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 -1/1 1/1 -7/1 -1/4 0/1 -20/3 0/1 -13/2 0/1 1/8 -6/1 0/1 1/2 1/1 -11/2 1/1 1/0 -5/1 0/1 -19/4 0/1 1/2 -14/3 0/1 1/2 1/1 -23/5 1/2 1/1 -9/2 1/2 1/1 -4/1 -1/1 1/1 -19/5 0/1 1/0 -15/4 1/1 -11/3 3/1 1/0 -7/2 0/1 1/0 -10/3 1/0 -13/4 -2/1 1/0 -3/1 -1/1 1/0 -20/7 -1/1 -17/6 -1/1 -1/2 -14/5 -1/1 -1/2 0/1 -11/4 -1/1 1/0 -30/11 1/0 -19/7 -2/1 1/0 -8/3 -1/1 -5/2 -1/1 -12/5 -1/1 -19/8 -2/3 -1/2 -26/11 -2/3 -3/5 -1/2 -7/3 -1/2 0/1 -16/7 -1/1 -1/3 -9/4 -1/1 1/0 -20/9 -1/1 -11/5 -1/1 -3/4 -2/1 -1/1 -1/2 0/1 -11/6 -5/9 -1/2 -20/11 -1/2 -9/5 -1/2 -3/7 -7/4 -1/2 0/1 -19/11 -1/2 0/1 -50/29 -1/2 -31/18 -1/2 -1/3 -43/25 -1/2 -1/3 -12/7 -1/1 -1/3 -41/24 -1/2 0/1 -29/17 -1/1 1/0 -17/10 -1/1 -1/2 -5/3 -1/2 -23/14 -1/2 -3/7 -41/25 -1/2 -2/5 -100/61 -1/2 -59/36 -1/2 -4/9 -18/11 -1/2 -4/9 -3/7 -13/8 -5/12 -2/5 -34/21 -2/5 -9/23 -7/18 -21/13 -2/5 -3/8 -8/5 -1/3 -19/12 -1/4 0/1 -30/19 0/1 -41/26 0/1 1/0 -11/7 -1/1 -1/2 -25/16 -1/3 -39/25 -1/4 0/1 -14/9 -1/1 -1/2 0/1 -17/11 -1/1 -1/2 -20/13 -1/2 -3/2 -1/2 -1/3 -10/7 -1/3 -17/12 -1/3 -1/4 -41/29 -1/4 0/1 -24/17 -1/3 -1/5 -7/5 -1/4 0/1 -32/23 -1/1 1/1 -25/18 -1/1 -68/49 -1/1 -3/5 -43/31 -1/1 -1/2 -18/13 -1/1 -2/3 -1/2 -11/8 -1/2 -3/7 -15/11 -2/5 -19/14 -2/5 -3/8 -4/3 -1/3 -21/16 -2/7 -1/4 -38/29 -1/3 -2/7 -1/4 -17/13 -1/3 -1/4 -13/10 -1/4 0/1 -22/17 -1/2 -2/5 -1/3 -9/7 -1/3 -1/4 -5/4 -1/3 -11/9 -1/3 -3/10 -17/14 -1/3 -1/4 -6/5 -1/3 -2/7 -1/4 -13/11 -4/15 -1/4 -20/17 -1/4 -7/6 -1/4 0/1 -1/1 -1/4 0/1 0/1 0/1 1/1 0/1 1/4 7/6 0/1 1/4 20/17 1/4 13/11 1/4 4/15 6/5 1/4 2/7 1/3 5/4 1/3 14/11 0/1 1/3 1/2 23/18 1/3 1/2 9/7 1/4 1/3 13/10 0/1 1/4 4/3 1/3 19/14 3/8 2/5 15/11 2/5 11/8 3/7 1/2 18/13 1/2 2/3 1/1 43/31 1/2 1/1 25/18 1/1 7/5 0/1 1/4 17/12 1/4 1/3 10/7 1/3 3/2 1/3 1/2 20/13 1/2 17/11 1/2 1/1 14/9 0/1 1/2 1/1 11/7 1/2 1/1 41/26 0/1 1/0 30/19 0/1 19/12 0/1 1/4 8/5 1/3 21/13 3/8 2/5 34/21 7/18 9/23 2/5 13/8 2/5 5/12 18/11 3/7 4/9 1/2 5/3 1/2 22/13 0/1 1/2 1/1 61/36 0/1 1/2 100/59 1/2 39/23 1/2 2/3 17/10 1/2 1/1 29/17 1/1 1/0 12/7 1/3 1/1 31/18 1/3 1/2 19/11 0/1 1/2 7/4 0/1 1/2 9/5 3/7 1/2 20/11 1/2 11/6 1/2 5/9 2/1 0/1 1/2 1/1 13/6 2/3 3/4 11/5 3/4 1/1 20/9 1/1 9/4 1/1 1/0 7/3 0/1 1/2 5/2 1/1 13/5 2/1 1/0 34/13 -1/1 0/1 1/0 21/8 0/1 1/0 8/3 1/1 19/7 2/1 1/0 49/18 3/1 1/0 30/11 1/0 11/4 1/1 1/0 14/5 0/1 1/2 1/1 17/6 1/2 1/1 20/7 1/1 3/1 1/1 1/0 22/7 0/1 1/1 1/0 19/6 0/1 1/0 16/5 1/1 13/4 2/1 1/0 10/3 1/0 7/2 0/1 1/0 11/3 -3/1 1/0 26/7 -3/1 -2/1 1/0 15/4 -1/1 34/9 -1/2 -1/3 0/1 19/5 0/1 1/0 4/1 -1/1 1/1 21/5 0/1 1/0 17/4 -1/1 1/0 13/3 0/1 1/0 9/2 -1/1 -1/2 32/7 -1/1 -1/3 23/5 -1/1 -1/2 14/3 -1/1 -1/2 0/1 19/4 -1/2 0/1 5/1 0/1 11/2 -1/1 1/0 6/1 -1/1 -1/2 0/1 13/2 -1/8 0/1 20/3 0/1 7/1 0/1 1/4 15/2 1/1 23/3 1/1 1/0 8/1 -1/1 1/1 9/1 -1/1 -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(1,0,-2,1) Matrix(39,280,-28,-201) -> Matrix(1,0,0,1) Matrix(41,280,6,41) -> Matrix(1,0,8,1) Matrix(79,520,12,79) -> Matrix(1,0,-16,1) Matrix(81,520,50,321) -> Matrix(11,-2,28,-5) Matrix(39,220,14,79) -> Matrix(1,0,0,1) Matrix(41,220,30,161) -> Matrix(1,2,2,5) Matrix(79,380,58,279) -> Matrix(7,-2,18,-5) Matrix(199,940,76,359) -> Matrix(1,0,-2,1) Matrix(121,560,78,361) -> Matrix(1,0,0,1) Matrix(241,1100,-140,-639) -> Matrix(1,0,-4,1) Matrix(41,180,-18,-79) -> Matrix(1,0,-2,1) Matrix(161,620,-114,-439) -> Matrix(1,0,-4,1) Matrix(281,1060,-180,-679) -> Matrix(1,0,-4,1) Matrix(119,440,-76,-281) -> Matrix(1,-2,-2,5) Matrix(39,140,22,79) -> Matrix(1,0,2,1) Matrix(41,140,12,41) -> Matrix(1,0,0,1) Matrix(79,260,24,79) -> Matrix(1,4,0,1) Matrix(81,260,-62,-199) -> Matrix(1,2,-4,-7) Matrix(41,120,14,41) -> Matrix(1,2,0,1) Matrix(239,680,84,239) -> Matrix(3,2,4,3) Matrix(199,560,156,439) -> Matrix(1,0,4,1) Matrix(79,220,14,39) -> Matrix(1,0,0,1) Matrix(241,660,88,241) -> Matrix(1,2,0,1) Matrix(559,1520,-324,-881) -> Matrix(1,2,-2,-3) Matrix(119,320,74,199) -> Matrix(3,4,8,11) Matrix(39,100,-16,-41) -> Matrix(1,2,-2,-3) Matrix(159,380,100,239) -> Matrix(3,2,10,7) Matrix(321,760,68,161) -> Matrix(3,2,-8,-5) Matrix(161,380,136,321) -> Matrix(7,4,26,15) Matrix(121,280,-86,-199) -> Matrix(1,0,-2,1) Matrix(161,360,72,161) -> Matrix(1,2,0,1) Matrix(199,440,90,199) -> Matrix(7,6,8,7) Matrix(119,260,-92,-201) -> Matrix(3,2,-8,-5) Matrix(119,220,86,159) -> Matrix(3,2,4,3) Matrix(241,440,132,241) -> Matrix(19,10,36,19) Matrix(199,360,110,199) -> Matrix(13,6,28,13) Matrix(79,140,22,39) -> Matrix(1,0,2,1) Matrix(81,140,70,121) -> Matrix(1,0,6,1) Matrix(1961,3380,720,1241) -> Matrix(9,4,2,1) Matrix(2001,3440,-1442,-2479) -> Matrix(5,2,-8,-3) Matrix(961,1640,610,1041) -> Matrix(1,0,2,1) Matrix(481,820,376,641) -> Matrix(1,0,4,1) Matrix(119,200,-72,-121) -> Matrix(7,4,-16,-9) Matrix(719,1180,170,279) -> Matrix(5,2,2,1) Matrix(4879,8000,2878,4719) -> Matrix(1,0,4,1) Matrix(7321,12000,4320,7081) -> Matrix(9,4,20,9) Matrix(1001,1640,318,521) -> Matrix(9,4,2,1) Matrix(319,520,-246,-401) -> Matrix(5,2,-8,-3) Matrix(321,520,50,81) -> Matrix(5,2,-28,-11) Matrix(1001,1620,-642,-1039) -> Matrix(5,2,-28,-11) Matrix(199,320,74,119) -> Matrix(11,4,8,3) Matrix(201,320,76,121) -> Matrix(1,0,4,1) Matrix(721,1140,456,721) -> Matrix(1,0,8,1) Matrix(1559,2460,988,1559) -> Matrix(1,0,0,1) Matrix(241,380,26,41) -> Matrix(1,0,0,1) Matrix(361,560,78,121) -> Matrix(1,0,0,1) Matrix(441,680,286,441) -> Matrix(3,2,4,3) Matrix(79,120,52,79) -> Matrix(5,2,12,5) Matrix(41,60,28,41) -> Matrix(5,2,12,5) Matrix(239,340,168,239) -> Matrix(7,2,24,7) Matrix(961,1360,566,801) -> Matrix(7,2,10,3) Matrix(1799,2500,-1296,-1801) -> Matrix(1,2,-2,-3) Matrix(1399,1940,-1068,-1481) -> Matrix(1,0,-2,1) Matrix(159,220,86,119) -> Matrix(3,2,4,3) Matrix(161,220,30,41) -> Matrix(5,2,2,1) Matrix(279,380,58,79) -> Matrix(5,2,-18,-7) Matrix(119,160,-90,-121) -> Matrix(11,4,-36,-13) Matrix(2121,2780,1252,1641) -> Matrix(7,2,10,3) Matrix(79,100,-64,-81) -> Matrix(5,2,-18,-7) Matrix(559,680,328,399) -> Matrix(7,2,10,3) Matrix(281,340,100,121) -> Matrix(7,2,10,3) Matrix(439,520,168,199) -> Matrix(7,2,-4,-1) Matrix(441,520,374,441) -> Matrix(31,8,120,31) Matrix(239,280,204,239) -> Matrix(1,0,8,1) Matrix(121,140,70,81) -> Matrix(1,0,6,1) Matrix(1,0,2,1) -> Matrix(1,0,8,1) Matrix(81,-100,64,-79) -> Matrix(7,-2,18,-5) Matrix(201,-260,92,-119) -> Matrix(5,-2,8,-3) Matrix(199,-260,62,-81) -> Matrix(7,-2,4,-1) Matrix(281,-380,88,-119) -> Matrix(5,-2,8,-3) Matrix(721,-1000,230,-319) -> Matrix(3,-2,2,-1) Matrix(879,-1220,116,-161) -> Matrix(3,-2,2,-1) Matrix(201,-280,28,-39) -> Matrix(1,0,0,1) Matrix(241,-340,56,-79) -> Matrix(1,0,-4,1) Matrix(281,-440,76,-119) -> Matrix(5,-2,-2,1) Matrix(841,-1360,222,-359) -> Matrix(5,-2,8,-3) Matrix(159,-260,74,-121) -> Matrix(9,-4,16,-7) Matrix(121,-200,72,-119) -> Matrix(9,-4,16,-7) Matrix(199,-340,24,-41) -> Matrix(1,0,-2,1) Matrix(639,-1100,140,-241) -> Matrix(1,0,-4,1) Matrix(881,-1520,324,-559) -> Matrix(3,-2,2,-1) Matrix(79,-180,18,-41) -> Matrix(1,0,-2,1) Matrix(41,-100,16,-39) -> Matrix(3,-2,2,-1) Matrix(241,-900,64,-239) -> Matrix(1,2,-2,-3) Matrix(41,-160,10,-39) -> Matrix(1,0,0,1) Matrix(201,-920,26,-119) -> Matrix(1,0,2,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 24 Degree of the the map X: 24 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 4 1 1/1 (0/1,1/4) 0 20 7/6 (0/1,1/4) 0 20 20/17 1/4 4 1 13/11 (1/4,4/15) 0 20 6/5 0 10 5/4 1/3 2 4 14/11 0 10 23/18 (1/3,1/2) 0 20 9/7 (1/4,1/3) 0 20 13/10 (0/1,1/4) 0 20 4/3 1/3 2 5 19/14 (3/8,2/5) 0 20 15/11 2/5 2 4 11/8 (3/7,1/2) 0 20 18/13 0 10 43/31 (1/2,1/1) 0 20 25/18 1/1 2 4 7/5 (0/1,1/4) 0 20 17/12 (1/4,1/3) 0 20 10/7 1/3 2 2 3/2 (1/3,1/2) 0 20 20/13 1/2 2 1 17/11 (1/2,1/1) 0 20 14/9 0 10 11/7 (1/2,1/1) 0 20 41/26 (0/1,1/0) 0 20 30/19 0/1 4 2 19/12 (0/1,1/4) 0 20 8/5 1/3 2 5 21/13 (3/8,2/5) 0 20 34/21 0 10 13/8 (2/5,5/12) 0 20 18/11 0 10 5/3 1/2 2 4 22/13 0 10 61/36 (0/1,1/2) 0 20 100/59 1/2 2 1 39/23 (1/2,2/3) 0 20 17/10 (1/2,1/1) 0 20 29/17 (1/1,1/0) 0 20 12/7 0 5 31/18 (1/3,1/2) 0 20 19/11 (0/1,1/2) 0 20 7/4 (0/1,1/2) 0 20 9/5 (3/7,1/2) 0 20 20/11 1/2 8 1 11/6 (1/2,5/9) 0 20 2/1 0 10 13/6 (2/3,3/4) 0 20 11/5 (3/4,1/1) 0 20 20/9 1/1 4 1 9/4 (1/1,1/0) 0 20 7/3 (0/1,1/2) 0 20 5/2 1/1 2 4 13/5 (2/1,1/0) 0 20 34/13 0 10 21/8 (0/1,1/0) 0 20 8/3 1/1 2 5 19/7 (2/1,1/0) 0 20 49/18 (3/1,1/0) 0 20 30/11 1/0 2 2 11/4 (1/1,1/0) 0 20 14/5 0 10 17/6 (1/2,1/1) 0 20 20/7 1/1 2 1 3/1 (1/1,1/0) 0 20 22/7 0 10 19/6 (0/1,1/0) 0 20 16/5 1/1 2 5 13/4 (2/1,1/0) 0 20 10/3 1/0 2 2 7/2 (0/1,1/0) 0 20 11/3 (-3/1,1/0) 0 20 26/7 0 10 15/4 -1/1 2 4 34/9 0 10 19/5 (0/1,1/0) 0 20 4/1 0 5 21/5 (0/1,1/0) 0 20 17/4 (-1/1,1/0) 0 20 13/3 (0/1,1/0) 0 20 9/2 (-1/1,-1/2) 0 20 32/7 0 5 23/5 (-1/1,-1/2) 0 20 14/3 0 10 19/4 (-1/2,0/1) 0 20 5/1 0/1 2 4 11/2 (-1/1,1/0) 0 20 6/1 0 10 13/2 (-1/8,0/1) 0 20 20/3 0/1 12 1 7/1 (0/1,1/4) 0 20 15/2 1/1 2 4 23/3 (1/1,1/0) 0 20 8/1 0 5 9/1 (-1/1,-1/2) 0 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(239,-280,204,-239) (7/6,20/17) -> (7/6,20/17) Reflection Matrix(441,-520,374,-441) (20/17,13/11) -> (20/17,13/11) Reflection Matrix(439,-520,168,-199) (13/11,6/5) -> (13/5,34/13) Glide Reflection Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(439,-560,156,-199) (14/11,23/18) -> (14/5,17/6) Glide Reflection Matrix(641,-820,376,-481) (23/18,9/7) -> (17/10,29/17) Glide Reflection 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(279,-380,58,-79) (19/14,15/11) -> (19/4,5/1) Glide Reflection Matrix(161,-220,30,-41) (15/11,11/8) -> (5/1,11/2) Glide Reflection Matrix(159,-220,86,-119) (11/8,18/13) -> (11/6,2/1) Glide Reflection 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(239,-340,168,-239) (17/12,10/7) -> (17/12,10/7) Reflection Matrix(41,-60,28,-41) (10/7,3/2) -> (10/7,3/2) Reflection Matrix(79,-120,52,-79) (3/2,20/13) -> (3/2,20/13) Reflection Matrix(441,-680,286,-441) (20/13,17/11) -> (20/13,17/11) Reflection Matrix(361,-560,78,-121) (17/11,14/9) -> (23/5,14/3) Glide Reflection Matrix(281,-440,76,-119) (14/9,11/7) -> (11/3,26/7) Hyperbolic Matrix(241,-380,26,-41) (11/7,41/26) -> (9/1,1/0) Glide Reflection Matrix(1559,-2460,988,-1559) (41/26,30/19) -> (41/26,30/19) 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(199,-320,74,-119) (8/5,21/13) -> (8/3,19/7) Glide Reflection Matrix(841,-1360,222,-359) (21/13,34/21) -> (34/9,19/5) Hyperbolic Matrix(321,-520,50,-81) (34/21,13/8) -> (6/1,13/2) Glide Reflection 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(1039,-1760,330,-559) (22/13,61/36) -> (22/7,19/6) Glide Reflection Matrix(7199,-12200,4248,-7199) (61/36,100/59) -> (61/36,100/59) Reflection Matrix(4601,-7800,2714,-4601) (100/59,39/23) -> (100/59,39/23) Reflection Matrix(601,-1020,142,-241) (39/23,17/10) -> (21/5,17/4) Glide Reflection 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,-140,22,-39) (7/4,9/5) -> (7/2,11/3) Glide Reflection Matrix(199,-360,110,-199) (9/5,20/11) -> (9/5,20/11) Reflection Matrix(241,-440,132,-241) (20/11,11/6) -> (20/11,11/6) Reflection 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(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(359,-940,76,-199) (34/13,21/8) -> (14/3,19/4) Glide Reflection Matrix(1079,-2940,396,-1079) (49/18,30/11) -> (49/18,30/11) 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(79,-260,24,-79) (13/4,10/3) -> (13/4,10/3) Reflection Matrix(41,-140,12,-41) (10/3,7/2) -> (10/3,7/2) Reflection 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 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(1,0,8,-1) (0/1,1/1) -> (0/1,1/4) Matrix(121,-140,70,-81) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(239,-280,204,-239) -> Matrix(1,0,8,-1) (7/6,20/17) -> (0/1,1/4) Matrix(441,-520,374,-441) -> Matrix(31,-8,120,-31) (20/17,13/11) -> (1/4,4/15) Matrix(439,-520,168,-199) -> Matrix(7,-2,-4,1) Matrix(81,-100,64,-79) -> Matrix(7,-2,18,-5) 1/3 Matrix(439,-560,156,-199) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(641,-820,376,-481) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(201,-260,92,-119) -> Matrix(5,-2,8,-3) 1/2 Matrix(199,-260,62,-81) -> Matrix(7,-2,4,-1) Matrix(281,-380,88,-119) -> Matrix(5,-2,8,-3) 1/2 Matrix(279,-380,58,-79) -> Matrix(5,-2,-18,7) Matrix(161,-220,30,-41) -> Matrix(5,-2,2,-1) Matrix(159,-220,86,-119) -> Matrix(3,-2,4,-3) *** -> (1/2,1/1) Matrix(721,-1000,230,-319) -> Matrix(3,-2,2,-1) 1/1 Matrix(879,-1220,116,-161) -> Matrix(3,-2,2,-1) 1/1 Matrix(201,-280,28,-39) -> Matrix(1,0,0,1) Matrix(241,-340,56,-79) -> Matrix(1,0,-4,1) 0/1 Matrix(239,-340,168,-239) -> Matrix(7,-2,24,-7) (17/12,10/7) -> (1/4,1/3) Matrix(41,-60,28,-41) -> Matrix(5,-2,12,-5) (10/7,3/2) -> (1/3,1/2) Matrix(79,-120,52,-79) -> Matrix(5,-2,12,-5) (3/2,20/13) -> (1/3,1/2) Matrix(441,-680,286,-441) -> Matrix(3,-2,4,-3) (20/13,17/11) -> (1/2,1/1) Matrix(361,-560,78,-121) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(281,-440,76,-119) -> Matrix(5,-2,-2,1) Matrix(241,-380,26,-41) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(1559,-2460,988,-1559) -> Matrix(1,0,0,-1) (41/26,30/19) -> (0/1,1/0) Matrix(721,-1140,456,-721) -> Matrix(1,0,8,-1) (30/19,19/12) -> (0/1,1/4) Matrix(201,-320,76,-121) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(199,-320,74,-119) -> Matrix(11,-4,8,-3) Matrix(841,-1360,222,-359) -> Matrix(5,-2,8,-3) 1/2 Matrix(321,-520,50,-81) -> Matrix(5,-2,-28,11) Matrix(159,-260,74,-121) -> Matrix(9,-4,16,-7) 1/2 Matrix(121,-200,72,-119) -> Matrix(9,-4,16,-7) 1/2 Matrix(1039,-1760,330,-559) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(7199,-12200,4248,-7199) -> Matrix(1,0,4,-1) (61/36,100/59) -> (0/1,1/2) Matrix(4601,-7800,2714,-4601) -> Matrix(7,-4,12,-7) (100/59,39/23) -> (1/2,2/3) Matrix(601,-1020,142,-241) -> Matrix(3,-2,-2,1) Matrix(199,-340,24,-41) -> Matrix(1,0,-2,1) 0/1 Matrix(639,-1100,140,-241) -> Matrix(1,0,-4,1) 0/1 Matrix(881,-1520,324,-559) -> Matrix(3,-2,2,-1) 1/1 Matrix(79,-140,22,-39) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(199,-360,110,-199) -> Matrix(13,-6,28,-13) (9/5,20/11) -> (3/7,1/2) Matrix(241,-440,132,-241) -> Matrix(19,-10,36,-19) (20/11,11/6) -> (1/2,5/9) Matrix(199,-440,90,-199) -> Matrix(7,-6,8,-7) (11/5,20/9) -> (3/4,1/1) Matrix(161,-360,72,-161) -> Matrix(-1,2,0,1) (20/9,9/4) -> (1/1,1/0) Matrix(79,-180,18,-41) -> Matrix(1,0,-2,1) 0/1 Matrix(41,-100,16,-39) -> Matrix(3,-2,2,-1) 1/1 Matrix(359,-940,76,-199) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(1079,-2940,396,-1079) -> Matrix(-1,6,0,1) (49/18,30/11) -> (3/1,1/0) Matrix(241,-660,88,-241) -> Matrix(-1,2,0,1) (30/11,11/4) -> (1/1,1/0) Matrix(79,-220,14,-39) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(239,-680,84,-239) -> Matrix(3,-2,4,-3) (17/6,20/7) -> (1/2,1/1) Matrix(41,-120,14,-41) -> Matrix(-1,2,0,1) (20/7,3/1) -> (1/1,1/0) Matrix(79,-260,24,-79) -> Matrix(-1,4,0,1) (13/4,10/3) -> (2/1,1/0) Matrix(41,-140,12,-41) -> Matrix(1,0,0,-1) (10/3,7/2) -> (0/1,1/0) Matrix(241,-900,64,-239) -> Matrix(1,2,-2,-3) -1/1 Matrix(41,-160,10,-39) -> Matrix(1,0,0,1) Matrix(201,-920,26,-119) -> Matrix(1,0,2,1) 0/1 Matrix(79,-520,12,-79) -> Matrix(-1,0,16,1) (13/2,20/3) -> (-1/8,0/1) Matrix(41,-280,6,-41) -> Matrix(1,0,8,-1) (20/3,7/1) -> (0/1,1/4) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.