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 -1/1 -2/3 -8/1 -2/3 0/1 -7/1 -1/2 0/1 -6/1 0/1 -11/2 -1/1 0/1 -16/3 0/1 -5/1 0/1 1/0 -14/3 -2/1 -9/2 -1/1 0/1 -13/3 -2/1 1/0 -4/1 -2/1 0/1 -19/5 -2/1 1/0 -15/4 -2/1 1/0 -11/3 -2/1 -1/1 -18/5 0/1 -7/2 -2/1 1/0 -24/7 -2/1 -41/12 -2/1 -9/5 -17/5 -7/4 -5/3 -10/3 -4/3 -13/4 -6/5 -1/1 -16/5 -1/1 -3/1 -1/1 0/1 -17/6 2/1 1/0 -14/5 -2/1 -11/4 -2/1 -1/1 -8/3 -2/1 0/1 -21/8 -2/1 -1/1 -13/5 -2/1 1/0 -31/12 -4/3 -1/1 -18/7 0/1 -5/2 0/1 1/0 -22/9 -6/1 -17/7 -7/2 -3/1 -29/12 -5/2 -2/1 -12/5 -2/1 -31/13 1/1 1/0 -19/8 -3/1 -2/1 -26/11 -2/1 -7/3 -2/1 1/0 -16/7 -2/1 -9/4 -2/1 -1/1 -20/9 -2/1 -4/3 -11/5 -1/1 0/1 -2/1 -2/1 -15/8 -3/2 -4/3 -13/7 -3/2 -4/3 -11/6 -4/3 -1/1 -31/17 1/1 1/0 -20/11 -2/1 -29/16 -2/1 1/0 -9/5 -2/1 -1/1 -16/9 -2/1 -7/4 -2/1 -3/2 -19/11 -2/1 -7/4 -31/18 -5/3 -8/5 -43/25 -5/3 -18/11 -12/7 -2/1 -8/5 -41/24 -5/3 -8/5 -29/17 -8/5 -11/7 -17/10 -2/1 -3/2 -5/3 -3/2 -4/3 -18/11 -2/1 -49/30 -14/9 -3/2 -80/49 -3/2 -31/19 -3/2 -13/9 -13/8 -4/3 -1/1 -8/5 -2/1 -4/3 -19/12 -4/3 -1/1 -30/19 -4/3 -11/7 -2/1 -1/1 -14/9 -4/3 -31/20 -8/7 -1/1 -48/31 -1/1 -17/11 -1/1 -1/2 -3/2 -2/1 -3/2 -16/11 -3/2 -13/9 -3/2 -16/11 -23/16 -13/9 -10/7 -33/23 -27/19 -17/12 -43/30 -38/27 -7/5 -10/7 -4/3 -37/26 -14/9 -3/2 -64/45 -3/2 -27/19 -3/2 -16/11 -17/12 -3/2 -10/7 -41/29 -10/7 -7/5 -24/17 -10/7 -4/3 -7/5 -3/2 -4/3 -32/23 -3/2 -25/18 -3/2 -10/7 -18/13 -4/3 -11/8 -7/5 -4/3 -26/19 -4/3 -67/49 -4/3 -1/1 -41/30 -7/5 -4/3 -15/11 -3/2 -1/1 -4/3 -4/3 -17/13 -9/7 -5/4 -64/49 -5/4 -47/36 -5/4 -16/13 -30/23 -6/5 -13/10 -6/5 -1/1 -22/17 -4/3 -75/58 -10/9 -1/1 -128/99 -1/1 -53/41 -1/1 0/1 -31/24 -2/1 -1/1 -9/7 -4/3 -1/1 -32/25 -1/1 -23/18 -2/1 -1/1 -14/11 -4/3 -5/4 -3/2 -4/3 -16/13 -4/3 -11/9 -4/3 -1/1 -17/14 -3/2 -4/3 -23/19 -7/5 -4/3 -29/24 -11/8 -4/3 -64/53 -4/3 -35/29 -4/3 -1/1 -6/5 -4/3 -19/16 -7/5 -4/3 -32/27 -4/3 -13/11 -4/3 -5/4 -7/6 -3/2 -4/3 -8/7 -4/3 -9/8 -4/3 -17/13 -1/1 -5/4 -1/1 0/1 -1/1 1/1 -1/1 -5/6 9/8 -17/21 -4/5 8/7 -4/5 7/6 -4/5 -3/4 6/5 -4/5 11/9 -1/1 -4/5 16/13 -4/5 5/4 -4/5 -3/4 14/11 -4/5 9/7 -1/1 -4/5 13/10 -1/1 -6/7 4/3 -4/5 19/14 -7/9 -10/13 15/11 -1/1 -3/4 11/8 -4/5 -7/9 18/13 -4/5 7/5 -4/5 -3/4 24/17 -4/5 -10/13 41/29 -7/9 -10/13 17/12 -10/13 -3/4 10/7 -4/5 13/9 -16/21 -3/4 16/11 -3/4 3/2 -3/4 -2/3 17/11 -1/1 1/0 14/9 -4/5 11/7 -1/1 -2/3 8/5 -4/5 -2/3 21/13 -1/1 -2/3 13/8 -1/1 -4/5 31/19 -13/17 -3/4 18/11 -2/3 5/3 -4/5 -3/4 22/13 -8/11 17/10 -3/4 -2/3 29/17 -11/15 -8/11 12/7 -8/11 -2/3 31/18 -8/11 -5/7 19/11 -7/10 -2/3 26/15 -2/3 7/4 -3/4 -2/3 16/9 -2/3 9/5 -1/1 -2/3 20/11 -2/3 11/6 -1/1 -4/5 2/1 -2/3 15/7 -1/2 -1/3 13/6 -1/1 -2/3 11/5 -1/1 0/1 31/14 -1/1 -2/3 20/9 -4/5 -2/3 29/13 -5/7 -2/3 9/4 -1/1 -2/3 16/7 -2/3 7/3 -2/3 -1/2 19/8 -2/3 -3/5 31/13 -1/2 -1/3 43/18 -1/1 -4/5 12/5 -2/3 41/17 -2/3 -3/5 29/12 -2/3 -5/8 17/7 -3/5 -7/12 5/2 -1/2 0/1 18/7 0/1 49/19 -5/4 -1/1 80/31 -1/1 31/12 -1/1 -4/5 13/5 -2/3 -1/2 8/3 -2/3 0/1 19/7 -2/3 -1/2 30/11 0/1 11/4 -1/1 -2/3 14/5 -2/3 31/11 -7/13 -1/2 48/17 -1/2 17/6 -1/2 -2/5 3/1 -1/1 0/1 16/5 -1/1 13/4 -1/1 -6/7 23/7 -1/1 -4/5 33/10 -1/1 -2/3 43/13 -1/1 0/1 10/3 -4/5 37/11 -16/21 -3/4 64/19 -3/4 27/8 -3/4 -14/19 17/5 -5/7 -7/10 41/12 -9/13 -2/3 24/7 -2/3 7/2 -2/3 -1/2 32/9 -1/2 25/7 -1/2 0/1 18/5 0/1 11/3 -1/1 -2/3 26/7 -2/3 67/18 -2/3 -5/8 41/11 -2/3 -3/5 15/4 -2/3 -1/2 4/1 -2/3 0/1 17/4 -2/3 -1/2 64/15 -1/2 47/11 -1/2 -3/7 30/7 0/1 13/3 -2/3 -1/2 22/5 -2/5 75/17 -6/17 -1/3 128/29 -1/3 53/12 -1/3 -4/13 31/7 -1/4 -1/5 9/2 -1/1 0/1 32/7 -1/1 23/5 -1/1 -2/3 14/3 -2/3 5/1 -1/2 0/1 16/3 0/1 11/2 -1/1 0/1 17/3 -1/2 -1/3 23/4 -1/1 0/1 29/5 -1/3 0/1 64/11 0/1 35/6 0/1 1/0 6/1 0/1 19/3 -1/2 0/1 32/5 0/1 13/2 0/1 1/1 7/1 0/1 1/0 8/1 -2/1 0/1 9/1 -2/1 -1/1 1/0 -1/1 0/1 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(1,2,-2,-3) Matrix(65,544,46,385) -> Matrix(11,4,-14,-5) Matrix(31,224,22,159) -> Matrix(11,4,-14,-5) Matrix(33,224,-14,-95) -> Matrix(3,2,-2,-1) Matrix(63,352,-46,-257) -> Matrix(3,-4,-2,3) Matrix(65,352,12,65) -> Matrix(1,0,0,1) Matrix(31,160,6,31) -> Matrix(1,0,-2,1) Matrix(33,160,20,97) -> Matrix(3,4,-4,-5) Matrix(97,448,-76,-351) -> Matrix(3,2,-2,-1) Matrix(95,416,-66,-289) -> Matrix(3,-10,-2,7) Matrix(31,128,-8,-33) -> Matrix(1,0,0,1) Matrix(127,480,-68,-257) -> Matrix(3,2,-2,-1) Matrix(95,352,-78,-289) -> Matrix(3,2,-2,-1) Matrix(97,352,62,225) -> Matrix(3,4,-4,-5) Matrix(161,576,-116,-415) -> Matrix(3,-4,-2,3) Matrix(65,224,56,193) -> Matrix(3,10,-4,-13) Matrix(159,544,140,479) -> Matrix(37,70,-46,-87) Matrix(319,1088,56,191) -> Matrix(1,2,-6,-11) Matrix(161,544,-66,-223) -> Matrix(9,14,-2,-3) Matrix(127,416,-98,-321) -> Matrix(1,0,0,1) Matrix(129,416,40,129) -> Matrix(11,12,-12,-13) Matrix(31,96,10,31) -> Matrix(1,0,0,1) Matrix(191,544,112,319) -> Matrix(3,-8,-4,11) Matrix(353,992,-216,-607) -> Matrix(3,8,-2,-5) Matrix(127,352,92,255) -> Matrix(3,10,-4,-13) Matrix(95,256,-36,-97) -> Matrix(1,0,0,1) Matrix(159,416,-86,-225) -> Matrix(3,2,-2,-1) Matrix(383,992,222,575) -> Matrix(7,12,-10,-17) Matrix(385,992,-248,-639) -> Matrix(5,4,-4,-3) Matrix(63,160,50,127) -> Matrix(3,4,-4,-5) Matrix(353,864,-248,-607) -> Matrix(3,14,-2,-9) Matrix(225,544,146,353) -> Matrix(1,4,-2,-7) Matrix(385,928,-212,-511) -> Matrix(3,8,-2,-5) Matrix(415,992,-228,-545) -> Matrix(1,0,0,1) Matrix(417,992,256,609) -> Matrix(3,10,-4,-13) Matrix(257,608,-216,-511) -> Matrix(3,2,-2,-1) Matrix(97,224,42,97) -> Matrix(1,4,-2,-7) Matrix(127,288,56,127) -> Matrix(3,4,-4,-5) Matrix(417,928,-244,-543) -> Matrix(13,18,-8,-11) Matrix(447,992,-260,-577) -> Matrix(13,18,-8,-11) Matrix(161,352,-102,-223) -> Matrix(3,2,-2,-1) Matrix(287,544,-220,-417) -> Matrix(17,28,-14,-23) Matrix(929,1696,-648,-1183) -> Matrix(17,10,-12,-7) Matrix(513,928,-424,-767) -> Matrix(11,18,-8,-13) Matrix(161,288,90,161) -> Matrix(3,4,-4,-5) Matrix(127,224,72,127) -> Matrix(7,12,-10,-17) Matrix(129,224,-110,-191) -> Matrix(11,18,-8,-13) Matrix(575,992,222,383) -> Matrix(7,12,-10,-17) Matrix(1729,2976,-1338,-2303) -> Matrix(11,18,-8,-13) Matrix(2305,3936,-1686,-2879) -> Matrix(23,36,-16,-25) Matrix(319,544,112,191) -> Matrix(5,8,-12,-19) Matrix(321,544,-226,-383) -> Matrix(23,36,-16,-25) Matrix(97,160,20,33) -> Matrix(3,4,-4,-5) Matrix(2273,3712,804,1313) -> Matrix(13,20,-28,-43) Matrix(2431,3968,862,1407) -> Matrix(23,34,-44,-65) Matrix(609,992,256,417) -> Matrix(7,10,-12,-17) Matrix(159,256,-100,-161) -> Matrix(1,0,0,1) Matrix(607,960,-466,-737) -> Matrix(3,2,-2,-1) Matrix(225,352,62,97) -> Matrix(3,4,-4,-5) Matrix(2561,3968,992,1537) -> Matrix(11,12,-12,-13) Matrix(2399,3712,930,1439) -> Matrix(7,6,-6,-5) Matrix(353,544,146,225) -> Matrix(1,4,-2,-7) Matrix(65,96,44,65) -> Matrix(7,12,-10,-17) Matrix(287,416,198,287) -> Matrix(65,96,-86,-127) Matrix(223,320,200,287) -> Matrix(83,118,-102,-145) Matrix(1185,1696,-916,-1311) -> Matrix(23,32,-18,-25) Matrix(2879,4096,854,1215) -> Matrix(55,84,-74,-113) Matrix(2881,4096,856,1217) -> Matrix(65,96,-86,-127) Matrix(927,1312,248,351) -> Matrix(11,16,-20,-29) Matrix(385,544,46,65) -> Matrix(3,4,2,3) Matrix(159,224,22,31) -> Matrix(3,4,2,3) Matrix(735,1024,206,287) -> Matrix(3,4,-4,-5) Matrix(737,1024,208,289) -> Matrix(11,16,-20,-29) Matrix(255,352,92,127) -> Matrix(7,10,-12,-17) Matrix(959,1312,-796,-1089) -> Matrix(1,0,0,1) Matrix(961,1312,282,385) -> Matrix(17,22,-24,-31) Matrix(95,128,-72,-97) -> Matrix(23,32,-18,-25) Matrix(3135,4096,734,959) -> Matrix(19,24,-42,-53) Matrix(3137,4096,736,961) -> Matrix(21,26,-38,-47) Matrix(12671,16384,2870,3711) -> Matrix(13,14,-40,-43) Matrix(12673,16384,2872,3713) -> Matrix(7,6,-20,-17) Matrix(223,288,24,31) -> Matrix(1,2,-2,-3) Matrix(799,1024,174,223) -> Matrix(5,6,-6,-7) Matrix(801,1024,176,225) -> Matrix(1,2,-2,-3) Matrix(127,160,50,63) -> Matrix(3,4,-4,-5) Matrix(129,160,104,129) -> Matrix(17,24,-22,-31) Matrix(287,352,234,287) -> Matrix(7,8,-8,-9) Matrix(897,1088,634,769) -> Matrix(29,42,-38,-55) Matrix(3391,4096,582,703) -> Matrix(3,4,8,11) Matrix(3393,4096,584,705) -> Matrix(3,4,-10,-13) Matrix(863,1024,134,159) -> Matrix(3,4,8,11) Matrix(865,1024,136,161) -> Matrix(3,4,-10,-13) Matrix(193,224,56,65) -> Matrix(7,10,-12,-17) Matrix(479,544,140,159) -> Matrix(53,70,-78,-103) Matrix(257,288,58,65) -> Matrix(3,4,-16,-21) Matrix(1,0,2,1) -> Matrix(9,10,-10,-11) Matrix(191,-224,110,-129) -> Matrix(23,18,-32,-25) Matrix(289,-352,78,-95) -> Matrix(3,2,-2,-1) Matrix(351,-448,76,-97) -> Matrix(3,2,-2,-1) Matrix(321,-416,98,-127) -> Matrix(1,0,0,1) Matrix(97,-128,72,-95) -> Matrix(39,32,-50,-41) Matrix(353,-480,164,-223) -> Matrix(5,4,-14,-11) Matrix(257,-352,46,-63) -> Matrix(5,4,-14,-11) Matrix(415,-576,116,-161) -> Matrix(5,4,-14,-11) Matrix(383,-544,226,-321) -> Matrix(47,36,-64,-49) Matrix(289,-416,66,-95) -> Matrix(13,10,-30,-23) Matrix(639,-992,248,-385) -> Matrix(5,4,-4,-3) Matrix(161,-256,100,-159) -> Matrix(1,0,0,1) Matrix(257,-416,118,-191) -> Matrix(3,2,-2,-1) Matrix(607,-992,216,-353) -> Matrix(11,8,-18,-13) Matrix(511,-864,152,-257) -> Matrix(49,36,-64,-47) Matrix(543,-928,244,-417) -> Matrix(25,18,-32,-23) Matrix(577,-992,260,-447) -> Matrix(25,18,-32,-23) Matrix(351,-608,56,-97) -> Matrix(3,2,4,3) Matrix(511,-928,212,-385) -> Matrix(11,8,-18,-13) Matrix(545,-992,228,-415) -> Matrix(1,0,0,1) Matrix(191,-352,70,-129) -> Matrix(3,2,-2,-1) Matrix(257,-544,60,-127) -> Matrix(3,2,-8,-5) Matrix(767,-1696,232,-513) -> Matrix(1,0,0,1) Matrix(415,-928,72,-161) -> Matrix(3,2,-2,-1) Matrix(95,-224,14,-33) -> Matrix(3,2,-2,-1) Matrix(1247,-2976,282,-673) -> Matrix(1,0,-2,1) Matrix(1631,-3936,438,-1057) -> Matrix(1,0,0,1) Matrix(223,-544,66,-161) -> Matrix(25,14,-34,-19) Matrix(97,-256,36,-95) -> Matrix(1,0,0,1) Matrix(353,-960,82,-223) -> Matrix(1,0,0,1) Matrix(511,-1696,116,-385) -> Matrix(7,6,-20,-17) Matrix(353,-1312,60,-223) -> Matrix(3,2,-8,-5) Matrix(33,-128,8,-31) -> 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: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 32 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): 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: 40 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -8/1 -6/1 -4/1 -10/3 -8/3 -12/5 -2/1 -12/7 -8/5 -4/3 -8/7 0/1 1/1 8/7 4/3 3/2 8/5 12/7 2/1 16/7 12/5 5/2 8/3 14/5 48/17 3/1 16/5 10/3 64/19 7/2 4/1 32/7 14/3 5/1 16/3 6/1 32/5 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 -2/3 0/1 -7/1 -1/2 0/1 -6/1 0/1 -5/1 0/1 1/0 -4/1 -2/1 0/1 -11/3 -2/1 -1/1 -18/5 0/1 -7/2 -2/1 1/0 -24/7 -2/1 -17/5 -7/4 -5/3 -10/3 -4/3 -3/1 -1/1 0/1 -8/3 -2/1 0/1 -13/5 -2/1 1/0 -31/12 -4/3 -1/1 -18/7 0/1 -5/2 0/1 1/0 -22/9 -6/1 -17/7 -7/2 -3/1 -12/5 -2/1 -31/13 1/1 1/0 -19/8 -3/1 -2/1 -26/11 -2/1 -7/3 -2/1 1/0 -2/1 -2/1 -9/5 -2/1 -1/1 -16/9 -2/1 -7/4 -2/1 -3/2 -19/11 -2/1 -7/4 -12/7 -2/1 -8/5 -29/17 -8/5 -11/7 -17/10 -2/1 -3/2 -5/3 -3/2 -4/3 -8/5 -2/1 -4/3 -11/7 -2/1 -1/1 -14/9 -4/3 -31/20 -8/7 -1/1 -48/31 -1/1 -17/11 -1/1 -1/2 -3/2 -2/1 -3/2 -16/11 -3/2 -13/9 -3/2 -16/11 -10/7 -4/3 -37/26 -14/9 -3/2 -64/45 -3/2 -27/19 -3/2 -16/11 -17/12 -3/2 -10/7 -24/17 -10/7 -4/3 -7/5 -3/2 -4/3 -4/3 -4/3 -9/7 -4/3 -1/1 -32/25 -1/1 -23/18 -2/1 -1/1 -14/11 -4/3 -5/4 -3/2 -4/3 -16/13 -4/3 -11/9 -4/3 -1/1 -6/5 -4/3 -19/16 -7/5 -4/3 -32/27 -4/3 -13/11 -4/3 -5/4 -7/6 -3/2 -4/3 -8/7 -4/3 -1/1 -5/4 -1/1 0/1 -1/1 1/1 -1/1 -5/6 8/7 -4/5 7/6 -4/5 -3/4 6/5 -4/5 5/4 -4/5 -3/4 4/3 -4/5 11/8 -4/5 -7/9 18/13 -4/5 7/5 -4/5 -3/4 24/17 -4/5 -10/13 17/12 -10/13 -3/4 10/7 -4/5 3/2 -3/4 -2/3 8/5 -4/5 -2/3 13/8 -1/1 -4/5 31/19 -13/17 -3/4 18/11 -2/3 5/3 -4/5 -3/4 22/13 -8/11 17/10 -3/4 -2/3 12/7 -8/11 -2/3 31/18 -8/11 -5/7 19/11 -7/10 -2/3 26/15 -2/3 7/4 -3/4 -2/3 2/1 -2/3 9/4 -1/1 -2/3 16/7 -2/3 7/3 -2/3 -1/2 19/8 -2/3 -3/5 12/5 -2/3 29/12 -2/3 -5/8 17/7 -3/5 -7/12 5/2 -1/2 0/1 8/3 -2/3 0/1 11/4 -1/1 -2/3 14/5 -2/3 31/11 -7/13 -1/2 48/17 -1/2 17/6 -1/2 -2/5 3/1 -1/1 0/1 16/5 -1/1 13/4 -1/1 -6/7 10/3 -4/5 37/11 -16/21 -3/4 64/19 -3/4 27/8 -3/4 -14/19 17/5 -5/7 -7/10 24/7 -2/3 7/2 -2/3 -1/2 4/1 -2/3 0/1 9/2 -1/1 0/1 32/7 -1/1 23/5 -1/1 -2/3 14/3 -2/3 5/1 -1/2 0/1 16/3 0/1 11/2 -1/1 0/1 6/1 0/1 19/3 -1/2 0/1 32/5 0/1 13/2 0/1 1/1 7/1 0/1 1/0 8/1 -2/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(17,160,12,113) (-8/1,1/0) -> (24/17,17/12) 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(17,96,-14,-79) (-6/1,-5/1) -> (-11/9,-6/5) Hyperbolic Matrix(15,64,-4,-17) (-5/1,-4/1) -> (-4/1,-11/3) Parabolic Matrix(79,288,48,175) (-11/3,-18/5) -> (18/11,5/3) Hyperbolic Matrix(143,512,-112,-401) (-18/5,-7/2) -> (-23/18,-14/11) Hyperbolic Matrix(65,224,56,193) (-7/2,-24/7) -> (8/7,7/6) Hyperbolic Matrix(47,160,42,143) (-24/7,-17/5) -> (1/1,8/7) Hyperbolic Matrix(161,544,-66,-223) (-17/5,-10/3) -> (-22/9,-17/7) Hyperbolic Matrix(49,160,-34,-111) (-10/3,-3/1) -> (-13/9,-10/7) Hyperbolic Matrix(47,128,-18,-49) (-3/1,-8/3) -> (-8/3,-13/5) Parabolic 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(113,288,82,209) (-18/7,-5/2) -> (11/8,18/13) Hyperbolic Matrix(353,864,-248,-607) (-5/2,-22/9) -> (-10/7,-37/26) Hyperbolic Matrix(239,576,-100,-241) (-17/7,-12/5) -> (-12/5,-31/13) Parabolic 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(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,62,111) (-9/5,-16/9) -> (16/7,7/3) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(129,224,-110,-191) (-7/4,-19/11) -> (-13/11,-7/6) Hyperbolic Matrix(335,576,-196,-337) (-19/11,-12/7) -> (-12/7,-29/17) Parabolic 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(79,128,-50,-81) (-5/3,-8/5) -> (-8/5,-11/7) Parabolic Matrix(143,224,30,47) (-11/7,-14/9) -> (14/3,5/1) 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(353,544,146,225) (-17/11,-3/2) -> (29/12,17/7) Hyperbolic Matrix(175,256,54,79) (-3/2,-16/11) -> (16/5,13/4) Hyperbolic Matrix(177,256,56,81) (-16/11,-13/9) -> (3/1,16/5) 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(113,160,12,17) (-17/12,-24/17) -> (8/1,1/0) Hyperbolic Matrix(159,224,22,31) (-24/17,-7/5) -> (7/1,8/1) Hyperbolic Matrix(47,64,-36,-49) (-7/5,-4/3) -> (-4/3,-9/7) Parabolic 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(177,224,64,81) (-14/11,-5/4) -> (11/4,14/5) Hyperbolic Matrix(207,256,38,47) (-5/4,-16/13) -> (16/3,11/2) Hyperbolic Matrix(209,256,40,49) (-16/13,-11/9) -> (5/1,16/3) 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(143,160,42,47) (-8/7,-1/1) -> (17/5,24/7) 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(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(369,-512,80,-111) (18/13,7/5) -> (23/5,14/3) Hyperbolic Matrix(383,-544,226,-321) (17/12,10/7) -> (22/13,17/10) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(81,-128,50,-79) (3/2,8/5) -> (8/5,13/8) Parabolic 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(337,-576,196,-335) (17/10,12/7) -> (12/7,31/18) Parabolic Matrix(351,-608,56,-97) (19/11,26/15) -> (6/1,19/3) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(95,-224,14,-33) (7/3,19/8) -> (13/2,7/1) Hyperbolic Matrix(241,-576,100,-239) (19/8,12/5) -> (12/5,29/12) Parabolic Matrix(223,-544,66,-161) (17/7,5/2) -> (27/8,17/5) Hyperbolic Matrix(49,-128,18,-47) (5/2,8/3) -> (8/3,11/4) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(17,160,12,113) -> Matrix(13,10,-17,-13) Matrix(31,224,22,159) -> Matrix(11,4,-14,-5) Matrix(33,224,-14,-95) -> Matrix(3,2,-2,-1) Matrix(17,96,-14,-79) -> Matrix(1,4,-1,-3) Matrix(15,64,-4,-17) -> Matrix(1,2,-1,-1) Matrix(79,288,48,175) -> Matrix(1,-2,-1,3) Matrix(143,512,-112,-401) -> Matrix(1,4,-1,-3) Matrix(65,224,56,193) -> Matrix(3,10,-4,-13) Matrix(47,160,42,143) -> Matrix(17,30,-21,-37) Matrix(161,544,-66,-223) -> Matrix(9,14,-2,-3) Matrix(49,160,-34,-111) -> Matrix(13,16,-9,-11) Matrix(47,128,-18,-49) -> Matrix(1,2,-1,-1) Matrix(383,992,222,575) -> Matrix(7,12,-10,-17) Matrix(385,992,-248,-639) -> Matrix(5,4,-4,-3) Matrix(113,288,82,209) -> Matrix(7,4,-9,-5) Matrix(353,864,-248,-607) -> Matrix(3,14,-2,-9) Matrix(239,576,-100,-241) -> Matrix(1,4,-1,-3) Matrix(417,992,256,609) -> Matrix(3,10,-4,-13) Matrix(257,608,-216,-511) -> Matrix(3,2,-2,-1) Matrix(15,32,-8,-17) -> Matrix(1,4,-1,-3) Matrix(143,256,62,111) -> Matrix(1,0,-1,1) Matrix(145,256,64,113) -> Matrix(5,8,-7,-11) Matrix(129,224,-110,-191) -> Matrix(11,18,-8,-13) Matrix(335,576,-196,-337) -> Matrix(21,34,-13,-21) Matrix(319,544,112,191) -> Matrix(5,8,-12,-19) Matrix(321,544,-226,-383) -> Matrix(23,36,-16,-25) Matrix(79,128,-50,-81) -> Matrix(7,10,-5,-7) Matrix(143,224,30,47) -> Matrix(1,2,-3,-5) Matrix(1487,2304,526,815) -> Matrix(9,10,-19,-21) Matrix(1489,2304,528,817) -> Matrix(9,8,-17,-15) Matrix(353,544,146,225) -> Matrix(1,4,-2,-7) Matrix(175,256,54,79) -> Matrix(13,20,-15,-23) Matrix(177,256,56,81) -> Matrix(11,16,-9,-13) Matrix(2879,4096,854,1215) -> Matrix(55,84,-74,-113) Matrix(2881,4096,856,1217) -> Matrix(65,96,-86,-127) Matrix(113,160,12,17) -> Matrix(7,10,-5,-7) Matrix(159,224,22,31) -> Matrix(3,4,2,3) Matrix(47,64,-36,-49) -> Matrix(11,16,-9,-13) Matrix(799,1024,174,223) -> Matrix(5,6,-6,-7) Matrix(801,1024,176,225) -> Matrix(1,2,-2,-3) Matrix(177,224,64,81) -> Matrix(1,2,-3,-5) Matrix(207,256,38,47) -> Matrix(3,4,-1,-1) Matrix(209,256,40,49) -> Matrix(3,4,-7,-9) Matrix(863,1024,134,159) -> Matrix(3,4,8,11) Matrix(865,1024,136,161) -> Matrix(3,4,-10,-13) Matrix(193,224,56,65) -> Matrix(7,10,-12,-17) Matrix(143,160,42,47) -> Matrix(23,30,-33,-43) Matrix(1,0,2,1) -> Matrix(9,10,-10,-11) Matrix(191,-224,110,-129) -> Matrix(23,18,-32,-25) Matrix(79,-96,14,-17) -> Matrix(5,4,-9,-7) Matrix(49,-64,36,-47) -> Matrix(19,16,-25,-21) Matrix(369,-512,80,-111) -> Matrix(13,10,-17,-13) Matrix(383,-544,226,-321) -> Matrix(47,36,-64,-49) Matrix(111,-160,34,-49) -> Matrix(21,16,-25,-19) Matrix(81,-128,50,-79) -> Matrix(13,10,-17,-13) Matrix(607,-992,216,-353) -> Matrix(11,8,-18,-13) Matrix(511,-864,152,-257) -> Matrix(49,36,-64,-47) Matrix(337,-576,196,-335) -> Matrix(47,34,-65,-47) Matrix(351,-608,56,-97) -> Matrix(3,2,4,3) Matrix(17,-32,8,-15) -> Matrix(5,4,-9,-7) Matrix(95,-224,14,-33) -> Matrix(3,2,-2,-1) Matrix(241,-576,100,-239) -> Matrix(5,4,-9,-7) Matrix(223,-544,66,-161) -> Matrix(25,14,-34,-19) Matrix(49,-128,18,-47) -> Matrix(3,2,-5,-3) Matrix(17,-64,4,-15) -> Matrix(3,2,-5,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 32 ----------------------------------------------------------------------- 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 -1/1 5 1 1/1 (-1/1,-5/6) 0 16 8/7 -4/5 5 2 7/6 (-4/5,-3/4) 0 16 6/5 -4/5 1 8 5/4 (-4/5,-3/4) 0 16 4/3 -4/5 1 4 11/8 (-4/5,-7/9) 0 16 18/13 -4/5 1 8 7/5 (-4/5,-3/4) 0 16 24/17 (-4/5,-10/13).(-7/9,-3/4) 0 2 17/12 (-10/13,-3/4) 0 16 10/7 -4/5 1 8 3/2 (-3/4,-2/3) 0 16 8/5 (-1/1,-3/4).(-4/5,-2/3) 0 2 13/8 (-1/1,-4/5) 0 16 31/19 (-13/17,-3/4) 0 16 18/11 -2/3 1 8 5/3 (-4/5,-3/4) 0 16 22/13 -8/11 1 8 17/10 (-3/4,-2/3) 0 16 12/7 (-3/4,-5/7).(-8/11,-2/3) 0 4 31/18 (-8/11,-5/7) 0 16 19/11 (-7/10,-2/3) 0 16 26/15 -2/3 1 8 7/4 (-3/4,-2/3) 0 16 2/1 -2/3 1 8 9/4 (-1/1,-2/3) 0 16 16/7 -2/3 1 1 7/3 (-2/3,-1/2) 0 16 19/8 (-2/3,-3/5) 0 16 12/5 -2/3 1 4 29/12 (-2/3,-5/8) 0 16 17/7 (-3/5,-7/12) 0 16 5/2 (-1/2,0/1) 0 16 8/3 (-1/1,-1/2).(-2/3,0/1) 0 2 11/4 (-1/1,-2/3) 0 16 14/5 -2/3 1 8 31/11 (-7/13,-1/2) 0 16 48/17 -1/2 9 1 17/6 (-1/2,-2/5) 0 16 3/1 (-1/1,0/1) 0 16 16/5 -1/1 6 1 13/4 (-1/1,-6/7) 0 16 10/3 -4/5 1 8 37/11 (-16/21,-3/4) 0 16 64/19 -3/4 10 1 27/8 (-3/4,-14/19) 0 16 17/5 (-5/7,-7/10) 0 16 24/7 -2/3 5 2 7/2 (-2/3,-1/2) 0 16 4/1 (-1/1,-1/2).(-2/3,0/1) 0 4 9/2 (-1/1,0/1) 0 16 32/7 -1/1 2 1 23/5 (-1/1,-2/3) 0 16 14/3 -2/3 1 8 5/1 (-1/2,0/1) 0 16 16/3 0/1 1 1 11/2 (-1/1,0/1) 0 16 6/1 0/1 1 8 19/3 (-1/2,0/1) 0 16 32/5 0/1 3 1 13/2 (0/1,1/1) 0 16 7/1 (0/1,1/0) 0 16 8/1 (-2/1,0/1).(-1/1,1/0) 0 2 1/0 (-1/1,0/1) 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(143,-160,42,-47) (1/1,8/7) -> (17/5,24/7) Glide Reflection Matrix(193,-224,56,-65) (8/7,7/6) -> (24/7,7/2) Glide Reflection Matrix(191,-224,110,-129) (7/6,6/5) -> (26/15,7/4) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(255,-352,92,-127) (11/8,18/13) -> (11/4,14/5) Glide Reflection Matrix(369,-512,80,-111) (18/13,7/5) -> (23/5,14/3) Hyperbolic Matrix(159,-224,22,-31) (7/5,24/17) -> (7/1,8/1) Glide Reflection Matrix(113,-160,12,-17) (24/17,17/12) -> (8/1,1/0) Glide Reflection Matrix(383,-544,226,-321) (17/12,10/7) -> (22/13,17/10) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(81,-128,50,-79) (3/2,8/5) -> (8/5,13/8) Parabolic Matrix(687,-1120,284,-463) (13/8,31/19) -> (29/12,17/7) Glide Reflection Matrix(607,-992,216,-353) (31/19,18/11) -> (14/5,31/11) Hyperbolic Matrix(97,-160,20,-33) (18/11,5/3) -> (14/3,5/1) Glide Reflection Matrix(511,-864,152,-257) (5/3,22/13) -> (10/3,37/11) Hyperbolic Matrix(337,-576,196,-335) (17/10,12/7) -> (12/7,31/18) Parabolic Matrix(241,-416,84,-145) (31/18,19/11) -> (17/6,3/1) Glide Reflection Matrix(351,-608,56,-97) (19/11,26/15) -> (6/1,19/3) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic 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(241,-576,100,-239) (19/8,12/5) -> (12/5,29/12) Parabolic Matrix(223,-544,66,-161) (17/7,5/2) -> (27/8,17/5) Hyperbolic Matrix(49,-128,18,-47) (5/2,8/3) -> (8/3,11/4) Parabolic 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(1407,-4736,418,-1407) (37/11,64/19) -> (37/11,64/19) Reflection Matrix(1025,-3456,304,-1025) (64/19,27/8) -> (64/19,27/8) Reflection Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic Matrix(127,-576,28,-127) (9/2,32/7) -> (9/2,32/7) Reflection Matrix(321,-1472,70,-321) (32/7,23/5) -> (32/7,23/5) 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(191,-1216,30,-191) (19/3,32/5) -> (19/3,32/5) 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) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(11,10,-12,-11) (0/1,1/1) -> (-1/1,-5/6) Matrix(143,-160,42,-47) -> Matrix(37,30,-53,-43) Matrix(193,-224,56,-65) -> Matrix(13,10,-22,-17) Matrix(191,-224,110,-129) -> Matrix(23,18,-32,-25) -3/4 Matrix(79,-96,14,-17) -> Matrix(5,4,-9,-7) -2/3 Matrix(49,-64,36,-47) -> Matrix(19,16,-25,-21) -4/5 Matrix(255,-352,92,-127) -> Matrix(13,10,-22,-17) Matrix(369,-512,80,-111) -> Matrix(13,10,-17,-13) (-1/1,-3/4).(-4/5,-2/3) Matrix(159,-224,22,-31) -> Matrix(5,4,4,3) Matrix(113,-160,12,-17) -> Matrix(13,10,-9,-7) Matrix(383,-544,226,-321) -> Matrix(47,36,-64,-49) -3/4 Matrix(111,-160,34,-49) -> Matrix(21,16,-25,-19) -4/5 Matrix(81,-128,50,-79) -> Matrix(13,10,-17,-13) (-1/1,-3/4).(-4/5,-2/3) Matrix(687,-1120,284,-463) -> Matrix(23,18,-37,-29) Matrix(607,-992,216,-353) -> Matrix(11,8,-18,-13) -2/3 Matrix(97,-160,20,-33) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(511,-864,152,-257) -> Matrix(49,36,-64,-47) -3/4 Matrix(337,-576,196,-335) -> Matrix(47,34,-65,-47) (-3/4,-5/7).(-8/11,-2/3) Matrix(241,-416,84,-145) -> Matrix(3,2,-13,-9) Matrix(351,-608,56,-97) -> Matrix(3,2,4,3) Matrix(17,-32,8,-15) -> Matrix(5,4,-9,-7) -2/3 Matrix(127,-288,56,-127) -> Matrix(5,4,-6,-5) (9/4,16/7) -> (-1/1,-2/3) Matrix(97,-224,42,-97) -> Matrix(7,4,-12,-7) (16/7,7/3) -> (-2/3,-1/2) Matrix(95,-224,14,-33) -> Matrix(3,2,-2,-1) -1/1 Matrix(241,-576,100,-239) -> Matrix(5,4,-9,-7) -2/3 Matrix(223,-544,66,-161) -> Matrix(25,14,-34,-19) Matrix(49,-128,18,-47) -> Matrix(3,2,-5,-3) (-1/1,-1/2).(-2/3,0/1) Matrix(1055,-2976,374,-1055) -> Matrix(27,14,-52,-27) (31/11,48/17) -> (-7/13,-1/2) Matrix(577,-1632,204,-577) -> Matrix(9,4,-20,-9) (48/17,17/6) -> (-1/2,-2/5) Matrix(31,-96,10,-31) -> Matrix(-1,0,2,1) (3/1,16/5) -> (-1/1,0/1) Matrix(129,-416,40,-129) -> Matrix(13,12,-14,-13) (16/5,13/4) -> (-1/1,-6/7) Matrix(1407,-4736,418,-1407) -> Matrix(127,96,-168,-127) (37/11,64/19) -> (-16/21,-3/4) Matrix(1025,-3456,304,-1025) -> Matrix(113,84,-152,-113) (64/19,27/8) -> (-3/4,-14/19) Matrix(17,-64,4,-15) -> Matrix(3,2,-5,-3) (-1/1,-1/2).(-2/3,0/1) Matrix(127,-576,28,-127) -> Matrix(-1,0,2,1) (9/2,32/7) -> (-1/1,0/1) Matrix(321,-1472,70,-321) -> Matrix(5,4,-6,-5) (32/7,23/5) -> (-1/1,-2/3) Matrix(31,-160,6,-31) -> Matrix(-1,0,4,1) (5/1,16/3) -> (-1/2,0/1) Matrix(65,-352,12,-65) -> Matrix(-1,0,2,1) (16/3,11/2) -> (-1/1,0/1) Matrix(191,-1216,30,-191) -> Matrix(-1,0,4,1) (19/3,32/5) -> (-1/2,0/1) Matrix(129,-832,20,-129) -> Matrix(1,0,2,-1) (32/5,13/2) -> (0/1,1/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.