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 0/1 -8/1 -2/3 0/1 -7/1 0/1 -6/1 0/1 -11/2 -2/1 -1/1 -16/3 -1/1 -5/1 -2/3 -14/3 -2/3 -9/2 -1/2 -1/3 -13/3 -2/3 -4/1 -1/1 -1/2 0/1 -19/5 -2/3 -15/4 -1/2 -2/5 -11/3 0/1 -18/5 -2/3 -7/2 -1/2 -1/3 -24/7 0/1 -41/12 1/2 1/1 -17/5 -2/1 -10/3 0/1 -13/4 -1/2 0/1 -16/5 -1/1 -1/3 -3/1 0/1 -17/6 -1/2 0/1 -14/5 0/1 -11/4 -1/3 0/1 -8/3 0/1 -21/8 0/1 1/3 -13/5 0/1 -31/12 4/5 1/1 -18/7 2/1 -5/2 0/1 1/0 -22/9 -2/1 -17/7 -4/3 -29/12 -1/1 0/1 -12/5 -1/1 0/1 1/0 -31/13 0/1 -19/8 0/1 1/0 -26/11 -2/1 -7/3 -2/1 -16/7 -1/1 -9/4 -1/1 -3/4 -20/9 -1/1 -2/3 -1/2 -11/5 -2/3 -2/1 0/1 -15/8 -6/1 1/0 -13/7 -2/1 -11/6 -2/1 -1/1 -31/17 -2/1 -20/11 -2/1 -3/2 -1/1 -29/16 -2/1 -1/1 -9/5 -4/3 -16/9 -1/1 -7/4 -1/1 -3/4 -19/11 0/1 -31/18 -1/1 -4/5 -43/25 -2/3 -12/7 -1/1 -2/3 -1/2 -41/24 -1/1 -3/4 -29/17 -2/3 -17/10 -2/3 -1/2 -5/3 0/1 -18/11 0/1 -49/30 -1/2 0/1 -80/49 -1/1 -1/3 -31/19 0/1 -13/8 -1/2 0/1 -8/5 0/1 -19/12 0/1 1/0 -30/19 -2/1 -11/7 0/1 -14/9 -2/1 -31/20 -8/7 -1/1 -48/31 -1/1 -17/11 -4/5 -3/2 -1/1 0/1 -16/11 -1/1 -13/9 -4/5 -23/16 -3/4 -5/7 -33/23 -2/3 -43/30 -1/1 -2/3 -10/7 -2/3 -37/26 -4/7 -1/2 -64/45 -1/2 -27/19 0/1 -17/12 -2/3 -1/2 -41/29 -2/3 -24/17 -2/3 0/1 -7/5 -2/3 -32/23 -1/2 -25/18 -1/2 -3/7 -18/13 0/1 -11/8 -1/1 0/1 -26/19 -2/3 -67/49 -2/3 -41/30 -1/1 -3/4 -15/11 -4/7 -4/3 -1/1 -1/2 0/1 -17/13 -4/7 -64/49 -1/2 -47/36 -1/2 -8/17 -30/23 -2/5 -13/10 -1/2 0/1 -22/17 0/1 -75/58 -1/1 0/1 -128/99 -1/2 1/0 -53/41 0/1 -31/24 -1/1 -2/3 -9/7 -2/3 -32/25 -1/2 -23/18 -1/2 -5/11 -14/11 -2/5 -5/4 -1/2 0/1 -16/13 -1/1 -1/3 -11/9 0/1 -17/14 -1/2 -2/5 -23/19 -2/5 -29/24 -1/3 0/1 -64/53 -1/2 -1/4 -35/29 0/1 -6/5 0/1 -19/16 -2/3 -1/2 -32/27 -1/2 -13/11 -2/5 -7/6 -1/3 -1/4 -8/7 0/1 -9/8 1/1 1/0 -1/1 0/1 0/1 -1/2 1/0 1/1 0/1 9/8 -1/2 -1/3 8/7 0/1 7/6 1/2 1/1 6/5 0/1 11/9 0/1 16/13 -1/1 1/1 5/4 0/1 1/0 14/11 2/1 9/7 -2/1 13/10 0/1 1/0 4/3 -1/1 0/1 1/0 19/14 0/1 1/0 15/11 -4/1 11/8 -1/1 0/1 18/13 0/1 7/5 -2/1 24/17 -2/1 0/1 41/29 -2/1 17/12 -2/1 1/0 10/7 -2/1 13/9 -4/3 16/11 -1/1 3/2 -1/1 0/1 17/11 -4/3 14/9 -2/3 11/7 0/1 8/5 0/1 21/13 0/1 13/8 0/1 1/0 31/19 0/1 18/11 0/1 5/3 0/1 22/13 -2/1 17/10 -2/1 1/0 29/17 -2/1 12/7 -2/1 -1/1 1/0 31/18 -4/3 -1/1 19/11 0/1 26/15 -2/1 7/4 -3/2 -1/1 16/9 -1/1 9/5 -4/5 20/11 -1/1 -3/4 -2/3 11/6 -1/1 -2/3 2/1 0/1 15/7 -6/1 13/6 -2/1 1/0 11/5 -2/1 31/14 -4/3 -1/1 20/9 -2/1 -1/1 1/0 29/13 -2/1 9/4 -3/2 -1/1 16/7 -1/1 7/3 -2/3 19/8 -1/2 0/1 31/13 0/1 43/18 -1/1 0/1 12/5 -1/1 -1/2 0/1 41/17 -2/3 29/12 -1/1 0/1 17/7 -4/5 5/2 -1/2 0/1 18/7 -2/5 49/19 -8/23 80/31 -1/3 31/12 -1/3 -4/13 13/5 0/1 8/3 0/1 19/7 0/1 30/11 2/3 11/4 0/1 1/1 14/5 0/1 31/11 0/1 48/17 -1/1 1/1 17/6 0/1 1/0 3/1 0/1 16/5 -1/1 1/1 13/4 0/1 1/0 23/7 -2/1 33/10 -2/1 -1/1 43/13 0/1 10/3 0/1 37/11 2/1 64/19 1/0 27/8 -2/1 1/0 17/5 -2/3 41/12 -1/3 -1/4 24/7 0/1 7/2 1/1 1/0 32/9 1/0 25/7 -4/1 18/5 -2/1 11/3 0/1 26/7 0/1 67/18 0/1 1/1 41/11 2/1 15/4 2/1 1/0 4/1 -1/1 0/1 1/0 17/4 2/1 1/0 64/15 1/0 47/11 -10/1 30/7 -4/1 13/3 -2/1 22/5 0/1 75/17 0/1 128/29 -1/2 1/0 53/12 -1/1 0/1 31/7 0/1 9/2 1/1 1/0 32/7 1/0 23/5 -6/1 14/3 -2/1 5/1 -2/1 16/3 -1/1 11/2 -1/1 -2/3 17/3 -2/5 23/4 -1/1 1/0 29/5 0/1 64/11 -1/2 1/0 35/6 -1/1 0/1 6/1 0/1 19/3 2/1 32/5 1/0 13/2 -2/1 1/0 7/1 0/1 8/1 -2/1 0/1 9/1 0/1 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(3,2,-2,-1) Matrix(65,544,46,385) -> Matrix(3,2,-2,-1) Matrix(31,224,22,159) -> Matrix(3,2,-2,-1) Matrix(33,224,-14,-95) -> Matrix(3,2,-2,-1) Matrix(63,352,-46,-257) -> Matrix(1,2,-2,-3) Matrix(65,352,12,65) -> Matrix(3,4,-4,-5) Matrix(31,160,6,31) -> Matrix(5,4,-4,-3) Matrix(33,160,20,97) -> Matrix(3,2,-2,-1) Matrix(97,448,-76,-351) -> Matrix(7,4,-16,-9) Matrix(95,416,-66,-289) -> Matrix(1,2,-2,-3) Matrix(31,128,-8,-33) -> Matrix(1,0,0,1) Matrix(127,480,-68,-257) -> Matrix(7,4,-2,-1) Matrix(95,352,-78,-289) -> Matrix(1,0,0,1) Matrix(97,352,62,225) -> Matrix(1,0,0,1) Matrix(161,576,-116,-415) -> Matrix(3,2,-8,-5) Matrix(65,224,56,193) -> Matrix(1,0,4,1) Matrix(159,544,140,479) -> Matrix(1,0,-4,1) Matrix(319,1088,56,191) -> Matrix(1,0,-2,1) Matrix(161,544,-66,-223) -> Matrix(3,2,-2,-1) Matrix(127,416,-98,-321) -> Matrix(1,0,0,1) Matrix(129,416,40,129) -> Matrix(1,0,2,1) Matrix(31,96,10,31) -> Matrix(1,0,2,1) Matrix(191,544,112,319) -> Matrix(3,2,-2,-1) Matrix(353,992,-216,-607) -> Matrix(1,0,0,1) Matrix(127,352,92,255) -> Matrix(1,0,2,1) Matrix(95,256,-36,-97) -> Matrix(1,0,6,1) Matrix(159,416,-86,-225) -> Matrix(5,-2,-2,1) Matrix(383,992,222,575) -> Matrix(1,0,-2,1) Matrix(385,992,-248,-639) -> Matrix(3,-4,-2,3) Matrix(63,160,50,127) -> Matrix(1,0,0,1) Matrix(353,864,-248,-607) -> Matrix(1,4,-2,-7) Matrix(225,544,146,353) -> Matrix(1,0,0,1) Matrix(385,928,-212,-511) -> Matrix(3,2,-2,-1) Matrix(415,992,-228,-545) -> Matrix(3,2,-2,-1) Matrix(417,992,256,609) -> Matrix(1,0,0,1) Matrix(257,608,-216,-511) -> Matrix(1,2,-2,-3) Matrix(97,224,42,97) -> Matrix(3,4,-4,-5) Matrix(127,288,56,127) -> Matrix(7,6,-6,-5) Matrix(417,928,-244,-543) -> Matrix(1,0,0,1) Matrix(447,992,-260,-577) -> Matrix(1,0,0,1) Matrix(161,352,-102,-223) -> Matrix(3,2,-2,-1) Matrix(287,544,-220,-417) -> Matrix(1,-2,-2,5) Matrix(929,1696,-648,-1183) -> Matrix(3,4,-4,-5) Matrix(513,928,-424,-767) -> Matrix(1,2,-4,-7) Matrix(161,288,90,161) -> Matrix(7,8,-8,-9) Matrix(127,224,72,127) -> Matrix(7,6,-6,-5) Matrix(129,224,-110,-191) -> Matrix(3,2,-8,-5) Matrix(575,992,222,383) -> Matrix(1,0,-2,1) Matrix(1729,2976,-1338,-2303) -> Matrix(3,2,-2,-1) Matrix(2305,3936,-1686,-2879) -> Matrix(1,0,0,1) Matrix(319,544,112,191) -> Matrix(3,2,-2,-1) Matrix(321,544,-226,-383) -> Matrix(1,0,0,1) Matrix(97,160,20,33) -> Matrix(3,2,-2,-1) Matrix(2273,3712,804,1313) -> Matrix(1,0,2,1) Matrix(2431,3968,862,1407) -> Matrix(1,0,2,1) Matrix(609,992,256,417) -> Matrix(1,0,0,1) Matrix(159,256,-100,-161) -> Matrix(1,0,2,1) Matrix(607,960,-466,-737) -> Matrix(1,0,-2,1) Matrix(225,352,62,97) -> Matrix(1,0,0,1) Matrix(2561,3968,992,1537) -> Matrix(11,12,-34,-37) Matrix(2399,3712,930,1439) -> Matrix(13,12,-38,-35) Matrix(353,544,146,225) -> Matrix(1,0,0,1) Matrix(65,96,44,65) -> Matrix(1,0,0,1) Matrix(287,416,198,287) -> Matrix(9,8,-8,-7) Matrix(223,320,200,287) -> Matrix(3,2,-2,-1) Matrix(1185,1696,-916,-1311) -> Matrix(3,2,-2,-1) Matrix(2879,4096,854,1215) -> Matrix(11,6,-2,-1) Matrix(2881,4096,856,1217) -> Matrix(5,2,2,1) Matrix(927,1312,248,351) -> Matrix(1,0,2,1) Matrix(385,544,46,65) -> Matrix(3,2,-2,-1) Matrix(159,224,22,31) -> Matrix(3,2,-2,-1) Matrix(735,1024,206,287) -> Matrix(11,6,-2,-1) Matrix(737,1024,208,289) -> Matrix(9,4,2,1) Matrix(255,352,92,127) -> Matrix(1,0,2,1) Matrix(959,1312,-796,-1089) -> Matrix(3,2,-8,-5) Matrix(961,1312,282,385) -> Matrix(3,2,-8,-5) Matrix(95,128,-72,-97) -> Matrix(1,0,0,1) Matrix(3135,4096,734,959) -> Matrix(27,14,-2,-1) Matrix(3137,4096,736,961) -> Matrix(21,10,2,1) Matrix(12671,16384,2870,3711) -> Matrix(1,0,0,1) Matrix(12673,16384,2872,3713) -> Matrix(1,0,0,1) Matrix(223,288,24,31) -> Matrix(3,2,-2,-1) Matrix(799,1024,174,223) -> Matrix(15,8,-2,-1) Matrix(801,1024,176,225) -> Matrix(13,6,2,1) Matrix(127,160,50,63) -> Matrix(1,0,0,1) Matrix(129,160,104,129) -> Matrix(1,0,2,1) Matrix(287,352,234,287) -> Matrix(1,0,2,1) Matrix(897,1088,634,769) -> Matrix(1,0,2,1) Matrix(3391,4096,582,703) -> Matrix(1,0,2,1) Matrix(3393,4096,584,705) -> Matrix(1,0,2,1) Matrix(863,1024,134,159) -> Matrix(7,4,-2,-1) Matrix(865,1024,136,161) -> Matrix(9,4,2,1) Matrix(193,224,56,65) -> Matrix(1,0,4,1) Matrix(479,544,140,159) -> Matrix(1,0,-4,1) Matrix(257,288,58,65) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(1,0,0,1) Matrix(191,-224,110,-129) -> Matrix(1,-2,0,1) Matrix(289,-352,78,-95) -> Matrix(1,0,0,1) Matrix(351,-448,76,-97) -> Matrix(1,-4,0,1) Matrix(321,-416,98,-127) -> Matrix(1,0,0,1) Matrix(97,-128,72,-95) -> Matrix(1,0,0,1) Matrix(353,-480,164,-223) -> Matrix(1,-2,0,1) Matrix(257,-352,46,-63) -> Matrix(1,2,-2,-3) Matrix(415,-576,116,-161) -> Matrix(1,-2,0,1) Matrix(383,-544,226,-321) -> Matrix(1,0,0,1) Matrix(289,-416,66,-95) -> Matrix(1,2,-2,-3) Matrix(639,-992,248,-385) -> Matrix(5,4,-14,-11) Matrix(161,-256,100,-159) -> Matrix(1,0,2,1) Matrix(257,-416,118,-191) -> Matrix(1,-2,0,1) Matrix(607,-992,216,-353) -> Matrix(1,0,0,1) Matrix(511,-864,152,-257) -> Matrix(1,2,0,1) Matrix(543,-928,244,-417) -> Matrix(1,0,0,1) Matrix(577,-992,260,-447) -> Matrix(1,0,0,1) Matrix(351,-608,56,-97) -> Matrix(1,2,0,1) Matrix(511,-928,212,-385) -> Matrix(3,2,-2,-1) Matrix(545,-992,228,-415) -> Matrix(3,2,-2,-1) Matrix(191,-352,70,-129) -> Matrix(3,2,4,3) Matrix(257,-544,60,-127) -> Matrix(1,-4,0,1) Matrix(767,-1696,232,-513) -> Matrix(1,2,-2,-3) Matrix(415,-928,72,-161) -> Matrix(1,2,-2,-3) Matrix(95,-224,14,-33) -> Matrix(3,2,-2,-1) Matrix(1247,-2976,282,-673) -> Matrix(1,0,0,1) Matrix(1631,-3936,438,-1057) -> Matrix(1,0,2,1) Matrix(223,-544,66,-161) -> Matrix(3,2,-2,-1) Matrix(97,-256,36,-95) -> Matrix(1,0,6,1) Matrix(353,-960,82,-223) -> Matrix(5,-2,-2,1) Matrix(511,-1696,116,-385) -> Matrix(1,0,0,1) Matrix(353,-1312,60,-223) -> Matrix(1,0,-2,1) 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: 32 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -4/1 -2/1 -48/31 -4/3 0/1 1/1 8/7 16/13 4/3 16/11 3/2 8/5 16/9 2/1 16/7 5/2 8/3 3/1 16/5 7/2 11/3 4/1 13/3 5/1 16/3 17/3 6/1 7/1 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 0/1 -11/2 -2/1 -1/1 -16/3 -1/1 -5/1 -2/3 -4/1 -1/1 -1/2 0/1 -15/4 -1/2 -2/5 -11/3 0/1 -7/2 -1/2 -1/3 -24/7 0/1 -17/5 -2/1 -10/3 0/1 -13/4 -1/2 0/1 -16/5 -1/1 -1/3 -3/1 0/1 -8/3 0/1 -13/5 0/1 -31/12 4/5 1/1 -18/7 2/1 -5/2 0/1 1/0 -7/3 -2/1 -16/7 -1/1 -9/4 -1/1 -3/4 -11/5 -2/3 -2/1 0/1 -9/5 -4/3 -16/9 -1/1 -7/4 -1/1 -3/4 -19/11 0/1 -31/18 -1/1 -4/5 -12/7 -1/1 -2/3 -1/2 -5/3 0/1 -8/5 0/1 -11/7 0/1 -14/9 -2/1 -31/20 -8/7 -1/1 -48/31 -1/1 -17/11 -4/5 -3/2 -1/1 0/1 -16/11 -1/1 -13/9 -4/5 -10/7 -2/3 -17/12 -2/3 -1/2 -41/29 -2/3 -24/17 -2/3 0/1 -7/5 -2/3 -11/8 -1/1 0/1 -26/19 -2/3 -15/11 -4/7 -4/3 -1/1 -1/2 0/1 -17/13 -4/7 -13/10 -1/2 0/1 -22/17 0/1 -9/7 -2/3 -14/11 -2/5 -5/4 -1/2 0/1 -16/13 -1/1 -1/3 -11/9 0/1 -17/14 -1/2 -2/5 -23/19 -2/5 -6/5 0/1 -13/11 -2/5 -7/6 -1/3 -1/4 -8/7 0/1 -1/1 0/1 0/1 -1/2 1/0 1/1 0/1 8/7 0/1 7/6 1/2 1/1 6/5 0/1 11/9 0/1 16/13 -1/1 1/1 5/4 0/1 1/0 9/7 -2/1 13/10 0/1 1/0 4/3 -1/1 0/1 1/0 7/5 -2/1 10/7 -2/1 13/9 -4/3 16/11 -1/1 3/2 -1/1 0/1 17/11 -4/3 14/9 -2/3 11/7 0/1 8/5 0/1 5/3 0/1 7/4 -3/2 -1/1 16/9 -1/1 9/5 -4/5 11/6 -1/1 -2/3 2/1 0/1 9/4 -3/2 -1/1 16/7 -1/1 7/3 -2/3 12/5 -1/1 -1/2 0/1 29/12 -1/1 0/1 17/7 -4/5 5/2 -1/2 0/1 18/7 -2/5 49/19 -8/23 80/31 -1/3 31/12 -1/3 -4/13 13/5 0/1 8/3 0/1 3/1 0/1 16/5 -1/1 1/1 13/4 0/1 1/0 23/7 -2/1 10/3 0/1 17/5 -2/3 24/7 0/1 7/2 1/1 1/0 18/5 -2/1 11/3 0/1 26/7 0/1 41/11 2/1 15/4 2/1 1/0 4/1 -1/1 0/1 1/0 17/4 2/1 1/0 13/3 -2/1 22/5 0/1 9/2 1/1 1/0 5/1 -2/1 16/3 -1/1 11/2 -1/1 -2/3 17/3 -2/5 6/1 0/1 7/1 0/1 8/1 -2/1 0/1 9/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(17,112,-12,-79) (-6/1,1/0) -> (-10/7,-17/12) 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(17,80,-10,-47) (-5/1,-4/1) -> (-12/7,-5/3) Hyperbolic Matrix(79,304,-46,-177) (-4/1,-15/4) -> (-31/18,-12/7) Hyperbolic Matrix(95,352,-78,-289) (-15/4,-11/3) -> (-11/9,-17/14) Hyperbolic Matrix(49,176,-22,-79) (-11/3,-7/2) -> (-9/4,-11/5) Hyperbolic Matrix(65,224,56,193) (-7/2,-24/7) -> (8/7,7/6) Hyperbolic Matrix(239,816,70,239) (-24/7,-17/5) -> (17/5,24/7) Hyperbolic Matrix(175,592,-128,-433) (-17/5,-10/3) -> (-26/19,-15/11) 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(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(79,208,30,79) (-8/3,-13/5) -> (13/5,8/3) Hyperbolic Matrix(241,624,56,145) (-13/5,-31/12) -> (17/4,13/3) Hyperbolic Matrix(385,992,-248,-639) (-31/12,-18/7) -> (-14/9,-31/20) Hyperbolic Matrix(81,208,44,113) (-18/7,-5/2) -> (11/6,2/1) Hyperbolic Matrix(47,112,-34,-81) (-5/2,-7/3) -> (-7/5,-11/8) 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(81,176,52,113) (-11/5,-2/1) -> (14/9,11/7) Hyperbolic Matrix(79,144,-62,-113) (-2/1,-9/5) -> (-9/7,-14/11) 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(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(111,176,70,111) (-8/5,-11/7) -> (11/7,8/5) 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(145,208,-122,-175) (-13/9,-10/7) -> (-6/5,-13/11) 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(95,128,-72,-97) (-15/11,-4/3) -> (-4/3,-17/13) Parabolic Matrix(209,272,136,177) (-17/13,-13/10) -> (3/2,17/11) Hyperbolic Matrix(433,560,-358,-463) (-22/17,-9/7) -> (-23/19,-6/5) 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(145,176,14,17) (-17/14,-23/19) -> (9/1,1/0) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(15,16,14,15) (-8/7,-1/1) -> (1/1,8/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(177,-208,40,-47) (7/6,6/5) -> (22/5,9/2) Hyperbolic Matrix(289,-352,78,-95) (6/5,11/9) -> (11/3,26/7) Hyperbolic Matrix(113,-144,62,-79) (5/4,9/7) -> (9/5,11/6) Hyperbolic Matrix(321,-416,98,-127) (9/7,13/10) -> (13/4,23/7) Hyperbolic Matrix(207,-272,86,-113) (13/10,4/3) -> (12/5,29/12) Hyperbolic Matrix(81,-112,34,-47) (4/3,7/5) -> (7/3,12/5) Hyperbolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) 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(47,-80,10,-17) (5/3,7/4) -> (9/2,5/1) Hyperbolic Matrix(79,-176,22,-49) (2/1,9/4) -> (7/2,18/5) Hyperbolic Matrix(111,-272,20,-49) (17/7,5/2) -> (11/2,17/3) Hyperbolic Matrix(305,-1008,82,-271) (23/7,10/3) -> (26/7,41/11) Hyperbolic Matrix(81,-272,14,-47) (10/3,17/5) -> (17/3,6/1) Hyperbolic Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(17,112,-12,-79) -> Matrix(3,2,-5,-3) Matrix(63,352,-46,-257) -> Matrix(1,2,-2,-3) Matrix(65,352,12,65) -> Matrix(3,4,-4,-5) Matrix(31,160,6,31) -> Matrix(5,4,-4,-3) Matrix(17,80,-10,-47) -> Matrix(3,2,-5,-3) Matrix(79,304,-46,-177) -> Matrix(3,2,-5,-3) Matrix(95,352,-78,-289) -> Matrix(1,0,0,1) Matrix(49,176,-22,-79) -> Matrix(3,2,-5,-3) Matrix(65,224,56,193) -> Matrix(1,0,4,1) Matrix(239,816,70,239) -> Matrix(1,0,-1,1) Matrix(175,592,-128,-433) -> Matrix(3,2,-5,-3) Matrix(127,416,-98,-321) -> Matrix(1,0,0,1) Matrix(129,416,40,129) -> Matrix(1,0,2,1) Matrix(31,96,10,31) -> Matrix(1,0,2,1) Matrix(17,48,6,17) -> Matrix(1,0,1,1) Matrix(79,208,30,79) -> Matrix(1,0,-5,1) Matrix(241,624,56,145) -> Matrix(3,-2,-1,1) Matrix(385,992,-248,-639) -> Matrix(3,-4,-2,3) Matrix(81,208,44,113) -> Matrix(1,-2,-1,3) Matrix(47,112,-34,-81) -> Matrix(1,0,-1,1) Matrix(97,224,42,97) -> Matrix(3,4,-4,-5) Matrix(127,288,56,127) -> Matrix(7,6,-6,-5) Matrix(81,176,52,113) -> Matrix(3,2,-5,-3) Matrix(79,144,-62,-113) -> Matrix(1,2,-3,-5) Matrix(161,288,90,161) -> Matrix(7,8,-8,-9) Matrix(127,224,72,127) -> Matrix(7,6,-6,-5) Matrix(129,224,-110,-191) -> Matrix(3,2,-8,-5) Matrix(575,992,222,383) -> Matrix(1,0,-2,1) Matrix(49,80,30,49) -> Matrix(1,0,1,1) Matrix(111,176,70,111) -> Matrix(1,0,-1,1) Matrix(225,352,62,97) -> Matrix(1,0,0,1) Matrix(2561,3968,992,1537) -> Matrix(11,12,-34,-37) Matrix(2399,3712,930,1439) -> Matrix(13,12,-38,-35) Matrix(353,544,146,225) -> Matrix(1,0,0,1) Matrix(65,96,44,65) -> Matrix(1,0,0,1) Matrix(287,416,198,287) -> Matrix(9,8,-8,-7) Matrix(145,208,-122,-175) -> Matrix(3,2,-5,-3) Matrix(927,1312,248,351) -> Matrix(1,0,2,1) Matrix(385,544,46,65) -> Matrix(3,2,-2,-1) Matrix(159,224,22,31) -> Matrix(3,2,-2,-1) Matrix(95,128,-72,-97) -> Matrix(1,0,0,1) Matrix(209,272,136,177) -> Matrix(1,0,1,1) Matrix(433,560,-358,-463) -> Matrix(1,0,-1,1) Matrix(127,160,50,63) -> Matrix(1,0,0,1) Matrix(129,160,104,129) -> Matrix(1,0,2,1) Matrix(287,352,234,287) -> Matrix(1,0,2,1) Matrix(145,176,14,17) -> Matrix(5,2,-3,-1) Matrix(193,224,56,65) -> Matrix(1,0,4,1) Matrix(15,16,14,15) -> Matrix(1,0,1,1) Matrix(1,0,2,1) -> Matrix(1,0,0,1) Matrix(177,-208,40,-47) -> Matrix(1,0,-1,1) Matrix(289,-352,78,-95) -> Matrix(1,0,0,1) Matrix(113,-144,62,-79) -> Matrix(1,-2,-1,3) Matrix(321,-416,98,-127) -> Matrix(1,0,0,1) Matrix(207,-272,86,-113) -> Matrix(1,0,-1,1) Matrix(81,-112,34,-47) -> Matrix(1,0,-1,1) Matrix(79,-112,12,-17) -> Matrix(1,2,-1,-1) Matrix(289,-416,66,-95) -> Matrix(1,2,-2,-3) Matrix(639,-992,248,-385) -> Matrix(5,4,-14,-11) Matrix(47,-80,10,-17) -> Matrix(1,2,-1,-1) Matrix(79,-176,22,-49) -> Matrix(1,2,-1,-1) Matrix(111,-272,20,-49) -> Matrix(3,2,-5,-3) Matrix(305,-1008,82,-271) -> Matrix(1,0,1,1) Matrix(81,-272,14,-47) -> Matrix(1,0,-1,1) 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): 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 (-1/1,0/1) 0 1 1/1 0/1 1 16 8/7 0/1 4 2 7/6 (1/2,1/1) 0 16 6/5 0/1 1 8 11/9 0/1 1 16 16/13 (0/1,1/0) 0 1 5/4 (0/1,1/0) 0 16 9/7 -2/1 1 16 13/10 (0/1,1/0) 0 16 4/3 0 4 7/5 -2/1 1 16 10/7 -2/1 1 8 13/9 -4/3 1 16 16/11 -1/1 4 1 3/2 (-1/1,0/1) 0 16 17/11 -4/3 1 16 14/9 -2/3 1 8 11/7 0/1 1 16 8/5 0/1 1 2 5/3 0/1 1 16 7/4 (-3/2,-1/1) 0 16 16/9 -1/1 7 1 9/5 -4/5 1 16 11/6 (-1/1,-2/3) 0 16 2/1 0/1 1 8 9/4 (-3/2,-1/1) 0 16 16/7 -1/1 5 1 7/3 -2/3 1 16 12/5 0 4 29/12 (-1/1,0/1) 0 16 17/7 -4/5 1 16 5/2 (-1/2,0/1) 0 16 18/7 -2/5 1 8 49/19 -8/23 1 16 80/31 -1/3 12 1 31/12 (-1/3,-4/13) 0 16 13/5 0/1 1 16 8/3 0/1 3 2 3/1 0/1 1 16 16/5 (0/1,1/0) 0 1 13/4 (0/1,1/0) 0 16 23/7 -2/1 1 16 10/3 0/1 1 8 17/5 -2/3 1 16 24/7 0/1 4 2 7/2 (1/1,1/0) 0 16 18/5 -2/1 1 8 11/3 0/1 1 16 26/7 0/1 1 8 41/11 2/1 1 16 15/4 (2/1,1/0) 0 16 4/1 0 4 17/4 (2/1,1/0) 0 16 13/3 -2/1 1 16 22/5 0/1 1 8 9/2 (1/1,1/0) 0 16 5/1 -2/1 1 16 16/3 -1/1 4 1 11/2 (-1/1,-2/3) 0 16 17/3 -2/5 1 16 6/1 0/1 1 8 7/1 0/1 1 16 8/1 (-2/1,0/1) 0 2 9/1 0/1 1 16 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(15,-16,14,-15) (1/1,8/7) -> (1/1,8/7) Reflection Matrix(193,-224,56,-65) (8/7,7/6) -> (24/7,7/2) Glide Reflection Matrix(177,-208,40,-47) (7/6,6/5) -> (22/5,9/2) Hyperbolic Matrix(289,-352,78,-95) (6/5,11/9) -> (11/3,26/7) Hyperbolic Matrix(287,-352,234,-287) (11/9,16/13) -> (11/9,16/13) Reflection Matrix(129,-160,104,-129) (16/13,5/4) -> (16/13,5/4) Reflection Matrix(113,-144,62,-79) (5/4,9/7) -> (9/5,11/6) Hyperbolic Matrix(321,-416,98,-127) (9/7,13/10) -> (13/4,23/7) Hyperbolic Matrix(207,-272,86,-113) (13/10,4/3) -> (12/5,29/12) Hyperbolic Matrix(81,-112,34,-47) (4/3,7/5) -> (7/3,12/5) Hyperbolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(289,-416,66,-95) (10/7,13/9) -> (13/3,22/5) Hyperbolic Matrix(287,-416,198,-287) (13/9,16/11) -> (13/9,16/11) Reflection Matrix(65,-96,44,-65) (16/11,3/2) -> (16/11,3/2) Reflection Matrix(353,-544,146,-225) (3/2,17/11) -> (29/12,17/7) Glide Reflection Matrix(639,-992,248,-385) (17/11,14/9) -> (18/7,49/19) Hyperbolic Matrix(225,-352,62,-97) (14/9,11/7) -> (18/5,11/3) Glide Reflection Matrix(111,-176,70,-111) (11/7,8/5) -> (11/7,8/5) Reflection Matrix(49,-80,30,-49) (8/5,5/3) -> (8/5,5/3) Reflection Matrix(47,-80,10,-17) (5/3,7/4) -> (9/2,5/1) Hyperbolic Matrix(127,-224,72,-127) (7/4,16/9) -> (7/4,16/9) Reflection Matrix(161,-288,90,-161) (16/9,9/5) -> (16/9,9/5) Reflection Matrix(113,-208,44,-81) (11/6,2/1) -> (5/2,18/7) Glide Reflection Matrix(79,-176,22,-49) (2/1,9/4) -> (7/2,18/5) Hyperbolic 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(111,-272,20,-49) (17/7,5/2) -> (11/2,17/3) Hyperbolic Matrix(3039,-7840,1178,-3039) (49/19,80/31) -> (49/19,80/31) Reflection Matrix(1921,-4960,744,-1921) (80/31,31/12) -> (80/31,31/12) Reflection Matrix(241,-624,56,-145) (31/12,13/5) -> (17/4,13/3) Glide Reflection Matrix(79,-208,30,-79) (13/5,8/3) -> (13/5,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) 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(305,-1008,82,-271) (23/7,10/3) -> (26/7,41/11) Hyperbolic Matrix(81,-272,14,-47) (10/3,17/5) -> (17/3,6/1) Hyperbolic Matrix(239,-816,70,-239) (17/5,24/7) -> (17/5,24/7) Reflection Matrix(47,-176,4,-15) (41/11,15/4) -> (9/1,1/0) Glide Reflection Matrix(33,-128,8,-31) (15/4,4/1) -> (4/1,17/4) Parabolic 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(15,-112,2,-15) (7/1,8/1) -> (7/1,8/1) Reflection Matrix(17,-144,2,-17) (8/1,9/1) -> (8/1,9/1) 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(-1,0,2,1) (0/1,1/1) -> (-1/1,0/1) Matrix(15,-16,14,-15) -> Matrix(-1,0,1,1) (1/1,8/7) -> (-2/1,0/1) Matrix(193,-224,56,-65) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(177,-208,40,-47) -> Matrix(1,0,-1,1) 0/1 Matrix(289,-352,78,-95) -> Matrix(1,0,0,1) Matrix(287,-352,234,-287) -> Matrix(1,0,0,-1) (11/9,16/13) -> (0/1,1/0) Matrix(129,-160,104,-129) -> Matrix(1,0,0,-1) (16/13,5/4) -> (0/1,1/0) Matrix(113,-144,62,-79) -> Matrix(1,-2,-1,3) Matrix(321,-416,98,-127) -> Matrix(1,0,0,1) Matrix(207,-272,86,-113) -> Matrix(1,0,-1,1) 0/1 Matrix(81,-112,34,-47) -> Matrix(1,0,-1,1) 0/1 Matrix(79,-112,12,-17) -> Matrix(1,2,-1,-1) (-2/1,0/1).(-1/1,1/0) Matrix(289,-416,66,-95) -> Matrix(1,2,-2,-3) -1/1 Matrix(287,-416,198,-287) -> Matrix(7,8,-6,-7) (13/9,16/11) -> (-4/3,-1/1) Matrix(65,-96,44,-65) -> Matrix(-1,0,2,1) (16/11,3/2) -> (-1/1,0/1) Matrix(353,-544,146,-225) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(639,-992,248,-385) -> Matrix(5,4,-14,-11) Matrix(225,-352,62,-97) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(111,-176,70,-111) -> Matrix(-1,0,3,1) (11/7,8/5) -> (-2/3,0/1) Matrix(49,-80,30,-49) -> Matrix(-1,0,1,1) (8/5,5/3) -> (-2/1,0/1) Matrix(47,-80,10,-17) -> Matrix(1,2,-1,-1) (-2/1,0/1).(-1/1,1/0) Matrix(127,-224,72,-127) -> Matrix(5,6,-4,-5) (7/4,16/9) -> (-3/2,-1/1) Matrix(161,-288,90,-161) -> Matrix(9,8,-10,-9) (16/9,9/5) -> (-1/1,-4/5) Matrix(113,-208,44,-81) -> Matrix(3,2,-7,-5) Matrix(79,-176,22,-49) -> Matrix(1,2,-1,-1) (-2/1,0/1).(-1/1,1/0) Matrix(127,-288,56,-127) -> Matrix(5,6,-4,-5) (9/4,16/7) -> (-3/2,-1/1) Matrix(97,-224,42,-97) -> Matrix(5,4,-6,-5) (16/7,7/3) -> (-1/1,-2/3) Matrix(111,-272,20,-49) -> Matrix(3,2,-5,-3) (-1/1,-1/2).(-2/3,0/1) Matrix(3039,-7840,1178,-3039) -> Matrix(47,16,-138,-47) (49/19,80/31) -> (-8/23,-1/3) Matrix(1921,-4960,744,-1921) -> Matrix(25,8,-78,-25) (80/31,31/12) -> (-1/3,-4/13) Matrix(241,-624,56,-145) -> Matrix(7,2,-3,-1) Matrix(79,-208,30,-79) -> Matrix(-1,0,7,1) (13/5,8/3) -> (-2/7,0/1) Matrix(17,-48,6,-17) -> Matrix(-1,0,1,1) (8/3,3/1) -> (-2/1,0/1) Matrix(31,-96,10,-31) -> Matrix(1,0,0,-1) (3/1,16/5) -> (0/1,1/0) Matrix(129,-416,40,-129) -> Matrix(1,0,0,-1) (16/5,13/4) -> (0/1,1/0) Matrix(305,-1008,82,-271) -> Matrix(1,0,1,1) 0/1 Matrix(81,-272,14,-47) -> Matrix(1,0,-1,1) 0/1 Matrix(239,-816,70,-239) -> Matrix(-1,0,3,1) (17/5,24/7) -> (-2/3,0/1) Matrix(47,-176,4,-15) -> Matrix(1,-2,-1,1) Matrix(33,-128,8,-31) -> Matrix(1,0,0,1) Matrix(31,-160,6,-31) -> Matrix(3,4,-2,-3) (5/1,16/3) -> (-2/1,-1/1) Matrix(65,-352,12,-65) -> Matrix(5,4,-6,-5) (16/3,11/2) -> (-1/1,-2/3) Matrix(15,-112,2,-15) -> Matrix(-1,0,1,1) (7/1,8/1) -> (-2/1,0/1) Matrix(17,-144,2,-17) -> Matrix(-1,0,1,1) (8/1,9/1) -> (-2/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.