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 -8/1 -6/1 -14/3 -4/1 -10/3 -8/3 -12/5 -2/1 -12/7 -8/5 -48/31 -4/3 -8/7 0/1 1/1 8/7 16/13 4/3 16/11 3/2 8/5 12/7 16/9 2/1 16/7 12/5 5/2 8/3 48/17 3/1 16/5 10/3 64/19 7/2 32/9 11/3 15/4 4/1 64/15 13/3 128/29 9/2 32/7 14/3 5/1 16/3 11/2 17/3 64/11 6/1 32/5 13/2 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 1/0 -8/1 1/0 -7/1 -2/1 -1/1 -6/1 -2/1 0/1 -11/2 -4/3 -1/1 -16/3 -1/1 -5/1 -1/1 -2/3 -14/3 -4/5 -2/3 -9/2 -2/3 -3/5 -13/3 -1/2 0/1 -4/1 -1/2 -19/5 -1/2 -4/9 -15/4 -1/2 -2/5 -11/3 -1/2 -4/9 -18/5 -8/19 -2/5 -7/2 -2/5 -3/8 -24/7 -1/3 -41/12 -2/5 -1/3 -17/5 -2/5 -1/3 -10/3 -2/5 -4/11 -13/4 -8/23 -1/3 -16/5 -1/3 -3/1 -1/3 -2/7 -17/6 -2/7 -1/4 -14/5 -2/7 0/1 -11/4 -3/11 -4/15 -8/3 -1/4 -21/8 -5/21 -4/17 -13/5 -1/4 -4/17 -31/12 -4/17 -3/13 -18/7 -4/17 -2/9 -5/2 -1/4 -2/9 -22/9 -2/9 -4/19 -17/7 -2/9 -1/5 -29/12 -1/4 -2/9 -12/5 -1/5 -31/13 -1/4 0/1 -19/8 -1/5 0/1 -26/11 -2/9 0/1 -7/3 -2/9 -1/5 -16/7 -1/5 -9/4 -1/5 -2/11 -20/9 -1/6 -11/5 -1/4 0/1 -2/1 -2/11 0/1 -15/8 -2/11 -1/6 -13/7 -1/6 0/1 -11/6 -1/7 0/1 -31/17 -1/4 0/1 -20/11 -1/5 -29/16 -3/16 -2/11 -9/5 -2/11 -1/6 -16/9 -1/6 -7/4 -1/6 -2/13 -19/11 -1/6 0/1 -31/18 -1/7 0/1 -43/25 -1/4 0/1 -12/7 -1/6 -41/24 -3/19 -2/13 -29/17 -2/13 -1/7 -17/10 -1/6 -2/13 -5/3 -2/13 -1/7 -18/11 -2/13 -4/27 -49/30 -2/13 -3/20 -80/49 -3/20 -31/19 -3/20 -4/27 -13/8 -4/27 -1/7 -8/5 -1/7 -19/12 -1/7 -4/29 -30/19 -6/43 -4/29 -11/7 -4/29 -3/22 -14/9 -2/15 0/1 -31/20 -1/7 0/1 -48/31 -1/7 -17/11 -1/7 -2/15 -3/2 -2/15 -1/8 -16/11 -1/8 -13/9 -1/8 -8/65 -23/16 -6/49 -5/41 -33/23 -1/8 -4/33 -43/30 -13/107 -4/33 -10/7 -4/33 -2/17 -37/26 -5/42 -2/17 -64/45 -2/17 -27/19 -2/17 -1/9 -17/12 -1/8 -2/17 -41/29 -1/8 -2/17 -24/17 -1/8 -7/5 -3/25 -2/17 -32/23 -2/17 -25/18 -2/17 -11/94 -18/13 -2/17 -8/69 -11/8 -4/35 -1/9 -26/19 -2/17 0/1 -67/49 -2/17 -5/43 -41/30 -2/17 -1/9 -15/11 -2/17 -1/9 -4/3 -1/9 -17/13 -1/9 -2/19 -64/49 -2/19 -47/36 -2/19 -1/10 -30/23 -2/19 0/1 -13/10 -1/9 0/1 -22/17 -6/55 -4/37 -75/58 -17/157 -4/37 -128/99 -4/37 -53/41 -4/37 -11/102 -31/24 -1/9 -4/37 -9/7 -3/28 -2/19 -32/25 -2/19 -23/18 -2/19 -7/67 -14/11 -2/19 -4/39 -5/4 -2/19 -1/10 -16/13 -1/10 -11/9 -1/10 -4/41 -17/14 -1/10 -2/21 -23/19 -1/10 -2/21 -29/24 -5/52 -2/21 -64/53 -2/21 -35/29 -2/21 -5/53 -6/5 -2/21 0/1 -19/16 -1/11 0/1 -32/27 0/1 -13/11 -1/10 0/1 -7/6 -1/10 -2/21 -8/7 -1/11 -9/8 -1/11 -2/23 -1/1 -1/12 0/1 0/1 0/1 1/1 0/1 1/14 9/8 2/27 1/13 8/7 1/13 7/6 2/25 1/12 6/5 0/1 2/25 11/9 4/49 1/12 16/13 1/12 5/4 1/12 2/23 14/11 4/47 2/23 9/7 2/23 3/34 13/10 0/1 1/11 4/3 1/11 19/14 1/11 4/43 15/11 1/11 2/21 11/8 1/11 4/43 18/13 8/85 2/21 7/5 2/21 3/31 24/17 1/10 41/29 2/21 1/10 17/12 2/21 1/10 10/7 2/21 4/41 13/9 8/81 1/10 16/11 1/10 3/2 1/10 2/19 17/11 2/19 1/9 14/9 0/1 2/19 11/7 3/28 4/37 8/5 1/9 21/13 5/44 4/35 13/8 1/9 4/35 31/19 4/35 3/26 18/11 4/35 2/17 5/3 1/9 2/17 22/13 2/17 4/33 17/10 2/17 1/8 29/17 1/9 2/17 12/7 1/8 31/18 0/1 1/9 19/11 0/1 1/8 26/15 0/1 2/17 7/4 2/17 1/8 16/9 1/8 9/5 1/8 2/15 20/11 1/7 11/6 0/1 1/9 2/1 0/1 2/15 15/7 2/15 1/7 13/6 0/1 1/7 11/5 0/1 1/6 31/14 0/1 1/9 20/9 1/8 29/13 3/23 2/15 9/4 2/15 1/7 16/7 1/7 7/3 1/7 2/13 19/8 0/1 1/7 31/13 0/1 1/6 43/18 0/1 1/9 12/5 1/7 41/17 3/20 2/13 29/12 2/13 1/6 17/7 1/7 2/13 5/2 2/13 1/6 18/7 2/13 4/25 49/19 2/13 3/19 80/31 3/19 31/12 3/19 4/25 13/5 4/25 1/6 8/3 1/6 19/7 1/6 4/23 30/11 6/35 4/23 11/4 4/23 3/17 14/5 0/1 2/11 31/11 0/1 1/6 48/17 1/6 17/6 1/6 2/11 3/1 2/11 1/5 16/5 1/5 13/4 1/5 8/39 23/7 6/29 5/24 33/10 1/5 4/19 43/13 13/62 4/19 10/3 4/19 2/9 37/11 5/23 2/9 64/19 2/9 27/8 2/9 1/4 17/5 1/5 2/9 41/12 1/5 2/9 24/7 1/5 7/2 3/14 2/9 32/9 2/9 25/7 2/9 11/49 18/5 2/9 8/35 11/3 4/17 1/4 26/7 0/1 2/9 67/18 2/9 5/22 41/11 2/9 1/4 15/4 2/9 1/4 4/1 1/4 17/4 1/4 2/7 64/15 2/7 47/11 2/7 1/3 30/7 0/1 2/7 13/3 0/1 1/4 22/5 6/23 4/15 75/17 17/64 4/15 128/29 4/15 53/12 4/15 11/41 31/7 1/4 4/15 9/2 3/11 2/7 32/7 2/7 23/5 2/7 7/24 14/3 2/7 4/13 5/1 2/7 1/3 16/3 1/3 11/2 1/3 4/11 17/3 1/3 2/5 23/4 1/3 2/5 29/5 5/13 2/5 64/11 2/5 35/6 2/5 5/12 6/1 0/1 2/5 19/3 0/1 1/2 32/5 0/1 13/2 0/1 1/3 7/1 1/3 2/5 8/1 1/2 9/1 1/2 2/3 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(33,320,10,97) (-9/1,1/0) -> (23/7,33/10) Hyperbolic Matrix(65,544,46,385) (-9/1,-8/1) -> (24/17,41/29) Hyperbolic Matrix(31,224,22,159) (-8/1,-7/1) -> (7/5,24/17) Hyperbolic Matrix(33,224,-14,-95) (-7/1,-6/1) -> (-26/11,-7/3) Hyperbolic Matrix(63,352,-46,-257) (-6/1,-11/2) -> (-11/8,-26/19) Hyperbolic Matrix(65,352,12,65) (-11/2,-16/3) -> (16/3,11/2) Hyperbolic Matrix(31,160,6,31) (-16/3,-5/1) -> (5/1,16/3) Hyperbolic Matrix(33,160,20,97) (-5/1,-14/3) -> (18/11,5/3) Hyperbolic Matrix(97,448,-76,-351) (-14/3,-9/2) -> (-23/18,-14/11) Hyperbolic Matrix(95,416,-66,-289) (-9/2,-13/3) -> (-13/9,-23/16) Hyperbolic Matrix(31,128,-8,-33) (-13/3,-4/1) -> (-4/1,-19/5) Parabolic Matrix(127,480,-68,-257) (-19/5,-15/4) -> (-15/8,-13/7) Hyperbolic Matrix(95,352,-78,-289) (-15/4,-11/3) -> (-11/9,-17/14) Hyperbolic Matrix(97,352,62,225) (-11/3,-18/5) -> (14/9,11/7) Hyperbolic Matrix(161,576,-116,-415) (-18/5,-7/2) -> (-25/18,-18/13) Hyperbolic Matrix(65,224,56,193) (-7/2,-24/7) -> (8/7,7/6) Hyperbolic Matrix(159,544,140,479) (-24/7,-41/12) -> (9/8,8/7) Hyperbolic Matrix(319,1088,56,191) (-41/12,-17/5) -> (17/3,23/4) Hyperbolic Matrix(161,544,-66,-223) (-17/5,-10/3) -> (-22/9,-17/7) Hyperbolic Matrix(127,416,-98,-321) (-10/3,-13/4) -> (-13/10,-22/17) Hyperbolic Matrix(129,416,40,129) (-13/4,-16/5) -> (16/5,13/4) Hyperbolic Matrix(31,96,10,31) (-16/5,-3/1) -> (3/1,16/5) Hyperbolic Matrix(191,544,112,319) (-3/1,-17/6) -> (17/10,29/17) Hyperbolic Matrix(353,992,-216,-607) (-17/6,-14/5) -> (-18/11,-49/30) Hyperbolic Matrix(127,352,92,255) (-14/5,-11/4) -> (11/8,18/13) Hyperbolic Matrix(95,256,-36,-97) (-11/4,-8/3) -> (-8/3,-21/8) Parabolic Matrix(159,416,-86,-225) (-21/8,-13/5) -> (-13/7,-11/6) Hyperbolic Matrix(383,992,222,575) (-13/5,-31/12) -> (31/18,19/11) Hyperbolic Matrix(385,992,-248,-639) (-31/12,-18/7) -> (-14/9,-31/20) Hyperbolic Matrix(63,160,50,127) (-18/7,-5/2) -> (5/4,14/11) Hyperbolic Matrix(353,864,-248,-607) (-5/2,-22/9) -> (-10/7,-37/26) Hyperbolic Matrix(225,544,146,353) (-17/7,-29/12) -> (3/2,17/11) Hyperbolic Matrix(385,928,-212,-511) (-29/12,-12/5) -> (-20/11,-29/16) Hyperbolic Matrix(415,992,-228,-545) (-12/5,-31/13) -> (-31/17,-20/11) Hyperbolic Matrix(417,992,256,609) (-31/13,-19/8) -> (13/8,31/19) Hyperbolic Matrix(257,608,-216,-511) (-19/8,-26/11) -> (-6/5,-19/16) Hyperbolic Matrix(97,224,42,97) (-7/3,-16/7) -> (16/7,7/3) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(417,928,-244,-543) (-9/4,-20/9) -> (-12/7,-41/24) Hyperbolic Matrix(447,992,-260,-577) (-20/9,-11/5) -> (-43/25,-12/7) Hyperbolic Matrix(161,352,-102,-223) (-11/5,-2/1) -> (-30/19,-11/7) Hyperbolic Matrix(287,544,-220,-417) (-2/1,-15/8) -> (-47/36,-30/23) Hyperbolic Matrix(929,1696,-648,-1183) (-11/6,-31/17) -> (-33/23,-43/30) Hyperbolic Matrix(513,928,-424,-767) (-29/16,-9/5) -> (-23/19,-29/24) Hyperbolic Matrix(161,288,90,161) (-9/5,-16/9) -> (16/9,9/5) Hyperbolic Matrix(127,224,72,127) (-16/9,-7/4) -> (7/4,16/9) Hyperbolic Matrix(129,224,-110,-191) (-7/4,-19/11) -> (-13/11,-7/6) Hyperbolic Matrix(575,992,222,383) (-19/11,-31/18) -> (31/12,13/5) Hyperbolic Matrix(1729,2976,-1338,-2303) (-31/18,-43/25) -> (-53/41,-31/24) Hyperbolic Matrix(2305,3936,-1686,-2879) (-41/24,-29/17) -> (-67/49,-41/30) Hyperbolic Matrix(319,544,112,191) (-29/17,-17/10) -> (17/6,3/1) Hyperbolic Matrix(321,544,-226,-383) (-17/10,-5/3) -> (-27/19,-17/12) Hyperbolic Matrix(97,160,20,33) (-5/3,-18/11) -> (14/3,5/1) Hyperbolic Matrix(2273,3712,804,1313) (-49/30,-80/49) -> (48/17,17/6) Hyperbolic Matrix(2431,3968,862,1407) (-80/49,-31/19) -> (31/11,48/17) Hyperbolic Matrix(609,992,256,417) (-31/19,-13/8) -> (19/8,31/13) Hyperbolic Matrix(159,256,-100,-161) (-13/8,-8/5) -> (-8/5,-19/12) Parabolic Matrix(607,960,-466,-737) (-19/12,-30/19) -> (-30/23,-13/10) Hyperbolic Matrix(225,352,62,97) (-11/7,-14/9) -> (18/5,11/3) Hyperbolic Matrix(2561,3968,992,1537) (-31/20,-48/31) -> (80/31,31/12) Hyperbolic Matrix(2399,3712,930,1439) (-48/31,-17/11) -> (49/19,80/31) Hyperbolic Matrix(353,544,146,225) (-17/11,-3/2) -> (29/12,17/7) Hyperbolic Matrix(65,96,44,65) (-3/2,-16/11) -> (16/11,3/2) Hyperbolic Matrix(287,416,198,287) (-16/11,-13/9) -> (13/9,16/11) Hyperbolic Matrix(223,320,200,287) (-23/16,-33/23) -> (1/1,9/8) Hyperbolic Matrix(1185,1696,-916,-1311) (-43/30,-10/7) -> (-22/17,-75/58) Hyperbolic Matrix(2879,4096,854,1215) (-37/26,-64/45) -> (64/19,27/8) Hyperbolic Matrix(2881,4096,856,1217) (-64/45,-27/19) -> (37/11,64/19) Hyperbolic Matrix(927,1312,248,351) (-17/12,-41/29) -> (41/11,15/4) Hyperbolic Matrix(385,544,46,65) (-41/29,-24/17) -> (8/1,9/1) Hyperbolic Matrix(159,224,22,31) (-24/17,-7/5) -> (7/1,8/1) Hyperbolic Matrix(735,1024,206,287) (-7/5,-32/23) -> (32/9,25/7) Hyperbolic Matrix(737,1024,208,289) (-32/23,-25/18) -> (7/2,32/9) Hyperbolic Matrix(255,352,92,127) (-18/13,-11/8) -> (11/4,14/5) Hyperbolic Matrix(959,1312,-796,-1089) (-26/19,-67/49) -> (-35/29,-6/5) Hyperbolic Matrix(961,1312,282,385) (-41/30,-15/11) -> (17/5,41/12) Hyperbolic Matrix(95,128,-72,-97) (-15/11,-4/3) -> (-4/3,-17/13) Parabolic Matrix(3135,4096,734,959) (-17/13,-64/49) -> (64/15,47/11) Hyperbolic Matrix(3137,4096,736,961) (-64/49,-47/36) -> (17/4,64/15) Hyperbolic Matrix(12671,16384,2870,3711) (-75/58,-128/99) -> (128/29,53/12) Hyperbolic Matrix(12673,16384,2872,3713) (-128/99,-53/41) -> (75/17,128/29) Hyperbolic Matrix(223,288,24,31) (-31/24,-9/7) -> (9/1,1/0) Hyperbolic Matrix(799,1024,174,223) (-9/7,-32/25) -> (32/7,23/5) Hyperbolic Matrix(801,1024,176,225) (-32/25,-23/18) -> (9/2,32/7) Hyperbolic Matrix(127,160,50,63) (-14/11,-5/4) -> (5/2,18/7) Hyperbolic Matrix(129,160,104,129) (-5/4,-16/13) -> (16/13,5/4) Hyperbolic Matrix(287,352,234,287) (-16/13,-11/9) -> (11/9,16/13) Hyperbolic Matrix(897,1088,634,769) (-17/14,-23/19) -> (41/29,17/12) Hyperbolic Matrix(3391,4096,582,703) (-29/24,-64/53) -> (64/11,35/6) Hyperbolic Matrix(3393,4096,584,705) (-64/53,-35/29) -> (29/5,64/11) Hyperbolic Matrix(863,1024,134,159) (-19/16,-32/27) -> (32/5,13/2) Hyperbolic Matrix(865,1024,136,161) (-32/27,-13/11) -> (19/3,32/5) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(479,544,140,159) (-8/7,-9/8) -> (41/12,24/7) Hyperbolic Matrix(257,288,58,65) (-9/8,-1/1) -> (31/7,9/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(191,-224,110,-129) (7/6,6/5) -> (26/15,7/4) Hyperbolic Matrix(289,-352,78,-95) (6/5,11/9) -> (11/3,26/7) Hyperbolic Matrix(351,-448,76,-97) (14/11,9/7) -> (23/5,14/3) Hyperbolic Matrix(321,-416,98,-127) (9/7,13/10) -> (13/4,23/7) Hyperbolic Matrix(97,-128,72,-95) (13/10,4/3) -> (4/3,19/14) Parabolic Matrix(353,-480,164,-223) (19/14,15/11) -> (15/7,13/6) Hyperbolic Matrix(257,-352,46,-63) (15/11,11/8) -> (11/2,17/3) Hyperbolic Matrix(415,-576,116,-161) (18/13,7/5) -> (25/7,18/5) Hyperbolic Matrix(383,-544,226,-321) (17/12,10/7) -> (22/13,17/10) Hyperbolic Matrix(289,-416,66,-95) (10/7,13/9) -> (13/3,22/5) Hyperbolic Matrix(639,-992,248,-385) (17/11,14/9) -> (18/7,49/19) Hyperbolic Matrix(161,-256,100,-159) (11/7,8/5) -> (8/5,21/13) Parabolic Matrix(257,-416,118,-191) (21/13,13/8) -> (13/6,11/5) Hyperbolic Matrix(607,-992,216,-353) (31/19,18/11) -> (14/5,31/11) Hyperbolic Matrix(511,-864,152,-257) (5/3,22/13) -> (10/3,37/11) Hyperbolic Matrix(543,-928,244,-417) (29/17,12/7) -> (20/9,29/13) Hyperbolic Matrix(577,-992,260,-447) (12/7,31/18) -> (31/14,20/9) Hyperbolic Matrix(351,-608,56,-97) (19/11,26/15) -> (6/1,19/3) Hyperbolic Matrix(511,-928,212,-385) (9/5,20/11) -> (12/5,41/17) Hyperbolic Matrix(545,-992,228,-415) (20/11,11/6) -> (43/18,12/5) Hyperbolic Matrix(191,-352,70,-129) (11/6,2/1) -> (30/11,11/4) Hyperbolic Matrix(257,-544,60,-127) (2/1,15/7) -> (47/11,30/7) Hyperbolic Matrix(767,-1696,232,-513) (11/5,31/14) -> (33/10,43/13) Hyperbolic Matrix(415,-928,72,-161) (29/13,9/4) -> (23/4,29/5) Hyperbolic Matrix(95,-224,14,-33) (7/3,19/8) -> (13/2,7/1) Hyperbolic Matrix(1247,-2976,282,-673) (31/13,43/18) -> (53/12,31/7) Hyperbolic Matrix(1631,-3936,438,-1057) (41/17,29/12) -> (67/18,41/11) Hyperbolic Matrix(223,-544,66,-161) (17/7,5/2) -> (27/8,17/5) Hyperbolic Matrix(97,-256,36,-95) (13/5,8/3) -> (8/3,19/7) Parabolic Matrix(353,-960,82,-223) (19/7,30/11) -> (30/7,13/3) Hyperbolic Matrix(511,-1696,116,-385) (43/13,10/3) -> (22/5,75/17) Hyperbolic Matrix(353,-1312,60,-223) (26/7,67/18) -> (35/6,6/1) Hyperbolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(33,320,10,97) -> Matrix(5,-4,24,-19) Matrix(65,544,46,385) -> Matrix(1,0,10,1) Matrix(31,224,22,159) -> Matrix(1,4,10,41) Matrix(33,224,-14,-95) -> Matrix(1,0,-4,1) Matrix(63,352,-46,-257) -> Matrix(1,0,-8,1) Matrix(65,352,12,65) -> Matrix(7,8,20,23) Matrix(31,160,6,31) -> Matrix(5,4,16,13) Matrix(33,160,20,97) -> Matrix(1,0,10,1) Matrix(97,448,-76,-351) -> Matrix(11,8,-106,-77) Matrix(95,416,-66,-289) -> Matrix(15,8,-122,-65) Matrix(31,128,-8,-33) -> Matrix(7,4,-16,-9) Matrix(127,480,-68,-257) -> Matrix(9,4,-52,-23) Matrix(95,352,-78,-289) -> Matrix(1,0,-8,1) Matrix(97,352,62,225) -> Matrix(19,8,178,75) Matrix(161,576,-116,-415) -> Matrix(39,16,-334,-137) Matrix(65,224,56,193) -> Matrix(11,4,140,51) Matrix(159,544,140,479) -> Matrix(1,0,16,1) Matrix(319,1088,56,191) -> Matrix(11,4,30,11) Matrix(161,544,-66,-223) -> Matrix(1,0,-2,1) Matrix(127,416,-98,-321) -> Matrix(23,8,-210,-73) Matrix(129,416,40,129) -> Matrix(47,16,232,79) Matrix(31,96,10,31) -> Matrix(13,4,68,21) Matrix(191,544,112,319) -> Matrix(1,0,12,1) Matrix(353,992,-216,-607) -> Matrix(13,4,-88,-27) Matrix(127,352,92,255) -> Matrix(29,8,308,85) Matrix(95,256,-36,-97) -> Matrix(31,8,-128,-33) Matrix(159,416,-86,-225) -> Matrix(17,4,-98,-23) Matrix(383,992,222,575) -> Matrix(17,4,140,33) Matrix(385,992,-248,-639) -> Matrix(17,4,-132,-31) Matrix(63,160,50,127) -> Matrix(1,0,16,1) Matrix(353,864,-248,-607) -> Matrix(37,8,-310,-67) Matrix(225,544,146,353) -> Matrix(1,0,14,1) Matrix(385,928,-212,-511) -> Matrix(19,4,-100,-21) Matrix(415,992,-228,-545) -> Matrix(1,0,0,1) Matrix(417,992,256,609) -> Matrix(19,4,166,35) Matrix(257,608,-216,-511) -> Matrix(1,0,-6,1) Matrix(97,224,42,97) -> Matrix(19,4,128,27) Matrix(127,288,56,127) -> Matrix(21,4,152,29) Matrix(417,928,-244,-543) -> Matrix(23,4,-144,-25) Matrix(447,992,-260,-577) -> Matrix(1,0,0,1) Matrix(161,352,-102,-223) -> Matrix(19,4,-138,-29) Matrix(287,544,-220,-417) -> Matrix(1,0,-4,1) Matrix(929,1696,-648,-1183) -> Matrix(15,4,-124,-33) Matrix(513,928,-424,-767) -> Matrix(23,4,-236,-41) Matrix(161,288,90,161) -> Matrix(23,4,178,31) Matrix(127,224,72,127) -> Matrix(25,4,206,33) Matrix(129,224,-110,-191) -> Matrix(1,0,-4,1) Matrix(575,992,222,383) -> Matrix(25,4,156,25) Matrix(1729,2976,-1338,-2303) -> Matrix(27,4,-250,-37) Matrix(2305,3936,-1686,-2879) -> Matrix(51,8,-440,-69) Matrix(319,544,112,191) -> Matrix(1,0,12,1) Matrix(321,544,-226,-383) -> Matrix(1,0,-2,1) Matrix(97,160,20,33) -> Matrix(1,0,10,1) Matrix(2273,3712,804,1313) -> Matrix(53,8,298,45) Matrix(2431,3968,862,1407) -> Matrix(27,4,182,27) Matrix(609,992,256,417) -> Matrix(27,4,182,27) Matrix(159,256,-100,-161) -> Matrix(55,8,-392,-57) Matrix(607,960,-466,-737) -> Matrix(29,4,-254,-35) Matrix(225,352,62,97) -> Matrix(59,8,258,35) Matrix(2561,3968,992,1537) -> Matrix(25,4,156,25) Matrix(2399,3712,930,1439) -> Matrix(59,8,376,51) Matrix(353,544,146,225) -> Matrix(1,0,14,1) Matrix(65,96,44,65) -> Matrix(31,4,302,39) Matrix(287,416,198,287) -> Matrix(129,16,1298,161) Matrix(223,320,200,287) -> Matrix(33,4,470,57) Matrix(1185,1696,-916,-1311) -> Matrix(133,16,-1222,-147) Matrix(2879,4096,854,1215) -> Matrix(101,12,446,53) Matrix(2881,4096,856,1217) -> Matrix(103,12,472,55) Matrix(927,1312,248,351) -> Matrix(33,4,140,17) Matrix(385,544,46,65) -> Matrix(1,0,10,1) Matrix(159,224,22,31) -> Matrix(33,4,74,9) Matrix(735,1024,206,287) -> Matrix(237,28,1058,125) Matrix(737,1024,208,289) -> Matrix(239,28,1084,127) Matrix(255,352,92,127) -> Matrix(69,8,388,45) Matrix(959,1312,-796,-1089) -> Matrix(1,0,-2,1) Matrix(961,1312,282,385) -> Matrix(35,4,166,19) Matrix(95,128,-72,-97) -> Matrix(35,4,-324,-37) Matrix(3135,4096,734,959) -> Matrix(37,4,120,13) Matrix(3137,4096,736,961) -> Matrix(39,4,146,15) Matrix(12671,16384,2870,3711) -> Matrix(1035,112,3872,419) Matrix(12673,16384,2872,3713) -> Matrix(1037,112,3898,421) Matrix(223,288,24,31) -> Matrix(37,4,46,5) Matrix(799,1024,174,223) -> Matrix(189,20,652,69) Matrix(801,1024,176,225) -> Matrix(191,20,678,71) Matrix(127,160,50,63) -> Matrix(1,0,16,1) Matrix(129,160,104,129) -> Matrix(39,4,458,47) Matrix(287,352,234,287) -> Matrix(81,8,982,97) Matrix(897,1088,634,769) -> Matrix(41,4,420,41) Matrix(3391,4096,582,703) -> Matrix(209,20,512,49) Matrix(3393,4096,584,705) -> Matrix(211,20,538,51) Matrix(863,1024,134,159) -> Matrix(1,0,14,1) Matrix(865,1024,136,161) -> Matrix(1,0,12,1) Matrix(193,224,56,65) -> Matrix(43,4,204,19) Matrix(479,544,140,159) -> Matrix(1,0,16,1) Matrix(257,288,58,65) -> Matrix(47,4,176,15) Matrix(1,0,2,1) -> Matrix(1,0,26,1) Matrix(191,-224,110,-129) -> Matrix(1,0,-4,1) Matrix(289,-352,78,-95) -> Matrix(1,0,-8,1) Matrix(351,-448,76,-97) -> Matrix(93,-8,314,-27) Matrix(321,-416,98,-127) -> Matrix(89,-8,434,-39) Matrix(97,-128,72,-95) -> Matrix(45,-4,484,-43) Matrix(353,-480,164,-223) -> Matrix(43,-4,312,-29) Matrix(257,-352,46,-63) -> Matrix(1,0,-8,1) Matrix(415,-576,116,-161) -> Matrix(169,-16,750,-71) Matrix(383,-544,226,-321) -> Matrix(1,0,-2,1) Matrix(289,-416,66,-95) -> Matrix(81,-8,314,-31) Matrix(639,-992,248,-385) -> Matrix(39,-4,244,-25) Matrix(161,-256,100,-159) -> Matrix(73,-8,648,-71) Matrix(257,-416,118,-191) -> Matrix(35,-4,254,-29) Matrix(607,-992,216,-353) -> Matrix(35,-4,184,-21) Matrix(511,-864,152,-257) -> Matrix(67,-8,310,-37) Matrix(543,-928,244,-417) -> Matrix(33,-4,256,-31) Matrix(577,-992,260,-447) -> Matrix(1,0,0,1) Matrix(351,-608,56,-97) -> Matrix(1,0,-6,1) Matrix(511,-928,212,-385) -> Matrix(29,-4,196,-27) Matrix(545,-992,228,-415) -> Matrix(1,0,0,1) Matrix(191,-352,70,-129) -> Matrix(33,-4,190,-23) Matrix(257,-544,60,-127) -> Matrix(1,0,-4,1) Matrix(767,-1696,232,-513) -> Matrix(37,-4,176,-19) Matrix(415,-928,72,-161) -> Matrix(29,-4,80,-11) Matrix(95,-224,14,-33) -> Matrix(1,0,-4,1) Matrix(1247,-2976,282,-673) -> Matrix(25,-4,94,-15) Matrix(1631,-3936,438,-1057) -> Matrix(53,-8,232,-35) Matrix(223,-544,66,-161) -> Matrix(1,0,-2,1) Matrix(97,-256,36,-95) -> Matrix(49,-8,288,-47) Matrix(353,-960,82,-223) -> Matrix(23,-4,98,-17) Matrix(511,-1696,116,-385) -> Matrix(75,-16,286,-61) Matrix(353,-1312,60,-223) -> Matrix(1,0,-2,1) Matrix(33,-128,8,-31) -> Matrix(17,-4,64,-15) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 21 Degree of the the map X: 42 Degree of the the map Y: 128 ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 384 Minimal number of generators: 65 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 32 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 8/7 4/3 8/5 12/7 2/1 16/7 12/5 8/3 48/17 3/1 16/5 10/3 64/19 32/9 4/1 64/15 128/29 32/7 14/3 5/1 16/3 11/2 17/3 64/11 6/1 32/5 13/2 7/1 8/1 9/1 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 1/14 9/8 2/27 1/13 8/7 1/13 7/6 2/25 1/12 6/5 0/1 2/25 11/9 4/49 1/12 16/13 1/12 5/4 1/12 2/23 14/11 4/47 2/23 9/7 2/23 3/34 13/10 0/1 1/11 4/3 1/11 19/14 1/11 4/43 15/11 1/11 2/21 11/8 1/11 4/43 18/13 8/85 2/21 7/5 2/21 3/31 24/17 1/10 41/29 2/21 1/10 17/12 2/21 1/10 10/7 2/21 4/41 13/9 8/81 1/10 16/11 1/10 3/2 1/10 2/19 17/11 2/19 1/9 14/9 0/1 2/19 11/7 3/28 4/37 8/5 1/9 21/13 5/44 4/35 13/8 1/9 4/35 31/19 4/35 3/26 18/11 4/35 2/17 5/3 1/9 2/17 22/13 2/17 4/33 17/10 2/17 1/8 29/17 1/9 2/17 12/7 1/8 31/18 0/1 1/9 19/11 0/1 1/8 26/15 0/1 2/17 7/4 2/17 1/8 16/9 1/8 9/5 1/8 2/15 20/11 1/7 11/6 0/1 1/9 2/1 0/1 2/15 15/7 2/15 1/7 13/6 0/1 1/7 11/5 0/1 1/6 31/14 0/1 1/9 20/9 1/8 29/13 3/23 2/15 9/4 2/15 1/7 16/7 1/7 7/3 1/7 2/13 19/8 0/1 1/7 31/13 0/1 1/6 43/18 0/1 1/9 12/5 1/7 41/17 3/20 2/13 29/12 2/13 1/6 17/7 1/7 2/13 5/2 2/13 1/6 18/7 2/13 4/25 49/19 2/13 3/19 80/31 3/19 31/12 3/19 4/25 13/5 4/25 1/6 8/3 1/6 19/7 1/6 4/23 30/11 6/35 4/23 11/4 4/23 3/17 14/5 0/1 2/11 31/11 0/1 1/6 48/17 1/6 17/6 1/6 2/11 3/1 2/11 1/5 16/5 1/5 13/4 1/5 8/39 23/7 6/29 5/24 33/10 1/5 4/19 43/13 13/62 4/19 10/3 4/19 2/9 37/11 5/23 2/9 64/19 2/9 27/8 2/9 1/4 17/5 1/5 2/9 41/12 1/5 2/9 24/7 1/5 7/2 3/14 2/9 32/9 2/9 25/7 2/9 11/49 18/5 2/9 8/35 11/3 4/17 1/4 26/7 0/1 2/9 67/18 2/9 5/22 41/11 2/9 1/4 15/4 2/9 1/4 4/1 1/4 17/4 1/4 2/7 64/15 2/7 47/11 2/7 1/3 30/7 0/1 2/7 13/3 0/1 1/4 22/5 6/23 4/15 75/17 17/64 4/15 128/29 4/15 53/12 4/15 11/41 31/7 1/4 4/15 9/2 3/11 2/7 32/7 2/7 23/5 2/7 7/24 14/3 2/7 4/13 5/1 2/7 1/3 16/3 1/3 11/2 1/3 4/11 17/3 1/3 2/5 23/4 1/3 2/5 29/5 5/13 2/5 64/11 2/5 35/6 2/5 5/12 6/1 0/1 2/5 19/3 0/1 1/2 32/5 0/1 13/2 0/1 1/3 7/1 1/3 2/5 8/1 1/2 9/1 1/2 2/3 1/0 0/1 1/1 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(287,-320,87,-97) (1/1,9/8) -> (23/7,33/10) Hyperbolic Matrix(479,-544,339,-385) (9/8,8/7) -> (24/17,41/29) Hyperbolic Matrix(193,-224,137,-159) (8/7,7/6) -> (7/5,24/17) Hyperbolic Matrix(191,-224,110,-129) (7/6,6/5) -> (26/15,7/4) Hyperbolic Matrix(289,-352,78,-95) (6/5,11/9) -> (11/3,26/7) Hyperbolic Matrix(287,-352,53,-65) (11/9,16/13) -> (16/3,11/2) Hyperbolic Matrix(129,-160,25,-31) (16/13,5/4) -> (5/1,16/3) Hyperbolic Matrix(127,-160,77,-97) (5/4,14/11) -> (18/11,5/3) Hyperbolic Matrix(351,-448,76,-97) (14/11,9/7) -> (23/5,14/3) Hyperbolic Matrix(321,-416,98,-127) (9/7,13/10) -> (13/4,23/7) Hyperbolic Matrix(97,-128,72,-95) (13/10,4/3) -> (4/3,19/14) Parabolic Matrix(353,-480,164,-223) (19/14,15/11) -> (15/7,13/6) Hyperbolic Matrix(257,-352,46,-63) (15/11,11/8) -> (11/2,17/3) Hyperbolic Matrix(255,-352,163,-225) (11/8,18/13) -> (14/9,11/7) Hyperbolic Matrix(415,-576,116,-161) (18/13,7/5) -> (25/7,18/5) Hyperbolic Matrix(769,-1088,135,-191) (41/29,17/12) -> (17/3,23/4) Hyperbolic Matrix(383,-544,226,-321) (17/12,10/7) -> (22/13,17/10) Hyperbolic Matrix(289,-416,66,-95) (10/7,13/9) -> (13/3,22/5) Hyperbolic Matrix(287,-416,89,-129) (13/9,16/11) -> (16/5,13/4) Hyperbolic Matrix(65,-96,21,-31) (16/11,3/2) -> (3/1,16/5) Hyperbolic Matrix(353,-544,207,-319) (3/2,17/11) -> (17/10,29/17) Hyperbolic Matrix(639,-992,248,-385) (17/11,14/9) -> (18/7,49/19) Hyperbolic Matrix(161,-256,100,-159) (11/7,8/5) -> (8/5,21/13) Parabolic Matrix(257,-416,118,-191) (21/13,13/8) -> (13/6,11/5) Hyperbolic Matrix(609,-992,353,-575) (13/8,31/19) -> (31/18,19/11) Hyperbolic Matrix(607,-992,216,-353) (31/19,18/11) -> (14/5,31/11) Hyperbolic Matrix(511,-864,152,-257) (5/3,22/13) -> (10/3,37/11) Hyperbolic Matrix(543,-928,244,-417) (29/17,12/7) -> (20/9,29/13) Hyperbolic Matrix(577,-992,260,-447) (12/7,31/18) -> (31/14,20/9) Hyperbolic Matrix(351,-608,56,-97) (19/11,26/15) -> (6/1,19/3) Hyperbolic Matrix(127,-224,55,-97) (7/4,16/9) -> (16/7,7/3) Hyperbolic Matrix(161,-288,71,-127) (16/9,9/5) -> (9/4,16/7) Hyperbolic Matrix(511,-928,212,-385) (9/5,20/11) -> (12/5,41/17) Hyperbolic Matrix(545,-992,228,-415) (20/11,11/6) -> (43/18,12/5) Hyperbolic Matrix(191,-352,70,-129) (11/6,2/1) -> (30/11,11/4) Hyperbolic Matrix(257,-544,60,-127) (2/1,15/7) -> (47/11,30/7) Hyperbolic Matrix(767,-1696,232,-513) (11/5,31/14) -> (33/10,43/13) Hyperbolic Matrix(415,-928,72,-161) (29/13,9/4) -> (23/4,29/5) Hyperbolic Matrix(95,-224,14,-33) (7/3,19/8) -> (13/2,7/1) Hyperbolic Matrix(417,-992,161,-383) (19/8,31/13) -> (31/12,13/5) Hyperbolic Matrix(1247,-2976,282,-673) (31/13,43/18) -> (53/12,31/7) Hyperbolic Matrix(1631,-3936,438,-1057) (41/17,29/12) -> (67/18,41/11) Hyperbolic Matrix(225,-544,79,-191) (29/12,17/7) -> (17/6,3/1) Hyperbolic Matrix(223,-544,66,-161) (17/7,5/2) -> (27/8,17/5) Hyperbolic Matrix(63,-160,13,-33) (5/2,18/7) -> (14/3,5/1) Hyperbolic Matrix(1439,-3712,509,-1313) (49/19,80/31) -> (48/17,17/6) Hyperbolic Matrix(1537,-3968,545,-1407) (80/31,31/12) -> (31/11,48/17) Hyperbolic Matrix(97,-256,36,-95) (13/5,8/3) -> (8/3,19/7) Parabolic Matrix(353,-960,82,-223) (19/7,30/11) -> (30/7,13/3) Hyperbolic Matrix(127,-352,35,-97) (11/4,14/5) -> (18/5,11/3) Hyperbolic Matrix(511,-1696,116,-385) (43/13,10/3) -> (22/5,75/17) Hyperbolic Matrix(1217,-4096,361,-1215) (37/11,64/19) -> (64/19,27/8) Parabolic Matrix(385,-1312,103,-351) (17/5,41/12) -> (41/11,15/4) Hyperbolic Matrix(159,-544,19,-65) (41/12,24/7) -> (8/1,9/1) Hyperbolic Matrix(65,-224,9,-31) (24/7,7/2) -> (7/1,8/1) Hyperbolic Matrix(289,-1024,81,-287) (7/2,32/9) -> (32/9,25/7) Parabolic Matrix(353,-1312,60,-223) (26/7,67/18) -> (35/6,6/1) Hyperbolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(961,-4096,225,-959) (17/4,64/15) -> (64/15,47/11) Parabolic Matrix(3713,-16384,841,-3711) (75/17,128/29) -> (128/29,53/12) Parabolic Matrix(65,-288,7,-31) (31/7,9/2) -> (9/1,1/0) Hyperbolic Matrix(225,-1024,49,-223) (9/2,32/7) -> (32/7,23/5) Parabolic Matrix(705,-4096,121,-703) (29/5,64/11) -> (64/11,35/6) Parabolic Matrix(161,-1024,25,-159) (19/3,32/5) -> (32/5,13/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,13,1) Matrix(287,-320,87,-97) -> Matrix(57,-4,271,-19) Matrix(479,-544,339,-385) -> Matrix(1,0,-3,1) Matrix(193,-224,137,-159) -> Matrix(51,-4,523,-41) Matrix(191,-224,110,-129) -> Matrix(1,0,-4,1) Matrix(289,-352,78,-95) -> Matrix(1,0,-8,1) Matrix(287,-352,53,-65) -> Matrix(97,-8,279,-23) Matrix(129,-160,25,-31) -> Matrix(47,-4,153,-13) Matrix(127,-160,77,-97) -> Matrix(1,0,-3,1) Matrix(351,-448,76,-97) -> Matrix(93,-8,314,-27) Matrix(321,-416,98,-127) -> Matrix(89,-8,434,-39) Matrix(97,-128,72,-95) -> Matrix(45,-4,484,-43) Matrix(353,-480,164,-223) -> Matrix(43,-4,312,-29) Matrix(257,-352,46,-63) -> Matrix(1,0,-8,1) Matrix(255,-352,163,-225) -> Matrix(85,-8,797,-75) Matrix(415,-576,116,-161) -> Matrix(169,-16,750,-71) Matrix(769,-1088,135,-191) -> Matrix(41,-4,113,-11) Matrix(383,-544,226,-321) -> Matrix(1,0,-2,1) Matrix(289,-416,66,-95) -> Matrix(81,-8,314,-31) Matrix(287,-416,89,-129) -> Matrix(161,-16,795,-79) Matrix(65,-96,21,-31) -> Matrix(39,-4,205,-21) Matrix(353,-544,207,-319) -> Matrix(1,0,-1,1) Matrix(639,-992,248,-385) -> Matrix(39,-4,244,-25) Matrix(161,-256,100,-159) -> Matrix(73,-8,648,-71) Matrix(257,-416,118,-191) -> Matrix(35,-4,254,-29) Matrix(609,-992,353,-575) -> Matrix(35,-4,289,-33) Matrix(607,-992,216,-353) -> Matrix(35,-4,184,-21) Matrix(511,-864,152,-257) -> Matrix(67,-8,310,-37) Matrix(543,-928,244,-417) -> Matrix(33,-4,256,-31) Matrix(577,-992,260,-447) -> Matrix(1,0,0,1) Matrix(351,-608,56,-97) -> Matrix(1,0,-6,1) Matrix(127,-224,55,-97) -> Matrix(33,-4,223,-27) Matrix(161,-288,71,-127) -> Matrix(31,-4,225,-29) Matrix(511,-928,212,-385) -> Matrix(29,-4,196,-27) Matrix(545,-992,228,-415) -> Matrix(1,0,0,1) Matrix(191,-352,70,-129) -> Matrix(33,-4,190,-23) Matrix(257,-544,60,-127) -> Matrix(1,0,-4,1) Matrix(767,-1696,232,-513) -> Matrix(37,-4,176,-19) Matrix(415,-928,72,-161) -> Matrix(29,-4,80,-11) Matrix(95,-224,14,-33) -> Matrix(1,0,-4,1) Matrix(417,-992,161,-383) -> Matrix(27,-4,169,-25) Matrix(1247,-2976,282,-673) -> Matrix(25,-4,94,-15) Matrix(1631,-3936,438,-1057) -> Matrix(53,-8,232,-35) Matrix(225,-544,79,-191) -> Matrix(1,0,-1,1) Matrix(223,-544,66,-161) -> Matrix(1,0,-2,1) Matrix(63,-160,13,-33) -> Matrix(1,0,-3,1) Matrix(1439,-3712,509,-1313) -> Matrix(51,-8,287,-45) Matrix(1537,-3968,545,-1407) -> Matrix(25,-4,169,-27) Matrix(97,-256,36,-95) -> Matrix(49,-8,288,-47) Matrix(353,-960,82,-223) -> Matrix(23,-4,98,-17) Matrix(127,-352,35,-97) -> Matrix(45,-8,197,-35) Matrix(511,-1696,116,-385) -> Matrix(75,-16,286,-61) Matrix(1217,-4096,361,-1215) -> Matrix(55,-12,243,-53) Matrix(385,-1312,103,-351) -> Matrix(19,-4,81,-17) Matrix(159,-544,19,-65) -> Matrix(1,0,-3,1) Matrix(65,-224,9,-31) -> Matrix(19,-4,43,-9) Matrix(289,-1024,81,-287) -> Matrix(127,-28,567,-125) Matrix(353,-1312,60,-223) -> Matrix(1,0,-2,1) Matrix(33,-128,8,-31) -> Matrix(17,-4,64,-15) Matrix(961,-4096,225,-959) -> Matrix(15,-4,49,-13) Matrix(3713,-16384,841,-3711) -> Matrix(421,-112,1575,-419) Matrix(65,-288,7,-31) -> Matrix(15,-4,19,-5) Matrix(225,-1024,49,-223) -> Matrix(71,-20,245,-69) Matrix(705,-4096,121,-703) -> Matrix(51,-20,125,-49) Matrix(161,-1024,25,-159) -> Matrix(1,0,1,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 21 ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 0/1 13 1 2/1 (0/1,2/15) 0 8 11/5 (0/1,1/6) 0 16 20/9 1/8 2 4 9/4 (2/15,1/7) 0 16 16/7 1/7 4 1 7/3 (1/7,2/13) 0 16 19/8 (0/1,1/7) 0 16 31/13 (0/1,1/6) 0 16 43/18 (0/1,1/9) 0 16 12/5 1/7 2 4 41/17 (3/20,2/13) 0 16 29/12 (2/13,1/6) 0 16 17/7 (1/7,2/13) 0 16 5/2 (2/13,1/6) 0 16 18/7 (2/13,4/25) 0 8 31/12 (3/19,4/25) 0 16 13/5 (4/25,1/6) 0 16 8/3 1/6 4 2 19/7 (1/6,4/23) 0 16 30/11 (6/35,4/23) 0 8 11/4 (4/23,3/17) 0 16 14/5 (0/1,2/11) 0 8 31/11 (0/1,1/6) 0 16 48/17 1/6 2 1 17/6 (1/6,2/11) 0 16 3/1 (2/11,1/5) 0 16 16/5 1/5 10 1 13/4 (1/5,8/39) 0 16 10/3 (4/19,2/9) 0 8 64/19 2/9 3 1 27/8 (2/9,1/4) 0 16 17/5 (1/5,2/9) 0 16 41/12 (1/5,2/9) 0 16 24/7 1/5 2 2 7/2 (3/14,2/9) 0 16 32/9 2/9 7 1 18/5 (2/9,8/35) 0 8 11/3 (4/17,1/4) 0 16 15/4 (2/9,1/4) 0 16 4/1 1/4 2 4 17/4 (1/4,2/7) 0 16 64/15 2/7 1 1 30/7 (0/1,2/7) 0 8 13/3 (0/1,1/4) 0 16 22/5 (6/23,4/15) 0 8 128/29 4/15 7 1 53/12 (4/15,11/41) 0 16 31/7 (1/4,4/15) 0 16 9/2 (3/11,2/7) 0 16 32/7 2/7 5 1 14/3 (2/7,4/13) 0 8 5/1 (2/7,1/3) 0 16 16/3 1/3 6 1 11/2 (1/3,4/11) 0 16 17/3 (1/3,2/5) 0 16 23/4 (1/3,2/5) 0 16 29/5 (5/13,2/5) 0 16 64/11 2/5 5 1 6/1 (0/1,2/5) 0 8 32/5 0/1 1 1 13/2 (0/1,1/3) 0 16 7/1 (1/3,2/5) 0 16 8/1 1/2 2 2 9/1 (1/2,2/3) 0 16 1/0 (0/1,1/1) 0 16 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(161,-352,59,-129) (2/1,11/5) -> (30/11,11/4) Glide Reflection Matrix(447,-992,187,-415) (11/5,20/9) -> (43/18,12/5) Glide Reflection Matrix(417,-928,173,-385) (20/9,9/4) -> (12/5,41/17) Glide Reflection Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(97,-224,42,-97) (16/7,7/3) -> (16/7,7/3) Reflection Matrix(95,-224,14,-33) (7/3,19/8) -> (13/2,7/1) Hyperbolic Matrix(417,-992,161,-383) (19/8,31/13) -> (31/12,13/5) Hyperbolic Matrix(1247,-2976,282,-673) (31/13,43/18) -> (53/12,31/7) Hyperbolic Matrix(769,-1856,133,-321) (41/17,29/12) -> (23/4,29/5) Glide Reflection Matrix(225,-544,79,-191) (29/12,17/7) -> (17/6,3/1) Hyperbolic Matrix(223,-544,66,-161) (17/7,5/2) -> (27/8,17/5) Hyperbolic Matrix(63,-160,13,-33) (5/2,18/7) -> (14/3,5/1) Hyperbolic Matrix(385,-992,137,-353) (18/7,31/12) -> (14/5,31/11) Glide Reflection Matrix(97,-256,36,-95) (13/5,8/3) -> (8/3,19/7) Parabolic Matrix(353,-960,82,-223) (19/7,30/11) -> (30/7,13/3) Hyperbolic Matrix(127,-352,35,-97) (11/4,14/5) -> (18/5,11/3) Hyperbolic Matrix(1055,-2976,374,-1055) (31/11,48/17) -> (31/11,48/17) Reflection Matrix(577,-1632,204,-577) (48/17,17/6) -> (48/17,17/6) Reflection Matrix(31,-96,10,-31) (3/1,16/5) -> (3/1,16/5) Reflection Matrix(129,-416,40,-129) (16/5,13/4) -> (16/5,13/4) Reflection Matrix(127,-416,29,-95) (13/4,10/3) -> (13/3,22/5) Glide Reflection Matrix(191,-640,57,-191) (10/3,64/19) -> (10/3,64/19) Reflection Matrix(1025,-3456,304,-1025) (64/19,27/8) -> (64/19,27/8) Reflection Matrix(319,-1088,56,-191) (17/5,41/12) -> (17/3,23/4) Glide Reflection Matrix(159,-544,19,-65) (41/12,24/7) -> (8/1,9/1) Hyperbolic Matrix(65,-224,9,-31) (24/7,7/2) -> (7/1,8/1) Hyperbolic Matrix(127,-448,36,-127) (7/2,32/9) -> (7/2,32/9) Reflection Matrix(161,-576,45,-161) (32/9,18/5) -> (32/9,18/5) Reflection Matrix(95,-352,17,-63) (11/3,15/4) -> (11/2,17/3) Glide Reflection Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic Matrix(511,-2176,120,-511) (17/4,64/15) -> (17/4,64/15) Reflection Matrix(449,-1920,105,-449) (64/15,30/7) -> (64/15,30/7) Reflection Matrix(639,-2816,145,-639) (22/5,128/29) -> (22/5,128/29) Reflection Matrix(3073,-13568,696,-3073) (128/29,53/12) -> (128/29,53/12) Reflection Matrix(65,-288,7,-31) (31/7,9/2) -> (9/1,1/0) Hyperbolic Matrix(127,-576,28,-127) (9/2,32/7) -> (9/2,32/7) Reflection Matrix(97,-448,21,-97) (32/7,14/3) -> (32/7,14/3) Reflection Matrix(31,-160,6,-31) (5/1,16/3) -> (5/1,16/3) Reflection Matrix(65,-352,12,-65) (16/3,11/2) -> (16/3,11/2) Reflection Matrix(639,-3712,110,-639) (29/5,64/11) -> (29/5,64/11) Reflection Matrix(65,-384,11,-65) (64/11,6/1) -> (64/11,6/1) Reflection Matrix(31,-192,5,-31) (6/1,32/5) -> (6/1,32/5) Reflection Matrix(129,-832,20,-129) (32/5,13/2) -> (32/5,13/2) Reflection 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) -> (0/1,1/1) Matrix(1,0,1,-1) -> Matrix(1,0,15,-1) (0/1,2/1) -> (0/1,2/15) Matrix(161,-352,59,-129) -> Matrix(27,-4,155,-23) Matrix(447,-992,187,-415) -> Matrix(1,0,15,-1) *** -> (0/1,2/15) Matrix(417,-928,173,-385) -> Matrix(31,-4,209,-27) Matrix(127,-288,56,-127) -> Matrix(29,-4,210,-29) (9/4,16/7) -> (2/15,1/7) Matrix(97,-224,42,-97) -> Matrix(27,-4,182,-27) (16/7,7/3) -> (1/7,2/13) Matrix(95,-224,14,-33) -> Matrix(1,0,-4,1) 0/1 Matrix(417,-992,161,-383) -> Matrix(27,-4,169,-25) 2/13 Matrix(1247,-2976,282,-673) -> Matrix(25,-4,94,-15) Matrix(769,-1856,133,-321) -> Matrix(53,-8,139,-21) Matrix(225,-544,79,-191) -> Matrix(1,0,-1,1) 0/1 Matrix(223,-544,66,-161) -> Matrix(1,0,-2,1) 0/1 Matrix(63,-160,13,-33) -> Matrix(1,0,-3,1) 0/1 Matrix(385,-992,137,-353) -> Matrix(25,-4,131,-21) Matrix(97,-256,36,-95) -> Matrix(49,-8,288,-47) 1/6 Matrix(353,-960,82,-223) -> Matrix(23,-4,98,-17) Matrix(127,-352,35,-97) -> Matrix(45,-8,197,-35) Matrix(1055,-2976,374,-1055) -> Matrix(1,0,12,-1) (31/11,48/17) -> (0/1,1/6) Matrix(577,-1632,204,-577) -> Matrix(23,-4,132,-23) (48/17,17/6) -> (1/6,2/11) Matrix(31,-96,10,-31) -> Matrix(21,-4,110,-21) (3/1,16/5) -> (2/11,1/5) Matrix(129,-416,40,-129) -> Matrix(79,-16,390,-79) (16/5,13/4) -> (1/5,8/39) Matrix(127,-416,29,-95) -> Matrix(39,-8,151,-31) Matrix(191,-640,57,-191) -> Matrix(37,-8,171,-37) (10/3,64/19) -> (4/19,2/9) Matrix(1025,-3456,304,-1025) -> Matrix(17,-4,72,-17) (64/19,27/8) -> (2/9,1/4) Matrix(319,-1088,56,-191) -> Matrix(19,-4,52,-11) Matrix(159,-544,19,-65) -> Matrix(1,0,-3,1) 0/1 Matrix(65,-224,9,-31) -> Matrix(19,-4,43,-9) Matrix(127,-448,36,-127) -> Matrix(55,-12,252,-55) (7/2,32/9) -> (3/14,2/9) Matrix(161,-576,45,-161) -> Matrix(71,-16,315,-71) (32/9,18/5) -> (2/9,8/35) Matrix(95,-352,17,-63) -> Matrix(1,0,7,-1) *** -> (0/1,2/7) Matrix(33,-128,8,-31) -> Matrix(17,-4,64,-15) 1/4 Matrix(511,-2176,120,-511) -> Matrix(15,-4,56,-15) (17/4,64/15) -> (1/4,2/7) Matrix(449,-1920,105,-449) -> Matrix(1,0,7,-1) (64/15,30/7) -> (0/1,2/7) Matrix(639,-2816,145,-639) -> Matrix(91,-24,345,-91) (22/5,128/29) -> (6/23,4/15) Matrix(3073,-13568,696,-3073) -> Matrix(329,-88,1230,-329) (128/29,53/12) -> (4/15,11/41) Matrix(65,-288,7,-31) -> Matrix(15,-4,19,-5) Matrix(127,-576,28,-127) -> Matrix(43,-12,154,-43) (9/2,32/7) -> (3/11,2/7) Matrix(97,-448,21,-97) -> Matrix(27,-8,91,-27) (32/7,14/3) -> (2/7,4/13) Matrix(31,-160,6,-31) -> Matrix(13,-4,42,-13) (5/1,16/3) -> (2/7,1/3) Matrix(65,-352,12,-65) -> Matrix(23,-8,66,-23) (16/3,11/2) -> (1/3,4/11) Matrix(639,-3712,110,-639) -> Matrix(51,-20,130,-51) (29/5,64/11) -> (5/13,2/5) Matrix(65,-384,11,-65) -> Matrix(1,0,5,-1) (64/11,6/1) -> (0/1,2/5) Matrix(31,-192,5,-31) -> Matrix(1,0,5,-1) (6/1,32/5) -> (0/1,2/5) Matrix(129,-832,20,-129) -> Matrix(1,0,6,-1) (32/5,13/2) -> (0/1,1/3) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.