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): 576 Minimal number of generators: 97 Number of equivalence classes of cusps: 48 Genus: 25 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -8/1 -6/1 -5/1 -4/1 -15/4 -8/3 -5/2 -2/1 -5/3 -25/18 -4/3 -5/4 0/1 1/1 20/17 5/4 4/3 10/7 3/2 20/13 30/19 5/3 100/59 20/11 2/1 20/9 5/2 8/3 30/11 20/7 3/1 10/3 7/2 15/4 4/1 17/4 9/2 19/4 5/1 11/2 6/1 13/2 20/3 7/1 15/2 8/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 -1/2 1/0 -7/1 -1/2 -20/3 0/1 -13/2 0/1 1/0 -6/1 -1/2 -11/2 -1/1 1/0 -5/1 -1/2 -19/4 -1/3 -1/4 -14/3 -1/2 -23/5 -3/10 -9/2 -1/4 0/1 -4/1 -1/2 -1/4 -19/5 -3/10 -15/4 -1/3 0/1 -11/3 -1/2 -7/2 -1/3 -1/4 -10/3 -1/4 -13/4 -1/4 -2/9 -3/1 -1/6 -20/7 0/1 -17/6 0/1 1/2 -14/5 -1/2 -11/4 -1/3 -1/4 -30/11 -1/4 -19/7 -3/14 -8/3 -1/4 -1/6 -5/2 -1/5 0/1 -12/5 -1/4 -1/6 -19/8 -1/4 -1/5 -26/11 -1/6 -7/3 -1/6 -16/7 -1/6 -1/8 -9/4 -1/8 0/1 -20/9 0/1 -11/5 -1/2 -2/1 -1/6 -11/6 -3/20 -1/7 -20/11 -1/7 -9/5 -3/22 -7/4 -1/7 -1/8 -19/11 -5/38 -50/29 -1/8 -31/18 -1/8 0/1 -43/25 -3/22 -12/7 -3/22 -1/8 -41/24 -2/15 -1/8 -29/17 -3/22 -17/10 -2/15 -5/38 -5/3 -1/8 -23/14 -4/33 -7/58 -41/25 -19/158 -100/61 -3/25 -59/36 -3/25 -23/192 -18/11 -5/42 -13/8 -1/8 -2/17 -34/21 -3/26 -21/13 -1/10 -8/5 -1/8 -3/26 -19/12 -1/8 -1/9 -30/19 -1/8 -41/26 -1/8 -2/17 -11/7 -3/26 -25/16 -2/17 -1/9 -39/25 -1/10 -14/9 -3/26 -17/11 -7/62 -20/13 -1/9 -3/2 -1/9 -1/10 -10/7 -1/10 -17/12 -1/10 -4/41 -41/29 -11/114 -24/17 -1/10 -5/52 -7/5 -1/10 -32/23 -1/10 -3/32 -25/18 -2/21 -1/11 -68/49 -1/10 -3/32 -43/31 -1/10 -18/13 -1/10 -11/8 -1/11 -1/12 -15/11 -1/10 -19/14 -3/32 -1/11 -4/3 -1/10 -1/12 -21/16 -3/32 -1/11 -38/29 -3/34 -17/13 -3/34 -13/10 -1/12 0/1 -22/17 -1/14 -9/7 -1/10 -5/4 -1/11 0/1 -11/9 -1/10 -17/14 -1/14 0/1 -6/5 -1/10 -13/11 -1/10 -20/17 -1/11 -7/6 -1/11 -1/12 -1/1 -1/14 0/1 0/1 1/1 1/14 7/6 1/12 1/11 20/17 1/11 13/11 1/10 6/5 1/10 5/4 0/1 1/11 14/11 1/10 23/18 0/1 1/14 9/7 1/10 13/10 0/1 1/12 4/3 1/12 1/10 19/14 1/11 3/32 15/11 1/10 11/8 1/12 1/11 18/13 1/10 43/31 1/10 25/18 1/11 2/21 7/5 1/10 17/12 4/41 1/10 10/7 1/10 3/2 1/10 1/9 20/13 1/9 17/11 7/62 14/9 3/26 11/7 3/26 41/26 2/17 1/8 30/19 1/8 19/12 1/9 1/8 8/5 3/26 1/8 21/13 1/10 34/21 3/26 13/8 2/17 1/8 18/11 5/42 5/3 1/8 22/13 7/54 61/36 25/192 3/23 100/59 3/23 39/23 17/130 17/10 5/38 2/15 29/17 3/22 12/7 1/8 3/22 31/18 0/1 1/8 19/11 5/38 7/4 1/8 1/7 9/5 3/22 20/11 1/7 11/6 1/7 3/20 2/1 1/6 13/6 0/1 1/4 11/5 1/2 20/9 0/1 9/4 0/1 1/8 7/3 1/6 5/2 0/1 1/5 13/5 1/6 34/13 1/6 21/8 1/5 1/4 8/3 1/6 1/4 19/7 3/14 49/18 4/17 1/4 30/11 1/4 11/4 1/4 1/3 14/5 1/2 17/6 -1/2 0/1 20/7 0/1 3/1 1/6 22/7 1/6 19/6 1/5 5/24 16/5 3/14 1/4 13/4 2/9 1/4 10/3 1/4 7/2 1/4 1/3 11/3 1/2 26/7 1/2 15/4 0/1 1/3 34/9 1/2 19/5 3/10 4/1 1/4 1/2 21/5 3/10 17/4 2/5 1/2 13/3 1/2 9/2 0/1 1/4 32/7 1/4 3/10 23/5 3/10 14/3 1/2 19/4 1/4 1/3 5/1 1/2 11/2 1/1 1/0 6/1 1/2 13/2 0/1 1/0 20/3 0/1 7/1 1/2 15/2 0/1 1/1 23/3 1/2 8/1 1/2 1/0 9/1 1/2 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(41,340,-24,-199) (-8/1,1/0) -> (-12/7,-41/24) Hyperbolic Matrix(39,280,-28,-201) (-8/1,-7/1) -> (-7/5,-32/23) Hyperbolic Matrix(41,280,6,41) (-7/1,-20/3) -> (20/3,7/1) Hyperbolic Matrix(79,520,12,79) (-20/3,-13/2) -> (13/2,20/3) Hyperbolic Matrix(81,520,50,321) (-13/2,-6/1) -> (34/21,13/8) Hyperbolic Matrix(39,220,14,79) (-6/1,-11/2) -> (11/4,14/5) Hyperbolic Matrix(41,220,30,161) (-11/2,-5/1) -> (15/11,11/8) Hyperbolic Matrix(79,380,58,279) (-5/1,-19/4) -> (19/14,15/11) Hyperbolic Matrix(199,940,76,359) (-19/4,-14/3) -> (34/13,21/8) Hyperbolic Matrix(121,560,78,361) (-14/3,-23/5) -> (17/11,14/9) Hyperbolic Matrix(241,1100,-140,-639) (-23/5,-9/2) -> (-31/18,-43/25) Hyperbolic Matrix(41,180,-18,-79) (-9/2,-4/1) -> (-16/7,-9/4) Hyperbolic Matrix(161,620,-114,-439) (-4/1,-19/5) -> (-41/29,-24/17) Hyperbolic Matrix(281,1060,-180,-679) (-19/5,-15/4) -> (-25/16,-39/25) Hyperbolic Matrix(119,440,-76,-281) (-15/4,-11/3) -> (-11/7,-25/16) Hyperbolic Matrix(39,140,22,79) (-11/3,-7/2) -> (7/4,9/5) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(79,260,24,79) (-10/3,-13/4) -> (13/4,10/3) Hyperbolic Matrix(81,260,-62,-199) (-13/4,-3/1) -> (-17/13,-13/10) Hyperbolic Matrix(41,120,14,41) (-3/1,-20/7) -> (20/7,3/1) Hyperbolic Matrix(239,680,84,239) (-20/7,-17/6) -> (17/6,20/7) Hyperbolic Matrix(199,560,156,439) (-17/6,-14/5) -> (14/11,23/18) Hyperbolic Matrix(79,220,14,39) (-14/5,-11/4) -> (11/2,6/1) Hyperbolic Matrix(241,660,88,241) (-11/4,-30/11) -> (30/11,11/4) Hyperbolic Matrix(559,1520,-324,-881) (-30/11,-19/7) -> (-19/11,-50/29) Hyperbolic Matrix(119,320,74,199) (-19/7,-8/3) -> (8/5,21/13) Hyperbolic Matrix(39,100,-16,-41) (-8/3,-5/2) -> (-5/2,-12/5) Parabolic Matrix(159,380,100,239) (-12/5,-19/8) -> (19/12,8/5) Hyperbolic Matrix(321,760,68,161) (-19/8,-26/11) -> (14/3,19/4) Hyperbolic Matrix(161,380,136,321) (-26/11,-7/3) -> (13/11,6/5) Hyperbolic Matrix(121,280,-86,-199) (-7/3,-16/7) -> (-24/17,-7/5) Hyperbolic Matrix(161,360,72,161) (-9/4,-20/9) -> (20/9,9/4) Hyperbolic Matrix(199,440,90,199) (-20/9,-11/5) -> (11/5,20/9) Hyperbolic Matrix(119,260,-92,-201) (-11/5,-2/1) -> (-22/17,-9/7) Hyperbolic Matrix(119,220,86,159) (-2/1,-11/6) -> (11/8,18/13) Hyperbolic Matrix(241,440,132,241) (-11/6,-20/11) -> (20/11,11/6) Hyperbolic Matrix(199,360,110,199) (-20/11,-9/5) -> (9/5,20/11) Hyperbolic Matrix(79,140,22,39) (-9/5,-7/4) -> (7/2,11/3) Hyperbolic Matrix(81,140,70,121) (-7/4,-19/11) -> (1/1,7/6) Hyperbolic Matrix(1961,3380,720,1241) (-50/29,-31/18) -> (49/18,30/11) Hyperbolic Matrix(2001,3440,-1442,-2479) (-43/25,-12/7) -> (-68/49,-43/31) Hyperbolic Matrix(961,1640,610,1041) (-41/24,-29/17) -> (11/7,41/26) Hyperbolic Matrix(481,820,376,641) (-29/17,-17/10) -> (23/18,9/7) Hyperbolic Matrix(119,200,-72,-121) (-17/10,-5/3) -> (-5/3,-23/14) Parabolic Matrix(719,1180,170,279) (-23/14,-41/25) -> (21/5,17/4) Hyperbolic Matrix(4879,8000,2878,4719) (-41/25,-100/61) -> (100/59,39/23) Hyperbolic Matrix(7321,12000,4320,7081) (-100/61,-59/36) -> (61/36,100/59) Hyperbolic Matrix(1001,1640,318,521) (-59/36,-18/11) -> (22/7,19/6) Hyperbolic Matrix(319,520,-246,-401) (-18/11,-13/8) -> (-13/10,-22/17) Hyperbolic Matrix(321,520,50,81) (-13/8,-34/21) -> (6/1,13/2) Hyperbolic Matrix(1001,1620,-642,-1039) (-34/21,-21/13) -> (-39/25,-14/9) Hyperbolic Matrix(199,320,74,119) (-21/13,-8/5) -> (8/3,19/7) Hyperbolic Matrix(201,320,76,121) (-8/5,-19/12) -> (21/8,8/3) Hyperbolic Matrix(721,1140,456,721) (-19/12,-30/19) -> (30/19,19/12) Hyperbolic Matrix(1559,2460,988,1559) (-30/19,-41/26) -> (41/26,30/19) Hyperbolic Matrix(241,380,26,41) (-41/26,-11/7) -> (9/1,1/0) Hyperbolic Matrix(361,560,78,121) (-14/9,-17/11) -> (23/5,14/3) Hyperbolic Matrix(441,680,286,441) (-17/11,-20/13) -> (20/13,17/11) Hyperbolic Matrix(79,120,52,79) (-20/13,-3/2) -> (3/2,20/13) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(239,340,168,239) (-10/7,-17/12) -> (17/12,10/7) Hyperbolic Matrix(961,1360,566,801) (-17/12,-41/29) -> (39/23,17/10) Hyperbolic Matrix(1799,2500,-1296,-1801) (-32/23,-25/18) -> (-25/18,-68/49) Parabolic Matrix(1399,1940,-1068,-1481) (-43/31,-18/13) -> (-38/29,-17/13) Hyperbolic Matrix(159,220,86,119) (-18/13,-11/8) -> (11/6,2/1) Hyperbolic Matrix(161,220,30,41) (-11/8,-15/11) -> (5/1,11/2) Hyperbolic Matrix(279,380,58,79) (-15/11,-19/14) -> (19/4,5/1) Hyperbolic Matrix(119,160,-90,-121) (-19/14,-4/3) -> (-4/3,-21/16) Parabolic Matrix(2121,2780,1252,1641) (-21/16,-38/29) -> (22/13,61/36) Hyperbolic Matrix(79,100,-64,-81) (-9/7,-5/4) -> (-5/4,-11/9) Parabolic Matrix(559,680,328,399) (-11/9,-17/14) -> (17/10,29/17) Hyperbolic Matrix(281,340,100,121) (-17/14,-6/5) -> (14/5,17/6) Hyperbolic Matrix(439,520,168,199) (-6/5,-13/11) -> (13/5,34/13) Hyperbolic Matrix(441,520,374,441) (-13/11,-20/17) -> (20/17,13/11) Hyperbolic Matrix(239,280,204,239) (-20/17,-7/6) -> (7/6,20/17) Hyperbolic Matrix(121,140,70,81) (-7/6,-1/1) -> (19/11,7/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(201,-260,92,-119) (9/7,13/10) -> (13/6,11/5) Hyperbolic Matrix(199,-260,62,-81) (13/10,4/3) -> (16/5,13/4) Hyperbolic Matrix(281,-380,88,-119) (4/3,19/14) -> (19/6,16/5) Hyperbolic Matrix(721,-1000,230,-319) (18/13,43/31) -> (3/1,22/7) Hyperbolic Matrix(879,-1220,116,-161) (43/31,25/18) -> (15/2,23/3) Hyperbolic Matrix(201,-280,28,-39) (25/18,7/5) -> (7/1,15/2) Hyperbolic Matrix(241,-340,56,-79) (7/5,17/12) -> (17/4,13/3) Hyperbolic Matrix(281,-440,76,-119) (14/9,11/7) -> (11/3,26/7) Hyperbolic Matrix(841,-1360,222,-359) (21/13,34/21) -> (34/9,19/5) Hyperbolic Matrix(159,-260,74,-121) (13/8,18/11) -> (2/1,13/6) Hyperbolic Matrix(121,-200,72,-119) (18/11,5/3) -> (5/3,22/13) Parabolic Matrix(199,-340,24,-41) (29/17,12/7) -> (8/1,9/1) Hyperbolic Matrix(639,-1100,140,-241) (12/7,31/18) -> (9/2,32/7) Hyperbolic Matrix(881,-1520,324,-559) (31/18,19/11) -> (19/7,49/18) Hyperbolic Matrix(79,-180,18,-41) (9/4,7/3) -> (13/3,9/2) Hyperbolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(241,-900,64,-239) (26/7,15/4) -> (15/4,34/9) Parabolic Matrix(41,-160,10,-39) (19/5,4/1) -> (4/1,21/5) Parabolic Matrix(201,-920,26,-119) (32/7,23/5) -> (23/3,8/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(41,340,-24,-199) -> Matrix(1,2,-8,-15) Matrix(39,280,-28,-201) -> Matrix(3,2,-32,-21) Matrix(41,280,6,41) -> Matrix(1,0,4,1) Matrix(79,520,12,79) -> Matrix(1,0,0,1) Matrix(81,520,50,321) -> Matrix(1,2,8,17) Matrix(39,220,14,79) -> Matrix(1,0,4,1) Matrix(41,220,30,161) -> Matrix(1,0,12,1) Matrix(79,380,58,279) -> Matrix(5,2,52,21) Matrix(199,940,76,359) -> Matrix(1,0,8,1) Matrix(121,560,78,361) -> Matrix(11,4,96,35) Matrix(241,1100,-140,-639) -> Matrix(1,0,-4,1) Matrix(41,180,-18,-79) -> Matrix(1,0,-4,1) Matrix(161,620,-114,-439) -> Matrix(3,2,-32,-21) Matrix(281,1060,-180,-679) -> Matrix(7,2,-60,-17) Matrix(119,440,-76,-281) -> Matrix(7,2,-60,-17) Matrix(39,140,22,79) -> Matrix(7,2,52,15) Matrix(41,140,12,41) -> Matrix(7,2,24,7) Matrix(79,260,24,79) -> Matrix(17,4,72,17) Matrix(81,260,-62,-199) -> Matrix(9,2,-104,-23) Matrix(41,120,14,41) -> Matrix(1,0,12,1) Matrix(239,680,84,239) -> Matrix(1,0,-4,1) Matrix(199,560,156,439) -> Matrix(1,0,12,1) Matrix(79,220,14,39) -> Matrix(1,0,4,1) Matrix(241,660,88,241) -> Matrix(7,2,24,7) Matrix(559,1520,-324,-881) -> Matrix(17,4,-132,-31) Matrix(119,320,74,199) -> Matrix(9,2,76,17) Matrix(39,100,-16,-41) -> Matrix(1,0,0,1) Matrix(159,380,100,239) -> Matrix(9,2,76,17) Matrix(321,760,68,161) -> Matrix(1,0,8,1) Matrix(161,380,136,321) -> Matrix(1,0,16,1) Matrix(121,280,-86,-199) -> Matrix(11,2,-116,-21) Matrix(161,360,72,161) -> Matrix(1,0,16,1) Matrix(199,440,90,199) -> Matrix(1,0,4,1) Matrix(119,260,-92,-201) -> Matrix(1,0,-8,1) Matrix(119,220,86,159) -> Matrix(13,2,136,21) Matrix(241,440,132,241) -> Matrix(41,6,280,41) Matrix(199,360,110,199) -> Matrix(43,6,308,43) Matrix(79,140,22,39) -> Matrix(15,2,52,7) Matrix(81,140,70,121) -> Matrix(15,2,172,23) Matrix(1961,3380,720,1241) -> Matrix(33,4,140,17) Matrix(2001,3440,-1442,-2479) -> Matrix(29,4,-312,-43) Matrix(961,1640,610,1041) -> Matrix(1,0,16,1) Matrix(481,820,376,641) -> Matrix(15,2,172,23) Matrix(119,200,-72,-121) -> Matrix(47,6,-384,-49) Matrix(719,1180,170,279) -> Matrix(83,10,224,27) Matrix(4879,8000,2878,4719) -> Matrix(899,108,6884,827) Matrix(7321,12000,4320,7081) -> Matrix(1201,144,9216,1105) Matrix(1001,1640,318,521) -> Matrix(67,8,360,43) Matrix(319,520,-246,-401) -> Matrix(17,2,-196,-23) Matrix(321,520,50,81) -> Matrix(17,2,8,1) Matrix(1001,1620,-642,-1039) -> Matrix(1,0,0,1) Matrix(199,320,74,119) -> Matrix(17,2,76,9) Matrix(201,320,76,121) -> Matrix(17,2,76,9) Matrix(721,1140,456,721) -> Matrix(17,2,144,17) Matrix(1559,2460,988,1559) -> Matrix(33,4,272,33) Matrix(241,380,26,41) -> Matrix(17,2,8,1) Matrix(361,560,78,121) -> Matrix(35,4,96,11) Matrix(441,680,286,441) -> Matrix(125,14,1116,125) Matrix(79,120,52,79) -> Matrix(19,2,180,19) Matrix(41,60,28,41) -> Matrix(19,2,180,19) Matrix(239,340,168,239) -> Matrix(81,8,820,81) Matrix(961,1360,566,801) -> Matrix(185,18,1408,137) Matrix(1799,2500,-1296,-1801) -> Matrix(1,0,0,1) Matrix(1399,1940,-1068,-1481) -> Matrix(43,4,-484,-45) Matrix(159,220,86,119) -> Matrix(21,2,136,13) Matrix(161,220,30,41) -> Matrix(1,0,12,1) Matrix(279,380,58,79) -> Matrix(21,2,52,5) Matrix(119,160,-90,-121) -> Matrix(1,0,0,1) Matrix(2121,2780,1252,1641) -> Matrix(179,16,1376,123) Matrix(79,100,-64,-81) -> Matrix(1,0,0,1) Matrix(559,680,328,399) -> Matrix(23,2,172,15) Matrix(281,340,100,121) -> Matrix(1,0,12,1) Matrix(439,520,168,199) -> Matrix(1,0,16,1) Matrix(441,520,374,441) -> Matrix(21,2,220,21) Matrix(239,280,204,239) -> Matrix(23,2,264,23) Matrix(121,140,70,81) -> Matrix(23,2,172,15) Matrix(1,0,2,1) -> Matrix(1,0,28,1) Matrix(81,-100,64,-79) -> Matrix(1,0,0,1) Matrix(201,-260,92,-119) -> Matrix(1,0,-8,1) Matrix(199,-260,62,-81) -> Matrix(23,-2,104,-9) Matrix(281,-380,88,-119) -> Matrix(23,-2,104,-9) Matrix(721,-1000,230,-319) -> Matrix(21,-2,116,-11) Matrix(879,-1220,116,-161) -> Matrix(21,-2,32,-3) Matrix(201,-280,28,-39) -> Matrix(21,-2,32,-3) Matrix(241,-340,56,-79) -> Matrix(21,-2,32,-3) Matrix(281,-440,76,-119) -> Matrix(17,-2,60,-7) Matrix(841,-1360,222,-359) -> Matrix(17,-2,60,-7) Matrix(159,-260,74,-121) -> Matrix(17,-2,60,-7) Matrix(121,-200,72,-119) -> Matrix(49,-6,384,-47) Matrix(199,-340,24,-41) -> Matrix(15,-2,8,-1) Matrix(639,-1100,140,-241) -> Matrix(1,0,-4,1) Matrix(881,-1520,324,-559) -> Matrix(31,-4,132,-17) Matrix(79,-180,18,-41) -> Matrix(1,0,-4,1) Matrix(41,-100,16,-39) -> Matrix(1,0,0,1) Matrix(241,-900,64,-239) -> Matrix(1,0,0,1) Matrix(41,-160,10,-39) -> Matrix(1,0,0,1) Matrix(201,-920,26,-119) -> Matrix(7,-2,4,-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): 15 Degree of the the map X: 30 Degree of the the map Y: 96 ----------------------------------------------------------------------- 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): 288 Minimal number of generators: 49 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: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -4/1 -8/3 -5/2 -2/1 -5/3 -5/4 0/1 1/1 5/4 10/7 3/2 30/19 5/3 2/1 20/9 5/2 8/3 30/11 20/7 3/1 10/3 7/2 11/3 4/1 9/2 5/1 6/1 20/3 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -1/2 -6/1 -1/2 -5/1 -1/2 -14/3 -1/2 -9/2 -1/4 0/1 -4/1 -1/2 -1/4 -11/3 -1/2 -7/2 -1/3 -1/4 -10/3 -1/4 -3/1 -1/6 -14/5 -1/2 -11/4 -1/3 -1/4 -30/11 -1/4 -19/7 -3/14 -8/3 -1/4 -1/6 -5/2 -1/5 0/1 -12/5 -1/4 -1/6 -19/8 -1/4 -1/5 -7/3 -1/6 -9/4 -1/8 0/1 -2/1 -1/6 -11/6 -3/20 -1/7 -20/11 -1/7 -9/5 -3/22 -7/4 -1/7 -1/8 -5/3 -1/8 -13/8 -1/8 -2/17 -21/13 -1/10 -8/5 -1/8 -3/26 -19/12 -1/8 -1/9 -30/19 -1/8 -11/7 -3/26 -14/9 -3/26 -17/11 -7/62 -20/13 -1/9 -3/2 -1/9 -1/10 -10/7 -1/10 -7/5 -1/10 -11/8 -1/11 -1/12 -4/3 -1/10 -1/12 -9/7 -1/10 -5/4 -1/11 0/1 -11/9 -1/10 -6/5 -1/10 -13/11 -1/10 -20/17 -1/11 -7/6 -1/11 -1/12 -1/1 -1/14 0/1 0/1 1/1 1/14 7/6 1/12 1/11 6/5 1/10 5/4 0/1 1/11 14/11 1/10 9/7 1/10 4/3 1/12 1/10 11/8 1/12 1/11 7/5 1/10 10/7 1/10 3/2 1/10 1/9 14/9 3/26 11/7 3/26 30/19 1/8 19/12 1/9 1/8 8/5 3/26 1/8 5/3 1/8 12/7 1/8 3/22 19/11 5/38 7/4 1/8 1/7 9/5 3/22 2/1 1/6 11/5 1/2 20/9 0/1 9/4 0/1 1/8 7/3 1/6 5/2 0/1 1/5 13/5 1/6 21/8 1/5 1/4 8/3 1/6 1/4 19/7 3/14 30/11 1/4 11/4 1/4 1/3 14/5 1/2 17/6 -1/2 0/1 20/7 0/1 3/1 1/6 10/3 1/4 7/2 1/4 1/3 11/3 1/2 4/1 1/4 1/2 9/2 0/1 1/4 5/1 1/2 11/2 1/1 1/0 6/1 1/2 13/2 0/1 1/0 20/3 0/1 7/1 1/2 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(21,160,8,61) (-7/1,1/0) -> (13/5,21/8) Hyperbolic Matrix(19,120,-16,-101) (-7/1,-6/1) -> (-6/5,-13/11) Hyperbolic Matrix(19,100,-4,-21) (-6/1,-5/1) -> (-5/1,-14/3) Parabolic Matrix(61,280,22,101) (-14/3,-9/2) -> (11/4,14/5) Hyperbolic Matrix(19,80,14,59) (-9/2,-4/1) -> (4/3,11/8) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(39,140,22,79) (-11/3,-7/2) -> (7/4,9/5) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(19,60,6,19) (-10/3,-3/1) -> (3/1,10/3) Hyperbolic Matrix(99,280,-64,-181) (-3/1,-14/5) -> (-14/9,-17/11) Hyperbolic Matrix(79,220,14,39) (-14/5,-11/4) -> (11/2,6/1) Hyperbolic Matrix(241,660,88,241) (-11/4,-30/11) -> (30/11,11/4) Hyperbolic Matrix(419,1140,154,419) (-30/11,-19/7) -> (19/7,30/11) Hyperbolic Matrix(141,380,82,221) (-19/7,-8/3) -> (12/7,19/11) Hyperbolic Matrix(39,100,-16,-41) (-8/3,-5/2) -> (-5/2,-12/5) Parabolic Matrix(159,380,100,239) (-12/5,-19/8) -> (19/12,8/5) Hyperbolic Matrix(59,140,8,19) (-19/8,-7/3) -> (7/1,1/0) Hyperbolic Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(19,40,-10,-21) (-9/4,-2/1) -> (-2/1,-11/6) Parabolic Matrix(219,400,98,179) (-11/6,-20/11) -> (20/9,9/4) Hyperbolic Matrix(221,400,100,181) (-20/11,-9/5) -> (11/5,20/9) Hyperbolic Matrix(79,140,22,39) (-9/5,-7/4) -> (7/2,11/3) Hyperbolic Matrix(59,100,-36,-61) (-7/4,-5/3) -> (-5/3,-13/8) Parabolic Matrix(99,160,86,139) (-13/8,-21/13) -> (1/1,7/6) Hyperbolic Matrix(199,320,74,119) (-21/13,-8/5) -> (8/3,19/7) Hyperbolic Matrix(201,320,76,121) (-8/5,-19/12) -> (21/8,8/3) Hyperbolic Matrix(721,1140,456,721) (-19/12,-30/19) -> (30/19,19/12) Hyperbolic Matrix(419,660,266,419) (-30/19,-11/7) -> (11/7,30/19) Hyperbolic Matrix(179,280,140,219) (-11/7,-14/9) -> (14/11,9/7) Hyperbolic Matrix(259,400,90,139) (-17/11,-20/13) -> (20/7,3/1) Hyperbolic Matrix(261,400,92,141) (-20/13,-3/2) -> (17/6,20/7) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(99,140,70,99) (-10/7,-7/5) -> (7/5,10/7) Hyperbolic Matrix(101,140,44,61) (-7/5,-11/8) -> (9/4,7/3) Hyperbolic Matrix(59,80,14,19) (-11/8,-4/3) -> (4/1,9/2) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(79,100,-64,-81) (-9/7,-5/4) -> (-5/4,-11/9) Parabolic Matrix(181,220,116,141) (-11/9,-6/5) -> (14/9,11/7) Hyperbolic Matrix(339,400,50,59) (-13/11,-20/17) -> (20/3,7/1) Hyperbolic Matrix(341,400,52,61) (-20/17,-7/6) -> (13/2,20/3) Hyperbolic Matrix(121,140,70,81) (-7/6,-1/1) -> (19/11,7/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(101,-120,16,-19) (7/6,6/5) -> (6/1,13/2) Hyperbolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(181,-280,64,-99) (3/2,14/9) -> (14/5,17/6) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(21,160,8,61) -> Matrix(1,1,4,5) Matrix(19,120,-16,-101) -> Matrix(1,1,-12,-11) Matrix(19,100,-4,-21) -> Matrix(1,1,-4,-3) Matrix(61,280,22,101) -> Matrix(3,1,8,3) Matrix(19,80,14,59) -> Matrix(3,1,32,11) Matrix(21,80,16,61) -> Matrix(3,1,32,11) Matrix(39,140,22,79) -> Matrix(7,2,52,15) Matrix(41,140,12,41) -> Matrix(7,2,24,7) Matrix(19,60,6,19) -> Matrix(5,1,24,5) Matrix(99,280,-64,-181) -> Matrix(1,-1,-8,9) Matrix(79,220,14,39) -> Matrix(1,0,4,1) Matrix(241,660,88,241) -> Matrix(7,2,24,7) Matrix(419,1140,154,419) -> Matrix(13,3,56,13) Matrix(141,380,82,221) -> Matrix(3,1,20,7) Matrix(39,100,-16,-41) -> Matrix(1,0,0,1) Matrix(159,380,100,239) -> Matrix(9,2,76,17) Matrix(59,140,8,19) -> Matrix(5,1,4,1) Matrix(61,140,44,101) -> Matrix(7,1,76,11) Matrix(19,40,-10,-21) -> Matrix(5,1,-36,-7) Matrix(219,400,98,179) -> Matrix(7,1,76,11) Matrix(221,400,100,181) -> Matrix(7,1,-8,-1) Matrix(79,140,22,39) -> Matrix(15,2,52,7) Matrix(59,100,-36,-61) -> Matrix(23,3,-192,-25) Matrix(99,160,86,139) -> Matrix(9,1,116,13) Matrix(199,320,74,119) -> Matrix(17,2,76,9) Matrix(201,320,76,121) -> Matrix(17,2,76,9) Matrix(721,1140,456,721) -> Matrix(17,2,144,17) Matrix(419,660,266,419) -> Matrix(25,3,208,25) Matrix(179,280,140,219) -> Matrix(9,1,116,13) Matrix(259,400,90,139) -> Matrix(9,1,116,13) Matrix(261,400,92,141) -> Matrix(9,1,-28,-3) Matrix(41,60,28,41) -> Matrix(19,2,180,19) Matrix(99,140,70,99) -> Matrix(31,3,320,31) Matrix(101,140,44,61) -> Matrix(11,1,76,7) Matrix(59,80,14,19) -> Matrix(11,1,32,3) Matrix(61,80,16,21) -> Matrix(11,1,32,3) Matrix(79,100,-64,-81) -> Matrix(1,0,0,1) Matrix(181,220,116,141) -> Matrix(13,1,116,9) Matrix(339,400,50,59) -> Matrix(11,1,32,3) Matrix(341,400,52,61) -> Matrix(11,1,-12,-1) Matrix(121,140,70,81) -> Matrix(23,2,172,15) Matrix(1,0,2,1) -> Matrix(1,0,28,1) Matrix(101,-120,16,-19) -> Matrix(11,-1,12,-1) Matrix(81,-100,64,-79) -> Matrix(1,0,0,1) Matrix(181,-280,64,-99) -> Matrix(9,-1,-8,1) Matrix(61,-100,36,-59) -> Matrix(25,-3,192,-23) Matrix(21,-40,10,-19) -> Matrix(7,-1,36,-5) Matrix(41,-100,16,-39) -> Matrix(1,0,0,1) Matrix(21,-100,4,-19) -> Matrix(3,-1,4,-1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of 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): 15 ----------------------------------------------------------------------- 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 0/1 14 1 1/1 1/14 1 20 7/6 (1/12,1/11) 0 20 6/5 1/10 1 10 5/4 0 4 14/11 1/10 1 10 9/7 1/10 1 20 4/3 0 5 11/8 (1/12,1/11) 0 20 7/5 1/10 1 20 10/7 1/10 5 2 3/2 (1/10,1/9) 0 20 14/9 3/26 1 10 11/7 3/26 1 20 30/19 1/8 1 2 19/12 (1/9,1/8) 0 20 8/5 0 5 5/3 1/8 3 4 12/7 0 5 19/11 5/38 1 20 7/4 (1/8,1/7) 0 20 9/5 3/22 1 20 2/1 1/6 1 10 11/5 1/2 1 20 20/9 0/1 6 1 9/4 (0/1,1/8) 0 20 7/3 1/6 1 20 5/2 0 4 13/5 1/6 1 20 21/8 (1/5,1/4) 0 20 8/3 0 5 19/7 3/14 1 20 30/11 1/4 5 2 11/4 (1/4,1/3) 0 20 14/5 1/2 1 10 17/6 (-1/2,0/1) 0 20 20/7 0/1 8 1 3/1 1/6 1 20 10/3 1/4 3 2 7/2 (1/4,1/3) 0 20 11/3 1/2 1 20 4/1 0 5 9/2 (0/1,1/4) 0 20 5/1 1/2 1 4 11/2 (1/1,1/0) 0 20 6/1 1/2 1 10 13/2 (0/1,1/0) 0 20 20/3 0/1 2 1 7/1 1/2 1 20 1/0 (0/1,1/0) 0 20 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(121,-140,70,-81) (1/1,7/6) -> (19/11,7/4) Glide Reflection Matrix(101,-120,16,-19) (7/6,6/5) -> (6/1,13/2) Hyperbolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(219,-280,140,-179) (14/11,9/7) -> (14/9,11/7) Glide Reflection Matrix(61,-80,16,-21) (9/7,4/3) -> (11/3,4/1) Glide Reflection Matrix(59,-80,14,-19) (4/3,11/8) -> (4/1,9/2) Glide Reflection Matrix(101,-140,44,-61) (11/8,7/5) -> (9/4,7/3) Glide Reflection Matrix(99,-140,70,-99) (7/5,10/7) -> (7/5,10/7) Reflection Matrix(41,-60,28,-41) (10/7,3/2) -> (10/7,3/2) Reflection Matrix(181,-280,64,-99) (3/2,14/9) -> (14/5,17/6) Hyperbolic Matrix(419,-660,266,-419) (11/7,30/19) -> (11/7,30/19) Reflection Matrix(721,-1140,456,-721) (30/19,19/12) -> (30/19,19/12) Reflection Matrix(201,-320,76,-121) (19/12,8/5) -> (21/8,8/3) Glide Reflection Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(221,-380,82,-141) (12/7,19/11) -> (8/3,19/7) Glide Reflection Matrix(79,-140,22,-39) (7/4,9/5) -> (7/2,11/3) Glide Reflection Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(199,-440,90,-199) (11/5,20/9) -> (11/5,20/9) Reflection Matrix(161,-360,72,-161) (20/9,9/4) -> (20/9,9/4) Reflection Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(61,-160,8,-21) (13/5,21/8) -> (7/1,1/0) Glide Reflection Matrix(419,-1140,154,-419) (19/7,30/11) -> (19/7,30/11) Reflection Matrix(241,-660,88,-241) (30/11,11/4) -> (30/11,11/4) Reflection Matrix(79,-220,14,-39) (11/4,14/5) -> (11/2,6/1) Glide Reflection Matrix(239,-680,84,-239) (17/6,20/7) -> (17/6,20/7) Reflection Matrix(41,-120,14,-41) (20/7,3/1) -> (20/7,3/1) Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) Reflection Matrix(41,-140,12,-41) (10/3,7/2) -> (10/3,7/2) Reflection Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic Matrix(79,-520,12,-79) (13/2,20/3) -> (13/2,20/3) Reflection Matrix(41,-280,6,-41) (20/3,7/1) -> (20/3,7/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,28,-1) (0/1,1/1) -> (0/1,1/14) Matrix(121,-140,70,-81) -> Matrix(23,-2,172,-15) Matrix(101,-120,16,-19) -> Matrix(11,-1,12,-1) Matrix(81,-100,64,-79) -> Matrix(1,0,0,1) Matrix(219,-280,140,-179) -> Matrix(13,-1,116,-9) Matrix(61,-80,16,-21) -> Matrix(11,-1,32,-3) Matrix(59,-80,14,-19) -> Matrix(11,-1,32,-3) Matrix(101,-140,44,-61) -> Matrix(11,-1,76,-7) Matrix(99,-140,70,-99) -> Matrix(31,-3,320,-31) (7/5,10/7) -> (3/32,1/10) Matrix(41,-60,28,-41) -> Matrix(19,-2,180,-19) (10/7,3/2) -> (1/10,1/9) Matrix(181,-280,64,-99) -> Matrix(9,-1,-8,1) Matrix(419,-660,266,-419) -> Matrix(25,-3,208,-25) (11/7,30/19) -> (3/26,1/8) Matrix(721,-1140,456,-721) -> Matrix(17,-2,144,-17) (30/19,19/12) -> (1/9,1/8) Matrix(201,-320,76,-121) -> Matrix(17,-2,76,-9) Matrix(61,-100,36,-59) -> Matrix(25,-3,192,-23) 1/8 Matrix(221,-380,82,-141) -> Matrix(7,-1,20,-3) Matrix(79,-140,22,-39) -> Matrix(15,-2,52,-7) Matrix(21,-40,10,-19) -> Matrix(7,-1,36,-5) 1/6 Matrix(199,-440,90,-199) -> Matrix(1,0,4,-1) (11/5,20/9) -> (0/1,1/2) Matrix(161,-360,72,-161) -> Matrix(1,0,16,-1) (20/9,9/4) -> (0/1,1/8) Matrix(41,-100,16,-39) -> Matrix(1,0,0,1) Matrix(61,-160,8,-21) -> Matrix(5,-1,4,-1) Matrix(419,-1140,154,-419) -> Matrix(13,-3,56,-13) (19/7,30/11) -> (3/14,1/4) Matrix(241,-660,88,-241) -> Matrix(7,-2,24,-7) (30/11,11/4) -> (1/4,1/3) Matrix(79,-220,14,-39) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(239,-680,84,-239) -> Matrix(-1,0,4,1) (17/6,20/7) -> (-1/2,0/1) Matrix(41,-120,14,-41) -> Matrix(1,0,12,-1) (20/7,3/1) -> (0/1,1/6) Matrix(19,-60,6,-19) -> Matrix(5,-1,24,-5) (3/1,10/3) -> (1/6,1/4) Matrix(41,-140,12,-41) -> Matrix(7,-2,24,-7) (10/3,7/2) -> (1/4,1/3) Matrix(21,-100,4,-19) -> Matrix(3,-1,4,-1) 1/2 Matrix(79,-520,12,-79) -> Matrix(1,0,0,-1) (13/2,20/3) -> (0/1,1/0) Matrix(41,-280,6,-41) -> Matrix(1,0,4,-1) (20/3,7/1) -> (0/1,1/2) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.