46) (-4/7,-5/9) -> (-5/9,-1/2) Parabolic Matrix(136,59,189,82) (-4/9,-3/7) -> (5/7,13/18) Hyperbolic Matrix(80,33,63,26) (-3/7,-2/5) -> (5/4,9/7) Hyperbolic Matrix(28,11,117,46) (-2/5,-5/13) -> (3/13,1/4) Hyperbolic Matrix(296,113,351,134) (-5/13,-8/21) -> (5/6,11/13) Hyperbolic Matrix(17,6,-54,-19) (-3/8,-1/3) -> (-1/3,-3/10) Parabolic Matrix(343,102,306,91) (-3/10,-11/37) -> (1/1,9/8) Hyperbolic Matrix(323,96,360,107) (-11/37,-8/27) -> (8/9,1/1) Hyperbolic Matrix(541,160,612,181) (-8/27,-13/44) -> (7/8,8/9) Hyperbolic Matrix(971,284,612,179) (-5/17,-7/24) -> (19/12,27/17) Hyperbolic Matrix(46,13,99,28) (-2/7,-3/11) -> (5/11,1/2) Hyperbolic Matrix(80,21,99,26) (-3/11,-1/4) -> (4/5,9/11) Hyperbolic Matrix(82,17,135,28) (-2/9,-1/5) -> (3/5,11/18) Hyperbolic Matrix(62,11,45,8) (-1/5,0/1) -> (11/8,7/5) Hyperbolic Matrix(44,-7,63,-10) (0/1,1/6) -> (2/3,7/10) Hyperbolic Matrix(161,-34,90,-19) (1/5,2/9) -> (16/9,9/5) Hyperbolic Matrix(415,-94,234,-53) (2/9,3/13) -> (23/13,16/9) Hyperbolic Matrix(10,-3,27,-8) (1/4,1/3) -> (1/3,2/5) Parabolic Matrix(730,-299,459,-188) (9/22,7/17) -> (27/17,8/5) Hyperbolic Matrix(98,-41,153,-64) (5/12,3/7) -> (7/11,2/3) Hyperbolic Matrix(197,-86,126,-55) (3/7,4/9) -> (14/9,11/7) Hyperbolic Matrix(307,-138,198,-89) (4/9,5/11) -> (17/11,14/9) Hyperbolic Matrix(91,-50,162,-89) (1/2,5/9) -> (5/9,9/16) Parabolic Matrix(314,-177,369,-208) (9/16,13/23) -> (11/13,6/7) Hyperbolic Matrix(289,-164,252,-143) (13/23,4/7) -> (8/7,15/13) Hyperbolic Matrix(64,-37,45,-26) (4/7,3/5) -> (7/5,3/2) Hyperbolic Matrix(460,-289,729,-458) (5/8,17/27) -> (17/27,12/19) Parabolic Matrix(244,-173,189,-134) (12/17,5/7) -> (9/7,13/10) Hyperbolic Matrix(64,-49,81,-62) (3/4,7/9) -> (7/9,4/5) Parabolic Matrix(100,-121,81,-98) (6/5,11/9) -> (11/9,5/4) Parabolic Matrix(116,-151,63,-82) (13/10,4/3) -> (11/6,2/1) Hyperbolic Matrix(242,-329,153,-208) (4/3,15/11) -> (11/7,19/12) Hyperbolic Matrix(1000,-1369,729,-998) (26/19,37/27) -> (37/27,11/8) Parabolic Matrix(46,-75,27,-44) (8/5,5/3) -> (5/3,7/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-2,1) Matrix(28,51,45,82) -> Matrix(3,2,-2,-1) Matrix(118,205,99,172) -> Matrix(1,0,-2,1) Matrix(152,261,99,170) -> Matrix(1,4,-2,-7) Matrix(89,150,-54,-91) -> Matrix(1,0,2,1) Matrix(197,322,-342,-559) -> Matrix(3,-2,-4,3) Matrix(305,494,-234,-379) -> Matrix(1,-2,0,1) Matrix(206,331,117,188) -> Matrix(1,2,-2,-3) Matrix(46,73,63,100) -> Matrix(3,2,-2,-1) Matrix(116,169,-81,-118) -> Matrix(1,0,2,1) Matrix(73,104,-252,-359) -> Matrix(3,-2,2,-1) Matrix(163,230,90,127) -> Matrix(1,-2,-2,5) Matrix(44,61,-189,-262) -> Matrix(1,0,-4,1) Matrix(352,487,-279,-386) -> Matrix(1,0,-2,1) Matrix(845,1166,-666,-919) -> Matrix(3,-2,2,-1) Matrix(188,257,-297,-406) -> Matrix(1,0,0,1) Matrix(116,157,99,134) -> Matrix(1,2,-2,-3) Matrix(136,179,117,154) -> Matrix(3,-2,-4,3) Matrix(332,429,243,314) -> Matrix(3,-2,-4,3) Matrix(62,79,-135,-172) -> Matrix(1,0,-2,1) Matrix(307,390,270,343) -> Matrix(5,-4,-6,5) Matrix(8,9,-9,-10) -> Matrix(1,0,0,1) Matrix(125,98,-162,-127) -> Matrix(1,0,2,1) Matrix(62,47,153,116) -> Matrix(1,-2,0,1) Matrix(172,127,-279,-206) -> Matrix(1,0,-2,1) Matrix(413,302,-666,-487) -> Matrix(1,2,-2,-3) Matrix(287,208,396,287) -> Matrix(7,2,-4,-1) Matrix(118,85,-261,-188) -> Matrix(1,0,-4,1) Matrix(154,109,243,172) -> Matrix(3,2,-2,-1) Matrix(89,62,-234,-163) -> Matrix(1,-2,0,1) Matrix(82,53,99,64) -> Matrix(1,-2,0,1) Matrix(685,432,972,613) -> Matrix(1,-4,0,1) Matrix(341,214,486,305) -> Matrix(1,0,0,1) Matrix(251,156,288,179) -> Matrix(7,6,-6,-5) Matrix(287,176,468,287) -> Matrix(11,2,-6,-1) Matrix(64,39,-279,-170) -> Matrix(1,0,-6,1) Matrix(17,10,90,53) -> Matrix(1,2,0,1) Matrix(179,104,432,251) -> Matrix(1,0,0,1) Matrix(377,218,-1278,-739) -> Matrix(5,6,4,5) Matrix(44,25,-81,-46) -> Matrix(1,0,2,1) Matrix(136,59,189,82) -> Matrix(5,-2,-2,1) Matrix(80,33,63,26) -> Matrix(3,-2,-4,3) Matrix(28,11,117,46) -> Matrix(1,-2,0,1) Matrix(296,113,351,134) -> Matrix(3,8,-2,-5) Matrix(17,6,-54,-19) -> Matrix(1,0,2,1) Matrix(343,102,306,91) -> Matrix(11,-10,-12,11) Matrix(323,96,360,107) -> Matrix(9,-8,-10,9) Matrix(541,160,612,181) -> Matrix(17,-18,-16,17) Matrix(971,284,612,179) -> Matrix(1,-2,0,1) Matrix(46,13,99,28) -> Matrix(1,-4,0,1) Matrix(80,21,99,26) -> Matrix(1,0,0,1) Matrix(82,17,135,28) -> Matrix(5,-2,-2,1) Matrix(62,11,45,8) -> Matrix(3,-2,-4,3) Matrix(44,-7,63,-10) -> Matrix(1,-2,0,1) Matrix(161,-34,90,-19) -> Matrix(1,-6,-2,13) Matrix(415,-94,234,-53) -> Matrix(1,10,-2,-19) Matrix(10,-3,27,-8) -> Matrix(1,-2,0,1) Matrix(730,-299,459,-188) -> Matrix(3,4,-4,-5) Matrix(98,-41,153,-64) -> Matrix(3,2,-2,-1) Matrix(197,-86,126,-55) -> Matrix(1,-2,-2,5) Matrix(307,-138,198,-89) -> Matrix(1,12,-2,-23) Matrix(91,-50,162,-89) -> Matrix(1,-2,0,1) Matrix(314,-177,369,-208) -> Matrix(3,8,-2,-5) Matrix(289,-164,252,-143) -> Matrix(3,10,-4,-13) Matrix(64,-37,45,-26) -> Matrix(1,4,-2,-7) Matrix(460,-289,729,-458) -> Matrix(3,8,-2,-5) Matrix(244,-173,189,-134) -> Matrix(1,4,-2,-7) Matrix(64,-49,81,-62) -> Matrix(3,4,-4,-5) Matrix(100,-121,81,-98) -> Matrix(3,4,-4,-5) Matrix(116,-151,63,-82) -> Matrix(3,2,-8,-5) Matrix(242,-329,153,-208) -> Matrix(3,2,-2,-1) Matrix(1000,-1369,729,-998) -> Matrix(11,8,-18,-13) Matrix(46,-75,27,-44) -> Matrix(3,2,-8,-5) 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): 22 ----------------------------------------------------------------------- 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 (0/1,1/0) 0 18 -7/9 0/1 2 2 -3/4 0 18 -11/15 1/0 4 6 -8/11 0 18 -5/7 1/1 2 18 -7/10 0 18 -2/3 0/1 1 6 -7/11 (0/1,1/0) 0 18 -17/27 1/0 4 2 -5/8 0 18 -18/29 0 18 -13/21 -1/2 4 6 -8/13 0 18 -3/5 1/1 2 18 -7/12 -1/1 1 6 -11/19 (-2/1,-1/1) 0 18 -4/7 0 18 -5/9 0/1 4 2 -1/2 0 18 -5/11 -1/3 2 18 -4/9 0/1 5 2 -3/7 (0/1,1/2) 0 18 -2/5 0 18 -5/13 (-3/1,1/0) 0 18 -8/21 -2/1 1 6 -3/8 0 18 -1/3 0/1 2 6 -3/10 0 18 -8/27 1/1 13 2 -5/17 (1/1,4/3) 0 18 -2/7 0 18 -3/11 (-2/1,1/0) 0 18 -1/4 0 18 -3/13 -1/5 2 18 -2/9 0/1 7 2 -1/5 (0/1,1/2) 0 18 0/1 0 18 1/6 1/1 1 6 1/5 3/1 2 18 2/9 1/0 8 2 3/13 (-5/1,1/0) 0 18 1/4 0 18 1/3 1/0 4 6 2/5 0 18 7/17 (-2/1,-1/1) 0 18 5/12 -1/1 1 6 3/7 1/1 2 18 4/9 1/0 7 2 5/11 (-6/1,1/0) 0 18 1/2 0 18 7/13 (-1/1,1/0) 0 18 5/9 1/0 2 2 3/5 (-2/1,1/0) 0 18 5/8 0 18 17/27 -2/1 4 2 12/19 0 18 7/11 -5/3 2 18 2/3 -1/1 1 6 7/10 0 18 19/27 1/0 4 2 5/7 (-2/1,1/0) 0 18 3/4 0 18 7/9 -1/1 8 2 4/5 0 18 9/11 (-2/1,1/0) 0 18 5/6 -2/1 1 6 11/13 (-3/2,-1/1) 0 18 6/7 0 18 7/8 0 18 8/9 -1/1 13 2 1/1 -1/1 2 18 1/0 0/1 1 2 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(8,7,-9,-8) (-1/1,-7/9) -> (-1/1,-7/9) Reflection Matrix(118,91,-153,-118) (-7/9,-13/17) -> (-7/9,-13/17) Reflection Matrix(62,47,153,116) (-10/13,-3/4) -> (2/5,9/22) Hyperbolic Matrix(172,127,-279,-206) (-3/4,-11/15) -> (-13/21,-8/13) Hyperbolic Matrix(413,302,-666,-487) (-11/15,-8/11) -> (-18/29,-13/21) Hyperbolic Matrix(62,45,-135,-98) (-8/11,-5/7) -> (-1/2,-5/11) Glide Reflection Matrix(154,109,243,172) (-5/7,-7/10) -> (12/19,7/11) Hyperbolic Matrix(89,62,-234,-163) (-7/10,-2/3) -> (-8/21,-3/8) Hyperbolic Matrix(82,53,99,64) (-2/3,-7/11) -> (9/11,5/6) Hyperbolic Matrix(188,119,-297,-188) (-7/11,-17/27) -> (-7/11,-17/27) Reflection Matrix(341,214,486,305) (-17/27,-5/8) -> (7/10,19/27) Hyperbolic Matrix(251,156,288,179) (-5/8,-18/29) -> (6/7,7/8) Hyperbolic Matrix(44,27,-189,-116) (-8/13,-3/5) -> (-1/4,-3/13) Glide Reflection Matrix(17,10,90,53) (-3/5,-7/12) -> (1/6,1/5) Hyperbolic Matrix(179,104,432,251) (-7/12,-11/19) -> (7/17,5/12) Hyperbolic Matrix(73,42,-252,-145) (-11/19,-4/7) -> (-5/17,-2/7) Glide Reflection Matrix(44,25,-81,-46) (-4/7,-5/9) -> (-5/9,-1/2) Parabolic Matrix(89,40,-198,-89) (-5/11,-4/9) -> (-5/11,-4/9) Reflection Matrix(55,24,-126,-55) (-4/9,-3/7) -> (-4/9,-3/7) Reflection Matrix(46,19,63,26) (-3/7,-2/5) -> (5/7,3/4) Glide Reflection Matrix(28,11,117,46) (-2/5,-5/13) -> (3/13,1/4) Hyperbolic Matrix(296,113,351,134) (-5/13,-8/21) -> (5/6,11/13) Hyperbolic Matrix(17,6,-54,-19) (-3/8,-1/3) -> (-1/3,-3/10) Parabolic Matrix(269,80,306,91) (-3/10,-8/27) -> (7/8,8/9) Glide Reflection Matrix(271,80,-918,-271) (-8/27,-5/17) -> (-8/27,-5/17) Reflection Matrix(46,13,99,28) (-2/7,-3/11) -> (5/11,1/2) Hyperbolic Matrix(80,21,99,26) (-3/11,-1/4) -> (4/5,9/11) Hyperbolic Matrix(53,12,-234,-53) (-3/13,-2/9) -> (-3/13,-2/9) Reflection Matrix(19,4,-90,-19) (-2/9,-1/5) -> (-2/9,-1/5) Reflection Matrix(28,5,45,8) (-1/5,0/1) -> (3/5,5/8) Glide Reflection Matrix(44,-7,63,-10) (0/1,1/6) -> (2/3,7/10) Hyperbolic Matrix(19,-4,90,-19) (1/5,2/9) -> (1/5,2/9) Reflection Matrix(53,-12,234,-53) (2/9,3/13) -> (2/9,3/13) Reflection Matrix(10,-3,27,-8) (1/4,1/3) -> (1/3,2/5) Parabolic Matrix(188,-77,459,-188) (11/27,7/17) -> (11/27,7/17) Reflection Matrix(98,-41,153,-64) (5/12,3/7) -> (7/11,2/3) Hyperbolic Matrix(55,-24,126,-55) (3/7,4/9) -> (3/7,4/9) Reflection Matrix(89,-40,198,-89) (4/9,5/11) -> (4/9,5/11) Reflection Matrix(100,-53,117,-62) (1/2,7/13) -> (11/13,6/7) Hyperbolic Matrix(64,-35,117,-64) (7/13,5/9) -> (7/13,5/9) Reflection Matrix(26,-15,45,-26) (5/9,3/5) -> (5/9,3/5) Reflection Matrix(460,-289,729,-458) (5/8,17/27) -> (17/27,12/19) Parabolic Matrix(134,-95,189,-134) (19/27,5/7) -> (19/27,5/7) Reflection Matrix(64,-49,81,-62) (3/4,7/9) -> (7/9,4/5) Parabolic Matrix(17,-16,18,-17) (8/9,1/1) -> (8/9,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(8,7,-9,-8) -> Matrix(1,0,0,-1) (-1/1,-7/9) -> (0/1,1/0) Matrix(118,91,-153,-118) -> Matrix(1,0,2,-1) (-7/9,-13/17) -> (0/1,1/1) Matrix(62,47,153,116) -> Matrix(1,-2,0,1) 1/0 Matrix(172,127,-279,-206) -> Matrix(1,0,-2,1) 0/1 Matrix(413,302,-666,-487) -> Matrix(1,2,-2,-3) -1/1 Matrix(62,45,-135,-98) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(154,109,243,172) -> Matrix(3,2,-2,-1) -1/1 Matrix(89,62,-234,-163) -> Matrix(1,-2,0,1) 1/0 Matrix(82,53,99,64) -> Matrix(1,-2,0,1) 1/0 Matrix(188,119,-297,-188) -> Matrix(1,0,0,-1) (-7/11,-17/27) -> (0/1,1/0) Matrix(341,214,486,305) -> Matrix(1,0,0,1) Matrix(251,156,288,179) -> Matrix(7,6,-6,-5) -1/1 Matrix(44,27,-189,-116) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(17,10,90,53) -> Matrix(1,2,0,1) 1/0 Matrix(179,104,432,251) -> Matrix(1,0,0,1) Matrix(73,42,-252,-145) -> Matrix(3,2,2,1) Matrix(44,25,-81,-46) -> Matrix(1,0,2,1) 0/1 Matrix(89,40,-198,-89) -> Matrix(-1,0,6,1) (-5/11,-4/9) -> (-1/3,0/1) Matrix(55,24,-126,-55) -> Matrix(1,0,4,-1) (-4/9,-3/7) -> (0/1,1/2) Matrix(46,19,63,26) -> Matrix(3,-2,-2,1) Matrix(28,11,117,46) -> Matrix(1,-2,0,1) 1/0 Matrix(296,113,351,134) -> Matrix(3,8,-2,-5) -2/1 Matrix(17,6,-54,-19) -> Matrix(1,0,2,1) 0/1 Matrix(269,80,306,91) -> Matrix(11,-10,-10,9) Matrix(271,80,-918,-271) -> Matrix(7,-8,6,-7) (-8/27,-5/17) -> (1/1,4/3) Matrix(46,13,99,28) -> Matrix(1,-4,0,1) 1/0 Matrix(80,21,99,26) -> Matrix(1,0,0,1) Matrix(53,12,-234,-53) -> Matrix(-1,0,10,1) (-3/13,-2/9) -> (-1/5,0/1) Matrix(19,4,-90,-19) -> Matrix(1,0,4,-1) (-2/9,-1/5) -> (0/1,1/2) Matrix(28,5,45,8) -> Matrix(3,-2,-2,1) Matrix(44,-7,63,-10) -> Matrix(1,-2,0,1) 1/0 Matrix(19,-4,90,-19) -> Matrix(-1,6,0,1) (1/5,2/9) -> (3/1,1/0) Matrix(53,-12,234,-53) -> Matrix(1,10,0,-1) (2/9,3/13) -> (-5/1,1/0) Matrix(10,-3,27,-8) -> Matrix(1,-2,0,1) 1/0 Matrix(188,-77,459,-188) -> Matrix(3,4,-2,-3) (11/27,7/17) -> (-2/1,-1/1) Matrix(98,-41,153,-64) -> Matrix(3,2,-2,-1) -1/1 Matrix(55,-24,126,-55) -> Matrix(-1,2,0,1) (3/7,4/9) -> (1/1,1/0) Matrix(89,-40,198,-89) -> Matrix(1,12,0,-1) (4/9,5/11) -> (-6/1,1/0) Matrix(100,-53,117,-62) -> Matrix(3,2,-2,-1) -1/1 Matrix(64,-35,117,-64) -> Matrix(1,2,0,-1) (7/13,5/9) -> (-1/1,1/0) Matrix(26,-15,45,-26) -> Matrix(1,4,0,-1) (5/9,3/5) -> (-2/1,1/0) Matrix(460,-289,729,-458) -> Matrix(3,8,-2,-5) -2/1 Matrix(134,-95,189,-134) -> Matrix(1,4,0,-1) (19/27,5/7) -> (-2/1,1/0) Matrix(64,-49,81,-62) -> Matrix(3,4,-4,-5) -1/1 Matrix(17,-16,18,-17) -> Matrix(1,2,0,-1) (8/9,1/1) -> (-1/1,1/0) Matrix(-1,2,0,1) -> Matrix(-1,0,2,1) (1/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.