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: 64 Genus: 33 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -9/1 -8/1 -6/1 -16/3 -14/3 -9/2 -4/1 -26/7 -18/5 -10/3 -36/11 -3/1 -20/7 -8/3 -12/5 -16/7 -2/1 -20/11 -12/7 -18/11 -36/23 -3/2 -4/3 -6/5 0/1 1/1 6/5 4/3 3/2 36/23 18/11 12/7 2/1 24/11 16/7 12/5 120/49 5/2 8/3 48/17 20/7 3/1 36/11 96/29 10/3 24/7 7/2 18/5 11/3 4/1 48/11 22/5 9/2 14/3 24/5 5/1 16/3 11/2 6/1 20/3 7/1 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -9/1 -2/1 0/1 -8/1 -1/1 -15/2 -4/5 -2/3 -7/1 -3/4 -2/3 -6/1 -2/3 0/1 -17/3 -3/4 -2/3 -11/2 -1/2 -2/5 -16/3 0/1 -5/1 -1/1 0/1 -14/3 -1/1 -2/3 -23/5 -1/1 0/1 -9/2 -2/3 0/1 -4/1 -2/3 0/1 -15/4 -2/3 0/1 -26/7 -1/1 -2/3 -11/3 -2/3 -1/2 -18/5 -2/3 0/1 -25/7 -1/1 -4/5 -7/2 -2/3 -1/2 -10/3 -1/2 0/1 -33/10 -2/1 0/1 -23/7 -1/1 0/1 -36/11 -2/3 0/1 -13/4 -2/3 -1/2 -16/5 0/1 -3/1 -2/3 0/1 -20/7 -4/5 -2/3 -17/6 -7/10 -2/3 -14/5 -2/3 -3/5 -25/9 -3/5 -4/7 -36/13 -2/3 -4/7 -11/4 -2/3 -1/2 -8/3 -1/2 -13/5 -1/2 -2/5 -18/7 -2/5 0/1 -23/9 -4/11 -1/3 -28/11 -2/7 0/1 -5/2 -1/1 0/1 -12/5 -2/3 0/1 -19/8 -1/1 0/1 -45/19 -2/1 0/1 -71/30 -8/7 -1/1 -26/11 -1/1 -2/3 -33/14 -2/3 0/1 -7/3 -2/3 -1/2 -23/10 -1/1 -4/5 -16/7 -2/3 -25/11 -3/5 -4/7 -9/4 -2/3 -4/7 -38/17 -7/12 -4/7 -29/13 -4/7 -13/23 -20/9 -4/7 -6/11 -11/5 -10/19 -1/2 -2/1 -1/2 0/1 -13/7 -10/19 -1/2 -24/13 -1/2 -11/6 -1/2 -10/21 -31/17 -13/28 -6/13 -20/11 -6/13 -4/9 -29/16 -13/29 -4/9 -9/5 -4/9 -2/5 -25/14 -4/9 -3/7 -16/9 -2/5 -23/13 -4/11 -1/3 -7/4 -1/2 -2/5 -12/7 -2/5 0/1 -17/10 -1/2 -2/5 -22/13 -2/5 -1/3 -71/42 -12/35 -1/3 -120/71 -1/3 -49/29 -1/3 -8/25 -27/16 -2/7 0/1 -5/3 -1/3 0/1 -28/17 -2/1 0/1 -51/31 -2/1 -4/3 -23/14 -1/1 -4/5 -18/11 -2/3 0/1 -13/8 -2/3 -1/2 -8/5 -1/2 -11/7 -1/2 -2/5 -36/23 -4/9 -2/5 -25/16 -4/9 -3/7 -14/9 -3/7 -2/5 -31/20 -13/32 -2/5 -48/31 -2/5 -17/11 -2/5 -7/18 -20/13 -2/5 -4/11 -23/15 -8/23 -1/3 -3/2 -2/5 0/1 -19/13 -1/3 0/1 -16/11 0/1 -13/9 -1/2 -2/5 -36/25 -2/5 0/1 -23/16 -1/3 0/1 -33/23 -2/7 0/1 -43/30 -1/7 0/1 -96/67 0/1 -53/37 0/1 1/3 -10/7 -1/2 0/1 -17/12 -2/3 -1/2 -24/17 -1/2 -7/5 -1/2 -2/5 -25/18 -4/11 -1/3 -18/13 -2/5 0/1 -11/8 -1/2 -2/5 -4/3 -2/5 0/1 -13/10 -1/2 -2/5 -48/37 -2/5 -35/27 -2/5 -3/8 -22/17 -2/5 -1/3 -31/24 -1/2 -2/5 -9/7 -2/5 0/1 -23/18 -1/3 0/1 -14/11 -2/5 -1/3 -19/15 -4/11 -1/3 -24/19 -1/3 -5/4 -1/3 0/1 -21/17 -2/7 0/1 -16/13 0/1 -11/9 -2/3 -1/2 -6/5 -2/5 0/1 -13/11 -2/3 -1/2 -20/17 -4/9 -2/5 -7/6 -2/5 -3/8 -15/13 -2/5 -4/11 -8/7 -1/3 -9/8 -2/7 0/1 -1/1 -1/3 0/1 0/1 0/1 1/1 0/1 1/1 9/8 0/1 2/3 8/7 1/1 15/13 4/3 2/1 7/6 3/2 2/1 6/5 0/1 2/1 17/14 3/2 2/1 11/9 -2/1 1/0 16/13 0/1 5/4 0/1 1/1 14/11 1/1 2/1 23/18 0/1 1/1 9/7 0/1 2/1 4/3 0/1 2/1 15/11 0/1 2/1 26/19 1/1 2/1 11/8 2/1 1/0 18/13 0/1 2/1 25/18 1/1 4/3 7/5 2/1 1/0 10/7 0/1 1/0 33/23 0/1 2/3 23/16 0/1 1/1 36/25 0/1 2/1 13/9 2/1 1/0 16/11 0/1 3/2 0/1 2/1 20/13 4/3 2/1 17/11 7/4 2/1 14/9 2/1 3/1 25/16 3/1 4/1 36/23 2/1 4/1 11/7 2/1 1/0 8/5 1/0 13/8 -2/1 1/0 18/11 -2/1 0/1 23/14 -4/3 -1/1 28/17 -2/3 0/1 5/3 0/1 1/1 12/7 0/1 2/1 19/11 0/1 1/1 45/26 0/1 2/3 71/41 8/9 1/1 26/15 1/1 2/1 33/19 0/1 2/1 7/4 2/1 1/0 23/13 1/1 4/3 16/9 2/1 25/14 3/1 4/1 9/5 2/1 4/1 38/21 7/2 4/1 29/16 4/1 13/3 20/11 4/1 6/1 11/6 10/1 1/0 2/1 0/1 1/0 13/6 10/1 1/0 24/11 1/0 11/5 -10/1 1/0 31/14 -13/2 -6/1 20/9 -6/1 -4/1 29/13 -13/3 -4/1 9/4 -4/1 -2/1 25/11 -4/1 -3/1 16/7 -2/1 23/10 -4/3 -1/1 7/3 -2/1 1/0 12/5 -2/1 0/1 17/7 -2/1 1/0 22/9 -2/1 -1/1 71/29 -12/11 -1/1 120/49 -1/1 49/20 -1/1 -8/9 27/11 -2/3 0/1 5/2 -1/1 0/1 28/11 0/1 2/3 51/20 2/3 4/5 23/9 1/1 4/3 18/7 0/1 2/1 13/5 2/1 1/0 8/3 1/0 11/4 -2/1 1/0 36/13 -4/1 -2/1 25/9 -4/1 -3/1 14/5 -3/1 -2/1 31/11 -13/6 -2/1 48/17 -2/1 17/6 -2/1 -7/4 20/7 -2/1 -4/3 23/8 -8/7 -1/1 3/1 -2/1 0/1 19/6 -1/1 0/1 16/5 0/1 13/4 -2/1 1/0 36/11 -2/1 0/1 23/7 -1/1 0/1 33/10 -2/3 0/1 43/13 -1/5 0/1 96/29 0/1 53/16 0/1 1/5 10/3 0/1 1/0 17/5 2/1 1/0 24/7 1/0 7/2 -2/1 1/0 25/7 -4/3 -1/1 18/5 -2/1 0/1 11/3 -2/1 1/0 4/1 -2/1 0/1 13/3 -2/1 1/0 48/11 -2/1 35/8 -2/1 -3/2 22/5 -2/1 -1/1 31/7 -2/1 1/0 9/2 -2/1 0/1 23/5 -1/1 0/1 14/3 -2/1 -1/1 19/4 -4/3 -1/1 24/5 -1/1 5/1 -1/1 0/1 21/4 -2/3 0/1 16/3 0/1 11/2 2/1 1/0 6/1 -2/1 0/1 13/2 2/1 1/0 20/3 -4/1 -2/1 7/1 -2/1 -3/2 15/2 -2/1 -4/3 8/1 -1/1 9/1 -2/3 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(23,240,16,167) (-9/1,1/0) -> (33/23,23/16) Hyperbolic Matrix(23,192,20,167) (-9/1,-8/1) -> (8/7,15/13) Hyperbolic Matrix(25,192,22,169) (-8/1,-15/2) -> (9/8,8/7) Hyperbolic Matrix(71,528,16,119) (-15/2,-7/1) -> (31/7,9/2) Hyperbolic Matrix(23,144,-4,-25) (-7/1,-6/1) -> (-6/1,-17/3) Parabolic Matrix(95,528,-52,-289) (-17/3,-11/2) -> (-11/6,-31/17) Hyperbolic Matrix(71,384,22,119) (-11/2,-16/3) -> (16/5,13/4) Hyperbolic Matrix(73,384,-50,-263) (-16/3,-5/1) -> (-19/13,-16/11) Hyperbolic Matrix(71,336,-56,-265) (-5/1,-14/3) -> (-14/11,-19/15) Hyperbolic Matrix(145,672,52,241) (-14/3,-23/5) -> (25/9,14/5) Hyperbolic Matrix(95,432,42,191) (-23/5,-9/2) -> (9/4,25/11) Hyperbolic Matrix(23,96,-6,-25) (-9/2,-4/1) -> (-4/1,-15/4) Parabolic Matrix(335,1248,-142,-529) (-15/4,-26/7) -> (-26/11,-33/14) Hyperbolic Matrix(311,1152,-240,-889) (-26/7,-11/3) -> (-35/27,-22/17) Hyperbolic Matrix(119,432,46,167) (-11/3,-18/5) -> (18/7,13/5) Hyperbolic Matrix(241,864,94,337) (-18/5,-25/7) -> (23/9,18/7) Hyperbolic Matrix(95,336,54,191) (-25/7,-7/2) -> (7/4,23/13) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(407,1344,-182,-601) (-10/3,-33/10) -> (-9/4,-38/17) Hyperbolic Matrix(73,240,66,217) (-33/10,-23/7) -> (1/1,9/8) Hyperbolic Matrix(527,1728,190,623) (-23/7,-36/11) -> (36/13,25/9) Hyperbolic Matrix(265,864,96,313) (-36/11,-13/4) -> (11/4,36/13) Hyperbolic Matrix(119,384,22,71) (-13/4,-16/5) -> (16/3,11/2) Hyperbolic Matrix(121,384,-98,-311) (-16/5,-3/1) -> (-21/17,-16/13) Hyperbolic Matrix(385,1104,-234,-671) (-3/1,-20/7) -> (-28/17,-51/31) Hyperbolic Matrix(337,960,152,433) (-20/7,-17/6) -> (31/14,20/9) Hyperbolic Matrix(239,672,-154,-433) (-17/6,-14/5) -> (-14/9,-31/20) Hyperbolic Matrix(241,672,52,145) (-14/5,-25/9) -> (23/5,14/3) Hyperbolic Matrix(623,1728,190,527) (-25/9,-36/13) -> (36/11,23/7) Hyperbolic Matrix(313,864,96,265) (-36/13,-11/4) -> (13/4,36/11) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(73,192,46,121) (-8/3,-13/5) -> (11/7,8/5) Hyperbolic Matrix(167,432,46,119) (-13/5,-18/7) -> (18/5,11/3) Hyperbolic Matrix(337,864,94,241) (-18/7,-23/9) -> (25/7,18/5) Hyperbolic Matrix(433,1104,-282,-719) (-23/9,-28/11) -> (-20/13,-23/15) Hyperbolic Matrix(359,912,198,503) (-28/11,-5/2) -> (29/16,20/11) Hyperbolic Matrix(119,288,-50,-121) (-5/2,-12/5) -> (-12/5,-19/8) Parabolic Matrix(385,912,122,289) (-19/8,-45/19) -> (3/1,19/6) Hyperbolic Matrix(527,1248,182,431) (-45/19,-71/30) -> (23/8,3/1) Hyperbolic Matrix(1441,3408,-852,-2015) (-71/30,-26/11) -> (-22/13,-71/42) Hyperbolic Matrix(143,336,20,47) (-33/14,-7/3) -> (7/1,15/2) Hyperbolic Matrix(145,336,104,241) (-7/3,-23/10) -> (25/18,7/5) Hyperbolic Matrix(335,768,188,431) (-23/10,-16/7) -> (16/9,25/14) Hyperbolic Matrix(337,768,190,433) (-16/7,-25/11) -> (23/13,16/9) Hyperbolic Matrix(191,432,42,95) (-25/11,-9/4) -> (9/2,23/5) Hyperbolic Matrix(1031,2304,-720,-1609) (-38/17,-29/13) -> (-53/37,-10/7) Hyperbolic Matrix(409,912,248,553) (-29/13,-20/9) -> (28/17,5/3) Hyperbolic Matrix(217,480,-184,-407) (-20/9,-11/5) -> (-13/11,-20/17) Hyperbolic Matrix(23,48,-12,-25) (-11/5,-2/1) -> (-2/1,-13/7) Parabolic Matrix(311,576,142,263) (-13/7,-24/13) -> (24/11,11/5) Hyperbolic Matrix(313,576,144,265) (-24/13,-11/6) -> (13/6,24/11) Hyperbolic Matrix(527,960,342,623) (-31/17,-20/11) -> (20/13,17/11) Hyperbolic Matrix(503,912,198,359) (-20/11,-29/16) -> (5/2,28/11) Hyperbolic Matrix(743,1344,-518,-937) (-29/16,-9/5) -> (-33/23,-43/30) Hyperbolic Matrix(241,432,188,337) (-9/5,-25/14) -> (23/18,9/7) Hyperbolic Matrix(431,768,188,335) (-25/14,-16/9) -> (16/7,23/10) Hyperbolic Matrix(433,768,190,337) (-16/9,-23/13) -> (25/11,16/7) Hyperbolic Matrix(191,336,54,95) (-23/13,-7/4) -> (7/2,25/7) Hyperbolic Matrix(167,288,-98,-169) (-7/4,-12/7) -> (-12/7,-17/10) Parabolic Matrix(623,1056,-482,-817) (-17/10,-22/13) -> (-22/17,-31/24) Hyperbolic Matrix(8519,14400,3478,5879) (-71/42,-120/71) -> (120/49,49/20) Hyperbolic Matrix(8521,14400,3480,5881) (-120/71,-49/29) -> (71/29,120/49) Hyperbolic Matrix(1847,3120,724,1223) (-49/29,-27/16) -> (51/20,23/9) Hyperbolic Matrix(143,240,28,47) (-27/16,-5/3) -> (5/1,21/4) Hyperbolic Matrix(553,912,248,409) (-5/3,-28/17) -> (20/9,29/13) Hyperbolic Matrix(1897,3120,774,1273) (-51/31,-23/14) -> (49/20,27/11) Hyperbolic Matrix(527,864,380,623) (-23/14,-18/11) -> (18/13,25/18) Hyperbolic Matrix(265,432,192,313) (-18/11,-13/8) -> (11/8,18/13) Hyperbolic Matrix(119,192,44,71) (-13/8,-8/5) -> (8/3,11/4) Hyperbolic Matrix(121,192,46,73) (-8/5,-11/7) -> (13/5,8/3) Hyperbolic Matrix(551,864,382,599) (-11/7,-36/23) -> (36/25,13/9) Hyperbolic Matrix(1105,1728,768,1201) (-36/23,-25/16) -> (23/16,36/25) Hyperbolic Matrix(431,672,338,527) (-25/16,-14/9) -> (14/11,23/18) Hyperbolic Matrix(1487,2304,526,815) (-31/20,-48/31) -> (48/17,17/6) Hyperbolic Matrix(1489,2304,528,817) (-48/31,-17/11) -> (31/11,48/17) Hyperbolic Matrix(311,480,46,71) (-17/11,-20/13) -> (20/3,7/1) Hyperbolic Matrix(817,1248,472,721) (-23/15,-3/2) -> (45/26,71/41) Hyperbolic Matrix(623,912,360,527) (-3/2,-19/13) -> (19/11,45/26) Hyperbolic Matrix(265,384,216,313) (-16/11,-13/9) -> (11/9,16/13) Hyperbolic Matrix(599,864,382,551) (-13/9,-36/25) -> (36/23,11/7) Hyperbolic Matrix(1201,1728,768,1105) (-36/25,-23/16) -> (25/16,36/23) Hyperbolic Matrix(167,240,16,23) (-23/16,-33/23) -> (9/1,1/0) Hyperbolic Matrix(6431,9216,1942,2783) (-43/30,-96/67) -> (96/29,53/16) Hyperbolic Matrix(6433,9216,1944,2785) (-96/67,-53/37) -> (43/13,96/29) Hyperbolic Matrix(407,576,118,167) (-17/12,-24/17) -> (24/7,7/2) Hyperbolic Matrix(409,576,120,169) (-24/17,-7/5) -> (17/5,24/7) Hyperbolic Matrix(241,336,104,145) (-7/5,-25/18) -> (23/10,7/3) Hyperbolic Matrix(623,864,380,527) (-25/18,-18/13) -> (18/11,23/14) Hyperbolic Matrix(313,432,192,265) (-18/13,-11/8) -> (13/8,18/11) Hyperbolic Matrix(71,96,-54,-73) (-11/8,-4/3) -> (-4/3,-13/10) Parabolic Matrix(1775,2304,406,527) (-13/10,-48/37) -> (48/11,35/8) Hyperbolic Matrix(1777,2304,408,529) (-48/37,-35/27) -> (13/3,48/11) Hyperbolic Matrix(409,528,354,457) (-31/24,-9/7) -> (15/13,7/6) Hyperbolic Matrix(337,432,188,241) (-9/7,-23/18) -> (25/14,9/5) Hyperbolic Matrix(527,672,338,431) (-23/18,-14/11) -> (14/9,25/16) Hyperbolic Matrix(455,576,94,119) (-19/15,-24/19) -> (24/5,5/1) Hyperbolic Matrix(457,576,96,121) (-24/19,-5/4) -> (19/4,24/5) Hyperbolic Matrix(193,240,78,97) (-5/4,-21/17) -> (27/11,5/2) Hyperbolic Matrix(313,384,216,265) (-16/13,-11/9) -> (13/9,16/11) Hyperbolic Matrix(119,144,-100,-121) (-11/9,-6/5) -> (-6/5,-13/11) Parabolic Matrix(409,480,144,169) (-20/17,-7/6) -> (17/6,20/7) Hyperbolic Matrix(289,336,166,193) (-7/6,-15/13) -> (33/19,7/4) Hyperbolic Matrix(167,192,20,23) (-15/13,-8/7) -> (8/1,9/1) Hyperbolic Matrix(169,192,22,25) (-8/7,-9/8) -> (15/2,8/1) Hyperbolic Matrix(217,240,66,73) (-9/8,-1/1) -> (23/7,33/10) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(121,-144,100,-119) (7/6,6/5) -> (6/5,17/14) Parabolic Matrix(433,-528,196,-239) (17/14,11/9) -> (11/5,31/14) Hyperbolic Matrix(311,-384,98,-121) (16/13,5/4) -> (19/6,16/5) Hyperbolic Matrix(265,-336,56,-71) (5/4,14/11) -> (14/3,19/4) Hyperbolic Matrix(73,-96,54,-71) (9/7,4/3) -> (4/3,15/11) Parabolic Matrix(913,-1248,526,-719) (15/11,26/19) -> (26/15,33/19) Hyperbolic Matrix(841,-1152,192,-263) (26/19,11/8) -> (35/8,22/5) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(937,-1344,518,-743) (10/7,33/23) -> (9/5,38/21) Hyperbolic Matrix(263,-384,50,-73) (16/11,3/2) -> (21/4,16/3) Hyperbolic Matrix(719,-1104,282,-433) (3/2,20/13) -> (28/11,51/20) Hyperbolic Matrix(433,-672,154,-239) (17/11,14/9) -> (14/5,31/11) Hyperbolic Matrix(671,-1104,234,-385) (23/14,28/17) -> (20/7,23/8) Hyperbolic Matrix(169,-288,98,-167) (5/3,12/7) -> (12/7,19/11) Parabolic Matrix(1967,-3408,804,-1393) (71/41,26/15) -> (22/9,71/29) Hyperbolic Matrix(1273,-2304,384,-695) (38/21,29/16) -> (53/16,10/3) Hyperbolic Matrix(263,-480,40,-73) (20/11,11/6) -> (13/2,20/3) Hyperbolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(601,-1344,182,-407) (29/13,9/4) -> (33/10,43/13) Hyperbolic Matrix(121,-288,50,-119) (7/3,12/5) -> (12/5,17/7) Parabolic Matrix(433,-1056,98,-239) (17/7,22/9) -> (22/5,31/7) Hyperbolic Matrix(25,-96,6,-23) (11/3,4/1) -> (4/1,13/3) Parabolic Matrix(25,-144,4,-23) (11/2,6/1) -> (6/1,13/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(23,240,16,167) -> Matrix(1,0,2,1) Matrix(23,192,20,167) -> Matrix(3,4,2,3) Matrix(25,192,22,169) -> Matrix(5,4,6,5) Matrix(71,528,16,119) -> Matrix(5,4,-4,-3) Matrix(23,144,-4,-25) -> Matrix(1,0,0,1) Matrix(95,528,-52,-289) -> Matrix(15,8,-32,-17) Matrix(71,384,22,119) -> Matrix(1,0,2,1) Matrix(73,384,-50,-263) -> Matrix(1,0,-2,1) Matrix(71,336,-56,-265) -> Matrix(5,4,-14,-11) Matrix(145,672,52,241) -> Matrix(7,4,-2,-1) Matrix(95,432,42,191) -> Matrix(7,4,-2,-1) Matrix(23,96,-6,-25) -> Matrix(1,0,0,1) Matrix(335,1248,-142,-529) -> Matrix(1,0,0,1) Matrix(311,1152,-240,-889) -> Matrix(5,4,-14,-11) Matrix(119,432,46,167) -> Matrix(1,0,2,1) Matrix(241,864,94,337) -> Matrix(1,0,2,1) Matrix(95,336,54,191) -> Matrix(1,0,2,1) Matrix(71,240,-50,-169) -> Matrix(1,0,0,1) Matrix(407,1344,-182,-601) -> Matrix(1,4,-2,-7) Matrix(73,240,66,217) -> Matrix(1,0,2,1) Matrix(527,1728,190,623) -> Matrix(7,4,-2,-1) Matrix(265,864,96,313) -> Matrix(7,4,-2,-1) Matrix(119,384,22,71) -> Matrix(1,0,2,1) Matrix(121,384,-98,-311) -> Matrix(1,0,-2,1) Matrix(385,1104,-234,-671) -> Matrix(5,4,-4,-3) Matrix(337,960,152,433) -> Matrix(21,16,-4,-3) Matrix(239,672,-154,-433) -> Matrix(19,12,-46,-29) Matrix(241,672,52,145) -> Matrix(7,4,-2,-1) Matrix(623,1728,190,527) -> Matrix(7,4,-2,-1) Matrix(313,864,96,265) -> Matrix(7,4,-2,-1) Matrix(71,192,44,119) -> Matrix(7,4,-2,-1) Matrix(73,192,46,121) -> Matrix(9,4,2,1) Matrix(167,432,46,119) -> Matrix(1,0,2,1) Matrix(337,864,94,241) -> Matrix(1,0,2,1) Matrix(433,1104,-282,-719) -> Matrix(13,4,-36,-11) Matrix(359,912,198,503) -> Matrix(17,4,4,1) Matrix(119,288,-50,-121) -> Matrix(1,0,0,1) Matrix(385,912,122,289) -> Matrix(1,0,0,1) Matrix(527,1248,182,431) -> Matrix(1,0,0,1) Matrix(1441,3408,-852,-2015) -> Matrix(5,4,-14,-11) Matrix(143,336,20,47) -> Matrix(5,4,-4,-3) Matrix(145,336,104,241) -> Matrix(1,0,2,1) Matrix(335,768,188,431) -> Matrix(11,8,4,3) Matrix(337,768,190,433) -> Matrix(13,8,8,5) Matrix(191,432,42,95) -> Matrix(7,4,-2,-1) Matrix(1031,2304,-720,-1609) -> Matrix(7,4,-2,-1) Matrix(409,912,248,553) -> Matrix(7,4,-16,-9) Matrix(217,480,-184,-407) -> Matrix(15,8,-32,-17) Matrix(23,48,-12,-25) -> Matrix(1,0,0,1) Matrix(311,576,142,263) -> Matrix(39,20,-2,-1) Matrix(313,576,144,265) -> Matrix(41,20,2,1) Matrix(527,960,342,623) -> Matrix(35,16,24,11) Matrix(503,912,198,359) -> Matrix(9,4,20,9) Matrix(743,1344,-518,-937) -> Matrix(9,4,-34,-15) Matrix(241,432,188,337) -> Matrix(9,4,2,1) Matrix(431,768,188,335) -> Matrix(19,8,-12,-5) Matrix(433,768,190,337) -> Matrix(21,8,-8,-3) Matrix(191,336,54,95) -> Matrix(1,0,2,1) Matrix(167,288,-98,-169) -> Matrix(1,0,0,1) Matrix(623,1056,-482,-817) -> Matrix(1,0,0,1) Matrix(8519,14400,3478,5879) -> Matrix(59,20,-62,-21) Matrix(8521,14400,3480,5881) -> Matrix(61,20,-58,-19) Matrix(1847,3120,724,1223) -> Matrix(13,4,16,5) Matrix(143,240,28,47) -> Matrix(1,0,2,1) Matrix(553,912,248,409) -> Matrix(1,-4,0,1) Matrix(1897,3120,774,1273) -> Matrix(3,4,-4,-5) Matrix(527,864,380,623) -> Matrix(1,0,2,1) Matrix(265,432,192,313) -> Matrix(1,0,2,1) Matrix(119,192,44,71) -> Matrix(7,4,-2,-1) Matrix(121,192,46,73) -> Matrix(9,4,2,1) Matrix(551,864,382,599) -> Matrix(9,4,2,1) Matrix(1105,1728,768,1201) -> Matrix(9,4,2,1) Matrix(431,672,338,527) -> Matrix(9,4,2,1) Matrix(1487,2304,526,815) -> Matrix(99,40,-52,-21) Matrix(1489,2304,528,817) -> Matrix(101,40,-48,-19) Matrix(311,480,46,71) -> Matrix(21,8,-8,-3) Matrix(817,1248,472,721) -> Matrix(1,0,4,1) Matrix(623,912,360,527) -> Matrix(1,0,4,1) Matrix(265,384,216,313) -> Matrix(1,0,2,1) Matrix(599,864,382,551) -> Matrix(9,4,2,1) Matrix(1201,1728,768,1105) -> Matrix(9,4,2,1) Matrix(167,240,16,23) -> Matrix(1,0,2,1) Matrix(6431,9216,1942,2783) -> Matrix(1,0,12,1) Matrix(6433,9216,1944,2785) -> Matrix(1,0,-8,1) Matrix(407,576,118,167) -> Matrix(7,4,-2,-1) Matrix(409,576,120,169) -> Matrix(9,4,2,1) Matrix(241,336,104,145) -> Matrix(1,0,2,1) Matrix(623,864,380,527) -> Matrix(1,0,2,1) Matrix(313,432,192,265) -> Matrix(1,0,2,1) Matrix(71,96,-54,-73) -> Matrix(1,0,0,1) Matrix(1775,2304,406,527) -> Matrix(19,8,-12,-5) Matrix(1777,2304,408,529) -> Matrix(21,8,-8,-3) Matrix(409,528,354,457) -> Matrix(11,4,8,3) Matrix(337,432,188,241) -> Matrix(9,4,2,1) Matrix(527,672,338,431) -> Matrix(9,4,2,1) Matrix(455,576,94,119) -> Matrix(11,4,-14,-5) Matrix(457,576,96,121) -> Matrix(13,4,-10,-3) Matrix(193,240,78,97) -> Matrix(1,0,2,1) Matrix(313,384,216,265) -> Matrix(1,0,2,1) Matrix(119,144,-100,-121) -> Matrix(1,0,0,1) Matrix(409,480,144,169) -> Matrix(19,8,-12,-5) Matrix(289,336,166,193) -> Matrix(11,4,8,3) Matrix(167,192,20,23) -> Matrix(11,4,-14,-5) Matrix(169,192,22,25) -> Matrix(13,4,-10,-3) Matrix(217,240,66,73) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(121,-144,100,-119) -> Matrix(1,0,0,1) Matrix(433,-528,196,-239) -> Matrix(1,-8,0,1) Matrix(311,-384,98,-121) -> Matrix(1,0,-2,1) Matrix(265,-336,56,-71) -> Matrix(3,-4,-2,3) Matrix(73,-96,54,-71) -> Matrix(1,0,0,1) Matrix(913,-1248,526,-719) -> Matrix(1,0,0,1) Matrix(841,-1152,192,-263) -> Matrix(3,-4,-2,3) Matrix(169,-240,50,-71) -> Matrix(1,0,0,1) Matrix(937,-1344,518,-743) -> Matrix(7,-4,2,-1) Matrix(263,-384,50,-73) -> Matrix(1,0,-2,1) Matrix(719,-1104,282,-433) -> Matrix(3,-4,4,-5) Matrix(433,-672,154,-239) -> Matrix(5,-12,-2,5) Matrix(671,-1104,234,-385) -> Matrix(5,4,-4,-3) Matrix(169,-288,98,-167) -> Matrix(1,0,0,1) Matrix(1967,-3408,804,-1393) -> Matrix(3,-4,-2,3) Matrix(1273,-2304,384,-695) -> Matrix(1,-4,2,-7) Matrix(263,-480,40,-73) -> Matrix(1,-8,0,1) Matrix(25,-48,12,-23) -> Matrix(1,0,0,1) Matrix(601,-1344,182,-407) -> Matrix(1,4,-2,-7) Matrix(121,-288,50,-119) -> Matrix(1,0,0,1) Matrix(433,-1056,98,-239) -> Matrix(1,0,0,1) Matrix(25,-96,6,-23) -> Matrix(1,0,0,1) Matrix(25,-144,4,-23) -> Matrix(1,0,0,1) 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): 16 Degree of the the map X: 32 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): 192 Minimal number of generators: 33 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: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 6/5 4/3 3/2 8/5 12/7 2/1 24/11 20/9 12/5 8/3 36/13 48/17 3/1 16/5 36/11 10/3 24/7 4/1 5/1 16/3 6/1 20/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 0/1 1/1 6/5 0/1 2/1 11/9 -2/1 1/0 5/4 0/1 1/1 4/3 0/1 2/1 7/5 2/1 1/0 10/7 0/1 1/0 13/9 2/1 1/0 3/2 0/1 2/1 17/11 7/4 2/1 14/9 2/1 3/1 11/7 2/1 1/0 8/5 1/0 13/8 -2/1 1/0 5/3 0/1 1/1 12/7 0/1 2/1 7/4 2/1 1/0 23/13 1/1 4/3 16/9 2/1 9/5 2/1 4/1 20/11 4/1 6/1 11/6 10/1 1/0 2/1 0/1 1/0 13/6 10/1 1/0 24/11 1/0 11/5 -10/1 1/0 20/9 -6/1 -4/1 29/13 -13/3 -4/1 9/4 -4/1 -2/1 25/11 -4/1 -3/1 16/7 -2/1 7/3 -2/1 1/0 12/5 -2/1 0/1 5/2 -1/1 0/1 28/11 0/1 2/3 23/9 1/1 4/3 18/7 0/1 2/1 13/5 2/1 1/0 8/3 1/0 11/4 -2/1 1/0 36/13 -4/1 -2/1 25/9 -4/1 -3/1 14/5 -3/1 -2/1 31/11 -13/6 -2/1 48/17 -2/1 17/6 -2/1 -7/4 20/7 -2/1 -4/3 3/1 -2/1 0/1 16/5 0/1 13/4 -2/1 1/0 36/11 -2/1 0/1 23/7 -1/1 0/1 10/3 0/1 1/0 17/5 2/1 1/0 24/7 1/0 7/2 -2/1 1/0 4/1 -2/1 0/1 5/1 -1/1 0/1 16/3 0/1 11/2 2/1 1/0 6/1 -2/1 0/1 13/2 2/1 1/0 20/3 -4/1 -2/1 7/1 -2/1 -3/2 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(83,-96,32,-37) (1/1,6/5) -> (18/7,13/5) Hyperbolic Matrix(277,-336,108,-131) (6/5,11/9) -> (23/9,18/7) Hyperbolic Matrix(155,-192,88,-109) (11/9,5/4) -> (7/4,23/13) Hyperbolic Matrix(37,-48,27,-35) (5/4,4/3) -> (4/3,7/5) Parabolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(301,-432,108,-155) (10/7,13/9) -> (25/9,14/5) Hyperbolic Matrix(131,-192,58,-85) (13/9,3/2) -> (9/4,25/11) Hyperbolic Matrix(157,-240,70,-107) (3/2,17/11) -> (29/13,9/4) Hyperbolic Matrix(433,-672,154,-239) (17/11,14/9) -> (14/5,31/11) Hyperbolic Matrix(277,-432,84,-131) (14/9,11/7) -> (23/7,10/3) Hyperbolic Matrix(121,-192,75,-119) (11/7,8/5) -> (8/5,13/8) Parabolic Matrix(59,-96,8,-13) (13/8,5/3) -> (7/1,1/0) Hyperbolic Matrix(85,-144,49,-83) (5/3,12/7) -> (12/7,7/4) Parabolic Matrix(325,-576,101,-179) (23/13,16/9) -> (16/5,13/4) Hyperbolic Matrix(107,-192,34,-61) (16/9,9/5) -> (3/1,16/5) Hyperbolic Matrix(133,-240,46,-83) (9/5,20/11) -> (20/7,3/1) Hyperbolic Matrix(263,-480,40,-73) (20/11,11/6) -> (13/2,20/3) Hyperbolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(265,-576,121,-263) (13/6,24/11) -> (24/11,11/5) Parabolic Matrix(347,-768,136,-301) (11/5,20/9) -> (28/11,23/9) Hyperbolic Matrix(409,-912,161,-359) (20/9,29/13) -> (5/2,28/11) Hyperbolic Matrix(253,-576,47,-107) (25/11,16/7) -> (16/3,11/2) Hyperbolic Matrix(83,-192,16,-37) (16/7,7/3) -> (5/1,16/3) Hyperbolic Matrix(61,-144,25,-59) (7/3,12/5) -> (12/5,5/2) Parabolic Matrix(73,-192,27,-71) (13/5,8/3) -> (8/3,11/4) Parabolic Matrix(469,-1296,169,-467) (11/4,36/13) -> (36/13,25/9) Parabolic Matrix(817,-2304,289,-815) (31/11,48/17) -> (48/17,17/6) Parabolic Matrix(169,-480,25,-71) (17/6,20/7) -> (20/3,7/1) Hyperbolic Matrix(397,-1296,121,-395) (13/4,36/11) -> (36/11,23/7) Parabolic Matrix(169,-576,49,-167) (17/5,24/7) -> (24/7,7/2) Parabolic Matrix(13,-48,3,-11) (7/2,4/1) -> (4/1,5/1) Parabolic Matrix(25,-144,4,-23) (11/2,6/1) -> (6/1,13/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,2,1) Matrix(83,-96,32,-37) -> Matrix(1,-2,1,-1) Matrix(277,-336,108,-131) -> Matrix(1,-2,1,-1) Matrix(155,-192,88,-109) -> Matrix(1,-2,1,-1) Matrix(37,-48,27,-35) -> Matrix(1,-2,1,-1) Matrix(169,-240,50,-71) -> Matrix(1,0,0,1) Matrix(301,-432,108,-155) -> Matrix(3,-2,-1,1) Matrix(131,-192,58,-85) -> Matrix(3,-2,-1,1) Matrix(157,-240,70,-107) -> Matrix(3,-2,-1,1) Matrix(433,-672,154,-239) -> Matrix(5,-12,-2,5) Matrix(277,-432,84,-131) -> Matrix(1,-2,-1,3) Matrix(121,-192,75,-119) -> Matrix(1,-4,0,1) Matrix(59,-96,8,-13) -> Matrix(1,2,-1,-1) Matrix(85,-144,49,-83) -> Matrix(1,-2,1,-1) Matrix(325,-576,101,-179) -> Matrix(1,-2,1,-1) Matrix(107,-192,34,-61) -> Matrix(1,-2,-1,3) Matrix(133,-240,46,-83) -> Matrix(1,-2,-1,3) Matrix(263,-480,40,-73) -> Matrix(1,-8,0,1) Matrix(25,-48,12,-23) -> Matrix(1,0,0,1) Matrix(265,-576,121,-263) -> Matrix(1,-20,0,1) Matrix(347,-768,136,-301) -> Matrix(1,6,1,7) Matrix(409,-912,161,-359) -> Matrix(1,4,2,9) Matrix(253,-576,47,-107) -> Matrix(1,2,1,3) Matrix(83,-192,16,-37) -> Matrix(1,2,-1,-1) Matrix(61,-144,25,-59) -> Matrix(1,2,-1,-1) Matrix(73,-192,27,-71) -> Matrix(1,-4,0,1) Matrix(469,-1296,169,-467) -> Matrix(3,10,-1,-3) Matrix(817,-2304,289,-815) -> Matrix(19,40,-10,-21) Matrix(169,-480,25,-71) -> Matrix(5,8,-2,-3) Matrix(397,-1296,121,-395) -> Matrix(1,2,-1,-1) Matrix(169,-576,49,-167) -> Matrix(1,-4,0,1) Matrix(13,-48,3,-11) -> Matrix(1,2,-1,-1) Matrix(25,-144,4,-23) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 3 Number of equivalence classes of elliptic points of order 2: 2 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): 8 ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 0/1 2 1 2/1 (0/1,1/0) 0 12 24/11 1/0 20 1 11/5 (-10/1,1/0) 0 24 20/9 (-6/1,-4/1).(-5/1,1/0) 0 6 9/4 0 8 16/7 -2/1 2 3 7/3 (-2/1,1/0) 0 24 12/5 (-2/1,0/1).(-1/1,1/0) 0 2 5/2 (-1/1,0/1) 0 24 13/5 (2/1,1/0) 0 24 8/3 1/0 4 3 11/4 (-2/1,1/0) 0 24 14/5 (-3/1,-2/1) 0 12 48/17 -2/1 10 1 17/6 (-2/1,-7/4) 0 24 20/7 (-2/1,-4/3).(-3/2,-1/1) 0 6 3/1 0 8 16/5 0/1 2 3 13/4 (-2/1,1/0) 0 24 36/11 (-2/1,0/1).(-1/1,1/0) 0 2 23/7 (-1/1,0/1) 0 24 10/3 (0/1,1/0) 0 12 24/7 1/0 4 1 7/2 (-2/1,1/0) 0 24 4/1 (-2/1,0/1).(-1/1,1/0) 0 6 5/1 (-1/1,0/1) 0 24 16/3 0/1 2 3 11/2 (2/1,1/0) 0 24 6/1 0 4 13/2 (2/1,1/0) 0 24 20/3 (-4/1,-2/1).(-3/1,1/0) 0 6 7/1 (-2/1,-3/2) 0 24 1/0 (-1/1,0/1) 0 24 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(23,-48,11,-23) (2/1,24/11) -> (2/1,24/11) Reflection Matrix(241,-528,110,-241) (24/11,11/5) -> (24/11,11/5) Reflection Matrix(217,-480,33,-73) (11/5,20/9) -> (13/2,20/3) Glide Reflection Matrix(107,-240,37,-83) (20/9,9/4) -> (20/7,3/1) Glide Reflection Matrix(85,-192,27,-61) (9/4,16/7) -> (3/1,16/5) Glide Reflection Matrix(83,-192,16,-37) (16/7,7/3) -> (5/1,16/3) Hyperbolic Matrix(61,-144,25,-59) (7/3,12/5) -> (12/5,5/2) Parabolic Matrix(37,-96,5,-13) (5/2,13/5) -> (7/1,1/0) Glide Reflection Matrix(73,-192,27,-71) (13/5,8/3) -> (8/3,11/4) Parabolic Matrix(155,-432,47,-131) (11/4,14/5) -> (23/7,10/3) Glide Reflection Matrix(239,-672,85,-239) (14/5,48/17) -> (14/5,48/17) Reflection Matrix(577,-1632,204,-577) (48/17,17/6) -> (48/17,17/6) Reflection Matrix(169,-480,25,-71) (17/6,20/7) -> (20/3,7/1) Hyperbolic Matrix(119,-384,22,-71) (16/5,13/4) -> (16/3,11/2) Glide Reflection Matrix(397,-1296,121,-395) (13/4,36/11) -> (36/11,23/7) Parabolic Matrix(71,-240,21,-71) (10/3,24/7) -> (10/3,24/7) Reflection Matrix(97,-336,28,-97) (24/7,7/2) -> (24/7,7/2) Reflection Matrix(13,-48,3,-11) (7/2,4/1) -> (4/1,5/1) Parabolic Matrix(25,-144,4,-23) (11/2,6/1) -> (6/1,13/2) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,1,-1) -> Matrix(1,0,0,-1) (0/1,2/1) -> (0/1,1/0) Matrix(23,-48,11,-23) -> Matrix(1,0,0,-1) (2/1,24/11) -> (0/1,1/0) Matrix(241,-528,110,-241) -> Matrix(1,20,0,-1) (24/11,11/5) -> (-10/1,1/0) Matrix(217,-480,33,-73) -> Matrix(1,8,0,-1) *** -> (-4/1,1/0) Matrix(107,-240,37,-83) -> Matrix(1,2,-1,-3) Matrix(85,-192,27,-61) -> Matrix(1,2,-1,-3) Matrix(83,-192,16,-37) -> Matrix(1,2,-1,-1) (-2/1,0/1).(-1/1,1/0) Matrix(61,-144,25,-59) -> Matrix(1,2,-1,-1) (-2/1,0/1).(-1/1,1/0) Matrix(37,-96,5,-13) -> Matrix(1,-2,-1,1) Matrix(73,-192,27,-71) -> Matrix(1,-4,0,1) 1/0 Matrix(155,-432,47,-131) -> Matrix(1,2,-1,-3) Matrix(239,-672,85,-239) -> Matrix(5,12,-2,-5) (14/5,48/17) -> (-3/1,-2/1) Matrix(577,-1632,204,-577) -> Matrix(15,28,-8,-15) (48/17,17/6) -> (-2/1,-7/4) Matrix(169,-480,25,-71) -> Matrix(5,8,-2,-3) -2/1 Matrix(119,-384,22,-71) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(397,-1296,121,-395) -> Matrix(1,2,-1,-1) (-2/1,0/1).(-1/1,1/0) Matrix(71,-240,21,-71) -> Matrix(1,0,0,-1) (10/3,24/7) -> (0/1,1/0) Matrix(97,-336,28,-97) -> Matrix(1,4,0,-1) (24/7,7/2) -> (-2/1,1/0) Matrix(13,-48,3,-11) -> Matrix(1,2,-1,-1) (-2/1,0/1).(-1/1,1/0) Matrix(25,-144,4,-23) -> Matrix(1,0,0,1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.