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 -7/8 -3/4 -5/8 -7/12 -1/2 -7/15 -5/12 -3/8 -17/48 -3/10 -19/64 -9/32 -1/4 -15/64 -29/128 -7/32 -3/14 -11/64 -1/6 -5/32 -2/13 -1/8 0/1 1/9 1/8 1/7 1/6 2/11 3/16 1/5 3/14 2/9 3/13 1/4 4/15 3/11 2/7 3/10 5/16 1/3 3/8 2/5 5/12 7/16 1/2 9/16 7/12 5/8 31/48 2/3 11/16 3/4 13/16 7/8 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -8/9 2/1 3/1 -7/8 1/0 -6/7 -4/1 -3/1 -5/6 1/0 -9/11 -3/2 -13/16 -1/1 -4/5 -1/1 0/1 -11/14 -1/2 -7/9 1/2 -10/13 1/1 4/3 -3/4 1/0 -14/19 -5/1 -14/3 -11/15 -15/4 -8/11 -3/1 -8/3 -13/18 -5/2 -5/7 -3/2 -17/24 1/0 -29/41 1/0 -12/17 -2/1 -1/1 -7/10 -3/2 -9/13 -5/4 -11/16 -1/1 -2/3 -1/1 0/1 -11/17 5/6 -9/14 3/2 -7/11 1/0 -5/8 1/0 -13/21 1/0 -8/13 -5/1 -4/1 -19/31 1/0 -11/18 1/0 -3/5 -5/2 -13/22 -7/4 -10/17 -2/1 -5/3 -17/29 1/0 -7/12 -3/2 -18/31 -6/5 -1/1 -11/19 1/0 -15/26 -3/2 -4/7 -4/3 -1/1 -9/16 -1/1 -5/9 -3/4 -11/20 -1/2 -6/11 -1/1 0/1 -1/2 1/0 -7/15 -15/8 -6/13 -5/3 -8/5 -5/11 -3/2 -14/31 -6/5 -1/1 -9/20 -3/2 -13/29 -11/8 -4/9 -6/5 -1/1 -7/16 -1/1 -3/7 -3/4 -8/19 -2/3 -3/5 -13/31 -1/2 -18/43 -2/3 -3/5 -5/12 -1/2 -17/41 -5/12 -12/29 -2/5 -5/13 -7/17 -3/10 -2/5 0/1 1/1 -7/18 1/2 -19/49 9/10 -31/80 1/1 -12/31 1/1 6/5 -5/13 1/0 -3/8 1/0 -7/19 1/0 -11/30 -7/2 -4/11 -4/1 -3/1 -5/14 1/0 -11/31 1/0 -17/48 -3/1 -6/17 -3/1 -2/1 -1/3 -3/2 -5/16 -1/1 -4/13 -1/1 -8/9 -7/23 -3/4 -10/33 -1/1 -2/3 -13/43 -5/6 -3/10 -3/4 -11/37 -15/22 -19/64 -2/3 -8/27 -2/3 -11/17 -5/17 -7/12 -12/41 -7/13 -8/15 -7/24 -1/2 -2/7 -1/3 0/1 -9/32 0/1 -7/25 1/6 -5/18 1/2 -3/11 1/0 -7/26 -1/2 -18/67 0/1 1/3 -11/41 -1/2 -4/15 -1/3 0/1 -1/4 1/0 -4/17 -7/3 -2/1 -15/64 -2/1 -11/47 -23/12 -7/30 -7/4 -3/13 -3/2 -5/22 1/0 -17/75 -1/2 -29/128 0/1 -12/53 0/1 1/1 -7/31 1/0 -2/9 -3/1 -2/1 -7/32 -2/1 -5/23 -11/6 -3/14 -3/2 -1/5 -5/4 -3/16 -1/1 -2/11 -1/1 -8/9 -3/17 -11/14 -4/23 -8/11 -5/7 -5/29 -11/16 -11/64 -2/3 -6/35 -2/3 -11/17 -1/6 -1/2 -3/19 -1/4 -5/32 0/1 -2/13 0/1 1/1 -1/7 -1/2 -1/8 1/0 -1/9 1/0 0/1 -1/1 0/1 1/9 -1/2 1/8 -1/2 1/7 1/0 1/6 1/0 2/11 -8/7 -1/1 3/16 -1/1 1/5 -5/6 3/14 -3/4 2/9 -2/3 -3/5 3/13 -3/4 1/4 -1/2 5/19 -1/4 4/15 0/1 1/1 3/11 -1/2 5/18 -1/4 2/7 0/1 1/1 7/24 1/0 12/41 -8/1 -7/1 5/17 -7/2 3/10 -3/2 4/13 -8/7 -1/1 5/16 -1/1 1/3 -3/4 6/17 -2/3 -3/5 5/14 -1/2 4/11 -3/5 -4/7 3/8 -1/2 8/21 -3/7 -2/5 5/13 -1/2 12/31 -6/17 -1/3 7/18 -1/4 2/5 -1/3 0/1 9/22 1/4 7/17 3/4 12/29 5/3 2/1 5/12 1/0 13/31 1/0 8/19 -3/1 -2/1 11/26 -3/2 3/7 -3/2 7/16 -1/1 4/9 -1/1 -6/7 9/20 -3/4 5/11 -3/4 1/2 -1/2 8/15 0/1 1/5 7/13 1/0 6/11 -1/1 0/1 17/31 1/0 11/20 1/0 16/29 -7/3 -2/1 5/9 -3/2 9/16 -1/1 4/7 -1/1 -4/5 11/19 -1/2 18/31 -1/1 -6/7 25/43 -3/4 7/12 -3/4 24/41 -5/7 -12/17 17/29 -1/2 10/17 -5/7 -2/3 3/5 -5/8 11/18 -1/2 30/49 -4/7 -5/9 49/80 -5/9 19/31 -1/2 8/13 -4/7 -5/9 5/8 -1/2 12/19 -6/13 -5/11 19/30 -7/16 7/11 -1/2 9/14 -3/8 20/31 -10/29 -1/3 31/48 -1/3 11/17 -5/16 2/3 -1/1 0/1 11/16 -1/1 9/13 -5/6 16/23 -4/5 -7/9 23/33 -3/4 30/43 -5/7 -2/3 7/10 -3/4 26/37 -17/25 -2/3 45/64 -2/3 19/27 -9/14 12/17 -1/1 -2/3 29/41 -1/2 17/24 -1/2 5/7 -3/4 23/32 -2/3 18/25 -2/3 -15/23 13/18 -5/8 8/11 -8/13 -3/5 19/26 -13/22 49/67 -47/80 30/41 -17/29 -24/41 11/15 -15/26 3/4 -1/2 13/17 -9/22 49/64 -2/5 36/47 -2/5 -21/53 23/30 -7/18 10/13 -4/11 -1/3 17/22 -1/4 58/75 -1/5 0/1 99/128 0/1 41/53 -1/2 24/31 -1/3 -2/7 7/9 -1/4 25/32 0/1 18/23 0/1 1/3 11/14 1/0 4/5 -1/1 0/1 13/16 -1/1 9/11 -3/4 14/17 -2/3 -3/5 19/23 -1/2 24/29 -9/13 -2/3 53/64 -2/3 29/35 -13/20 5/6 -1/2 16/19 -5/7 -2/3 27/32 -2/3 11/13 -5/8 6/7 -3/5 -4/7 7/8 -1/2 8/9 -3/7 -2/5 1/1 -1/2 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(223,200,320,287) (-1/1,-8/9) -> (16/23,23/33) Hyperbolic Matrix(159,140,544,479) (-8/9,-7/8) -> (7/24,12/41) Hyperbolic Matrix(65,56,224,193) (-7/8,-6/7) -> (2/7,7/24) Hyperbolic Matrix(129,110,-224,-191) (-6/7,-5/6) -> (-15/26,-4/7) Hyperbolic Matrix(95,78,-352,-289) (-5/6,-9/11) -> (-3/11,-7/26) Hyperbolic Matrix(287,234,352,287) (-9/11,-13/16) -> (13/16,9/11) Hyperbolic Matrix(129,104,160,129) (-13/16,-4/5) -> (4/5,13/16) Hyperbolic Matrix(63,50,160,127) (-4/5,-11/14) -> (7/18,2/5) Hyperbolic Matrix(97,76,-448,-351) (-11/14,-7/9) -> (-5/23,-3/14) Hyperbolic Matrix(127,98,-416,-321) (-7/9,-10/13) -> (-4/13,-7/23) Hyperbolic Matrix(95,72,-128,-97) (-10/13,-3/4) -> (-3/4,-14/19) Parabolic Matrix(223,164,-480,-353) (-14/19,-11/15) -> (-7/15,-6/13) Hyperbolic Matrix(63,46,-352,-257) (-11/15,-8/11) -> (-2/11,-3/17) Hyperbolic Matrix(127,92,352,255) (-8/11,-13/18) -> (5/14,4/11) Hyperbolic Matrix(161,116,-576,-415) (-13/18,-5/7) -> (-7/25,-5/18) Hyperbolic Matrix(31,22,224,159) (-5/7,-17/24) -> (1/8,1/7) Hyperbolic Matrix(65,46,544,385) (-17/24,-29/41) -> (1/9,1/8) Hyperbolic Matrix(897,634,1088,769) (-29/41,-12/17) -> (14/17,19/23) Hyperbolic Matrix(321,226,-544,-383) (-12/17,-7/10) -> (-13/22,-10/17) Hyperbolic Matrix(95,66,-416,-289) (-7/10,-9/13) -> (-3/13,-5/22) Hyperbolic Matrix(287,198,416,287) (-9/13,-11/16) -> (11/16,9/13) Hyperbolic Matrix(65,44,96,65) (-11/16,-2/3) -> (2/3,11/16) Hyperbolic Matrix(225,146,544,353) (-2/3,-11/17) -> (7/17,12/29) Hyperbolic Matrix(385,248,-992,-639) (-11/17,-9/14) -> (-7/18,-19/49) Hyperbolic Matrix(97,62,352,225) (-9/14,-7/11) -> (3/11,5/18) Hyperbolic Matrix(159,100,-256,-161) (-7/11,-5/8) -> (-5/8,-13/21) Parabolic Matrix(191,118,-416,-257) (-13/21,-8/13) -> (-6/13,-5/11) Hyperbolic Matrix(417,256,992,609) (-8/13,-19/31) -> (13/31,8/19) Hyperbolic Matrix(353,216,-992,-607) (-19/31,-11/18) -> (-5/14,-11/31) Hyperbolic Matrix(33,20,160,97) (-11/18,-3/5) -> (1/5,3/14) Hyperbolic Matrix(257,152,-864,-511) (-3/5,-13/22) -> (-3/10,-11/37) Hyperbolic Matrix(191,112,544,319) (-10/17,-17/29) -> (1/3,6/17) Hyperbolic Matrix(417,244,-928,-543) (-17/29,-7/12) -> (-9/20,-13/29) Hyperbolic Matrix(447,260,-992,-577) (-7/12,-18/31) -> (-14/31,-9/20) Hyperbolic Matrix(383,222,992,575) (-18/31,-11/19) -> (5/13,12/31) Hyperbolic Matrix(97,56,-608,-351) (-11/19,-15/26) -> (-1/6,-3/19) Hyperbolic Matrix(127,72,224,127) (-4/7,-9/16) -> (9/16,4/7) Hyperbolic Matrix(161,90,288,161) (-9/16,-5/9) -> (5/9,9/16) Hyperbolic Matrix(385,212,-928,-511) (-5/9,-11/20) -> (-5/12,-17/41) Hyperbolic Matrix(415,228,-992,-545) (-11/20,-6/11) -> (-18/43,-5/12) Hyperbolic Matrix(129,70,-352,-191) (-6/11,-1/2) -> (-11/30,-4/11) Hyperbolic Matrix(127,60,-544,-257) (-1/2,-7/15) -> (-11/47,-7/30) Hyperbolic Matrix(513,232,-1696,-767) (-5/11,-14/31) -> (-10/33,-13/43) Hyperbolic Matrix(161,72,-928,-415) (-13/29,-4/9) -> (-4/23,-5/29) Hyperbolic Matrix(127,56,288,127) (-4/9,-7/16) -> (7/16,4/9) Hyperbolic Matrix(97,42,224,97) (-7/16,-3/7) -> (3/7,7/16) Hyperbolic Matrix(33,14,-224,-95) (-3/7,-8/19) -> (-2/13,-1/7) Hyperbolic Matrix(609,256,992,417) (-8/19,-13/31) -> (19/31,8/13) Hyperbolic Matrix(673,282,-2976,-1247) (-13/31,-18/43) -> (-12/53,-7/31) Hyperbolic Matrix(1057,438,-3936,-1631) (-17/41,-12/29) -> (-18/67,-11/41) Hyperbolic Matrix(353,146,544,225) (-12/29,-7/17) -> (11/17,2/3) Hyperbolic Matrix(161,66,-544,-223) (-7/17,-2/5) -> (-8/27,-5/17) Hyperbolic Matrix(127,50,160,63) (-2/5,-7/18) -> (11/14,4/5) Hyperbolic Matrix(2399,930,3712,1439) (-19/49,-31/80) -> (31/48,11/17) Hyperbolic Matrix(2561,992,3968,1537) (-31/80,-12/31) -> (20/31,31/48) Hyperbolic Matrix(575,222,992,383) (-12/31,-5/13) -> (11/19,18/31) Hyperbolic Matrix(95,36,-256,-97) (-5/13,-3/8) -> (-3/8,-7/19) Parabolic Matrix(223,82,-960,-353) (-7/19,-11/30) -> (-7/30,-3/13) Hyperbolic Matrix(255,92,352,127) (-4/11,-5/14) -> (13/18,8/11) Hyperbolic Matrix(2431,862,3968,1407) (-11/31,-17/48) -> (49/80,19/31) Hyperbolic Matrix(2273,804,3712,1313) (-17/48,-6/17) -> (30/49,49/80) Hyperbolic Matrix(319,112,544,191) (-6/17,-1/3) -> (17/29,10/17) Hyperbolic Matrix(31,10,96,31) (-1/3,-5/16) -> (5/16,1/3) Hyperbolic Matrix(129,40,416,129) (-5/16,-4/13) -> (4/13,5/16) Hyperbolic Matrix(33,10,320,97) (-7/23,-10/33) -> (0/1,1/9) Hyperbolic Matrix(385,116,-1696,-511) (-13/43,-3/10) -> (-5/22,-17/75) Hyperbolic Matrix(2881,856,4096,1217) (-11/37,-19/64) -> (45/64,19/27) Hyperbolic Matrix(2879,854,4096,1215) (-19/64,-8/27) -> (26/37,45/64) Hyperbolic Matrix(961,282,1312,385) (-5/17,-12/41) -> (30/41,11/15) Hyperbolic Matrix(479,140,544,159) (-12/41,-7/24) -> (7/8,8/9) Hyperbolic Matrix(193,56,224,65) (-7/24,-2/7) -> (6/7,7/8) Hyperbolic Matrix(737,208,1024,289) (-2/7,-9/32) -> (23/32,18/25) Hyperbolic Matrix(735,206,1024,287) (-9/32,-7/25) -> (5/7,23/32) Hyperbolic Matrix(225,62,352,97) (-5/18,-3/11) -> (7/11,9/14) Hyperbolic Matrix(223,60,-1312,-353) (-7/26,-18/67) -> (-6/35,-1/6) Hyperbolic Matrix(927,248,1312,351) (-11/41,-4/15) -> (12/17,29/41) Hyperbolic Matrix(31,8,-128,-33) (-4/15,-1/4) -> (-1/4,-4/17) Parabolic Matrix(3137,736,4096,961) (-4/17,-15/64) -> (49/64,36/47) Hyperbolic Matrix(3135,734,4096,959) (-15/64,-11/47) -> (13/17,49/64) Hyperbolic Matrix(12673,2872,16384,3713) (-17/75,-29/128) -> (99/128,41/53) Hyperbolic Matrix(12671,2870,16384,3711) (-29/128,-12/53) -> (58/75,99/128) Hyperbolic Matrix(257,58,288,65) (-7/31,-2/9) -> (8/9,1/1) Hyperbolic Matrix(801,176,1024,225) (-2/9,-7/32) -> (25/32,18/23) Hyperbolic Matrix(799,174,1024,223) (-7/32,-5/23) -> (7/9,25/32) Hyperbolic Matrix(97,20,160,33) (-3/14,-1/5) -> (3/5,11/18) Hyperbolic Matrix(31,6,160,31) (-1/5,-3/16) -> (3/16,1/5) Hyperbolic Matrix(65,12,352,65) (-3/16,-2/11) -> (2/11,3/16) Hyperbolic Matrix(319,56,1088,191) (-3/17,-4/23) -> (12/41,5/17) Hyperbolic Matrix(3393,584,4096,705) (-5/29,-11/64) -> (53/64,29/35) Hyperbolic Matrix(3391,582,4096,703) (-11/64,-6/35) -> (24/29,53/64) Hyperbolic Matrix(865,136,1024,161) (-3/19,-5/32) -> (27/32,11/13) Hyperbolic Matrix(863,134,1024,159) (-5/32,-2/13) -> (16/19,27/32) Hyperbolic Matrix(159,22,224,31) (-1/7,-1/8) -> (17/24,5/7) Hyperbolic Matrix(385,46,544,65) (-1/8,-1/9) -> (29/41,17/24) Hyperbolic Matrix(223,24,288,31) (-1/9,0/1) -> (24/31,7/9) Hyperbolic Matrix(95,-14,224,-33) (1/7,1/6) -> (11/26,3/7) Hyperbolic Matrix(257,-46,352,-63) (1/6,2/11) -> (8/11,19/26) Hyperbolic Matrix(351,-76,448,-97) (3/14,2/9) -> (18/23,11/14) Hyperbolic Matrix(289,-66,416,-95) (2/9,3/13) -> (9/13,16/23) Hyperbolic Matrix(33,-8,128,-31) (3/13,1/4) -> (1/4,5/19) Parabolic Matrix(257,-68,480,-127) (5/19,4/15) -> (8/15,7/13) Hyperbolic Matrix(289,-78,352,-95) (4/15,3/11) -> (9/11,14/17) Hyperbolic Matrix(415,-116,576,-161) (5/18,2/7) -> (18/25,13/18) Hyperbolic Matrix(223,-66,544,-161) (5/17,3/10) -> (9/22,7/17) Hyperbolic Matrix(321,-98,416,-127) (3/10,4/13) -> (10/13,17/22) Hyperbolic Matrix(607,-216,992,-353) (6/17,5/14) -> (11/18,30/49) Hyperbolic Matrix(97,-36,256,-95) (4/11,3/8) -> (3/8,8/21) Parabolic Matrix(225,-86,416,-159) (8/21,5/13) -> (7/13,6/11) Hyperbolic Matrix(639,-248,992,-385) (12/31,7/18) -> (9/14,20/31) Hyperbolic Matrix(607,-248,864,-353) (2/5,9/22) -> (7/10,26/37) Hyperbolic Matrix(511,-212,928,-385) (12/29,5/12) -> (11/20,16/29) Hyperbolic Matrix(545,-228,992,-415) (5/12,13/31) -> (17/31,11/20) Hyperbolic Matrix(511,-216,608,-257) (8/19,11/26) -> (5/6,16/19) Hyperbolic Matrix(543,-244,928,-417) (4/9,9/20) -> (7/12,24/41) Hyperbolic Matrix(577,-260,992,-447) (9/20,5/11) -> (25/43,7/12) Hyperbolic Matrix(223,-102,352,-161) (5/11,1/2) -> (19/30,7/11) Hyperbolic Matrix(417,-220,544,-287) (1/2,8/15) -> (36/47,23/30) Hyperbolic Matrix(1183,-648,1696,-929) (6/11,17/31) -> (23/33,30/43) Hyperbolic Matrix(767,-424,928,-513) (16/29,5/9) -> (19/23,24/29) Hyperbolic Matrix(191,-110,224,-129) (4/7,11/19) -> (11/13,6/7) Hyperbolic Matrix(2303,-1338,2976,-1729) (18/31,25/43) -> (41/53,24/31) Hyperbolic Matrix(2879,-1686,3936,-2305) (24/41,17/29) -> (49/67,30/41) Hyperbolic Matrix(383,-226,544,-321) (10/17,3/5) -> (19/27,12/17) Hyperbolic Matrix(161,-100,256,-159) (8/13,5/8) -> (5/8,12/19) Parabolic Matrix(737,-466,960,-607) (12/19,19/30) -> (23/30,10/13) Hyperbolic Matrix(1311,-916,1696,-1185) (30/43,7/10) -> (17/22,58/75) Hyperbolic Matrix(1089,-796,1312,-959) (19/26,49/67) -> (29/35,5/6) Hyperbolic Matrix(97,-72,128,-95) (11/15,3/4) -> (3/4,13/17) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(223,200,320,287) -> Matrix(3,-2,-4,3) Matrix(159,140,544,479) -> Matrix(1,-10,0,1) Matrix(65,56,224,193) -> Matrix(1,4,0,1) Matrix(129,110,-224,-191) -> Matrix(3,8,-2,-5) Matrix(95,78,-352,-289) -> Matrix(1,2,-2,-3) Matrix(287,234,352,287) -> Matrix(5,6,-6,-7) Matrix(129,104,160,129) -> Matrix(1,0,0,1) Matrix(63,50,160,127) -> Matrix(1,0,-2,1) Matrix(97,76,-448,-351) -> Matrix(7,2,-4,-1) Matrix(127,98,-416,-321) -> Matrix(5,-4,-6,5) Matrix(95,72,-128,-97) -> Matrix(1,-6,0,1) Matrix(223,164,-480,-353) -> Matrix(7,30,-4,-17) Matrix(63,46,-352,-257) -> Matrix(5,16,-6,-19) Matrix(127,92,352,255) -> Matrix(5,12,-8,-19) Matrix(161,116,-576,-415) -> Matrix(1,2,4,9) Matrix(31,22,224,159) -> Matrix(1,2,-2,-3) Matrix(65,46,544,385) -> Matrix(1,4,-2,-7) Matrix(897,634,1088,769) -> Matrix(1,4,-2,-7) Matrix(321,226,-544,-383) -> Matrix(3,8,-2,-5) Matrix(95,66,-416,-289) -> Matrix(1,2,-2,-3) Matrix(287,198,416,287) -> Matrix(9,10,-10,-11) Matrix(65,44,96,65) -> Matrix(1,0,0,1) Matrix(225,146,544,353) -> Matrix(3,-2,2,-1) Matrix(385,248,-992,-639) -> Matrix(3,-4,4,-5) Matrix(97,62,352,225) -> Matrix(1,-2,-2,5) Matrix(159,100,-256,-161) -> Matrix(1,-10,0,1) Matrix(191,118,-416,-257) -> Matrix(3,20,-2,-13) Matrix(417,256,992,609) -> Matrix(1,2,0,1) Matrix(353,216,-992,-607) -> Matrix(1,2,0,1) Matrix(33,20,160,97) -> Matrix(3,10,-4,-13) Matrix(257,152,-864,-511) -> Matrix(11,20,-16,-29) Matrix(191,112,544,319) -> Matrix(3,4,-4,-5) Matrix(417,244,-928,-543) -> Matrix(11,18,-8,-13) Matrix(447,260,-992,-577) -> Matrix(1,0,0,1) Matrix(383,222,992,575) -> Matrix(1,0,-2,1) Matrix(97,56,-608,-351) -> Matrix(1,2,-4,-7) Matrix(127,72,224,127) -> Matrix(7,8,-8,-9) Matrix(161,90,288,161) -> Matrix(7,6,-6,-5) Matrix(385,212,-928,-511) -> Matrix(7,4,-16,-9) Matrix(415,228,-992,-545) -> Matrix(5,2,-8,-3) Matrix(129,70,-352,-191) -> Matrix(7,4,-2,-1) Matrix(127,60,-544,-257) -> Matrix(7,16,-4,-9) Matrix(513,232,-1696,-767) -> Matrix(7,8,-8,-9) Matrix(161,72,-928,-415) -> Matrix(17,22,-24,-31) Matrix(127,56,288,127) -> Matrix(11,12,-12,-13) Matrix(97,42,224,97) -> Matrix(7,6,-6,-5) Matrix(33,14,-224,-95) -> Matrix(3,2,-2,-1) Matrix(609,256,992,417) -> Matrix(5,2,-8,-3) Matrix(673,282,-2976,-1247) -> Matrix(3,2,-2,-1) Matrix(1057,438,-3936,-1631) -> Matrix(5,2,2,1) Matrix(353,146,544,225) -> Matrix(5,2,-18,-7) Matrix(161,66,-544,-223) -> Matrix(9,2,-14,-3) Matrix(127,50,160,63) -> Matrix(1,0,-2,1) Matrix(2399,930,3712,1439) -> Matrix(15,-14,-46,43) Matrix(2561,992,3968,1537) -> Matrix(15,-16,-44,47) Matrix(575,222,992,383) -> Matrix(1,0,-2,1) Matrix(95,36,-256,-97) -> Matrix(1,-6,0,1) Matrix(223,82,-960,-353) -> Matrix(3,14,-2,-9) Matrix(255,92,352,127) -> Matrix(5,12,-8,-19) Matrix(2431,862,3968,1407) -> Matrix(1,8,-2,-15) Matrix(2273,804,3712,1313) -> Matrix(9,22,-16,-39) Matrix(319,112,544,191) -> Matrix(3,4,-4,-5) Matrix(31,10,96,31) -> Matrix(5,6,-6,-7) Matrix(129,40,416,129) -> Matrix(17,16,-16,-15) Matrix(33,10,320,97) -> Matrix(3,2,-2,-1) Matrix(385,116,-1696,-511) -> Matrix(5,4,-4,-3) Matrix(2881,856,4096,1217) -> Matrix(71,48,-108,-73) Matrix(2879,854,4096,1215) -> Matrix(85,56,-126,-83) Matrix(961,282,1312,385) -> Matrix(57,32,-98,-55) Matrix(479,140,544,159) -> Matrix(19,10,-40,-21) Matrix(193,56,224,65) -> Matrix(9,4,-16,-7) Matrix(737,208,1024,289) -> Matrix(21,2,-32,-3) Matrix(735,206,1024,287) -> Matrix(15,-2,-22,3) Matrix(225,62,352,97) -> Matrix(1,-2,-2,5) Matrix(223,60,-1312,-353) -> Matrix(5,2,-8,-3) Matrix(927,248,1312,351) -> Matrix(5,2,-8,-3) Matrix(31,8,-128,-33) -> Matrix(1,-2,0,1) Matrix(3137,736,4096,961) -> Matrix(27,56,-68,-141) Matrix(3135,734,4096,959) -> Matrix(33,64,-82,-159) Matrix(12673,2872,16384,3713) -> Matrix(1,0,0,1) Matrix(12671,2870,16384,3711) -> Matrix(1,0,-6,1) Matrix(257,58,288,65) -> Matrix(1,0,-2,1) Matrix(801,176,1024,225) -> Matrix(1,2,4,9) Matrix(799,174,1024,223) -> Matrix(1,2,-10,-19) Matrix(97,20,160,33) -> Matrix(7,10,-12,-17) Matrix(31,6,160,31) -> Matrix(9,10,-10,-11) Matrix(65,12,352,65) -> Matrix(17,16,-16,-15) Matrix(319,56,1088,191) -> Matrix(21,16,-4,-3) Matrix(3393,584,4096,705) -> Matrix(71,48,-108,-73) Matrix(3391,582,4096,703) -> Matrix(61,40,-90,-59) Matrix(865,136,1024,161) -> Matrix(13,2,-20,-3) Matrix(863,134,1024,159) -> Matrix(7,-2,-10,3) Matrix(159,22,224,31) -> Matrix(1,2,-2,-3) Matrix(385,46,544,65) -> Matrix(1,4,-2,-7) Matrix(223,24,288,31) -> Matrix(1,2,-4,-7) Matrix(95,-14,224,-33) -> Matrix(3,2,-2,-1) Matrix(257,-46,352,-63) -> Matrix(13,16,-22,-27) Matrix(351,-76,448,-97) -> Matrix(3,2,4,3) Matrix(289,-66,416,-95) -> Matrix(1,2,-2,-3) Matrix(33,-8,128,-31) -> Matrix(3,2,-8,-5) Matrix(257,-68,480,-127) -> Matrix(1,0,4,1) Matrix(289,-78,352,-95) -> Matrix(1,2,-2,-3) Matrix(415,-116,576,-161) -> Matrix(13,2,-20,-3) Matrix(223,-66,544,-161) -> Matrix(1,2,2,5) Matrix(321,-98,416,-127) -> Matrix(3,4,-10,-13) Matrix(607,-216,992,-353) -> Matrix(5,2,-8,-3) Matrix(97,-36,256,-95) -> Matrix(11,6,-24,-13) Matrix(225,-86,416,-159) -> Matrix(5,2,2,1) Matrix(639,-248,992,-385) -> Matrix(13,4,-36,-11) Matrix(607,-248,864,-353) -> Matrix(11,-2,-16,3) Matrix(511,-212,928,-385) -> Matrix(1,-4,0,1) Matrix(545,-228,992,-415) -> Matrix(1,2,0,1) Matrix(511,-216,608,-257) -> Matrix(3,4,-4,-5) Matrix(543,-244,928,-417) -> Matrix(23,18,-32,-25) Matrix(577,-260,992,-447) -> Matrix(1,0,0,1) Matrix(223,-102,352,-161) -> Matrix(13,10,-30,-23) Matrix(417,-220,544,-287) -> Matrix(11,2,-28,-5) Matrix(1183,-648,1696,-929) -> Matrix(3,-2,-4,3) Matrix(767,-424,928,-513) -> Matrix(3,4,-4,-5) Matrix(191,-110,224,-129) -> Matrix(11,8,-18,-13) Matrix(2303,-1338,2976,-1729) -> Matrix(5,4,-14,-11) Matrix(2879,-1686,3936,-2305) -> Matrix(121,84,-206,-143) Matrix(383,-226,544,-321) -> Matrix(11,8,-18,-13) Matrix(161,-100,256,-159) -> Matrix(19,10,-40,-21) Matrix(737,-466,960,-607) -> Matrix(31,14,-82,-37) Matrix(1311,-916,1696,-1185) -> Matrix(3,2,-8,-5) Matrix(1089,-796,1312,-959) -> Matrix(61,36,-100,-59) Matrix(97,-72,128,-95) -> Matrix(11,6,-24,-13) 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): 64 Degree of the the map X: 64 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 -3/4 -1/2 -1/4 -3/13 -1/6 -1/7 -1/8 -1/9 0/1 1/6 3/16 1/5 1/4 3/11 2/7 5/16 1/3 3/8 2/5 7/16 1/2 9/16 5/8 31/48 2/3 11/16 11/15 3/4 13/16 7/8 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -7/8 1/0 -6/7 -4/1 -3/1 -5/6 1/0 -9/11 -3/2 -13/16 -1/1 -4/5 -1/1 0/1 -7/9 1/2 -10/13 1/1 4/3 -3/4 1/0 -5/7 -3/2 -7/10 -3/2 -9/13 -5/4 -11/16 -1/1 -2/3 -1/1 0/1 -11/17 5/6 -9/14 3/2 -7/11 1/0 -5/8 1/0 -3/5 -5/2 -4/7 -4/3 -1/1 -9/16 -1/1 -5/9 -3/4 -6/11 -1/1 0/1 -1/2 1/0 -4/9 -6/5 -1/1 -7/16 -1/1 -3/7 -3/4 -5/12 -1/2 -12/29 -2/5 -5/13 -7/17 -3/10 -2/5 0/1 1/1 -7/18 1/2 -19/49 9/10 -31/80 1/1 -12/31 1/1 6/5 -5/13 1/0 -3/8 1/0 -1/3 -3/2 -5/16 -1/1 -4/13 -1/1 -8/9 -7/23 -3/4 -3/10 -3/4 -5/17 -7/12 -7/24 -1/2 -2/7 -1/3 0/1 -5/18 1/2 -3/11 1/0 -7/26 -1/2 -11/41 -1/2 -4/15 -1/3 0/1 -1/4 1/0 -4/17 -7/3 -2/1 -3/13 -3/2 -5/22 1/0 -2/9 -3/1 -2/1 -1/5 -5/4 -3/16 -1/1 -2/11 -1/1 -8/9 -3/17 -11/14 -1/6 -1/2 -1/7 -1/2 -1/8 1/0 -1/9 1/0 0/1 -1/1 0/1 1/6 1/0 2/11 -8/7 -1/1 3/16 -1/1 1/5 -5/6 1/4 -1/2 4/15 0/1 1/1 3/11 -1/2 2/7 0/1 1/1 7/24 1/0 5/17 -7/2 3/10 -3/2 4/13 -8/7 -1/1 5/16 -1/1 1/3 -3/4 3/8 -1/2 5/13 -1/2 12/31 -6/17 -1/3 7/18 -1/4 2/5 -1/3 0/1 3/7 -3/2 7/16 -1/1 4/9 -1/1 -6/7 5/11 -3/4 1/2 -1/2 5/9 -3/2 9/16 -1/1 4/7 -1/1 -4/5 11/19 -1/2 18/31 -1/1 -6/7 7/12 -3/4 3/5 -5/8 5/8 -1/2 7/11 -1/2 9/14 -3/8 20/31 -10/29 -1/3 31/48 -1/3 11/17 -5/16 2/3 -1/1 0/1 11/16 -1/1 9/13 -5/6 7/10 -3/4 12/17 -1/1 -2/3 29/41 -1/2 17/24 -1/2 5/7 -3/4 8/11 -8/13 -3/5 19/26 -13/22 11/15 -15/26 3/4 -1/2 13/17 -9/22 10/13 -4/11 -1/3 17/22 -1/4 7/9 -1/4 11/14 1/0 4/5 -1/1 0/1 13/16 -1/1 9/11 -3/4 14/17 -2/3 -3/5 19/23 -1/2 5/6 -1/2 11/13 -5/8 6/7 -3/5 -4/7 7/8 -1/2 1/1 -1/2 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(15,14,16,15) (-1/1,-7/8) -> (7/8,1/1) Hyperbolic Matrix(65,56,224,193) (-7/8,-6/7) -> (2/7,7/24) Hyperbolic Matrix(47,40,-208,-177) (-6/7,-5/6) -> (-5/22,-2/9) Hyperbolic Matrix(95,78,-352,-289) (-5/6,-9/11) -> (-3/11,-7/26) Hyperbolic Matrix(287,234,352,287) (-9/11,-13/16) -> (13/16,9/11) Hyperbolic Matrix(129,104,160,129) (-13/16,-4/5) -> (4/5,13/16) Hyperbolic Matrix(79,62,-144,-113) (-4/5,-7/9) -> (-5/9,-6/11) Hyperbolic Matrix(127,98,-416,-321) (-7/9,-10/13) -> (-4/13,-7/23) Hyperbolic Matrix(113,86,-272,-207) (-10/13,-3/4) -> (-5/12,-12/29) Hyperbolic Matrix(47,34,-112,-81) (-3/4,-5/7) -> (-3/7,-5/12) Hyperbolic Matrix(17,12,-112,-79) (-5/7,-7/10) -> (-1/6,-1/7) Hyperbolic Matrix(95,66,-416,-289) (-7/10,-9/13) -> (-3/13,-5/22) Hyperbolic Matrix(287,198,416,287) (-9/13,-11/16) -> (11/16,9/13) Hyperbolic Matrix(65,44,96,65) (-11/16,-2/3) -> (2/3,11/16) Hyperbolic Matrix(209,136,272,177) (-2/3,-11/17) -> (13/17,10/13) Hyperbolic Matrix(385,248,-992,-639) (-11/17,-9/14) -> (-7/18,-19/49) Hyperbolic Matrix(81,52,176,113) (-9/14,-7/11) -> (5/11,1/2) Hyperbolic Matrix(111,70,176,111) (-7/11,-5/8) -> (5/8,7/11) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(17,10,-80,-47) (-3/5,-4/7) -> (-2/9,-1/5) Hyperbolic Matrix(127,72,224,127) (-4/7,-9/16) -> (9/16,4/7) Hyperbolic Matrix(161,90,288,161) (-9/16,-5/9) -> (5/9,9/16) Hyperbolic Matrix(81,44,208,113) (-6/11,-1/2) -> (7/18,2/5) Hyperbolic Matrix(49,22,-176,-79) (-1/2,-4/9) -> (-2/7,-5/18) Hyperbolic Matrix(127,56,288,127) (-4/9,-7/16) -> (7/16,4/9) Hyperbolic Matrix(97,42,224,97) (-7/16,-3/7) -> (3/7,7/16) Hyperbolic Matrix(353,146,544,225) (-12/29,-7/17) -> (11/17,2/3) Hyperbolic Matrix(49,20,-272,-111) (-7/17,-2/5) -> (-2/11,-3/17) Hyperbolic Matrix(127,50,160,63) (-2/5,-7/18) -> (11/14,4/5) Hyperbolic Matrix(2399,930,3712,1439) (-19/49,-31/80) -> (31/48,11/17) Hyperbolic Matrix(2561,992,3968,1537) (-31/80,-12/31) -> (20/31,31/48) Hyperbolic Matrix(575,222,992,383) (-12/31,-5/13) -> (11/19,18/31) Hyperbolic Matrix(79,30,208,79) (-5/13,-3/8) -> (3/8,5/13) Hyperbolic Matrix(17,6,48,17) (-3/8,-1/3) -> (1/3,3/8) Hyperbolic Matrix(31,10,96,31) (-1/3,-5/16) -> (5/16,1/3) Hyperbolic Matrix(129,40,416,129) (-5/16,-4/13) -> (4/13,5/16) Hyperbolic Matrix(271,82,-1008,-305) (-7/23,-3/10) -> (-7/26,-11/41) Hyperbolic Matrix(47,14,-272,-81) (-3/10,-5/17) -> (-3/17,-1/6) Hyperbolic Matrix(239,70,816,239) (-5/17,-7/24) -> (7/24,5/17) Hyperbolic Matrix(193,56,224,65) (-7/24,-2/7) -> (6/7,7/8) Hyperbolic Matrix(225,62,352,97) (-5/18,-3/11) -> (7/11,9/14) Hyperbolic Matrix(927,248,1312,351) (-11/41,-4/15) -> (12/17,29/41) Hyperbolic Matrix(31,8,-128,-33) (-4/15,-1/4) -> (-1/4,-4/17) Parabolic Matrix(241,56,624,145) (-4/17,-3/13) -> (5/13,12/31) Hyperbolic Matrix(31,6,160,31) (-1/5,-3/16) -> (3/16,1/5) Hyperbolic Matrix(65,12,352,65) (-3/16,-2/11) -> (2/11,3/16) Hyperbolic Matrix(159,22,224,31) (-1/7,-1/8) -> (17/24,5/7) Hyperbolic Matrix(385,46,544,65) (-1/8,-1/9) -> (29/41,17/24) Hyperbolic Matrix(145,14,176,17) (-1/9,0/1) -> (14/17,19/23) Hyperbolic Matrix(79,-12,112,-17) (0/1,1/6) -> (7/10,12/17) Hyperbolic Matrix(257,-46,352,-63) (1/6,2/11) -> (8/11,19/26) Hyperbolic Matrix(47,-10,80,-17) (1/5,1/4) -> (7/12,3/5) Hyperbolic Matrix(177,-46,304,-79) (1/4,4/15) -> (18/31,7/12) Hyperbolic Matrix(289,-78,352,-95) (4/15,3/11) -> (9/11,14/17) Hyperbolic Matrix(79,-22,176,-49) (3/11,2/7) -> (4/9,5/11) Hyperbolic Matrix(433,-128,592,-175) (5/17,3/10) -> (19/26,11/15) Hyperbolic Matrix(321,-98,416,-127) (3/10,4/13) -> (10/13,17/22) Hyperbolic Matrix(639,-248,992,-385) (12/31,7/18) -> (9/14,20/31) Hyperbolic Matrix(81,-34,112,-47) (2/5,3/7) -> (5/7,8/11) Hyperbolic Matrix(113,-62,144,-79) (1/2,5/9) -> (7/9,11/14) Hyperbolic Matrix(191,-110,224,-129) (4/7,11/19) -> (11/13,6/7) Hyperbolic Matrix(175,-122,208,-145) (9/13,7/10) -> (5/6,11/13) Hyperbolic Matrix(97,-72,128,-95) (11/15,3/4) -> (3/4,13/17) Parabolic Matrix(463,-358,560,-433) (17/22,7/9) -> (19/23,5/6) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(15,14,16,15) -> Matrix(1,1,-2,-1) Matrix(65,56,224,193) -> Matrix(1,4,0,1) Matrix(47,40,-208,-177) -> Matrix(1,1,0,1) Matrix(95,78,-352,-289) -> Matrix(1,2,-2,-3) Matrix(287,234,352,287) -> Matrix(5,6,-6,-7) Matrix(129,104,160,129) -> Matrix(1,0,0,1) Matrix(79,62,-144,-113) -> Matrix(1,1,-2,-1) Matrix(127,98,-416,-321) -> Matrix(5,-4,-6,5) Matrix(113,86,-272,-207) -> Matrix(1,-3,-2,7) Matrix(47,34,-112,-81) -> Matrix(1,3,-2,-5) Matrix(17,12,-112,-79) -> Matrix(1,1,0,1) Matrix(95,66,-416,-289) -> Matrix(1,2,-2,-3) Matrix(287,198,416,287) -> Matrix(9,10,-10,-11) Matrix(65,44,96,65) -> Matrix(1,0,0,1) Matrix(209,136,272,177) -> Matrix(3,-1,-8,3) Matrix(385,248,-992,-639) -> Matrix(3,-4,4,-5) Matrix(81,52,176,113) -> Matrix(3,-5,-4,7) Matrix(111,70,176,111) -> Matrix(1,-5,-2,11) Matrix(49,30,80,49) -> Matrix(1,5,-2,-9) Matrix(17,10,-80,-47) -> Matrix(3,5,-2,-3) Matrix(127,72,224,127) -> Matrix(7,8,-8,-9) Matrix(161,90,288,161) -> Matrix(7,6,-6,-5) Matrix(81,44,208,113) -> Matrix(1,1,-4,-3) Matrix(49,22,-176,-79) -> Matrix(1,1,2,3) Matrix(127,56,288,127) -> Matrix(11,12,-12,-13) Matrix(97,42,224,97) -> Matrix(7,6,-6,-5) Matrix(353,146,544,225) -> Matrix(5,2,-18,-7) Matrix(49,20,-272,-111) -> Matrix(7,1,-8,-1) Matrix(127,50,160,63) -> Matrix(1,0,-2,1) Matrix(2399,930,3712,1439) -> Matrix(15,-14,-46,43) Matrix(2561,992,3968,1537) -> Matrix(15,-16,-44,47) Matrix(575,222,992,383) -> Matrix(1,0,-2,1) Matrix(79,30,208,79) -> Matrix(1,-3,-2,7) Matrix(17,6,48,17) -> Matrix(1,3,-2,-5) Matrix(31,10,96,31) -> Matrix(5,6,-6,-7) Matrix(129,40,416,129) -> Matrix(17,16,-16,-15) Matrix(271,82,-1008,-305) -> Matrix(1,1,-6,-5) Matrix(47,14,-272,-81) -> Matrix(7,5,-10,-7) Matrix(239,70,816,239) -> Matrix(13,7,-2,-1) Matrix(193,56,224,65) -> Matrix(9,4,-16,-7) Matrix(225,62,352,97) -> Matrix(1,-2,-2,5) Matrix(927,248,1312,351) -> Matrix(5,2,-8,-3) Matrix(31,8,-128,-33) -> Matrix(1,-2,0,1) Matrix(241,56,624,145) -> Matrix(3,5,-8,-13) Matrix(31,6,160,31) -> Matrix(9,10,-10,-11) Matrix(65,12,352,65) -> Matrix(17,16,-16,-15) Matrix(159,22,224,31) -> Matrix(1,2,-2,-3) Matrix(385,46,544,65) -> Matrix(1,4,-2,-7) Matrix(145,14,176,17) -> Matrix(1,3,-2,-5) Matrix(79,-12,112,-17) -> Matrix(3,1,-4,-1) Matrix(257,-46,352,-63) -> Matrix(13,16,-22,-27) Matrix(47,-10,80,-17) -> Matrix(7,5,-10,-7) Matrix(177,-46,304,-79) -> Matrix(5,1,-6,-1) Matrix(289,-78,352,-95) -> Matrix(1,2,-2,-3) Matrix(79,-22,176,-49) -> Matrix(5,1,-6,-1) Matrix(433,-128,592,-175) -> Matrix(7,17,-12,-29) Matrix(321,-98,416,-127) -> Matrix(3,4,-10,-13) Matrix(639,-248,992,-385) -> Matrix(13,4,-36,-11) Matrix(81,-34,112,-47) -> Matrix(1,3,-2,-5) Matrix(113,-62,144,-79) -> Matrix(1,1,-2,-1) Matrix(191,-110,224,-129) -> Matrix(11,8,-18,-13) Matrix(175,-122,208,-145) -> Matrix(7,5,-10,-7) Matrix(97,-72,128,-95) -> Matrix(11,6,-24,-13) Matrix(463,-358,560,-433) -> Matrix(5,1,-6,-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): 64 ----------------------------------------------------------------------- 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 16 1/6 1/0 1 8 2/11 (-8/7,-1/1) 0 16 3/16 -1/1 13 1 1/5 -5/6 1 16 1/4 -1/2 1 4 4/15 (0/1,1/1) 0 16 3/11 -1/2 1 16 2/7 (0/1,1/1) 0 16 7/24 1/0 7 2 5/17 -7/2 1 16 3/10 -3/2 1 8 4/13 (-8/7,-1/1) 0 16 5/16 -1/1 11 1 1/3 -3/4 1 16 3/8 -1/2 3 2 5/13 -1/2 1 16 12/31 (-6/17,-1/3) 0 16 7/18 -1/4 1 8 2/5 (-1/3,0/1) 0 16 3/7 -3/2 1 16 7/16 -1/1 9 1 4/9 (-1/1,-6/7) 0 16 5/11 -3/4 1 16 1/2 -1/2 1 8 5/9 -3/2 1 16 9/16 -1/1 7 1 4/7 (-1/1,-4/5) 0 16 11/19 -1/2 1 16 18/31 (-1/1,-6/7) 0 16 7/12 -3/4 1 4 3/5 -5/8 1 16 5/8 -1/2 5 2 7/11 -1/2 1 16 9/14 -3/8 1 8 20/31 (-10/29,-1/3) 0 16 31/48 -1/3 15 1 11/17 -5/16 1 16 2/3 (-1/1,0/1) 0 16 11/16 -1/1 5 1 9/13 -5/6 1 16 7/10 -3/4 1 8 12/17 (-1/1,-2/3) 0 16 29/41 -1/2 1 16 17/24 -1/2 1 2 5/7 -3/4 1 16 8/11 (-8/13,-3/5) 0 16 19/26 -13/22 1 8 11/15 -15/26 1 16 3/4 -1/2 3 4 13/17 -9/22 1 16 10/13 (-4/11,-1/3) 0 16 17/22 -1/4 1 8 7/9 -1/4 1 16 11/14 1/0 1 8 4/5 (-1/1,0/1) 0 16 13/16 -1/1 3 1 9/11 -3/4 1 16 14/17 (-2/3,-3/5) 0 16 19/23 -1/2 1 16 5/6 -1/2 1 8 11/13 -5/8 1 16 6/7 (-3/5,-4/7) 0 16 7/8 -1/2 7 2 1/1 -1/2 1 16 1/0 0/1 1 1 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(79,-12,112,-17) (0/1,1/6) -> (7/10,12/17) Hyperbolic Matrix(257,-46,352,-63) (1/6,2/11) -> (8/11,19/26) Hyperbolic Matrix(65,-12,352,-65) (2/11,3/16) -> (2/11,3/16) Reflection Matrix(31,-6,160,-31) (3/16,1/5) -> (3/16,1/5) Reflection Matrix(47,-10,80,-17) (1/5,1/4) -> (7/12,3/5) Hyperbolic Matrix(177,-46,304,-79) (1/4,4/15) -> (18/31,7/12) Hyperbolic Matrix(289,-78,352,-95) (4/15,3/11) -> (9/11,14/17) Hyperbolic Matrix(79,-22,176,-49) (3/11,2/7) -> (4/9,5/11) Hyperbolic Matrix(193,-56,224,-65) (2/7,7/24) -> (6/7,7/8) Glide Reflection Matrix(239,-70,816,-239) (7/24,5/17) -> (7/24,5/17) Reflection Matrix(433,-128,592,-175) (5/17,3/10) -> (19/26,11/15) Hyperbolic Matrix(321,-98,416,-127) (3/10,4/13) -> (10/13,17/22) Hyperbolic Matrix(129,-40,416,-129) (4/13,5/16) -> (4/13,5/16) Reflection Matrix(31,-10,96,-31) (5/16,1/3) -> (5/16,1/3) Reflection Matrix(17,-6,48,-17) (1/3,3/8) -> (1/3,3/8) Reflection Matrix(79,-30,208,-79) (3/8,5/13) -> (3/8,5/13) Reflection Matrix(575,-222,992,-383) (5/13,12/31) -> (11/19,18/31) Glide Reflection Matrix(639,-248,992,-385) (12/31,7/18) -> (9/14,20/31) Hyperbolic Matrix(127,-50,160,-63) (7/18,2/5) -> (11/14,4/5) Glide Reflection Matrix(81,-34,112,-47) (2/5,3/7) -> (5/7,8/11) Hyperbolic Matrix(97,-42,224,-97) (3/7,7/16) -> (3/7,7/16) Reflection Matrix(127,-56,288,-127) (7/16,4/9) -> (7/16,4/9) Reflection Matrix(113,-52,176,-81) (5/11,1/2) -> (7/11,9/14) Glide Reflection Matrix(113,-62,144,-79) (1/2,5/9) -> (7/9,11/14) Hyperbolic Matrix(161,-90,288,-161) (5/9,9/16) -> (5/9,9/16) Reflection Matrix(127,-72,224,-127) (9/16,4/7) -> (9/16,4/7) Reflection Matrix(191,-110,224,-129) (4/7,11/19) -> (11/13,6/7) Hyperbolic Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(111,-70,176,-111) (5/8,7/11) -> (5/8,7/11) Reflection Matrix(1921,-1240,2976,-1921) (20/31,31/48) -> (20/31,31/48) Reflection Matrix(1055,-682,1632,-1055) (31/48,11/17) -> (31/48,11/17) Reflection Matrix(209,-136,272,-177) (11/17,2/3) -> (13/17,10/13) Glide Reflection Matrix(65,-44,96,-65) (2/3,11/16) -> (2/3,11/16) Reflection Matrix(287,-198,416,-287) (11/16,9/13) -> (11/16,9/13) Reflection Matrix(175,-122,208,-145) (9/13,7/10) -> (5/6,11/13) Hyperbolic Matrix(897,-634,1088,-769) (12/17,29/41) -> (14/17,19/23) Glide Reflection Matrix(1393,-986,1968,-1393) (29/41,17/24) -> (29/41,17/24) Reflection Matrix(239,-170,336,-239) (17/24,5/7) -> (17/24,5/7) Reflection Matrix(97,-72,128,-95) (11/15,3/4) -> (3/4,13/17) Parabolic Matrix(463,-358,560,-433) (17/22,7/9) -> (19/23,5/6) Hyperbolic Matrix(129,-104,160,-129) (4/5,13/16) -> (4/5,13/16) Reflection Matrix(287,-234,352,-287) (13/16,9/11) -> (13/16,9/11) Reflection Matrix(15,-14,16,-15) (7/8,1/1) -> (7/8,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(79,-12,112,-17) -> Matrix(3,1,-4,-1) -1/2 Matrix(257,-46,352,-63) -> Matrix(13,16,-22,-27) Matrix(65,-12,352,-65) -> Matrix(15,16,-14,-15) (2/11,3/16) -> (-8/7,-1/1) Matrix(31,-6,160,-31) -> Matrix(11,10,-12,-11) (3/16,1/5) -> (-1/1,-5/6) Matrix(47,-10,80,-17) -> Matrix(7,5,-10,-7) (-1/1,-2/3).(-3/4,-1/2) Matrix(177,-46,304,-79) -> Matrix(5,1,-6,-1) Matrix(289,-78,352,-95) -> Matrix(1,2,-2,-3) -1/1 Matrix(79,-22,176,-49) -> Matrix(5,1,-6,-1) Matrix(193,-56,224,-65) -> Matrix(1,-4,-2,7) Matrix(239,-70,816,-239) -> Matrix(1,7,0,-1) (7/24,5/17) -> (-7/2,1/0) Matrix(433,-128,592,-175) -> Matrix(7,17,-12,-29) Matrix(321,-98,416,-127) -> Matrix(3,4,-10,-13) Matrix(129,-40,416,-129) -> Matrix(15,16,-14,-15) (4/13,5/16) -> (-8/7,-1/1) Matrix(31,-10,96,-31) -> Matrix(7,6,-8,-7) (5/16,1/3) -> (-1/1,-3/4) Matrix(17,-6,48,-17) -> Matrix(5,3,-8,-5) (1/3,3/8) -> (-3/4,-1/2) Matrix(79,-30,208,-79) -> Matrix(7,3,-16,-7) (3/8,5/13) -> (-1/2,-3/8) Matrix(575,-222,992,-383) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(639,-248,992,-385) -> Matrix(13,4,-36,-11) -1/3 Matrix(127,-50,160,-63) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(81,-34,112,-47) -> Matrix(1,3,-2,-5) Matrix(97,-42,224,-97) -> Matrix(5,6,-4,-5) (3/7,7/16) -> (-3/2,-1/1) Matrix(127,-56,288,-127) -> Matrix(13,12,-14,-13) (7/16,4/9) -> (-1/1,-6/7) Matrix(113,-52,176,-81) -> Matrix(7,5,-18,-13) Matrix(113,-62,144,-79) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(161,-90,288,-161) -> Matrix(5,6,-4,-5) (5/9,9/16) -> (-3/2,-1/1) Matrix(127,-72,224,-127) -> Matrix(9,8,-10,-9) (9/16,4/7) -> (-1/1,-4/5) Matrix(191,-110,224,-129) -> Matrix(11,8,-18,-13) -2/3 Matrix(49,-30,80,-49) -> Matrix(9,5,-16,-9) (3/5,5/8) -> (-5/8,-1/2) Matrix(111,-70,176,-111) -> Matrix(11,5,-24,-11) (5/8,7/11) -> (-1/2,-5/12) Matrix(1921,-1240,2976,-1921) -> Matrix(59,20,-174,-59) (20/31,31/48) -> (-10/29,-1/3) Matrix(1055,-682,1632,-1055) -> Matrix(31,10,-96,-31) (31/48,11/17) -> (-1/3,-5/16) Matrix(209,-136,272,-177) -> Matrix(5,1,-14,-3) Matrix(65,-44,96,-65) -> Matrix(-1,0,2,1) (2/3,11/16) -> (-1/1,0/1) Matrix(287,-198,416,-287) -> Matrix(11,10,-12,-11) (11/16,9/13) -> (-1/1,-5/6) Matrix(175,-122,208,-145) -> Matrix(7,5,-10,-7) (-1/1,-2/3).(-3/4,-1/2) Matrix(897,-634,1088,-769) -> Matrix(7,4,-12,-7) *** -> (-2/3,-1/2) Matrix(1393,-986,1968,-1393) -> Matrix(9,5,-16,-9) (29/41,17/24) -> (-5/8,-1/2) Matrix(239,-170,336,-239) -> Matrix(5,3,-8,-5) (17/24,5/7) -> (-3/4,-1/2) Matrix(97,-72,128,-95) -> Matrix(11,6,-24,-13) -1/2 Matrix(463,-358,560,-433) -> Matrix(5,1,-6,-1) Matrix(129,-104,160,-129) -> Matrix(-1,0,2,1) (4/5,13/16) -> (-1/1,0/1) Matrix(287,-234,352,-287) -> Matrix(7,6,-8,-7) (13/16,9/11) -> (-1/1,-3/4) Matrix(15,-14,16,-15) -> Matrix(1,1,0,-1) (7/8,1/1) -> (-1/2,1/0) Matrix(-1,2,0,1) -> Matrix(-1,0,4,1) (1/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.