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 -9/1 -8/1 -15/2 -6/1 -9/2 -4/1 -3/1 -8/3 -21/8 -16/7 -2/1 -8/5 -3/2 -4/3 -6/5 -8/7 0/1 1/1 6/5 24/19 4/3 24/17 3/2 36/23 8/5 12/7 24/13 2/1 24/11 12/5 5/2 21/8 8/3 192/71 3/1 36/11 10/3 24/7 7/2 11/3 23/6 4/1 9/2 19/4 24/5 5/1 11/2 6/1 13/2 7/1 15/2 8/1 17/2 9/1 10/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -9/1 -2/1 0/1 -8/1 -2/1 0/1 -15/2 -2/1 0/1 -7/1 -3/2 -1/1 -6/1 -1/1 -11/2 -1/1 0/1 -5/1 -2/1 -1/1 -24/5 -1/1 -19/4 -1/1 -4/5 -14/3 -2/3 0/1 -23/5 -1/1 1/0 -9/2 -2/1 0/1 -13/3 -2/1 1/0 -17/4 0/1 1/0 -4/1 -2/1 -19/5 -2/1 -1/1 -15/4 -2/1 -4/3 -26/7 -2/1 -4/3 -11/3 -2/1 -3/2 -18/5 -3/2 -7/2 -4/3 -1/1 -24/7 -1/1 -17/5 -1/1 -1/2 -27/8 -2/1 0/1 -10/3 -2/1 0/1 -33/10 -2/1 0/1 -56/17 -2/1 0/1 -23/7 -3/1 1/0 -36/11 -2/1 -13/4 -2/1 -3/2 -3/1 -2/1 -4/3 -14/5 -10/7 -4/3 -25/9 -7/5 -11/8 -36/13 -4/3 -11/4 -4/3 -1/1 -19/7 -2/1 -1/1 -8/3 -2/1 -4/3 -29/11 -2/1 -1/1 -21/8 -2/1 -4/3 -34/13 -2/1 -4/3 -47/18 -2/1 -1/1 -13/5 -2/1 -3/2 -5/2 -3/2 -4/3 -12/5 -4/3 -19/8 -4/3 -17/13 -26/11 -4/3 -22/17 -33/14 -4/3 -22/17 -7/3 -9/7 -5/4 -23/10 -3/2 -4/3 -16/7 -4/3 -25/11 -9/7 -5/4 -9/4 -4/3 -14/11 -11/5 -14/11 -5/4 -24/11 -5/4 -13/6 -5/4 -16/13 -2/1 -4/3 -6/5 -13/7 -5/4 -6/5 -24/13 -6/5 -11/6 -6/5 -1/1 -9/5 -4/3 -6/5 -25/14 -5/4 -6/5 -16/9 -6/5 -7/4 -6/5 -1/1 -12/7 -6/5 -17/10 -6/5 -7/6 -22/13 -6/5 -8/7 -49/29 -7/6 -15/13 -27/16 -6/5 -8/7 -5/3 -6/5 -1/1 -28/17 -6/5 -23/14 -7/6 -8/7 -18/11 -1/1 -31/19 -5/4 -1/1 -13/8 -5/4 -6/5 -21/13 -4/3 -6/5 -8/5 -6/5 -27/17 -6/5 -20/17 -73/46 -20/17 -7/6 -192/121 -7/6 -119/75 -7/6 -15/13 -46/29 -6/5 -8/7 -19/12 -6/5 -13/11 -49/31 -13/11 -7/6 -30/19 -7/6 -11/7 -6/5 -7/6 -3/2 -6/5 -8/7 -13/9 -6/5 -7/6 -49/34 -6/5 -1/1 -36/25 -6/5 -23/16 -6/5 -7/6 -10/7 -6/5 -20/17 -27/19 -20/17 -34/29 -71/50 -20/17 -7/6 -44/31 -34/29 -17/12 -48/41 -7/6 -24/17 -7/6 -7/5 -7/6 -29/25 -25/18 -22/19 -37/32 -18/13 -15/13 -11/8 -15/13 -8/7 -15/11 -22/19 -8/7 -49/36 -15/13 -8/7 -34/25 -22/19 -8/7 -19/14 -15/13 -38/33 -23/17 -15/13 -23/20 -4/3 -8/7 -25/19 -33/29 -25/22 -21/16 -8/7 -42/37 -17/13 -17/15 -9/8 -13/10 -26/23 -9/8 -22/17 -8/7 -10/9 -31/24 -8/7 -9/8 -40/31 -8/7 -10/9 -9/7 -8/7 -10/9 -23/18 -8/7 -9/8 -14/11 -8/7 -10/9 -33/26 -8/7 -10/9 -19/15 -8/7 -1/1 -24/19 -8/7 -5/4 -8/7 -9/8 -11/9 -26/23 -9/8 -17/14 -8/7 -9/8 -40/33 -8/7 -26/23 -23/19 -35/31 -9/8 -6/5 -9/8 -7/6 -19/17 -10/9 -8/7 -10/9 -1/1 -11/10 -1/1 0/1 -1/1 1/1 -1/1 -11/12 7/6 -10/11 -19/21 6/5 -9/10 5/4 -9/10 -8/9 24/19 -8/9 19/15 -1/1 -8/9 14/11 -10/11 -8/9 23/18 -9/10 -8/9 9/7 -10/11 -8/9 22/17 -10/11 -8/9 13/10 -9/10 -26/29 17/13 -9/10 -17/19 4/3 -8/9 23/17 -23/26 -15/17 19/14 -38/43 -15/17 15/11 -8/9 -22/25 11/8 -8/9 -15/17 18/13 -15/17 25/18 -37/42 -22/25 7/5 -29/33 -7/8 24/17 -7/8 17/12 -7/8 -48/55 10/7 -20/23 -6/7 3/2 -8/9 -6/7 14/9 -20/23 -6/7 25/16 -7/8 -6/7 36/23 -6/7 11/7 -7/8 -6/7 19/12 -13/15 -6/7 8/5 -6/7 29/18 -1/1 -6/7 21/13 -6/7 -4/5 13/8 -6/7 -5/6 18/11 -1/1 23/14 -8/9 -7/8 5/3 -1/1 -6/7 22/13 -8/9 -6/7 39/23 -8/9 -6/7 17/10 -7/8 -6/7 12/7 -6/7 7/4 -1/1 -6/7 23/13 -13/15 -19/22 16/9 -6/7 25/14 -6/7 -5/6 9/5 -6/7 -4/5 11/6 -1/1 -6/7 24/13 -6/7 13/7 -6/7 -5/6 2/1 -6/7 -4/5 13/6 -16/19 -5/6 24/11 -5/6 11/5 -5/6 -14/17 9/4 -14/17 -4/5 25/11 -5/6 -9/11 16/7 -4/5 23/10 -4/5 -3/4 7/3 -5/6 -9/11 26/11 -22/27 -4/5 71/30 -13/16 -30/37 45/19 -30/37 -4/5 19/8 -17/21 -4/5 12/5 -4/5 5/2 -4/5 -3/4 18/7 -3/4 49/19 -3/4 -5/7 80/31 -4/5 -2/3 31/12 -4/5 -3/4 13/5 -3/4 -2/3 60/23 -2/3 47/18 -1/1 -2/3 34/13 -4/5 -2/3 21/8 -4/5 -2/3 8/3 -4/5 -2/3 27/10 -4/5 -2/3 73/27 -7/9 -3/4 192/71 -3/4 119/44 -3/4 -8/11 46/17 -4/5 -2/3 19/7 -1/1 -2/3 11/4 -1/1 -4/5 36/13 -4/5 25/9 -11/14 -7/9 14/5 -4/5 -10/13 17/6 -3/4 -8/11 3/1 -4/5 -2/3 13/4 -3/4 -2/3 36/11 -2/3 23/7 -3/5 -1/2 33/10 -2/3 0/1 10/3 -2/3 0/1 27/8 -2/3 0/1 71/21 -1/2 -1/3 44/13 0/1 17/5 -1/1 1/0 24/7 -1/1 7/2 -1/1 -4/5 18/5 -3/4 47/13 -3/4 -5/7 29/8 -5/7 -2/3 11/3 -3/4 -2/3 26/7 -4/5 -2/3 41/11 -1/1 -1/2 15/4 -4/5 -2/3 34/9 -4/5 -2/3 19/5 -1/1 -2/3 23/6 -4/5 -3/4 4/1 -2/3 25/6 -4/7 -1/2 21/5 -2/3 -4/7 17/4 -1/2 0/1 13/3 -2/3 -1/2 9/2 -2/3 0/1 23/5 -1/1 -1/2 14/3 -2/1 0/1 33/7 -2/1 0/1 19/4 -4/3 -1/1 24/5 -1/1 5/1 -1/1 -2/3 11/2 -1/1 0/1 6/1 -1/1 13/2 -4/5 -3/4 7/1 -1/1 -3/4 15/2 -2/3 0/1 23/3 -3/5 -1/2 8/1 -2/3 0/1 17/2 -1/2 0/1 9/1 -2/3 0/1 10/1 -2/3 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(49,456,-36,-335) (-9/1,1/0) -> (-15/11,-49/36) Hyperbolic Matrix(49,432,-38,-335) (-9/1,-8/1) -> (-40/31,-9/7) Hyperbolic Matrix(145,1104,-44,-335) (-8/1,-15/2) -> (-33/10,-56/17) Hyperbolic Matrix(97,720,26,193) (-15/2,-7/1) -> (41/11,15/4) Hyperbolic Matrix(49,312,-30,-191) (-7/1,-6/1) -> (-18/11,-31/19) Hyperbolic Matrix(47,264,34,191) (-6/1,-11/2) -> (11/8,18/13) Hyperbolic Matrix(49,264,18,97) (-11/2,-5/1) -> (19/7,11/4) Hyperbolic Matrix(49,240,10,49) (-5/1,-24/5) -> (24/5,5/1) Hyperbolic Matrix(191,912,40,191) (-24/5,-19/4) -> (19/4,24/5) Hyperbolic Matrix(193,912,-142,-671) (-19/4,-14/3) -> (-34/25,-19/14) Hyperbolic Matrix(145,672,52,241) (-14/3,-23/5) -> (25/9,14/5) Hyperbolic Matrix(95,432,42,191) (-23/5,-9/2) -> (9/4,25/11) Hyperbolic Matrix(49,216,22,97) (-9/2,-13/3) -> (11/5,9/4) Hyperbolic Matrix(145,624,56,241) (-13/3,-17/4) -> (31/12,13/5) Hyperbolic Matrix(193,816,-136,-575) (-17/4,-4/1) -> (-44/31,-17/12) Hyperbolic Matrix(145,552,-88,-335) (-4/1,-19/5) -> (-5/3,-28/17) Hyperbolic Matrix(241,912,-190,-719) (-19/5,-15/4) -> (-33/26,-19/15) Hyperbolic Matrix(335,1248,-142,-529) (-15/4,-26/7) -> (-26/11,-33/14) Hyperbolic Matrix(97,360,52,193) (-26/7,-11/3) -> (13/7,2/1) Hyperbolic Matrix(145,528,-92,-335) (-11/3,-18/5) -> (-30/19,-11/7) Hyperbolic Matrix(47,168,40,143) (-18/5,-7/2) -> (7/6,6/5) Hyperbolic Matrix(97,336,28,97) (-7/2,-24/7) -> (24/7,7/2) Hyperbolic Matrix(239,816,70,239) (-24/7,-17/5) -> (17/5,24/7) Hyperbolic Matrix(241,816,-184,-623) (-17/5,-27/8) -> (-21/16,-17/13) Hyperbolic Matrix(335,1128,128,431) (-27/8,-10/3) -> (34/13,21/8) Hyperbolic Matrix(385,1272,102,337) (-10/3,-33/10) -> (15/4,34/9) Hyperbolic Matrix(1393,4584,540,1777) (-56/17,-23/7) -> (49/19,80/31) Hyperbolic Matrix(527,1728,190,623) (-23/7,-36/11) -> (36/13,25/9) Hyperbolic Matrix(287,936,88,287) (-36/11,-13/4) -> (13/4,36/11) Hyperbolic Matrix(97,312,60,193) (-13/4,-3/1) -> (21/13,13/8) Hyperbolic Matrix(145,408,-102,-287) (-3/1,-14/5) -> (-10/7,-27/19) Hyperbolic Matrix(241,672,52,145) (-14/5,-25/9) -> (23/5,14/3) Hyperbolic Matrix(623,1728,190,527) (-25/9,-36/13) -> (36/11,23/7) Hyperbolic Matrix(287,792,104,287) (-36/13,-11/4) -> (11/4,36/13) Hyperbolic Matrix(97,264,18,49) (-11/4,-19/7) -> (5/1,11/2) Hyperbolic Matrix(143,384,-54,-145) (-19/7,-8/3) -> (-8/3,-29/11) Parabolic Matrix(337,888,-200,-527) (-29/11,-21/8) -> (-27/16,-5/3) Hyperbolic Matrix(431,1128,128,335) (-21/8,-34/13) -> (10/3,27/8) Hyperbolic Matrix(1249,3264,-918,-2399) (-34/13,-47/18) -> (-49/36,-34/25) Hyperbolic Matrix(479,1248,-332,-865) (-47/18,-13/5) -> (-13/9,-49/34) Hyperbolic Matrix(47,120,-38,-97) (-13/5,-5/2) -> (-5/4,-11/9) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(191,456,80,191) (-12/5,-19/8) -> (19/8,12/5) Hyperbolic Matrix(577,1368,-364,-863) (-19/8,-26/11) -> (-46/29,-19/12) Hyperbolic Matrix(143,336,20,47) (-33/14,-7/3) -> (7/1,15/2) Hyperbolic Matrix(145,336,104,241) (-7/3,-23/10) -> (25/18,7/5) Hyperbolic Matrix(335,768,188,431) (-23/10,-16/7) -> (16/9,25/14) Hyperbolic Matrix(337,768,190,433) (-16/7,-25/11) -> (23/13,16/9) Hyperbolic Matrix(191,432,42,95) (-25/11,-9/4) -> (9/2,23/5) Hyperbolic Matrix(97,216,22,49) (-9/4,-11/5) -> (13/3,9/2) Hyperbolic Matrix(241,528,110,241) (-11/5,-24/11) -> (24/11,11/5) Hyperbolic Matrix(287,624,132,287) (-24/11,-13/6) -> (13/6,24/11) Hyperbolic Matrix(145,312,112,241) (-13/6,-2/1) -> (22/17,13/10) Hyperbolic Matrix(193,360,52,97) (-2/1,-13/7) -> (11/3,26/7) Hyperbolic Matrix(337,624,182,337) (-13/7,-24/13) -> (24/13,13/7) Hyperbolic Matrix(287,528,156,287) (-24/13,-11/6) -> (11/6,24/13) Hyperbolic Matrix(145,264,106,193) (-11/6,-9/5) -> (15/11,11/8) Hyperbolic Matrix(241,432,188,337) (-9/5,-25/14) -> (23/18,9/7) Hyperbolic Matrix(431,768,188,335) (-25/14,-16/9) -> (16/7,23/10) Hyperbolic Matrix(95,168,-82,-145) (-16/9,-7/4) -> (-7/6,-8/7) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(239,408,140,239) (-12/7,-17/10) -> (17/10,12/7) Hyperbolic Matrix(623,1056,-482,-817) (-17/10,-22/13) -> (-22/17,-31/24) Hyperbolic Matrix(2881,4872,-1816,-3071) (-22/13,-49/29) -> (-119/75,-46/29) Hyperbolic Matrix(1919,3240,568,959) (-49/29,-27/16) -> (27/8,71/21) Hyperbolic Matrix(1823,3000,-1284,-2113) (-28/17,-23/14) -> (-71/50,-44/31) Hyperbolic Matrix(527,864,380,623) (-23/14,-18/11) -> (18/13,25/18) Hyperbolic Matrix(383,624,294,479) (-31/19,-13/8) -> (13/10,17/13) Hyperbolic Matrix(193,312,60,97) (-13/8,-21/13) -> (3/1,13/4) Hyperbolic Matrix(239,384,-150,-241) (-21/13,-8/5) -> (-8/5,-27/17) Parabolic Matrix(1391,2208,332,527) (-27/17,-73/46) -> (25/6,21/5) Hyperbolic Matrix(23231,36864,8590,13631) (-73/46,-192/121) -> (192/71,119/44) Hyperbolic Matrix(23233,36864,8592,13633) (-192/121,-119/75) -> (73/27,192/71) Hyperbolic Matrix(865,1368,638,1009) (-19/12,-49/31) -> (23/17,19/14) Hyperbolic Matrix(1489,2352,578,913) (-49/31,-30/19) -> (18/7,49/19) Hyperbolic Matrix(47,72,-32,-49) (-11/7,-3/2) -> (-3/2,-13/9) Parabolic Matrix(3215,4632,1232,1775) (-49/34,-36/25) -> (60/23,47/18) Hyperbolic Matrix(1201,1728,768,1105) (-36/25,-23/16) -> (25/16,36/23) Hyperbolic Matrix(385,552,302,433) (-23/16,-10/7) -> (14/11,23/18) Hyperbolic Matrix(3599,5112,1520,2159) (-27/19,-71/50) -> (71/30,45/19) Hyperbolic Matrix(577,816,408,577) (-17/12,-24/17) -> (24/17,17/12) Hyperbolic Matrix(239,336,170,239) (-24/17,-7/5) -> (7/5,24/17) Hyperbolic Matrix(241,336,104,145) (-7/5,-25/18) -> (23/10,7/3) Hyperbolic Matrix(623,864,380,527) (-25/18,-18/13) -> (18/11,23/14) Hyperbolic Matrix(191,264,34,47) (-18/13,-11/8) -> (11/2,6/1) Hyperbolic Matrix(193,264,106,145) (-11/8,-15/11) -> (9/5,11/6) Hyperbolic Matrix(1151,1560,318,431) (-19/14,-23/17) -> (47/13,29/8) Hyperbolic Matrix(143,192,-108,-145) (-23/17,-4/3) -> (-4/3,-25/19) Parabolic Matrix(1681,2208,622,817) (-25/19,-21/16) -> (27/10,73/27) Hyperbolic Matrix(239,312,36,47) (-17/13,-13/10) -> (13/2,7/1) Hyperbolic Matrix(241,312,112,145) (-13/10,-22/17) -> (2/1,13/6) Hyperbolic Matrix(2881,3720,1116,1441) (-31/24,-40/31) -> (80/31,31/12) Hyperbolic Matrix(337,432,188,241) (-9/7,-23/18) -> (25/14,9/5) Hyperbolic Matrix(527,672,338,431) (-23/18,-14/11) -> (14/9,25/16) Hyperbolic Matrix(623,792,188,239) (-14/11,-33/26) -> (33/10,10/3) Hyperbolic Matrix(721,912,570,721) (-19/15,-24/19) -> (24/19,19/15) Hyperbolic Matrix(191,240,152,191) (-24/19,-5/4) -> (5/4,24/19) Hyperbolic Matrix(335,408,78,95) (-11/9,-17/14) -> (17/4,13/3) Hyperbolic Matrix(673,816,80,97) (-17/14,-40/33) -> (8/1,17/2) Hyperbolic Matrix(911,1104,118,143) (-40/33,-23/19) -> (23/3,8/1) Hyperbolic Matrix(577,696,160,193) (-23/19,-6/5) -> (18/5,47/13) Hyperbolic Matrix(143,168,40,47) (-6/5,-7/6) -> (7/2,18/5) Hyperbolic Matrix(191,216,84,95) (-8/7,-1/1) -> (25/11,16/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(145,-168,82,-95) (1/1,7/6) -> (7/4,23/13) Hyperbolic Matrix(97,-120,38,-47) (6/5,5/4) -> (5/2,18/7) Hyperbolic Matrix(719,-912,190,-241) (19/15,14/11) -> (34/9,19/5) Hyperbolic Matrix(817,-1056,482,-623) (9/7,22/17) -> (22/13,39/23) Hyperbolic Matrix(623,-816,184,-241) (17/13,4/3) -> (44/13,17/5) Hyperbolic Matrix(961,-1296,284,-383) (4/3,23/17) -> (71/21,44/13) Hyperbolic Matrix(671,-912,142,-193) (19/14,15/11) -> (33/7,19/4) Hyperbolic Matrix(287,-408,102,-145) (17/12,10/7) -> (14/5,17/6) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(719,-1128,276,-433) (36/23,11/7) -> (13/5,60/23) Hyperbolic Matrix(335,-528,92,-145) (11/7,19/12) -> (29/8,11/3) Hyperbolic Matrix(241,-384,150,-239) (19/12,8/5) -> (8/5,29/18) Parabolic Matrix(1057,-1704,446,-719) (29/18,21/13) -> (45/19,19/8) Hyperbolic Matrix(191,-312,30,-49) (13/8,18/11) -> (6/1,13/2) Hyperbolic Matrix(335,-552,88,-145) (23/14,5/3) -> (19/5,23/6) Hyperbolic Matrix(385,-648,142,-239) (5/3,22/13) -> (46/17,19/7) Hyperbolic Matrix(481,-816,56,-95) (39/23,17/10) -> (17/2,9/1) Hyperbolic Matrix(529,-1248,142,-335) (7/3,26/11) -> (26/7,41/11) Hyperbolic Matrix(2689,-6360,994,-2351) (26/11,71/30) -> (119/44,46/17) Hyperbolic Matrix(193,-504,18,-47) (47/18,34/13) -> (10/1,1/0) Hyperbolic Matrix(145,-384,54,-143) (21/8,8/3) -> (8/3,27/10) Parabolic Matrix(143,-408,34,-97) (17/6,3/1) -> (21/5,17/4) Hyperbolic Matrix(335,-1104,44,-145) (23/7,33/10) -> (15/2,23/3) Hyperbolic Matrix(49,-192,12,-47) (23/6,4/1) -> (4/1,25/6) Parabolic Matrix(97,-456,10,-47) (14/3,33/7) -> (9/1,10/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(49,456,-36,-335) -> Matrix(7,-8,-6,7) Matrix(49,432,-38,-335) -> Matrix(9,8,-8,-7) Matrix(145,1104,-44,-335) -> Matrix(1,0,0,1) Matrix(97,720,26,193) -> Matrix(3,4,-4,-5) Matrix(49,312,-30,-191) -> Matrix(3,2,-2,-1) Matrix(47,264,34,191) -> Matrix(23,8,-26,-9) Matrix(49,264,18,97) -> Matrix(3,4,-4,-5) Matrix(49,240,10,49) -> Matrix(3,4,-4,-5) Matrix(191,912,40,191) -> Matrix(9,8,-8,-7) Matrix(193,912,-142,-671) -> Matrix(37,22,-32,-19) Matrix(145,672,52,241) -> Matrix(11,4,-14,-5) Matrix(95,432,42,191) -> Matrix(5,-4,-6,5) Matrix(49,216,22,97) -> Matrix(5,-4,-6,5) Matrix(145,624,56,241) -> Matrix(3,4,-4,-5) Matrix(193,816,-136,-575) -> Matrix(7,48,-6,-41) Matrix(145,552,-88,-335) -> Matrix(5,4,-4,-3) Matrix(241,912,-190,-719) -> Matrix(7,6,-6,-5) Matrix(335,1248,-142,-529) -> Matrix(5,14,-4,-11) Matrix(97,360,52,193) -> Matrix(7,8,-8,-9) Matrix(145,528,-92,-335) -> Matrix(5,4,-4,-3) Matrix(47,168,40,143) -> Matrix(47,66,-52,-73) Matrix(97,336,28,97) -> Matrix(7,8,-8,-9) Matrix(239,816,70,239) -> Matrix(3,2,-2,-1) Matrix(241,816,-184,-623) -> Matrix(25,8,-22,-7) Matrix(335,1128,128,431) -> Matrix(3,4,-4,-5) Matrix(385,1272,102,337) -> Matrix(3,4,-4,-5) Matrix(1393,4584,540,1777) -> Matrix(3,4,-4,-5) Matrix(527,1728,190,623) -> Matrix(11,26,-14,-33) Matrix(287,936,88,287) -> Matrix(7,12,-10,-17) Matrix(97,312,60,193) -> Matrix(7,8,-8,-9) Matrix(145,408,-102,-287) -> Matrix(61,88,-52,-75) Matrix(241,672,52,145) -> Matrix(3,4,2,3) Matrix(623,1728,190,527) -> Matrix(19,26,-30,-41) Matrix(287,792,104,287) -> Matrix(7,8,-8,-9) Matrix(97,264,18,49) -> Matrix(3,4,-4,-5) Matrix(143,384,-54,-145) -> Matrix(1,0,0,1) Matrix(337,888,-200,-527) -> Matrix(5,4,-4,-3) Matrix(431,1128,128,335) -> Matrix(3,4,-4,-5) Matrix(1249,3264,-918,-2399) -> Matrix(23,38,-20,-33) Matrix(479,1248,-332,-865) -> Matrix(5,4,-4,-3) Matrix(47,120,-38,-97) -> Matrix(43,60,-38,-53) Matrix(49,120,20,49) -> Matrix(17,24,-22,-31) Matrix(191,456,80,191) -> Matrix(103,136,-128,-169) Matrix(577,1368,-364,-863) -> Matrix(39,50,-32,-41) Matrix(143,336,20,47) -> Matrix(17,22,-24,-31) Matrix(145,336,104,241) -> Matrix(67,82,-76,-93) Matrix(335,768,188,431) -> Matrix(27,38,-32,-45) Matrix(337,768,190,433) -> Matrix(81,106,-94,-123) Matrix(191,432,42,95) -> Matrix(3,4,-10,-13) Matrix(97,216,22,49) -> Matrix(3,4,-10,-13) Matrix(241,528,110,241) -> Matrix(111,140,-134,-169) Matrix(287,624,132,287) -> Matrix(129,160,-154,-191) Matrix(145,312,112,241) -> Matrix(13,14,-14,-15) Matrix(193,360,52,97) -> Matrix(7,8,-8,-9) Matrix(337,624,182,337) -> Matrix(49,60,-58,-71) Matrix(287,528,156,287) -> Matrix(11,12,-12,-13) Matrix(145,264,106,193) -> Matrix(67,82,-76,-93) Matrix(241,432,188,337) -> Matrix(13,14,-14,-15) Matrix(431,768,188,335) -> Matrix(31,38,-40,-49) Matrix(95,168,-82,-145) -> Matrix(105,124,-94,-111) Matrix(97,168,56,97) -> Matrix(11,12,-12,-13) Matrix(239,408,140,239) -> Matrix(71,84,-82,-97) Matrix(623,1056,-482,-817) -> Matrix(3,2,-2,-1) Matrix(2881,4872,-1816,-3071) -> Matrix(1,0,0,1) Matrix(1919,3240,568,959) -> Matrix(7,8,-8,-9) Matrix(1823,3000,-1284,-2113) -> Matrix(169,196,-144,-167) Matrix(527,864,380,623) -> Matrix(127,142,-144,-161) Matrix(383,624,294,479) -> Matrix(59,76,-66,-85) Matrix(193,312,60,97) -> Matrix(7,8,-8,-9) Matrix(239,384,-150,-241) -> Matrix(59,72,-50,-61) Matrix(1391,2208,332,527) -> Matrix(7,8,-8,-9) Matrix(23231,36864,8590,13631) -> Matrix(99,116,-134,-157) Matrix(23233,36864,8592,13633) -> Matrix(81,94,-106,-123) Matrix(865,1368,638,1009) -> Matrix(343,404,-388,-457) Matrix(1489,2352,578,913) -> Matrix(63,74,-86,-101) Matrix(47,72,-32,-49) -> Matrix(1,0,0,1) Matrix(3215,4632,1232,1775) -> Matrix(7,8,-8,-9) Matrix(1201,1728,768,1105) -> Matrix(71,84,-82,-97) Matrix(385,552,302,433) -> Matrix(93,110,-104,-123) Matrix(3599,5112,1520,2159) -> Matrix(401,470,-494,-579) Matrix(577,816,408,577) -> Matrix(575,672,-658,-769) Matrix(239,336,170,239) -> Matrix(349,406,-398,-463) Matrix(241,336,104,145) -> Matrix(71,82,-84,-97) Matrix(623,864,380,527) -> Matrix(123,142,-136,-157) Matrix(191,264,34,47) -> Matrix(7,8,6,7) Matrix(193,264,106,145) -> Matrix(71,82,-84,-97) Matrix(1151,1560,318,431) -> Matrix(139,160,-192,-221) Matrix(143,192,-108,-145) -> Matrix(335,384,-294,-337) Matrix(1681,2208,622,817) -> Matrix(67,76,-82,-93) Matrix(239,312,36,47) -> Matrix(37,42,-52,-59) Matrix(241,312,112,145) -> Matrix(13,14,-14,-15) Matrix(2881,3720,1116,1441) -> Matrix(11,12,-12,-13) Matrix(337,432,188,241) -> Matrix(13,14,-14,-15) Matrix(527,672,338,431) -> Matrix(97,110,-112,-127) Matrix(623,792,188,239) -> Matrix(9,10,-10,-11) Matrix(721,912,570,721) -> Matrix(15,16,-16,-17) Matrix(191,240,152,191) -> Matrix(127,144,-142,-161) Matrix(335,408,78,95) -> Matrix(7,8,-22,-25) Matrix(673,816,80,97) -> Matrix(7,8,-22,-25) Matrix(911,1104,118,143) -> Matrix(7,8,-22,-25) Matrix(577,696,160,193) -> Matrix(133,150,-180,-203) Matrix(143,168,40,47) -> Matrix(59,66,-76,-85) Matrix(191,216,84,95) -> Matrix(85,94,-104,-115) Matrix(1,0,2,1) -> Matrix(21,22,-22,-23) Matrix(145,-168,82,-95) -> Matrix(137,124,-158,-143) Matrix(97,-120,38,-47) -> Matrix(67,60,-86,-77) Matrix(719,-912,190,-241) -> Matrix(7,6,-6,-5) Matrix(817,-1056,482,-623) -> Matrix(3,2,-2,-1) Matrix(623,-816,184,-241) -> Matrix(9,8,10,9) Matrix(961,-1296,284,-383) -> Matrix(9,8,-44,-39) Matrix(671,-912,142,-193) -> Matrix(25,22,-8,-7) Matrix(287,-408,102,-145) -> Matrix(101,88,-132,-115) Matrix(49,-72,32,-47) -> Matrix(1,0,0,1) Matrix(719,-1128,276,-433) -> Matrix(5,4,-4,-3) Matrix(335,-528,92,-145) -> Matrix(5,4,-4,-3) Matrix(241,-384,150,-239) -> Matrix(83,72,-98,-85) Matrix(1057,-1704,446,-719) -> Matrix(115,98,-142,-121) Matrix(191,-312,30,-49) -> Matrix(3,2,-2,-1) Matrix(335,-552,88,-145) -> Matrix(5,4,-4,-3) Matrix(385,-648,142,-239) -> Matrix(5,4,-4,-3) Matrix(481,-816,56,-95) -> Matrix(7,6,-6,-5) Matrix(529,-1248,142,-335) -> Matrix(17,14,-28,-23) Matrix(2689,-6360,994,-2351) -> Matrix(17,14,-28,-23) Matrix(193,-504,18,-47) -> Matrix(3,2,-2,-1) Matrix(145,-384,54,-143) -> Matrix(1,0,0,1) Matrix(143,-408,34,-97) -> Matrix(11,8,-18,-13) Matrix(335,-1104,44,-145) -> Matrix(1,0,0,1) Matrix(49,-192,12,-47) -> Matrix(11,8,-18,-13) Matrix(97,-456,10,-47) -> Matrix(1,2,-2,-3) 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): 30 Degree of the the map X: 30 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: 28 Genus: 19 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -2/1 -8/7 0/1 1/1 4/3 3/2 8/5 12/7 24/13 2/1 12/5 5/2 21/8 8/3 3/1 24/7 7/2 4/1 9/2 24/5 5/1 6/1 13/2 15/2 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -9/1 -2/1 0/1 -8/1 -2/1 0/1 -15/2 -2/1 0/1 -7/1 -3/2 -1/1 -6/1 -1/1 -5/1 -2/1 -1/1 -9/2 -2/1 0/1 -4/1 -2/1 -15/4 -2/1 -4/3 -26/7 -2/1 -4/3 -11/3 -2/1 -3/2 -18/5 -3/2 -7/2 -4/3 -1/1 -24/7 -1/1 -17/5 -1/1 -1/2 -27/8 -2/1 0/1 -10/3 -2/1 0/1 -13/4 -2/1 -3/2 -3/1 -2/1 -4/3 -8/3 -2/1 -4/3 -21/8 -2/1 -4/3 -34/13 -2/1 -4/3 -13/5 -2/1 -3/2 -5/2 -3/2 -4/3 -12/5 -4/3 -19/8 -4/3 -17/13 -26/11 -4/3 -22/17 -33/14 -4/3 -22/17 -7/3 -9/7 -5/4 -16/7 -4/3 -9/4 -4/3 -14/11 -11/5 -14/11 -5/4 -24/11 -5/4 -13/6 -5/4 -16/13 -2/1 -4/3 -6/5 -9/5 -4/3 -6/5 -16/9 -6/5 -7/4 -6/5 -1/1 -12/7 -6/5 -17/10 -6/5 -7/6 -22/13 -6/5 -8/7 -5/3 -6/5 -1/1 -18/11 -1/1 -31/19 -5/4 -1/1 -13/8 -5/4 -6/5 -21/13 -4/3 -6/5 -8/5 -6/5 -3/2 -6/5 -8/7 -4/3 -8/7 -21/16 -8/7 -42/37 -17/13 -17/15 -9/8 -13/10 -26/23 -9/8 -22/17 -8/7 -10/9 -31/24 -8/7 -9/8 -9/7 -8/7 -10/9 -14/11 -8/7 -10/9 -19/15 -8/7 -1/1 -24/19 -8/7 -5/4 -8/7 -9/8 -11/9 -26/23 -9/8 -6/5 -9/8 -7/6 -19/17 -10/9 -8/7 -10/9 -1/1 -11/10 -1/1 0/1 -1/1 1/1 -1/1 -11/12 7/6 -10/11 -19/21 6/5 -9/10 5/4 -9/10 -8/9 9/7 -10/11 -8/9 22/17 -10/11 -8/9 13/10 -9/10 -26/29 17/13 -9/10 -17/19 4/3 -8/9 3/2 -8/9 -6/7 8/5 -6/7 21/13 -6/7 -4/5 13/8 -6/7 -5/6 18/11 -1/1 5/3 -1/1 -6/7 22/13 -8/9 -6/7 39/23 -8/9 -6/7 17/10 -7/8 -6/7 12/7 -6/7 7/4 -1/1 -6/7 16/9 -6/7 9/5 -6/7 -4/5 11/6 -1/1 -6/7 24/13 -6/7 13/7 -6/7 -5/6 2/1 -6/7 -4/5 9/4 -14/17 -4/5 16/7 -4/5 23/10 -4/5 -3/4 7/3 -5/6 -9/11 19/8 -17/21 -4/5 12/5 -4/5 5/2 -4/5 -3/4 13/5 -3/4 -2/3 34/13 -4/5 -2/3 21/8 -4/5 -2/3 8/3 -4/5 -2/3 3/1 -4/5 -2/3 13/4 -3/4 -2/3 23/7 -3/5 -1/2 10/3 -2/3 0/1 27/8 -2/3 0/1 44/13 0/1 17/5 -1/1 1/0 24/7 -1/1 7/2 -1/1 -4/5 18/5 -3/4 11/3 -3/4 -2/3 26/7 -4/5 -2/3 41/11 -1/1 -1/2 15/4 -4/5 -2/3 4/1 -2/3 9/2 -2/3 0/1 14/3 -2/1 0/1 19/4 -4/3 -1/1 24/5 -1/1 5/1 -1/1 -2/3 11/2 -1/1 0/1 6/1 -1/1 13/2 -4/5 -3/4 7/1 -1/1 -3/4 15/2 -2/3 0/1 8/1 -2/3 0/1 9/1 -2/3 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(17,192,10,113) (-9/1,1/0) -> (39/23,17/10) Hyperbolic Matrix(17,144,2,17) (-9/1,-8/1) -> (8/1,9/1) Hyperbolic Matrix(31,240,4,31) (-8/1,-15/2) -> (15/2,8/1) Hyperbolic Matrix(97,720,26,193) (-15/2,-7/1) -> (41/11,15/4) Hyperbolic Matrix(49,312,-30,-191) (-7/1,-6/1) -> (-18/11,-31/19) Hyperbolic Matrix(17,96,-14,-79) (-6/1,-5/1) -> (-11/9,-6/5) Hyperbolic Matrix(31,144,-14,-65) (-5/1,-9/2) -> (-9/4,-11/5) Hyperbolic Matrix(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(31,120,8,31) (-4/1,-15/4) -> (15/4,4/1) Hyperbolic Matrix(335,1248,-142,-529) (-15/4,-26/7) -> (-26/11,-33/14) Hyperbolic Matrix(97,360,52,193) (-26/7,-11/3) -> (13/7,2/1) Hyperbolic Matrix(79,288,48,175) (-11/3,-18/5) -> (18/11,5/3) Hyperbolic Matrix(47,168,40,143) (-18/5,-7/2) -> (7/6,6/5) Hyperbolic Matrix(97,336,28,97) (-7/2,-24/7) -> (24/7,7/2) Hyperbolic Matrix(239,816,70,239) (-24/7,-17/5) -> (17/5,24/7) Hyperbolic Matrix(241,816,-184,-623) (-17/5,-27/8) -> (-21/16,-17/13) Hyperbolic Matrix(335,1128,128,431) (-27/8,-10/3) -> (34/13,21/8) Hyperbolic Matrix(161,528,-68,-223) (-10/3,-13/4) -> (-19/8,-26/11) Hyperbolic Matrix(97,312,60,193) (-13/4,-3/1) -> (21/13,13/8) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(127,336,48,127) (-8/3,-21/8) -> (21/8,8/3) Hyperbolic Matrix(431,1128,128,335) (-21/8,-34/13) -> (10/3,27/8) Hyperbolic Matrix(175,456,104,271) (-34/13,-13/5) -> (5/3,22/13) Hyperbolic Matrix(47,120,-38,-97) (-13/5,-5/2) -> (-5/4,-11/9) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(191,456,80,191) (-12/5,-19/8) -> (19/8,12/5) Hyperbolic Matrix(143,336,20,47) (-33/14,-7/3) -> (7/1,15/2) Hyperbolic Matrix(31,72,-28,-65) (-7/3,-16/7) -> (-8/7,-1/1) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(175,384,36,79) (-11/5,-24/11) -> (24/5,5/1) Hyperbolic Matrix(353,768,74,161) (-24/11,-13/6) -> (19/4,24/5) Hyperbolic Matrix(145,312,112,241) (-13/6,-2/1) -> (22/17,13/10) 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(95,168,-82,-145) (-16/9,-7/4) -> (-7/6,-8/7) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(239,408,140,239) (-12/7,-17/10) -> (17/10,12/7) Hyperbolic Matrix(623,1056,-482,-817) (-17/10,-22/13) -> (-22/17,-31/24) Hyperbolic Matrix(271,456,104,175) (-22/13,-5/3) -> (13/5,34/13) Hyperbolic Matrix(175,288,48,79) (-5/3,-18/11) -> (18/5,11/3) Hyperbolic Matrix(383,624,294,479) (-31/19,-13/8) -> (13/10,17/13) Hyperbolic Matrix(193,312,60,97) (-13/8,-21/13) -> (3/1,13/4) Hyperbolic Matrix(209,336,130,209) (-21/13,-8/5) -> (8/5,21/13) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(785,1032,232,305) (-4/3,-21/16) -> (27/8,44/13) Hyperbolic Matrix(239,312,36,47) (-17/13,-13/10) -> (13/2,7/1) Hyperbolic Matrix(463,600,98,127) (-13/10,-22/17) -> (14/3,19/4) Hyperbolic Matrix(223,288,24,31) (-31/24,-9/7) -> (9/1,1/0) Hyperbolic Matrix(511,648,138,175) (-14/11,-19/15) -> (11/3,26/7) Hyperbolic Matrix(607,768,328,415) (-19/15,-24/19) -> (24/13,13/7) Hyperbolic Matrix(305,384,166,209) (-24/19,-5/4) -> (11/6,24/13) Hyperbolic Matrix(143,168,40,47) (-6/5,-7/6) -> (7/2,18/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(65,-72,28,-31) (1/1,7/6) -> (23/10,7/3) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(113,-144,62,-79) (5/4,9/7) -> (9/5,11/6) Hyperbolic Matrix(817,-1056,482,-623) (9/7,22/17) -> (22/13,39/23) Hyperbolic Matrix(623,-816,184,-241) (17/13,4/3) -> (44/13,17/5) Hyperbolic Matrix(191,-312,30,-49) (13/8,18/11) -> (6/1,13/2) Hyperbolic Matrix(271,-480,118,-209) (7/4,16/9) -> (16/7,23/10) Hyperbolic Matrix(65,-144,14,-31) (2/1,9/4) -> (9/2,14/3) Hyperbolic Matrix(223,-528,68,-161) (7/3,19/8) -> (13/4,23/7) Hyperbolic Matrix(65,-168,12,-31) (5/2,13/5) -> (5/1,11/2) Hyperbolic Matrix(305,-1008,82,-271) (23/7,10/3) -> (26/7,41/11) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(17,192,10,113) -> Matrix(1,-6,-1,7) Matrix(17,144,2,17) -> Matrix(1,0,-1,1) Matrix(31,240,4,31) -> Matrix(1,0,-1,1) Matrix(97,720,26,193) -> Matrix(3,4,-4,-5) Matrix(49,312,-30,-191) -> Matrix(3,2,-2,-1) Matrix(17,96,-14,-79) -> Matrix(17,8,-15,-7) Matrix(31,144,-14,-65) -> Matrix(9,4,-7,-3) Matrix(17,72,4,17) -> Matrix(1,0,-1,1) Matrix(31,120,8,31) -> Matrix(5,8,-7,-11) Matrix(335,1248,-142,-529) -> Matrix(5,14,-4,-11) Matrix(97,360,52,193) -> Matrix(7,8,-8,-9) Matrix(79,288,48,175) -> Matrix(13,20,-15,-23) Matrix(47,168,40,143) -> Matrix(47,66,-52,-73) Matrix(97,336,28,97) -> Matrix(7,8,-8,-9) Matrix(239,816,70,239) -> Matrix(3,2,-2,-1) Matrix(241,816,-184,-623) -> Matrix(25,8,-22,-7) Matrix(335,1128,128,431) -> Matrix(3,4,-4,-5) Matrix(161,528,-68,-223) -> Matrix(9,22,-7,-17) Matrix(97,312,60,193) -> Matrix(7,8,-8,-9) Matrix(17,48,6,17) -> Matrix(5,8,-7,-11) Matrix(127,336,48,127) -> Matrix(5,8,-7,-11) Matrix(431,1128,128,335) -> Matrix(3,4,-4,-5) Matrix(175,456,104,271) -> Matrix(13,20,-15,-23) Matrix(47,120,-38,-97) -> Matrix(43,60,-38,-53) Matrix(49,120,20,49) -> Matrix(17,24,-22,-31) Matrix(191,456,80,191) -> Matrix(103,136,-128,-169) Matrix(143,336,20,47) -> Matrix(17,22,-24,-31) Matrix(31,72,-28,-65) -> Matrix(37,46,-33,-41) Matrix(127,288,56,127) -> Matrix(43,56,-53,-69) Matrix(175,384,36,79) -> Matrix(19,24,-23,-29) Matrix(353,768,74,161) -> Matrix(29,36,-25,-31) Matrix(145,312,112,241) -> Matrix(13,14,-14,-15) Matrix(79,144,-62,-113) -> Matrix(37,46,-33,-41) Matrix(161,288,90,161) -> Matrix(19,24,-23,-29) Matrix(95,168,-82,-145) -> Matrix(105,124,-94,-111) Matrix(97,168,56,97) -> Matrix(11,12,-12,-13) Matrix(239,408,140,239) -> Matrix(71,84,-82,-97) Matrix(623,1056,-482,-817) -> Matrix(3,2,-2,-1) Matrix(271,456,104,175) -> Matrix(17,20,-23,-27) Matrix(175,288,48,79) -> Matrix(17,20,-23,-27) Matrix(383,624,294,479) -> Matrix(59,76,-66,-85) Matrix(193,312,60,97) -> Matrix(7,8,-8,-9) Matrix(209,336,130,209) -> Matrix(19,24,-23,-29) Matrix(31,48,20,31) -> Matrix(41,48,-47,-55) Matrix(17,24,12,17) -> Matrix(41,48,-47,-55) Matrix(785,1032,232,305) -> Matrix(7,8,-29,-33) Matrix(239,312,36,47) -> Matrix(37,42,-52,-59) Matrix(463,600,98,127) -> Matrix(9,10,-1,-1) Matrix(223,288,24,31) -> Matrix(7,8,-15,-17) Matrix(511,648,138,175) -> Matrix(23,26,-31,-35) Matrix(607,768,328,415) -> Matrix(29,34,-35,-41) Matrix(305,384,166,209) -> Matrix(55,62,-63,-71) Matrix(143,168,40,47) -> Matrix(59,66,-76,-85) Matrix(1,0,2,1) -> Matrix(21,22,-22,-23) Matrix(65,-72,28,-31) -> Matrix(51,46,-61,-55) Matrix(79,-96,14,-17) -> Matrix(9,8,1,1) Matrix(113,-144,62,-79) -> Matrix(51,46,-61,-55) Matrix(817,-1056,482,-623) -> Matrix(3,2,-2,-1) Matrix(623,-816,184,-241) -> Matrix(9,8,10,9) Matrix(191,-312,30,-49) -> Matrix(3,2,-2,-1) Matrix(271,-480,118,-209) -> Matrix(25,22,-33,-29) Matrix(65,-144,14,-31) -> Matrix(5,4,1,1) Matrix(223,-528,68,-161) -> Matrix(27,22,-43,-35) Matrix(65,-168,12,-31) -> Matrix(5,4,-9,-7) Matrix(305,-1008,82,-271) -> Matrix(7,4,-9,-5) 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): 30 ----------------------------------------------------------------------- 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 11 1 1/1 (-1/1,-11/12) 0 24 7/6 (-10/11,-19/21) 0 24 6/5 -9/10 4 4 5/4 (-9/10,-8/9) 0 24 9/7 (-10/11,-8/9) 0 8 22/17 0 12 13/10 (-9/10,-26/29) 0 24 17/13 (-9/10,-17/19) 0 24 4/3 -8/9 3 6 3/2 (-8/9,-6/7) 0 8 8/5 -6/7 1 3 21/13 (-6/7,-4/5) 0 8 13/8 (-6/7,-5/6) 0 24 18/11 -1/1 4 4 5/3 (-1/1,-6/7) 0 24 22/13 0 12 39/23 (-8/9,-6/7) 0 8 17/10 (-7/8,-6/7) 0 24 12/7 -6/7 1 2 7/4 (-1/1,-6/7) 0 24 16/9 -6/7 3 3 9/5 (-6/7,-4/5) 0 8 11/6 (-1/1,-6/7) 0 24 24/13 -6/7 1 1 13/7 (-6/7,-5/6) 0 24 2/1 0 12 9/4 (-14/17,-4/5) 0 8 16/7 -4/5 3 3 23/10 (-4/5,-3/4) 0 24 7/3 (-5/6,-9/11) 0 24 19/8 (-17/21,-4/5) 0 24 12/5 -4/5 5 2 5/2 (-4/5,-3/4) 0 24 13/5 (-3/4,-2/3) 0 24 34/13 0 12 21/8 (-4/5,-2/3) 0 8 8/3 (-4/5,-2/3) 0 3 3/1 (-4/5,-2/3) 0 8 13/4 (-3/4,-2/3) 0 24 23/7 (-3/5,-1/2) 0 24 10/3 0 12 27/8 (-2/3,0/1) 0 8 44/13 0/1 3 6 17/5 (-1/1,1/0) 0 24 24/7 -1/1 5 1 7/2 (-1/1,-4/5) 0 24 18/5 -3/4 4 4 11/3 (-3/4,-2/3) 0 24 26/7 0 12 41/11 (-1/1,-1/2) 0 24 15/4 (-4/5,-2/3) 0 8 4/1 -2/3 1 6 9/2 (-2/3,0/1) 0 8 14/3 0 12 19/4 (-4/3,-1/1) 0 24 24/5 -1/1 6 1 5/1 (-1/1,-2/3) 0 24 11/2 (-1/1,0/1) 0 24 6/1 -1/1 4 4 13/2 (-4/5,-3/4) 0 24 7/1 (-1/1,-3/4) 0 24 15/2 (-2/3,0/1) 0 8 8/1 (-2/3,0/1) 0 3 9/1 (-2/3,0/1) 0 8 1/0 (-1/1,0/1) 0 24 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(65,-72,28,-31) (1/1,7/6) -> (23/10,7/3) Hyperbolic Matrix(143,-168,40,-47) (7/6,6/5) -> (7/2,18/5) Glide Reflection Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(113,-144,62,-79) (5/4,9/7) -> (9/5,11/6) Hyperbolic Matrix(817,-1056,482,-623) (9/7,22/17) -> (22/13,39/23) Hyperbolic Matrix(463,-600,98,-127) (22/17,13/10) -> (14/3,19/4) Glide Reflection Matrix(239,-312,36,-47) (13/10,17/13) -> (13/2,7/1) Glide Reflection Matrix(623,-816,184,-241) (17/13,4/3) -> (44/13,17/5) Hyperbolic Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(31,-48,20,-31) (3/2,8/5) -> (3/2,8/5) Reflection Matrix(209,-336,130,-209) (8/5,21/13) -> (8/5,21/13) Reflection Matrix(193,-312,60,-97) (21/13,13/8) -> (3/1,13/4) Glide Reflection Matrix(191,-312,30,-49) (13/8,18/11) -> (6/1,13/2) Hyperbolic Matrix(175,-288,48,-79) (18/11,5/3) -> (18/5,11/3) Glide Reflection Matrix(271,-456,104,-175) (5/3,22/13) -> (13/5,34/13) Glide Reflection Matrix(113,-192,10,-17) (39/23,17/10) -> (9/1,1/0) Glide Reflection Matrix(239,-408,140,-239) (17/10,12/7) -> (17/10,12/7) Reflection Matrix(97,-168,56,-97) (12/7,7/4) -> (12/7,7/4) Reflection Matrix(271,-480,118,-209) (7/4,16/9) -> (16/7,23/10) Hyperbolic Matrix(161,-288,90,-161) (16/9,9/5) -> (16/9,9/5) Reflection Matrix(287,-528,156,-287) (11/6,24/13) -> (11/6,24/13) Reflection Matrix(337,-624,182,-337) (24/13,13/7) -> (24/13,13/7) Reflection Matrix(193,-360,52,-97) (13/7,2/1) -> (11/3,26/7) Glide Reflection Matrix(65,-144,14,-31) (2/1,9/4) -> (9/2,14/3) Hyperbolic Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(223,-528,68,-161) (7/3,19/8) -> (13/4,23/7) Hyperbolic Matrix(191,-456,80,-191) (19/8,12/5) -> (19/8,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(65,-168,12,-31) (5/2,13/5) -> (5/1,11/2) Hyperbolic Matrix(431,-1128,128,-335) (34/13,21/8) -> (10/3,27/8) Glide Reflection Matrix(127,-336,48,-127) (21/8,8/3) -> (21/8,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(305,-1008,82,-271) (23/7,10/3) -> (26/7,41/11) Hyperbolic Matrix(703,-2376,208,-703) (27/8,44/13) -> (27/8,44/13) Reflection Matrix(239,-816,70,-239) (17/5,24/7) -> (17/5,24/7) Reflection Matrix(97,-336,28,-97) (24/7,7/2) -> (24/7,7/2) Reflection Matrix(193,-720,26,-97) (41/11,15/4) -> (7/1,15/2) Glide Reflection Matrix(31,-120,8,-31) (15/4,4/1) -> (15/4,4/1) Reflection Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(191,-912,40,-191) (19/4,24/5) -> (19/4,24/5) Reflection Matrix(49,-240,10,-49) (24/5,5/1) -> (24/5,5/1) Reflection Matrix(31,-240,4,-31) (15/2,8/1) -> (15/2,8/1) 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(23,22,-24,-23) (0/1,1/1) -> (-1/1,-11/12) Matrix(65,-72,28,-31) -> Matrix(51,46,-61,-55) Matrix(143,-168,40,-47) -> Matrix(73,66,-94,-85) Matrix(79,-96,14,-17) -> Matrix(9,8,1,1) Matrix(113,-144,62,-79) -> Matrix(51,46,-61,-55) Matrix(817,-1056,482,-623) -> Matrix(3,2,-2,-1) -1/1 Matrix(463,-600,98,-127) -> Matrix(11,10,-1,-1) Matrix(239,-312,36,-47) -> Matrix(47,42,-66,-59) Matrix(623,-816,184,-241) -> Matrix(9,8,10,9) Matrix(17,-24,12,-17) -> Matrix(55,48,-63,-55) (4/3,3/2) -> (-8/9,-6/7) Matrix(31,-48,20,-31) -> Matrix(55,48,-63,-55) (3/2,8/5) -> (-8/9,-6/7) Matrix(209,-336,130,-209) -> Matrix(29,24,-35,-29) (8/5,21/13) -> (-6/7,-4/5) Matrix(193,-312,60,-97) -> Matrix(9,8,-10,-9) *** -> (-1/1,-4/5) Matrix(191,-312,30,-49) -> Matrix(3,2,-2,-1) -1/1 Matrix(175,-288,48,-79) -> Matrix(23,20,-31,-27) Matrix(271,-456,104,-175) -> Matrix(23,20,-31,-27) Matrix(113,-192,10,-17) -> Matrix(7,6,-15,-13) Matrix(239,-408,140,-239) -> Matrix(97,84,-112,-97) (17/10,12/7) -> (-7/8,-6/7) Matrix(97,-168,56,-97) -> Matrix(13,12,-14,-13) (12/7,7/4) -> (-1/1,-6/7) Matrix(271,-480,118,-209) -> Matrix(25,22,-33,-29) Matrix(161,-288,90,-161) -> Matrix(29,24,-35,-29) (16/9,9/5) -> (-6/7,-4/5) Matrix(287,-528,156,-287) -> Matrix(13,12,-14,-13) (11/6,24/13) -> (-1/1,-6/7) Matrix(337,-624,182,-337) -> Matrix(71,60,-84,-71) (24/13,13/7) -> (-6/7,-5/6) Matrix(193,-360,52,-97) -> Matrix(9,8,-10,-9) *** -> (-1/1,-4/5) Matrix(65,-144,14,-31) -> Matrix(5,4,1,1) Matrix(127,-288,56,-127) -> Matrix(69,56,-85,-69) (9/4,16/7) -> (-14/17,-4/5) Matrix(223,-528,68,-161) -> Matrix(27,22,-43,-35) Matrix(191,-456,80,-191) -> Matrix(169,136,-210,-169) (19/8,12/5) -> (-17/21,-4/5) Matrix(49,-120,20,-49) -> Matrix(31,24,-40,-31) (12/5,5/2) -> (-4/5,-3/4) Matrix(65,-168,12,-31) -> Matrix(5,4,-9,-7) -2/3 Matrix(431,-1128,128,-335) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(127,-336,48,-127) -> Matrix(11,8,-15,-11) (21/8,8/3) -> (-4/5,-2/3) Matrix(17,-48,6,-17) -> Matrix(11,8,-15,-11) (8/3,3/1) -> (-4/5,-2/3) Matrix(305,-1008,82,-271) -> Matrix(7,4,-9,-5) -2/3 Matrix(703,-2376,208,-703) -> Matrix(-1,0,3,1) (27/8,44/13) -> (-2/3,0/1) Matrix(239,-816,70,-239) -> Matrix(1,2,0,-1) (17/5,24/7) -> (-1/1,1/0) Matrix(97,-336,28,-97) -> Matrix(9,8,-10,-9) (24/7,7/2) -> (-1/1,-4/5) Matrix(193,-720,26,-97) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(31,-120,8,-31) -> Matrix(11,8,-15,-11) (15/4,4/1) -> (-4/5,-2/3) Matrix(17,-72,4,-17) -> Matrix(-1,0,3,1) (4/1,9/2) -> (-2/3,0/1) Matrix(191,-912,40,-191) -> Matrix(7,8,-6,-7) (19/4,24/5) -> (-4/3,-1/1) Matrix(49,-240,10,-49) -> Matrix(5,4,-6,-5) (24/5,5/1) -> (-1/1,-2/3) Matrix(31,-240,4,-31) -> Matrix(-1,0,3,1) (15/2,8/1) -> (-2/3,0/1) Matrix(17,-144,2,-17) -> Matrix(-1,0,3,1) (8/1,9/1) -> (-2/3,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.