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 0/1 1/1 -8/9 0/1 1/1 -7/8 0/1 -6/7 1/2 1/1 -5/6 1/3 1/1 -9/11 2/3 1/1 -13/16 1/1 -4/5 1/1 1/0 -11/14 -1/1 -1/3 -7/9 0/1 1/1 -10/13 -1/1 0/1 -3/4 0/1 -14/19 0/1 1/3 -11/15 1/4 1/3 -8/11 1/3 2/5 -13/18 1/3 1/1 -5/7 1/2 1/1 -17/24 1/2 -29/41 2/3 1/1 -12/17 1/2 1/1 -7/10 3/5 1/1 -9/13 4/5 1/1 -11/16 1/1 -2/3 1/1 1/0 -11/17 -5/4 -1/1 -9/14 -1/1 -3/5 -7/11 -1/3 0/1 -5/8 0/1 -13/21 0/1 1/5 -8/13 0/1 1/3 -19/31 0/1 1/5 -11/18 1/5 1/3 -3/5 1/3 1/2 -13/22 3/5 1/1 -10/17 1/2 1/1 -17/29 1/3 1/2 -7/12 1/2 -18/31 10/17 3/5 -11/19 7/11 2/3 -15/26 5/7 1/1 -4/7 3/4 1/1 -9/16 1/1 -5/9 1/1 4/3 -11/20 2/1 -6/11 2/1 3/1 -1/2 -1/1 1/1 -7/15 -1/1 1/0 -6/13 -1/1 0/1 -5/11 0/1 1/1 -14/31 2/7 1/3 -9/20 1/2 -13/29 1/3 1/2 -4/9 2/3 1/1 -7/16 1/1 -3/7 1/1 3/2 -8/19 2/1 3/1 -13/31 1/1 2/1 -18/43 2/1 3/1 -5/12 2/1 -17/41 7/3 12/5 -12/29 5/2 13/5 -7/17 3/1 7/2 -2/5 1/1 1/0 -7/18 7/3 3/1 -19/49 23/8 3/1 -31/80 3/1 -12/31 3/1 16/5 -5/13 4/1 5/1 -3/8 1/0 -7/19 -4/1 -3/1 -11/30 -5/1 -3/1 -4/11 -2/1 -1/1 -5/14 -1/1 1/1 -11/31 0/1 1/1 -17/48 1/1 -6/17 1/1 1/0 -1/3 -1/1 1/0 -5/16 -1/1 -4/13 -1/1 0/1 -7/23 0/1 1/1 -10/33 -2/1 -1/1 -13/43 -1/3 0/1 -3/10 -1/1 1/1 -11/37 3/1 1/0 -19/64 1/0 -8/27 -3/1 1/0 -5/17 -1/1 -1/2 -12/41 -1/3 0/1 -7/24 0/1 -2/7 1/1 1/0 -9/32 1/0 -7/25 -3/1 1/0 -5/18 -3/1 -1/1 -3/11 0/1 1/1 -7/26 1/1 3/1 -18/67 -3/1 1/0 -11/41 1/1 2/1 -4/15 1/1 1/0 -1/4 1/0 -4/17 -3/1 1/0 -15/64 1/0 -11/47 -5/1 1/0 -7/30 -5/1 -3/1 -3/13 -4/1 -3/1 -5/22 -3/1 -7/3 -17/75 -11/5 -2/1 -29/128 -2/1 -12/53 -2/1 -1/1 -7/31 -3/1 -2/1 -2/9 -3/1 -2/1 -7/32 -2/1 -5/23 -2/1 -5/3 -3/14 -3/1 -1/1 -1/5 -3/2 -1/1 -3/16 -1/1 -2/11 -1/1 -4/5 -3/17 -1/1 -1/2 -4/23 -2/3 -3/5 -5/29 -5/9 -1/2 -11/64 -1/2 -6/35 -1/2 -5/11 -1/6 -1/1 -1/3 -3/19 -1/7 0/1 -5/32 0/1 -2/13 0/1 1/3 -1/7 1/1 1/0 -1/8 1/0 -1/9 -3/1 -2/1 0/1 -1/1 0/1 1/9 -2/3 -3/5 1/8 -1/2 1/7 -1/2 -1/3 1/6 -1/1 1/1 2/11 -4/3 -1/1 3/16 -1/1 1/5 -1/1 -3/4 3/14 -1/1 -3/5 2/9 -2/3 -3/5 3/13 -3/5 -4/7 1/4 -1/2 5/19 -1/3 0/1 4/15 -1/2 -1/3 3/11 -1/3 0/1 5/18 -1/1 -3/5 2/7 -1/2 -1/3 7/24 0/1 12/41 0/1 1/1 5/17 -1/1 1/0 3/10 -1/1 -1/3 4/13 -1/1 0/1 5/16 -1/1 1/3 -1/1 -1/2 6/17 -1/2 -1/3 5/14 -1/1 -1/3 4/11 -1/1 -2/3 3/8 -1/2 8/21 -7/15 -6/13 5/13 -5/11 -4/9 12/31 -16/37 -3/7 7/18 -3/7 -7/17 2/5 -1/2 -1/3 9/22 -5/11 -3/7 7/17 -7/16 -3/7 12/29 -13/31 -5/12 5/12 -2/5 13/31 -2/5 -1/3 8/19 -3/7 -2/5 11/26 -9/23 -5/13 3/7 -3/8 -1/3 7/16 -1/3 4/9 -1/3 -2/7 9/20 -1/4 5/11 -1/3 0/1 1/2 -1/1 -1/3 8/15 -1/2 -1/3 7/13 -1/3 0/1 6/11 -3/7 -2/5 17/31 -2/5 -1/3 11/20 -2/5 16/29 -11/29 -3/8 5/9 -4/11 -1/3 9/16 -1/3 4/7 -1/3 -3/10 11/19 -2/7 -7/25 18/31 -3/11 -10/37 25/43 -4/15 -5/19 7/12 -1/4 24/41 -1/3 -2/7 17/29 -1/4 -1/5 10/17 -1/3 -1/4 3/5 -1/4 -1/5 11/18 -1/5 -1/7 30/49 -1/6 -1/7 49/80 -1/7 19/31 -1/7 0/1 8/13 -1/5 0/1 5/8 0/1 12/19 0/1 1/3 19/30 1/3 1/1 7/11 0/1 1/1 9/14 -3/1 -1/1 20/31 -8/7 -1/1 31/48 -1/1 11/17 -1/1 -5/6 2/3 -1/2 -1/3 11/16 -1/3 9/13 -1/3 -4/13 16/23 -1/3 -2/7 23/33 -1/3 -2/7 30/43 -5/17 -2/7 7/10 -1/3 -3/11 26/37 -5/19 -1/4 45/64 -1/4 19/27 -1/4 -1/5 12/17 -1/3 -1/4 29/41 -1/3 -2/7 17/24 -1/4 5/7 -1/3 -1/4 23/32 -1/4 18/25 -1/4 -3/13 13/18 -1/3 -1/5 8/11 -2/9 -1/5 19/26 -1/5 -3/17 49/67 -1/6 -3/19 30/41 -1/7 0/1 11/15 -1/5 -1/6 3/4 0/1 13/17 -1/1 -1/2 49/64 -1/2 36/47 -1/2 -1/3 23/30 -1/1 -1/3 10/13 -1/1 0/1 17/22 -1/1 -1/3 58/75 -1/5 0/1 99/128 0/1 41/53 0/1 1/1 24/31 -1/1 -2/3 7/9 -1/3 0/1 25/32 0/1 18/23 0/1 1/1 11/14 -1/1 1/1 4/5 -1/2 -1/3 13/16 -1/3 9/11 -1/3 -2/7 14/17 -1/3 -1/4 19/23 -1/3 0/1 24/29 -5/19 -1/4 53/64 -1/4 29/35 -1/4 -5/21 5/6 -1/3 -1/5 16/19 -1/9 0/1 27/32 0/1 11/13 0/1 1/1 6/7 -1/3 -1/4 7/8 0/1 8/9 -1/3 0/1 1/1 -1/3 0/1 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,-4,1) Matrix(223,200,320,287) -> Matrix(3,-2,-10,7) Matrix(159,140,544,479) -> Matrix(1,0,0,1) Matrix(65,56,224,193) -> Matrix(1,0,-4,1) Matrix(129,110,-224,-191) -> Matrix(1,-2,2,-3) Matrix(95,78,-352,-289) -> Matrix(3,-2,2,-1) Matrix(287,234,352,287) -> Matrix(5,-4,-16,13) Matrix(129,104,160,129) -> Matrix(1,-2,-2,5) Matrix(63,50,160,127) -> Matrix(1,-2,-2,5) Matrix(97,76,-448,-351) -> Matrix(3,2,-2,-1) Matrix(127,98,-416,-321) -> Matrix(1,0,0,1) Matrix(95,72,-128,-97) -> Matrix(1,0,4,1) Matrix(223,164,-480,-353) -> Matrix(1,0,-4,1) Matrix(63,46,-352,-257) -> Matrix(7,-2,-10,3) Matrix(127,92,352,255) -> Matrix(1,0,-4,1) Matrix(161,116,-576,-415) -> Matrix(5,-2,-2,1) Matrix(31,22,224,159) -> Matrix(1,0,-4,1) Matrix(65,46,544,385) -> Matrix(7,-4,-12,7) Matrix(897,634,1088,769) -> Matrix(3,-2,-10,7) Matrix(321,226,-544,-383) -> Matrix(1,0,0,1) Matrix(95,66,-416,-289) -> Matrix(11,-8,-4,3) Matrix(287,198,416,287) -> Matrix(9,-8,-28,25) Matrix(65,44,96,65) -> Matrix(1,-2,-2,5) Matrix(225,146,544,353) -> Matrix(5,8,-12,-19) Matrix(385,248,-992,-639) -> Matrix(11,8,4,3) Matrix(97,62,352,225) -> Matrix(1,0,0,1) Matrix(159,100,-256,-161) -> Matrix(1,0,8,1) Matrix(191,118,-416,-257) -> Matrix(1,0,-4,1) Matrix(417,256,992,609) -> Matrix(9,-2,-22,5) Matrix(353,216,-992,-607) -> Matrix(1,0,-4,1) Matrix(33,20,160,97) -> Matrix(7,-2,-10,3) Matrix(257,152,-864,-511) -> Matrix(3,-2,2,-1) Matrix(191,112,544,319) -> Matrix(1,0,-4,1) Matrix(417,244,-928,-543) -> Matrix(1,0,0,1) Matrix(447,260,-992,-577) -> Matrix(7,-4,16,-9) Matrix(383,222,992,575) -> Matrix(29,-18,-66,41) Matrix(97,56,-608,-351) -> Matrix(3,-2,-10,7) Matrix(127,72,224,127) -> Matrix(7,-6,-22,19) Matrix(161,90,288,161) -> Matrix(7,-8,-20,23) Matrix(385,212,-928,-511) -> Matrix(9,-16,4,-7) Matrix(415,228,-992,-545) -> Matrix(1,0,0,1) Matrix(129,70,-352,-191) -> Matrix(1,-4,0,1) Matrix(127,60,-544,-257) -> Matrix(1,-4,0,1) Matrix(513,232,-1696,-767) -> Matrix(1,0,-4,1) Matrix(161,72,-928,-415) -> Matrix(7,-4,-12,7) Matrix(127,56,288,127) -> Matrix(5,-4,-16,13) Matrix(97,42,224,97) -> Matrix(5,-6,-14,17) Matrix(33,14,-224,-95) -> Matrix(1,-2,2,-3) Matrix(609,256,992,417) -> Matrix(1,-2,-6,13) Matrix(673,282,-2976,-1247) -> Matrix(1,-4,0,1) Matrix(1057,438,-3936,-1631) -> Matrix(1,-2,-2,5) Matrix(353,146,544,225) -> Matrix(3,-8,-4,11) Matrix(161,66,-544,-223) -> Matrix(1,-4,0,1) Matrix(127,50,160,63) -> Matrix(1,-2,-2,5) Matrix(2399,930,3712,1439) -> Matrix(13,-38,-14,41) Matrix(2561,992,3968,1537) -> Matrix(13,-40,-12,37) Matrix(575,222,992,383) -> Matrix(5,-18,-18,65) Matrix(95,36,-256,-97) -> Matrix(1,-8,0,1) Matrix(223,82,-960,-353) -> Matrix(1,0,0,1) Matrix(255,92,352,127) -> Matrix(1,0,-4,1) Matrix(2431,862,3968,1407) -> Matrix(1,0,-8,1) Matrix(2273,804,3712,1313) -> Matrix(1,-2,-6,13) Matrix(319,112,544,191) -> Matrix(1,0,-4,1) Matrix(31,10,96,31) -> Matrix(1,2,-2,-3) Matrix(129,40,416,129) -> Matrix(1,0,0,1) Matrix(33,10,320,97) -> Matrix(1,2,-2,-3) Matrix(385,116,-1696,-511) -> Matrix(5,-2,-2,1) Matrix(2881,856,4096,1217) -> Matrix(1,-4,-4,17) Matrix(2879,854,4096,1215) -> Matrix(1,8,-4,-31) Matrix(961,282,1312,385) -> Matrix(1,0,-4,1) Matrix(479,140,544,159) -> Matrix(1,0,0,1) Matrix(193,56,224,65) -> Matrix(1,0,-4,1) Matrix(737,208,1024,289) -> Matrix(1,-4,-4,17) Matrix(735,206,1024,287) -> Matrix(1,4,-4,-15) Matrix(225,62,352,97) -> Matrix(1,0,0,1) Matrix(223,60,-1312,-353) -> Matrix(1,-2,-2,5) Matrix(927,248,1312,351) -> Matrix(1,0,-4,1) Matrix(31,8,-128,-33) -> Matrix(1,-4,0,1) Matrix(3137,736,4096,961) -> Matrix(1,2,-2,-3) Matrix(3135,734,4096,959) -> Matrix(1,6,-2,-11) Matrix(12673,2872,16384,3713) -> Matrix(1,2,6,13) Matrix(12671,2870,16384,3711) -> Matrix(1,2,-6,-11) Matrix(257,58,288,65) -> Matrix(1,2,-2,-3) Matrix(801,176,1024,225) -> Matrix(1,2,2,5) Matrix(799,174,1024,223) -> Matrix(1,2,-6,-11) Matrix(97,20,160,33) -> Matrix(1,2,-6,-11) Matrix(31,6,160,31) -> Matrix(5,6,-6,-7) Matrix(65,12,352,65) -> Matrix(9,8,-8,-7) Matrix(319,56,1088,191) -> Matrix(3,2,-2,-1) Matrix(3393,584,4096,705) -> Matrix(19,10,-78,-41) Matrix(3391,582,4096,703) -> Matrix(21,10,-82,-39) Matrix(865,136,1024,161) -> Matrix(1,0,8,1) Matrix(863,134,1024,159) -> Matrix(1,0,-12,1) Matrix(159,22,224,31) -> Matrix(1,0,-4,1) Matrix(385,46,544,65) -> Matrix(1,4,-4,-15) Matrix(223,24,288,31) -> Matrix(1,2,-2,-3) Matrix(95,-14,224,-33) -> Matrix(7,2,-18,-5) Matrix(257,-46,352,-63) -> Matrix(1,2,-6,-11) Matrix(351,-76,448,-97) -> Matrix(3,2,-2,-1) Matrix(289,-66,416,-95) -> Matrix(13,8,-44,-27) Matrix(33,-8,128,-31) -> Matrix(7,4,-16,-9) Matrix(257,-68,480,-127) -> Matrix(1,0,0,1) Matrix(289,-78,352,-95) -> Matrix(5,2,-18,-7) Matrix(415,-116,576,-161) -> Matrix(3,2,-14,-9) Matrix(223,-66,544,-161) -> Matrix(7,4,-16,-9) Matrix(321,-98,416,-127) -> Matrix(1,0,0,1) Matrix(607,-216,992,-353) -> Matrix(1,0,-4,1) Matrix(97,-36,256,-95) -> Matrix(15,8,-32,-17) Matrix(225,-86,416,-159) -> Matrix(9,4,-16,-7) Matrix(639,-248,992,-385) -> Matrix(19,8,-12,-5) Matrix(607,-248,864,-353) -> Matrix(13,6,-50,-23) Matrix(511,-212,928,-385) -> Matrix(39,16,-100,-41) Matrix(545,-228,992,-415) -> Matrix(1,0,0,1) Matrix(511,-216,608,-257) -> Matrix(5,2,-38,-15) Matrix(543,-244,928,-417) -> Matrix(1,0,0,1) Matrix(577,-260,992,-447) -> Matrix(17,4,-64,-15) Matrix(223,-102,352,-161) -> Matrix(1,0,4,1) Matrix(417,-220,544,-287) -> Matrix(1,0,0,1) Matrix(1183,-648,1696,-929) -> Matrix(11,4,-36,-13) Matrix(767,-424,928,-513) -> Matrix(11,4,-36,-13) Matrix(191,-110,224,-129) -> Matrix(7,2,-18,-5) Matrix(2303,-1338,2976,-1729) -> Matrix(15,4,-4,-1) Matrix(2879,-1686,3936,-2305) -> Matrix(7,2,-46,-13) Matrix(383,-226,544,-321) -> Matrix(1,0,0,1) Matrix(161,-100,256,-159) -> Matrix(1,0,8,1) Matrix(737,-466,960,-607) -> Matrix(1,0,-4,1) Matrix(1311,-916,1696,-1185) -> Matrix(7,2,-18,-5) Matrix(1089,-796,1312,-959) -> Matrix(11,2,-50,-9) Matrix(97,-72,128,-95) -> Matrix(1,0,4,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 21 Degree of the the map X: 42 Degree of the the map Y: 128 ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 384 Minimal number of generators: 65 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 32 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/12 -3/8 -19/64 -1/4 -5/32 -1/8 0/1 1/8 1/7 1/6 3/16 1/5 3/14 1/4 2/7 3/10 5/16 1/3 3/8 2/5 5/12 7/16 1/2 9/16 49/80 31/48 2/3 11/16 23/32 13/16 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -7/8 0/1 -6/7 1/2 1/1 -5/6 1/3 1/1 -4/5 1/1 1/0 -3/4 0/1 -5/7 1/2 1/1 -17/24 1/2 -12/17 1/2 1/1 -7/10 3/5 1/1 -2/3 1/1 1/0 -5/8 0/1 -3/5 1/3 1/2 -13/22 3/5 1/1 -10/17 1/2 1/1 -7/12 1/2 -11/19 7/11 2/3 -15/26 5/7 1/1 -4/7 3/4 1/1 -1/2 -1/1 1/1 -3/7 1/1 3/2 -8/19 2/1 3/1 -5/12 2/1 -7/17 3/1 7/2 -2/5 1/1 1/0 -3/8 1/0 -1/3 -1/1 1/0 -3/10 -1/1 1/1 -11/37 3/1 1/0 -19/64 1/0 -8/27 -3/1 1/0 -5/17 -1/1 -1/2 -7/24 0/1 -2/7 1/1 1/0 -1/4 1/0 -1/5 -3/2 -1/1 -1/6 -1/1 -1/3 -3/19 -1/7 0/1 -5/32 0/1 -2/13 0/1 1/3 -1/7 1/1 1/0 -1/8 1/0 0/1 -1/1 0/1 1/8 -1/2 1/7 -1/2 -1/3 1/6 -1/1 1/1 2/11 -4/3 -1/1 3/16 -1/1 1/5 -1/1 -3/4 3/14 -1/1 -3/5 2/9 -2/3 -3/5 1/4 -1/2 3/11 -1/3 0/1 5/18 -1/1 -3/5 2/7 -1/2 -1/3 7/24 0/1 5/17 -1/1 1/0 3/10 -1/1 -1/3 4/13 -1/1 0/1 5/16 -1/1 1/3 -1/1 -1/2 6/17 -1/2 -1/3 5/14 -1/1 -1/3 4/11 -1/1 -2/3 3/8 -1/2 5/13 -5/11 -4/9 12/31 -16/37 -3/7 7/18 -3/7 -7/17 2/5 -1/2 -1/3 9/22 -5/11 -3/7 7/17 -7/16 -3/7 12/29 -13/31 -5/12 5/12 -2/5 13/31 -2/5 -1/3 8/19 -3/7 -2/5 11/26 -9/23 -5/13 3/7 -3/8 -1/3 7/16 -1/3 4/9 -1/3 -2/7 1/2 -1/1 -1/3 5/9 -4/11 -1/3 9/16 -1/3 4/7 -1/3 -3/10 11/19 -2/7 -7/25 18/31 -3/11 -10/37 7/12 -1/4 17/29 -1/4 -1/5 10/17 -1/3 -1/4 3/5 -1/4 -1/5 11/18 -1/5 -1/7 30/49 -1/6 -1/7 49/80 -1/7 19/31 -1/7 0/1 8/13 -1/5 0/1 5/8 0/1 7/11 0/1 1/1 9/14 -3/1 -1/1 20/31 -8/7 -1/1 31/48 -1/1 11/17 -1/1 -5/6 2/3 -1/2 -1/3 11/16 -1/3 9/13 -1/3 -4/13 7/10 -1/3 -3/11 26/37 -5/19 -1/4 45/64 -1/4 19/27 -1/4 -1/5 12/17 -1/3 -1/4 17/24 -1/4 5/7 -1/3 -1/4 23/32 -1/4 18/25 -1/4 -3/13 13/18 -1/3 -1/5 8/11 -2/9 -1/5 3/4 0/1 7/9 -1/3 0/1 25/32 0/1 18/23 0/1 1/1 11/14 -1/1 1/1 4/5 -1/2 -1/3 13/16 -1/3 9/11 -1/3 -2/7 5/6 -1/3 -1/5 16/19 -1/9 0/1 27/32 0/1 11/13 0/1 1/1 6/7 -1/3 -1/4 7/8 0/1 1/1 -1/3 0/1 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(113,101,-160,-143) (-1/1,-7/8) -> (-17/24,-12/17) 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(79,65,96,79) (-5/6,-4/5) -> (9/11,5/6) Hyperbolic Matrix(17,13,64,49) (-4/5,-3/4) -> (1/4,3/11) Hyperbolic Matrix(15,11,64,47) (-3/4,-5/7) -> (2/9,1/4) Hyperbolic Matrix(31,22,224,159) (-5/7,-17/24) -> (1/8,1/7) Hyperbolic Matrix(321,226,-544,-383) (-12/17,-7/10) -> (-13/22,-10/17) Hyperbolic Matrix(111,77,160,111) (-7/10,-2/3) -> (9/13,7/10) Hyperbolic Matrix(49,31,128,81) (-2/3,-5/8) -> (3/8,5/13) Hyperbolic Matrix(47,29,128,79) (-5/8,-3/5) -> (4/11,3/8) Hyperbolic Matrix(257,152,-864,-511) (-3/5,-13/22) -> (-3/10,-11/37) Hyperbolic Matrix(241,141,576,337) (-10/17,-7/12) -> (5/12,13/31) Hyperbolic Matrix(239,139,576,335) (-7/12,-11/19) -> (12/29,5/12) Hyperbolic Matrix(97,56,-608,-351) (-11/19,-15/26) -> (-1/6,-3/19) Hyperbolic Matrix(17,9,32,17) (-4/7,-1/2) -> (1/2,5/9) Hyperbolic Matrix(15,7,32,15) (-1/2,-3/7) -> (4/9,1/2) Hyperbolic Matrix(33,14,-224,-95) (-3/7,-8/19) -> (-2/13,-1/7) Hyperbolic Matrix(337,141,576,241) (-8/19,-5/12) -> (7/12,17/29) Hyperbolic Matrix(335,139,576,239) (-5/12,-7/17) -> (18/31,7/12) Hyperbolic Matrix(161,66,-544,-223) (-7/17,-2/5) -> (-8/27,-5/17) Hyperbolic Matrix(81,31,128,49) (-2/5,-3/8) -> (5/8,7/11) Hyperbolic Matrix(79,29,128,47) (-3/8,-1/3) -> (8/13,5/8) Hyperbolic Matrix(49,15,160,49) (-1/3,-3/10) -> (3/10,4/13) 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(17,5,-160,-47) (-5/17,-7/24) -> (-1/8,0/1) Hyperbolic Matrix(193,56,224,65) (-7/24,-2/7) -> (6/7,7/8) Hyperbolic Matrix(49,13,64,17) (-2/7,-1/4) -> (3/4,7/9) Hyperbolic Matrix(47,11,64,15) (-1/4,-1/5) -> (8/11,3/4) Hyperbolic Matrix(17,3,96,17) (-1/5,-1/6) -> (1/6,2/11) 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(47,-5,160,-17) (0/1,1/8) -> (7/24,5/17) Hyperbolic Matrix(95,-14,224,-33) (1/7,1/6) -> (11/26,3/7) Hyperbolic Matrix(49,-9,256,-47) (2/11,3/16) -> (3/16,1/5) Parabolic Matrix(81,-17,224,-47) (1/5,3/14) -> (5/14,4/11) Hyperbolic Matrix(351,-76,448,-97) (3/14,2/9) -> (18/23,11/14) Hyperbolic Matrix(113,-31,288,-79) (3/11,5/18) -> (7/18,2/5) 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(81,-25,256,-79) (4/13,5/16) -> (5/16,1/3) Parabolic Matrix(175,-61,416,-145) (1/3,6/17) -> (13/31,8/19) Hyperbolic Matrix(607,-216,992,-353) (6/17,5/14) -> (11/18,30/49) Hyperbolic Matrix(463,-179,1120,-433) (5/13,12/31) -> (7/17,12/29) 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,-216,608,-257) (8/19,11/26) -> (5/6,16/19) Hyperbolic Matrix(113,-49,256,-111) (3/7,7/16) -> (7/16,4/9) Parabolic Matrix(145,-81,256,-143) (5/9,9/16) -> (9/16,4/7) Parabolic Matrix(191,-110,224,-129) (4/7,11/19) -> (11/13,6/7) Hyperbolic Matrix(271,-157,416,-241) (11/19,18/31) -> (11/17,2/3) Hyperbolic Matrix(687,-403,1120,-657) (17/29,10/17) -> (19/31,8/13) Hyperbolic Matrix(383,-226,544,-321) (10/17,3/5) -> (19/27,12/17) Hyperbolic Matrix(209,-127,288,-175) (3/5,11/18) -> (13/18,8/11) Hyperbolic Matrix(3921,-2401,6400,-3919) (30/49,49/80) -> (49/80,19/31) Parabolic Matrix(177,-113,224,-143) (7/11,9/14) -> (11/14,4/5) Hyperbolic Matrix(1489,-961,2304,-1487) (20/31,31/48) -> (31/48,11/17) Parabolic Matrix(177,-121,256,-175) (2/3,11/16) -> (11/16,9/13) Parabolic Matrix(143,-101,160,-113) (12/17,17/24) -> (7/8,1/1) Hyperbolic Matrix(751,-539,960,-689) (5/7,23/32) -> (25/32,18/23) Hyperbolic Matrix(849,-611,1088,-783) (23/32,18/25) -> (7/9,25/32) Hyperbolic Matrix(209,-169,256,-207) (4/5,13/16) -> (13/16,9/11) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(113,101,-160,-143) -> Matrix(0,1,-1,2) Matrix(65,56,224,193) -> Matrix(1,0,-4,1) Matrix(129,110,-224,-191) -> Matrix(1,-2,2,-3) Matrix(79,65,96,79) -> Matrix(2,-1,-7,4) Matrix(17,13,64,49) -> Matrix(0,-1,1,2) Matrix(15,11,64,47) -> Matrix(4,-1,-7,2) Matrix(31,22,224,159) -> Matrix(1,0,-4,1) Matrix(321,226,-544,-383) -> Matrix(1,0,0,1) Matrix(111,77,160,111) -> Matrix(4,-3,-13,10) Matrix(49,31,128,81) -> Matrix(4,1,-9,-2) Matrix(47,29,128,79) -> Matrix(4,-1,-7,2) Matrix(257,152,-864,-511) -> Matrix(3,-2,2,-1) Matrix(241,141,576,337) -> Matrix(0,-1,1,2) Matrix(239,139,576,335) -> Matrix(16,-9,-39,22) Matrix(97,56,-608,-351) -> Matrix(3,-2,-10,7) Matrix(17,9,32,17) -> Matrix(0,-1,1,2) Matrix(15,7,32,15) -> Matrix(0,-1,1,2) Matrix(33,14,-224,-95) -> Matrix(1,-2,2,-3) Matrix(337,141,576,241) -> Matrix(0,-1,1,2) Matrix(335,139,576,239) -> Matrix(4,-9,-15,34) Matrix(161,66,-544,-223) -> Matrix(1,-4,0,1) Matrix(81,31,128,49) -> Matrix(0,1,-1,2) Matrix(79,29,128,47) -> Matrix(0,-1,1,6) Matrix(49,15,160,49) -> Matrix(0,-1,1,2) Matrix(2881,856,4096,1217) -> Matrix(1,-4,-4,17) Matrix(2879,854,4096,1215) -> Matrix(1,8,-4,-31) Matrix(17,5,-160,-47) -> Matrix(2,1,-1,0) Matrix(193,56,224,65) -> Matrix(1,0,-4,1) Matrix(49,13,64,17) -> Matrix(0,-1,1,2) Matrix(47,11,64,15) -> Matrix(0,-1,1,6) Matrix(17,3,96,17) -> Matrix(2,1,-1,0) Matrix(865,136,1024,161) -> Matrix(1,0,8,1) Matrix(863,134,1024,159) -> Matrix(1,0,-12,1) Matrix(159,22,224,31) -> Matrix(1,0,-4,1) Matrix(47,-5,160,-17) -> Matrix(2,1,-1,0) Matrix(95,-14,224,-33) -> Matrix(7,2,-18,-5) Matrix(49,-9,256,-47) -> Matrix(6,7,-7,-8) Matrix(81,-17,224,-47) -> Matrix(2,1,-1,0) Matrix(351,-76,448,-97) -> Matrix(3,2,-2,-1) Matrix(113,-31,288,-79) -> Matrix(4,1,-9,-2) Matrix(415,-116,576,-161) -> Matrix(3,2,-14,-9) Matrix(223,-66,544,-161) -> Matrix(7,4,-16,-9) Matrix(81,-25,256,-79) -> Matrix(0,-1,1,2) Matrix(175,-61,416,-145) -> Matrix(4,1,-9,-2) Matrix(607,-216,992,-353) -> Matrix(1,0,-4,1) Matrix(463,-179,1120,-433) -> Matrix(62,27,-147,-64) Matrix(639,-248,992,-385) -> Matrix(19,8,-12,-5) Matrix(607,-248,864,-353) -> Matrix(13,6,-50,-23) Matrix(511,-216,608,-257) -> Matrix(5,2,-38,-15) Matrix(113,-49,256,-111) -> Matrix(14,5,-45,-16) Matrix(145,-81,256,-143) -> Matrix(20,7,-63,-22) Matrix(191,-110,224,-129) -> Matrix(7,2,-18,-5) Matrix(271,-157,416,-241) -> Matrix(18,5,-29,-8) Matrix(687,-403,1120,-657) -> Matrix(4,1,-25,-6) Matrix(383,-226,544,-321) -> Matrix(1,0,0,1) Matrix(209,-127,288,-175) -> Matrix(6,1,-25,-4) Matrix(3921,-2401,6400,-3919) -> Matrix(6,1,-49,-8) Matrix(177,-113,224,-143) -> Matrix(0,-1,1,2) Matrix(1489,-961,2304,-1487) -> Matrix(12,13,-13,-14) Matrix(177,-121,256,-175) -> Matrix(14,5,-45,-16) Matrix(143,-101,160,-113) -> Matrix(4,1,-9,-2) Matrix(751,-539,960,-689) -> Matrix(4,1,7,2) Matrix(849,-611,1088,-783) -> Matrix(4,1,-25,-6) Matrix(209,-169,256,-207) -> Matrix(8,3,-27,-10) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 21 ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 (-1/1,0/1) 0 16 1/8 -1/2 2 2 1/7 (-1/2,-1/3) 0 16 1/6 (-1/1,1/1) 0 8 3/16 -1/1 7 1 1/5 (-1/1,-3/4) 0 16 1/4 -1/2 2 4 2/7 (-1/2,-1/3) 0 16 7/24 0/1 2 2 5/17 (-1/1,1/0) 0 16 3/10 (-1/1,-1/3) 0 8 5/16 -1/1 1 1 1/3 (-1/1,-1/2) 0 16 3/8 -1/2 4 2 2/5 (-1/2,-1/3) 0 16 9/22 (-5/11,-3/7) 0 8 7/17 (-7/16,-3/7) 0 16 5/12 -2/5 2 4 8/19 (-3/7,-2/5) 0 16 11/26 (-9/23,-5/13) 0 8 3/7 (-3/8,-1/3) 0 16 7/16 -1/3 5 1 1/2 (-1/1,-1/3) 0 8 9/16 -1/3 7 1 4/7 (-1/3,-3/10) 0 16 11/19 (-2/7,-7/25) 0 16 18/31 (-3/11,-10/37) 0 16 7/12 -1/4 2 4 17/29 (-1/4,-1/5) 0 16 10/17 (-1/3,-1/4) 0 16 3/5 (-1/4,-1/5) 0 16 11/18 (-1/5,-1/7) 0 8 49/80 -1/7 1 1 19/31 (-1/7,0/1) 0 16 8/13 (-1/5,0/1) 0 16 5/8 0/1 4 2 7/11 (0/1,1/1) 0 16 9/14 (-3/1,-1/1) 0 8 31/48 -1/1 13 1 11/17 (-1/1,-5/6) 0 16 2/3 (-1/2,-1/3) 0 16 11/16 -1/3 5 1 7/10 (-1/3,-3/11) 0 8 26/37 (-5/19,-1/4) 0 16 45/64 -1/4 6 1 19/27 (-1/4,-1/5) 0 16 12/17 (-1/3,-1/4) 0 16 17/24 -1/4 2 2 5/7 (-1/3,-1/4) 0 16 13/18 (-1/3,-1/5) 0 8 8/11 (-2/9,-1/5) 0 16 3/4 0/1 2 4 7/9 (-1/3,0/1) 0 16 25/32 0/1 4 1 18/23 (0/1,1/1) 0 16 11/14 (-1/1,1/1) 0 8 4/5 (-1/2,-1/3) 0 16 13/16 -1/3 3 1 5/6 (-1/3,-1/5) 0 8 16/19 (-1/9,0/1) 0 16 27/32 0/1 10 1 11/13 (0/1,1/1) 0 16 6/7 (-1/3,-1/4) 0 16 7/8 0/1 2 2 1/1 (-1/3,0/1) 0 16 1/0 0/1 2 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(47,-5,160,-17) (0/1,1/8) -> (7/24,5/17) Hyperbolic Matrix(159,-22,224,-31) (1/8,1/7) -> (17/24,5/7) Glide Reflection Matrix(95,-14,224,-33) (1/7,1/6) -> (11/26,3/7) Hyperbolic Matrix(17,-3,96,-17) (1/6,3/16) -> (1/6,3/16) Reflection Matrix(31,-6,160,-31) (3/16,1/5) -> (3/16,1/5) Reflection Matrix(47,-11,64,-15) (1/5,1/4) -> (8/11,3/4) Glide Reflection Matrix(49,-13,64,-17) (1/4,2/7) -> (3/4,7/9) Glide Reflection Matrix(193,-56,224,-65) (2/7,7/24) -> (6/7,7/8) Glide Reflection Matrix(223,-66,544,-161) (5/17,3/10) -> (9/22,7/17) Hyperbolic Matrix(49,-15,160,-49) (3/10,5/16) -> (3/10,5/16) Reflection Matrix(31,-10,96,-31) (5/16,1/3) -> (5/16,1/3) Reflection Matrix(79,-29,128,-47) (1/3,3/8) -> (8/13,5/8) Glide Reflection Matrix(81,-31,128,-49) (3/8,2/5) -> (5/8,7/11) Glide Reflection Matrix(607,-248,864,-353) (2/5,9/22) -> (7/10,26/37) Hyperbolic Matrix(335,-139,576,-239) (7/17,5/12) -> (18/31,7/12) Glide Reflection Matrix(337,-141,576,-241) (5/12,8/19) -> (7/12,17/29) Glide Reflection Matrix(511,-216,608,-257) (8/19,11/26) -> (5/6,16/19) Hyperbolic Matrix(97,-42,224,-97) (3/7,7/16) -> (3/7,7/16) Reflection Matrix(15,-7,32,-15) (7/16,1/2) -> (7/16,1/2) Reflection Matrix(17,-9,32,-17) (1/2,9/16) -> (1/2,9/16) 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(271,-157,416,-241) (11/19,18/31) -> (11/17,2/3) Hyperbolic Matrix(687,-403,1120,-657) (17/29,10/17) -> (19/31,8/13) Hyperbolic Matrix(383,-226,544,-321) (10/17,3/5) -> (19/27,12/17) Hyperbolic Matrix(209,-127,288,-175) (3/5,11/18) -> (13/18,8/11) Hyperbolic Matrix(881,-539,1440,-881) (11/18,49/80) -> (11/18,49/80) Reflection Matrix(3039,-1862,4960,-3039) (49/80,19/31) -> (49/80,19/31) Reflection Matrix(177,-113,224,-143) (7/11,9/14) -> (11/14,4/5) Hyperbolic Matrix(433,-279,672,-433) (9/14,31/48) -> (9/14,31/48) Reflection Matrix(1055,-682,1632,-1055) (31/48,11/17) -> (31/48,11/17) Reflection Matrix(65,-44,96,-65) (2/3,11/16) -> (2/3,11/16) Reflection Matrix(111,-77,160,-111) (11/16,7/10) -> (11/16,7/10) Reflection Matrix(3329,-2340,4736,-3329) (26/37,45/64) -> (26/37,45/64) Reflection Matrix(2431,-1710,3456,-2431) (45/64,19/27) -> (45/64,19/27) Reflection Matrix(143,-101,160,-113) (12/17,17/24) -> (7/8,1/1) Hyperbolic Matrix(401,-289,512,-369) (5/7,13/18) -> (18/23,11/14) Glide Reflection Matrix(449,-350,576,-449) (7/9,25/32) -> (7/9,25/32) Reflection Matrix(1151,-900,1472,-1151) (25/32,18/23) -> (25/32,18/23) Reflection Matrix(129,-104,160,-129) (4/5,13/16) -> (4/5,13/16) Reflection Matrix(79,-65,96,-79) (13/16,5/6) -> (13/16,5/6) Reflection Matrix(1025,-864,1216,-1025) (16/19,27/32) -> (16/19,27/32) Reflection Matrix(703,-594,832,-703) (27/32,11/13) -> (27/32,11/13) 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(47,-5,160,-17) -> Matrix(2,1,-1,0) -1/1 Matrix(159,-22,224,-31) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(95,-14,224,-33) -> Matrix(7,2,-18,-5) -1/3 Matrix(17,-3,96,-17) -> Matrix(0,1,1,0) (1/6,3/16) -> (-1/1,1/1) Matrix(31,-6,160,-31) -> Matrix(7,6,-8,-7) (3/16,1/5) -> (-1/1,-3/4) Matrix(47,-11,64,-15) -> Matrix(2,1,-11,-6) Matrix(49,-13,64,-17) -> Matrix(2,1,-3,-2) *** -> (-1/1,-1/3) Matrix(193,-56,224,-65) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(223,-66,544,-161) -> Matrix(7,4,-16,-9) -1/2 Matrix(49,-15,160,-49) -> Matrix(2,1,-3,-2) (3/10,5/16) -> (-1/1,-1/3) Matrix(31,-10,96,-31) -> Matrix(3,2,-4,-3) (5/16,1/3) -> (-1/1,-1/2) Matrix(79,-29,128,-47) -> Matrix(2,1,-11,-6) Matrix(81,-31,128,-49) -> Matrix(2,1,5,2) Matrix(607,-248,864,-353) -> Matrix(13,6,-50,-23) Matrix(335,-139,576,-239) -> Matrix(22,9,-83,-34) Matrix(337,-141,576,-241) -> Matrix(2,1,-3,-2) *** -> (-1/1,-1/3) Matrix(511,-216,608,-257) -> Matrix(5,2,-38,-15) Matrix(97,-42,224,-97) -> Matrix(17,6,-48,-17) (3/7,7/16) -> (-3/8,-1/3) Matrix(15,-7,32,-15) -> Matrix(2,1,-3,-2) (7/16,1/2) -> (-1/1,-1/3) Matrix(17,-9,32,-17) -> Matrix(2,1,-3,-2) (1/2,9/16) -> (-1/1,-1/3) Matrix(127,-72,224,-127) -> Matrix(19,6,-60,-19) (9/16,4/7) -> (-1/3,-3/10) Matrix(191,-110,224,-129) -> Matrix(7,2,-18,-5) -1/3 Matrix(271,-157,416,-241) -> Matrix(18,5,-29,-8) Matrix(687,-403,1120,-657) -> Matrix(4,1,-25,-6) -1/5 Matrix(383,-226,544,-321) -> Matrix(1,0,0,1) Matrix(209,-127,288,-175) -> Matrix(6,1,-25,-4) -1/5 Matrix(881,-539,1440,-881) -> Matrix(6,1,-35,-6) (11/18,49/80) -> (-1/5,-1/7) Matrix(3039,-1862,4960,-3039) -> Matrix(-1,0,14,1) (49/80,19/31) -> (-1/7,0/1) Matrix(177,-113,224,-143) -> Matrix(0,-1,1,2) -1/1 Matrix(433,-279,672,-433) -> Matrix(2,3,-1,-2) (9/14,31/48) -> (-3/1,-1/1) Matrix(1055,-682,1632,-1055) -> Matrix(11,10,-12,-11) (31/48,11/17) -> (-1/1,-5/6) Matrix(65,-44,96,-65) -> Matrix(5,2,-12,-5) (2/3,11/16) -> (-1/2,-1/3) Matrix(111,-77,160,-111) -> Matrix(10,3,-33,-10) (11/16,7/10) -> (-1/3,-3/11) Matrix(3329,-2340,4736,-3329) -> Matrix(39,10,-152,-39) (26/37,45/64) -> (-5/19,-1/4) Matrix(2431,-1710,3456,-2431) -> Matrix(9,2,-40,-9) (45/64,19/27) -> (-1/4,-1/5) Matrix(143,-101,160,-113) -> Matrix(4,1,-9,-2) -1/3 Matrix(401,-289,512,-369) -> Matrix(4,1,1,0) Matrix(449,-350,576,-449) -> Matrix(-1,0,6,1) (7/9,25/32) -> (-1/3,0/1) Matrix(1151,-900,1472,-1151) -> Matrix(1,0,2,-1) (25/32,18/23) -> (0/1,1/1) Matrix(129,-104,160,-129) -> Matrix(5,2,-12,-5) (4/5,13/16) -> (-1/2,-1/3) Matrix(79,-65,96,-79) -> Matrix(4,1,-15,-4) (13/16,5/6) -> (-1/3,-1/5) Matrix(1025,-864,1216,-1025) -> Matrix(-1,0,18,1) (16/19,27/32) -> (-1/9,0/1) Matrix(703,-594,832,-703) -> Matrix(1,0,2,-1) (27/32,11/13) -> (0/1,1/1) Matrix(-1,2,0,1) -> Matrix(-1,0,6,1) (1/1,1/0) -> (-1/3,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.