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): 1152 Minimal number of generators: 193 Number of equivalence classes of cusps: 80 Genus: 57 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/5 -3/4 -11/15 -7/10 -3/5 -59/100 -7/12 -8/15 -1/2 -7/15 -7/16 -17/40 -67/160 -5/12 -2/5 -39/100 -3/8 -11/30 -3/10 -5/18 -11/40 -43/160 -4/15 -1/4 -19/80 -9/40 -3/14 -1/5 -3/16 -7/40 -1/6 -2/15 -1/8 -1/10 0/1 1/9 1/8 1/7 3/20 2/13 1/6 3/17 2/11 3/16 1/5 4/19 3/14 2/9 3/13 4/17 1/4 5/19 4/15 3/11 5/18 2/7 3/10 1/3 7/20 3/8 2/5 5/12 7/16 9/20 7/15 1/2 11/20 7/12 3/5 19/30 13/20 2/3 7/10 11/15 3/4 4/5 17/20 9/10 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 0/1 -8/9 -1/1 0/1 -7/8 -1/2 1/0 -6/7 -4/3 -1/1 -17/20 -1/1 -11/13 -1/1 -10/11 -5/6 -3/4 -14/17 -1/1 -2/3 -9/11 -1/1 -2/3 -13/16 -3/4 -1/2 -4/5 -1/2 -19/24 -1/2 -1/4 -15/19 0/1 1/1 -11/14 -1/2 -18/23 0/1 1/1 -25/32 -1/2 1/0 -7/9 -1/1 0/1 -10/13 0/1 1/1 -13/17 -2/1 -1/1 -3/4 -1/1 -17/23 -1/1 -2/3 -31/42 -3/4 -14/19 -1/1 -2/3 -11/15 -3/4 -8/11 -1/1 -2/3 -13/18 -1/2 -31/43 -1/1 -4/5 -18/25 -3/4 -5/7 -2/3 -3/5 -12/17 -3/5 -4/7 -7/10 -1/2 -16/23 -3/7 -2/5 -9/13 -1/3 0/1 -11/16 -1/2 -1/4 -13/19 -1/7 0/1 -2/3 -1/1 0/1 -13/20 -1/1 -11/17 -1/1 -6/7 -9/14 -3/4 -7/11 -1/1 -2/3 -26/41 -2/3 -3/5 -19/30 -1/2 -12/19 -1/1 -2/3 -5/8 -3/4 -1/2 -13/21 -1/1 -2/3 -21/34 -7/10 -8/13 -2/3 -7/11 -19/31 -2/3 -3/5 -11/18 -5/8 -3/5 -1/2 -13/22 -3/8 -36/61 -12/35 -1/3 -59/100 -1/3 -23/39 -1/3 -6/19 -10/17 -1/3 0/1 -17/29 -1/3 0/1 -24/41 -1/5 0/1 -31/53 -1/3 0/1 -7/12 0/1 -32/55 1/0 -25/43 -1/1 0/1 -18/31 -1/1 0/1 -11/19 -1/3 0/1 -15/26 1/0 -4/7 -1/1 0/1 -9/16 -1/2 1/0 -23/41 -1/1 -4/5 -14/25 -1/2 -19/34 1/0 -5/9 -1/1 0/1 -11/20 -1/1 -6/11 -1/1 -2/3 -7/13 -1/1 -2/3 -15/28 -2/3 -8/15 -1/2 -9/17 -1/1 -2/3 -1/2 -1/2 -8/17 -1/3 0/1 -7/15 -1/2 -13/28 -1/3 -6/13 -1/3 0/1 -5/11 -1/3 0/1 -9/20 0/1 -4/9 -1/1 0/1 -19/43 -1/1 0/1 -15/34 1/0 -11/25 -1/2 -18/41 -1/5 0/1 -7/16 -1/2 1/0 -17/39 -4/3 -1/1 -10/23 -1/1 -2/3 -3/7 -1/1 0/1 -17/40 -1/2 1/0 -14/33 -1/1 0/1 -11/26 1/0 -8/19 -1/1 -2/3 -21/50 -1/2 -13/31 -1/1 0/1 -31/74 1/0 -49/117 -1/1 0/1 -67/160 -1/2 1/0 -18/43 -1/1 0/1 -5/12 -1/1 -22/53 -1/1 -2/3 -17/41 -1/1 -4/5 -12/29 -1/1 -2/3 -7/17 -1/1 -2/3 -2/5 -1/2 -9/23 -3/7 -2/5 -16/41 -15/37 -2/5 -39/100 -2/5 -23/59 -2/5 -21/53 -7/18 -3/8 -5/13 -4/11 -1/3 -13/34 -3/10 -21/55 -1/4 -8/21 -1/3 0/1 -3/8 -1/2 -1/4 -7/19 -1/3 0/1 -18/49 -1/1 0/1 -11/30 -1/2 -26/71 -2/5 -1/3 -15/41 -2/5 -1/3 -4/11 -1/3 0/1 -9/25 -1/4 -14/39 -1/1 0/1 -19/53 -1/3 0/1 -5/14 -1/4 -6/17 -1/7 0/1 -7/20 0/1 -1/3 -1/1 0/1 -7/22 -3/2 -6/19 -1/1 -6/7 -5/16 -3/4 -1/2 -9/29 -1/1 -2/3 -4/13 -1/1 -2/3 -3/10 -1/2 -8/27 -5/11 -4/9 -5/17 -3/7 -2/5 -12/41 -2/5 -9/23 -7/24 -1/2 -3/8 -2/7 -2/5 -1/3 -9/32 -3/10 -1/4 -16/57 -2/7 -3/11 -7/25 -1/4 -19/68 -1/5 -12/43 -1/5 0/1 -17/61 -1/11 0/1 -5/18 -1/2 -8/29 -1/3 0/1 -11/40 -1/2 -1/4 -3/11 -1/3 0/1 -7/26 -1/4 -25/93 -1/3 0/1 -43/160 -1/2 -1/4 -18/67 -1/3 0/1 -11/41 -2/5 -1/3 -4/15 -1/4 -9/34 -1/6 -5/19 -1/3 0/1 -1/4 0/1 -5/21 0/1 1/1 -19/80 1/2 1/0 -14/59 0/1 1/1 -9/38 1/0 -4/17 0/1 1/1 -3/13 -2/1 -1/1 -5/22 1/0 -7/31 -1/1 0/1 -9/40 -1/2 1/0 -2/9 -1/1 0/1 -7/32 -1/2 1/0 -5/23 -2/1 -1/1 -3/14 -1/2 -4/19 -2/1 -1/1 -1/5 -1/2 -4/21 -4/11 -1/3 -3/16 -1/2 -1/4 -5/27 -1/3 0/1 -2/11 -1/3 0/1 -3/17 -1/3 0/1 -7/40 -1/2 -1/4 -4/23 -1/3 0/1 -1/6 -1/4 -2/13 -1/11 0/1 -3/20 0/1 -1/7 0/1 1/3 -2/15 1/0 -3/23 -2/1 -1/1 -1/8 -1/2 1/0 -1/9 -1/1 0/1 -1/10 -1/2 0/1 -1/1 0/1 1/9 -1/1 0/1 1/8 -1/2 1/0 1/7 -1/5 0/1 3/20 0/1 2/13 0/1 1/9 1/6 1/2 3/17 0/1 1/1 2/11 0/1 1/1 3/16 1/2 1/0 1/5 1/0 5/24 -3/2 1/0 4/19 -1/1 -2/3 3/14 1/0 5/23 -1/1 -2/3 7/32 -1/2 1/0 2/9 -1/1 0/1 3/13 -1/1 -2/3 4/17 -1/3 0/1 1/4 0/1 6/23 0/1 1/1 11/42 1/2 5/19 0/1 1/1 4/15 1/2 3/11 0/1 1/1 5/18 1/0 12/43 0/1 1/3 7/25 1/2 2/7 1/1 2/1 5/17 2/1 3/1 3/10 1/0 7/23 -4/1 -3/1 4/13 -2/1 -1/1 5/16 -3/2 1/0 6/19 -6/5 -1/1 1/3 -1/1 0/1 7/20 0/1 6/17 0/1 1/5 5/14 1/2 4/11 0/1 1/1 15/41 1/1 2/1 11/30 1/0 7/19 0/1 1/1 3/8 1/2 1/0 8/21 0/1 1/1 13/34 3/4 5/13 1/1 4/3 12/31 1/1 2/1 7/18 3/2 2/5 1/0 9/22 -5/2 25/61 -23/11 -2/1 41/100 -2/1 16/39 -2/1 -13/7 7/17 -2/1 -1/1 12/29 -2/1 -1/1 17/41 -4/3 -1/1 22/53 -2/1 -1/1 5/12 -1/1 23/55 -1/2 18/43 -1/1 0/1 13/31 -1/1 0/1 8/19 -2/1 -1/1 11/26 -1/2 3/7 -1/1 0/1 7/16 -1/2 1/0 18/41 0/1 1/3 11/25 1/0 15/34 -1/2 4/9 -1/1 0/1 9/20 0/1 5/11 0/1 1/1 6/13 0/1 1/1 13/28 1/1 7/15 1/0 8/17 0/1 1/1 1/2 1/0 9/17 -2/1 -1/1 8/15 1/0 15/28 -2/1 7/13 -2/1 -1/1 6/11 -2/1 -1/1 11/20 -1/1 5/9 -1/1 0/1 24/43 -1/1 0/1 19/34 -1/2 14/25 1/0 23/41 -4/3 -1/1 9/16 -1/2 1/0 22/39 -1/5 0/1 13/23 0/1 1/1 4/7 -1/1 0/1 23/40 -1/2 1/0 19/33 -1/1 0/1 15/26 -1/2 11/19 0/1 1/1 29/50 1/0 18/31 -1/1 0/1 43/74 -1/2 68/117 -1/1 0/1 93/160 -1/2 1/0 25/43 -1/1 0/1 7/12 0/1 31/53 0/1 1/1 24/41 0/1 1/3 17/29 0/1 1/1 10/17 0/1 1/1 3/5 1/0 14/23 -4/1 -3/1 25/41 -22/7 -3/1 61/100 -3/1 36/59 -3/1 -32/11 11/18 -5/2 8/13 -7/3 -2/1 21/34 -7/4 34/55 -3/2 13/21 -2/1 -1/1 5/8 -3/2 1/0 12/19 -2/1 -1/1 31/49 -1/1 0/1 19/30 1/0 45/71 -3/1 -2/1 26/41 -3/1 -2/1 7/11 -2/1 -1/1 16/25 -3/2 25/39 -1/1 0/1 34/53 -2/1 -1/1 9/14 -3/2 11/17 -6/5 -1/1 13/20 -1/1 2/3 -1/1 0/1 15/22 -1/4 13/19 0/1 1/5 11/16 1/2 1/0 20/29 0/1 1/1 9/13 0/1 1/1 7/10 1/0 19/27 -6/1 -5/1 12/17 -4/1 -3/1 29/41 -3/1 -14/5 17/24 -5/2 1/0 5/7 -3/1 -2/1 23/32 -7/4 -3/2 41/57 -5/3 -8/5 18/25 -3/2 49/68 -4/3 31/43 -4/3 -1/1 44/61 -10/9 -1/1 13/18 1/0 21/29 -2/1 -1/1 29/40 -3/2 1/0 8/11 -2/1 -1/1 19/26 -3/2 68/93 -2/1 -1/1 117/160 -3/2 1/0 49/67 -2/1 -1/1 30/41 -3/1 -2/1 11/15 -3/2 25/34 -5/4 14/19 -2/1 -1/1 3/4 -1/1 16/21 -1/1 -2/3 61/80 -3/4 -1/2 45/59 -1/1 -2/3 29/38 -1/2 13/17 -1/1 -2/3 10/13 -1/3 0/1 17/22 -1/2 24/31 -1/1 0/1 31/40 -1/2 1/0 7/9 -1/1 0/1 25/32 -1/2 1/0 18/23 -1/3 0/1 11/14 1/0 15/19 -1/3 0/1 4/5 1/0 17/21 -7/3 -2/1 13/16 -3/2 1/0 22/27 -2/1 -1/1 9/11 -2/1 -1/1 14/17 -2/1 -1/1 33/40 -3/2 1/0 19/23 -2/1 -1/1 5/6 -3/2 11/13 -10/9 -1/1 17/20 -1/1 6/7 -1/1 -4/5 13/15 -1/2 20/23 -1/3 0/1 7/8 -1/2 1/0 8/9 -1/1 0/1 9/10 1/0 1/1 -1/1 0/1 1/0 -1/2 1/0 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(161,146,-440,-399) (-1/1,-8/9) -> (-26/71,-15/41) Hyperbolic Matrix(79,70,360,319) (-8/9,-7/8) -> (7/32,2/9) Hyperbolic Matrix(79,68,-280,-241) (-7/8,-6/7) -> (-2/7,-9/32) Hyperbolic Matrix(239,204,280,239) (-6/7,-17/20) -> (17/20,6/7) Hyperbolic Matrix(441,374,520,441) (-17/20,-11/13) -> (11/13,17/20) Hyperbolic Matrix(199,168,520,439) (-11/13,-5/6) -> (13/34,5/13) Hyperbolic Matrix(41,34,-240,-199) (-5/6,-14/17) -> (-4/23,-1/6) Hyperbolic Matrix(399,328,680,559) (-14/17,-9/11) -> (17/29,10/17) Hyperbolic Matrix(199,162,-640,-521) (-9/11,-13/16) -> (-5/16,-9/29) Hyperbolic Matrix(159,128,-200,-161) (-13/16,-4/5) -> (-4/5,-19/24) Parabolic Matrix(521,412,760,601) (-19/24,-15/19) -> (13/19,11/16) Hyperbolic Matrix(439,346,760,599) (-15/19,-11/14) -> (15/26,11/19) Hyperbolic Matrix(199,156,560,439) (-11/14,-18/23) -> (6/17,5/14) Hyperbolic Matrix(719,562,-2560,-2001) (-18/23,-25/32) -> (-9/32,-16/57) Hyperbolic Matrix(41,32,360,281) (-25/32,-7/9) -> (1/9,1/8) Hyperbolic Matrix(319,246,-520,-401) (-7/9,-10/13) -> (-8/13,-19/31) Hyperbolic Matrix(159,122,520,399) (-10/13,-13/17) -> (7/23,4/13) Hyperbolic Matrix(119,90,-160,-121) (-13/17,-3/4) -> (-3/4,-17/23) Parabolic Matrix(401,296,760,561) (-17/23,-31/42) -> (1/2,9/17) Hyperbolic Matrix(759,560,-3200,-2361) (-31/42,-14/19) -> (-14/59,-9/38) Hyperbolic Matrix(519,382,-1360,-1001) (-14/19,-11/15) -> (-21/55,-8/21) Hyperbolic Matrix(159,116,-440,-321) (-11/15,-8/11) -> (-4/11,-9/25) Hyperbolic Matrix(199,144,-720,-521) (-8/11,-13/18) -> (-5/18,-8/29) Hyperbolic Matrix(319,230,-1000,-721) (-13/18,-31/43) -> (-1/3,-7/22) Hyperbolic Matrix(2001,1442,-3440,-2479) (-31/43,-18/25) -> (-32/55,-25/43) Hyperbolic Matrix(39,28,-280,-201) (-18/25,-5/7) -> (-1/7,-2/15) Hyperbolic Matrix(121,86,-280,-199) (-5/7,-12/17) -> (-10/23,-3/7) Hyperbolic Matrix(279,196,-400,-281) (-12/17,-7/10) -> (-7/10,-16/23) Parabolic Matrix(121,84,520,361) (-16/23,-9/13) -> (3/13,4/17) Hyperbolic Matrix(119,82,-640,-441) (-9/13,-11/16) -> (-3/16,-5/27) Hyperbolic Matrix(519,356,640,439) (-11/16,-13/19) -> (17/21,13/16) Hyperbolic Matrix(279,190,-1000,-681) (-13/19,-2/3) -> (-12/43,-17/61) Hyperbolic Matrix(79,52,120,79) (-2/3,-13/20) -> (13/20,2/3) Hyperbolic Matrix(441,286,680,441) (-13/20,-11/17) -> (11/17,13/20) Hyperbolic Matrix(121,78,560,361) (-11/17,-9/14) -> (3/14,5/23) Hyperbolic Matrix(119,76,-440,-281) (-9/14,-7/11) -> (-3/11,-7/26) Hyperbolic Matrix(961,610,1640,1041) (-7/11,-26/41) -> (24/41,17/29) Hyperbolic Matrix(41,26,-440,-279) (-26/41,-19/30) -> (-1/10,0/1) Hyperbolic Matrix(639,404,-1520,-961) (-19/30,-12/19) -> (-8/19,-21/50) Hyperbolic Matrix(121,76,320,201) (-12/19,-5/8) -> (3/8,8/21) Hyperbolic Matrix(119,74,320,199) (-5/8,-13/21) -> (7/19,3/8) Hyperbolic Matrix(359,222,-1360,-841) (-13/21,-21/34) -> (-9/34,-5/19) Hyperbolic Matrix(81,50,520,321) (-21/34,-8/13) -> (2/13,1/6) Hyperbolic Matrix(281,172,-1240,-759) (-19/31,-11/18) -> (-5/22,-7/31) Hyperbolic Matrix(119,72,-200,-121) (-11/18,-3/5) -> (-3/5,-13/22) Parabolic Matrix(1761,1040,2440,1441) (-13/22,-36/61) -> (44/61,13/18) Hyperbolic Matrix(7321,4320,12000,7081) (-36/61,-59/100) -> (61/100,36/59) Hyperbolic Matrix(4879,2878,8000,4719) (-59/100,-23/39) -> (25/41,61/100) Hyperbolic Matrix(961,566,1360,801) (-23/39,-10/17) -> (12/17,29/41) Hyperbolic Matrix(559,328,680,399) (-10/17,-17/29) -> (9/11,14/17) Hyperbolic Matrix(1041,610,1640,961) (-17/29,-24/41) -> (26/41,7/11) Hyperbolic Matrix(3599,2106,4920,2879) (-24/41,-31/53) -> (49/67,30/41) Hyperbolic Matrix(879,514,1640,959) (-31/53,-7/12) -> (15/28,7/13) Hyperbolic Matrix(921,536,1720,1001) (-7/12,-32/55) -> (8/15,15/28) Hyperbolic Matrix(761,442,-1720,-999) (-25/43,-18/31) -> (-4/9,-19/43) Hyperbolic Matrix(559,324,-1520,-881) (-18/31,-11/19) -> (-7/19,-18/49) Hyperbolic Matrix(599,346,760,439) (-11/19,-15/26) -> (11/14,15/19) Hyperbolic Matrix(441,254,-1040,-599) (-15/26,-4/7) -> (-14/33,-11/26) Hyperbolic Matrix(81,46,-280,-159) (-4/7,-9/16) -> (-7/24,-2/7) Hyperbolic Matrix(1161,652,1640,921) (-9/16,-23/41) -> (29/41,17/24) Hyperbolic Matrix(1281,718,2000,1121) (-23/41,-14/25) -> (16/25,25/39) Hyperbolic Matrix(1681,940,2720,1521) (-14/25,-19/34) -> (21/34,34/55) Hyperbolic Matrix(721,402,-1720,-959) (-19/34,-5/9) -> (-13/31,-31/74) Hyperbolic Matrix(199,110,360,199) (-5/9,-11/20) -> (11/20,5/9) Hyperbolic Matrix(241,132,440,241) (-11/20,-6/11) -> (6/11,11/20) Hyperbolic Matrix(81,44,-440,-239) (-6/11,-7/13) -> (-5/27,-2/11) Hyperbolic Matrix(959,514,1640,879) (-7/13,-15/28) -> (7/12,31/53) Hyperbolic Matrix(1239,662,1720,919) (-15/28,-8/15) -> (18/25,49/68) Hyperbolic Matrix(921,490,1280,681) (-8/15,-9/17) -> (41/57,18/25) Hyperbolic Matrix(519,274,680,359) (-9/17,-1/2) -> (29/38,13/17) Hyperbolic Matrix(199,94,760,359) (-1/2,-8/17) -> (6/23,11/42) Hyperbolic Matrix(521,244,600,281) (-8/17,-7/15) -> (13/15,20/23) Hyperbolic Matrix(719,334,1720,799) (-7/15,-13/28) -> (5/12,23/55) Hyperbolic Matrix(681,316,1640,761) (-13/28,-6/13) -> (22/53,5/12) Hyperbolic Matrix(161,74,-520,-239) (-6/13,-5/11) -> (-9/29,-4/13) Hyperbolic Matrix(199,90,440,199) (-5/11,-9/20) -> (9/20,5/11) Hyperbolic Matrix(161,72,360,161) (-9/20,-4/9) -> (4/9,9/20) Hyperbolic Matrix(2999,1324,-7160,-3161) (-19/43,-15/34) -> (-31/74,-49/117) Hyperbolic Matrix(999,440,1360,599) (-15/34,-11/25) -> (11/15,25/34) Hyperbolic Matrix(1201,528,1640,721) (-11/25,-18/41) -> (30/41,11/15) Hyperbolic Matrix(721,316,1280,561) (-18/41,-7/16) -> (9/16,22/39) Hyperbolic Matrix(719,314,1280,559) (-7/16,-17/39) -> (23/41,9/16) Hyperbolic Matrix(1121,488,1840,801) (-17/39,-10/23) -> (14/23,25/41) Hyperbolic Matrix(921,392,1600,681) (-3/7,-17/40) -> (23/40,19/33) Hyperbolic Matrix(919,390,1600,679) (-17/40,-14/33) -> (4/7,23/40) Hyperbolic Matrix(161,68,760,321) (-11/26,-8/19) -> (4/19,3/14) Hyperbolic Matrix(81,34,-760,-319) (-21/50,-13/31) -> (-1/9,-1/10) Hyperbolic Matrix(14881,6232,25600,10721) (-49/117,-67/160) -> (93/160,25/43) Hyperbolic Matrix(14879,6230,25600,10719) (-67/160,-18/43) -> (68/117,93/160) Hyperbolic Matrix(961,402,-3440,-1439) (-18/43,-5/12) -> (-19/68,-12/43) Hyperbolic Matrix(761,316,1640,681) (-5/12,-22/53) -> (6/13,13/28) Hyperbolic Matrix(2719,1128,4240,1759) (-22/53,-17/41) -> (25/39,34/53) Hyperbolic Matrix(599,248,1640,679) (-17/41,-12/29) -> (4/11,15/41) Hyperbolic Matrix(121,50,680,281) (-12/29,-7/17) -> (3/17,2/11) Hyperbolic Matrix(79,32,-200,-81) (-7/17,-2/5) -> (-2/5,-9/23) Parabolic Matrix(1039,406,1840,719) (-9/23,-16/41) -> (22/39,13/23) Hyperbolic Matrix(3281,1280,8000,3121) (-16/41,-39/100) -> (41/100,16/39) Hyperbolic Matrix(4919,1918,12000,4679) (-39/100,-23/59) -> (25/61,41/100) Hyperbolic Matrix(1119,436,1640,639) (-23/59,-7/18) -> (15/22,13/19) Hyperbolic Matrix(119,46,-520,-201) (-7/18,-5/13) -> (-3/13,-5/22) Hyperbolic Matrix(439,168,520,199) (-5/13,-13/34) -> (5/6,11/13) Hyperbolic Matrix(1199,458,2720,1039) (-13/34,-21/55) -> (11/25,15/34) Hyperbolic Matrix(201,76,320,121) (-8/21,-3/8) -> (5/8,12/19) Hyperbolic Matrix(199,74,320,119) (-3/8,-7/19) -> (13/21,5/8) Hyperbolic Matrix(1319,484,-3600,-1321) (-18/49,-11/30) -> (-11/30,-26/71) Parabolic Matrix(679,248,1640,599) (-15/41,-4/11) -> (12/29,17/41) Hyperbolic Matrix(879,316,2000,719) (-9/25,-14/39) -> (18/41,11/25) Hyperbolic Matrix(2481,890,4240,1521) (-14/39,-19/53) -> (31/53,24/41) Hyperbolic Matrix(721,258,-2680,-959) (-19/53,-5/14) -> (-7/26,-25/93) Hyperbolic Matrix(439,156,560,199) (-5/14,-6/17) -> (18/23,11/14) Hyperbolic Matrix(239,84,680,239) (-6/17,-7/20) -> (7/20,6/17) Hyperbolic Matrix(41,14,120,41) (-7/20,-1/3) -> (1/3,7/20) Hyperbolic Matrix(1001,318,1640,521) (-7/22,-6/19) -> (36/59,11/18) Hyperbolic Matrix(159,50,760,239) (-6/19,-5/16) -> (5/24,4/19) Hyperbolic Matrix(119,36,-400,-121) (-4/13,-3/10) -> (-3/10,-8/27) Parabolic Matrix(521,154,680,201) (-8/27,-5/17) -> (13/17,10/13) Hyperbolic Matrix(559,164,1360,399) (-5/17,-12/41) -> (16/39,7/17) Hyperbolic Matrix(719,210,1640,479) (-12/41,-7/24) -> (7/16,18/41) Hyperbolic Matrix(599,168,1280,359) (-16/57,-7/25) -> (7/15,8/17) Hyperbolic Matrix(801,224,1720,481) (-7/25,-19/68) -> (13/28,7/15) Hyperbolic Matrix(999,278,2440,679) (-17/61,-5/18) -> (9/22,25/61) Hyperbolic Matrix(1161,320,1600,441) (-8/29,-11/40) -> (29/40,8/11) Hyperbolic Matrix(1159,318,1600,439) (-11/40,-3/11) -> (21/29,29/40) Hyperbolic Matrix(18721,5032,25600,6881) (-25/93,-43/160) -> (117/160,49/67) Hyperbolic Matrix(18719,5030,25600,6879) (-43/160,-18/67) -> (68/93,117/160) Hyperbolic Matrix(2041,548,4920,1321) (-18/67,-11/41) -> (17/41,22/53) Hyperbolic Matrix(919,246,1640,439) (-11/41,-4/15) -> (14/25,23/41) Hyperbolic Matrix(761,202,1360,361) (-4/15,-9/34) -> (19/34,14/25) Hyperbolic Matrix(39,10,-160,-41) (-5/19,-1/4) -> (-1/4,-5/21) Parabolic Matrix(4881,1160,6400,1521) (-5/21,-19/80) -> (61/80,45/59) Hyperbolic Matrix(4879,1158,6400,1519) (-19/80,-14/59) -> (16/21,61/80) Hyperbolic Matrix(321,76,680,161) (-9/38,-4/17) -> (8/17,1/2) Hyperbolic Matrix(479,112,680,159) (-4/17,-3/13) -> (19/27,12/17) Hyperbolic Matrix(1241,280,1600,361) (-7/31,-9/40) -> (31/40,7/9) Hyperbolic Matrix(1239,278,1600,359) (-9/40,-2/9) -> (24/31,31/40) Hyperbolic Matrix(319,70,360,79) (-2/9,-7/32) -> (7/8,8/9) Hyperbolic Matrix(119,26,-920,-201) (-7/32,-5/23) -> (-3/23,-1/8) Hyperbolic Matrix(361,78,560,121) (-5/23,-3/14) -> (9/14,11/17) Hyperbolic Matrix(321,68,760,161) (-3/14,-4/19) -> (8/19,11/26) Hyperbolic Matrix(39,8,-200,-41) (-4/19,-1/5) -> (-1/5,-4/21) Parabolic Matrix(201,38,640,121) (-4/21,-3/16) -> (5/16,6/19) Hyperbolic Matrix(281,50,680,121) (-2/11,-3/17) -> (7/17,12/29) Hyperbolic Matrix(1321,232,1600,281) (-3/17,-7/40) -> (33/40,19/23) Hyperbolic Matrix(1319,230,1600,279) (-7/40,-4/23) -> (14/17,33/40) Hyperbolic Matrix(321,50,520,81) (-1/6,-2/13) -> (8/13,21/34) Hyperbolic Matrix(79,12,520,79) (-2/13,-3/20) -> (3/20,2/13) Hyperbolic Matrix(41,6,280,41) (-3/20,-1/7) -> (1/7,3/20) Hyperbolic Matrix(319,42,600,79) (-2/15,-3/23) -> (9/17,8/15) Hyperbolic Matrix(281,32,360,41) (-1/8,-1/9) -> (7/9,25/32) Hyperbolic Matrix(279,-26,440,-41) (0/1,1/9) -> (45/71,26/41) Hyperbolic Matrix(201,-28,280,-39) (1/8,1/7) -> (5/7,23/32) Hyperbolic Matrix(199,-34,240,-41) (1/6,3/17) -> (19/23,5/6) Hyperbolic Matrix(441,-82,640,-119) (2/11,3/16) -> (11/16,20/29) Hyperbolic Matrix(41,-8,200,-39) (3/16,1/5) -> (1/5,5/24) Parabolic Matrix(1841,-402,2560,-559) (5/23,7/32) -> (23/32,41/57) Hyperbolic Matrix(201,-46,520,-119) (2/9,3/13) -> (5/13,12/31) Hyperbolic Matrix(41,-10,160,-39) (4/17,1/4) -> (1/4,6/23) Parabolic Matrix(2441,-640,3200,-839) (11/42,5/19) -> (45/59,29/38) Hyperbolic Matrix(841,-222,1360,-359) (5/19,4/15) -> (34/55,13/21) Hyperbolic Matrix(281,-76,440,-119) (4/15,3/11) -> (7/11,16/25) Hyperbolic Matrix(521,-144,720,-199) (3/11,5/18) -> (13/18,21/29) Hyperbolic Matrix(681,-190,1000,-279) (5/18,12/43) -> (2/3,15/22) Hyperbolic Matrix(1439,-402,3440,-961) (12/43,7/25) -> (23/55,18/43) Hyperbolic Matrix(241,-68,280,-79) (7/25,2/7) -> (6/7,13/15) Hyperbolic Matrix(159,-46,280,-81) (2/7,5/17) -> (13/23,4/7) Hyperbolic Matrix(121,-36,400,-119) (5/17,3/10) -> (3/10,7/23) Parabolic Matrix(521,-162,640,-199) (4/13,5/16) -> (13/16,22/27) Hyperbolic Matrix(721,-230,1000,-319) (6/19,1/3) -> (31/43,44/61) Hyperbolic Matrix(321,-116,440,-159) (5/14,4/11) -> (8/11,19/26) Hyperbolic Matrix(399,-146,440,-161) (15/41,11/30) -> (9/10,1/1) Hyperbolic Matrix(881,-324,1520,-559) (11/30,7/19) -> (11/19,29/50) Hyperbolic Matrix(1001,-382,1360,-519) (8/21,13/34) -> (25/34,14/19) Hyperbolic Matrix(959,-372,1240,-481) (12/31,7/18) -> (17/22,24/31) Hyperbolic Matrix(81,-32,200,-79) (7/18,2/5) -> (2/5,9/22) Parabolic Matrix(959,-402,1720,-721) (18/43,13/31) -> (5/9,24/43) Hyperbolic Matrix(961,-404,1520,-639) (13/31,8/19) -> (12/19,31/49) Hyperbolic Matrix(599,-254,1040,-441) (11/26,3/7) -> (19/33,15/26) Hyperbolic Matrix(199,-86,280,-121) (3/7,7/16) -> (17/24,5/7) Hyperbolic Matrix(999,-442,1720,-761) (15/34,4/9) -> (18/31,43/74) Hyperbolic Matrix(359,-164,440,-201) (5/11,6/13) -> (22/27,9/11) Hyperbolic Matrix(359,-194,520,-281) (7/13,6/11) -> (20/29,9/13) Hyperbolic Matrix(4161,-2324,7160,-3999) (24/43,19/34) -> (43/74,68/117) Hyperbolic Matrix(679,-394,760,-441) (29/50,18/31) -> (8/9,9/10) Hyperbolic Matrix(2479,-1442,3440,-2001) (25/43,7/12) -> (49/68,31/43) Hyperbolic Matrix(121,-72,200,-119) (10/17,3/5) -> (3/5,14/23) Parabolic Matrix(401,-246,520,-319) (11/18,8/13) -> (10/13,17/22) Hyperbolic Matrix(2281,-1444,3600,-2279) (31/49,19/30) -> (19/30,45/71) Parabolic Matrix(1959,-1258,2680,-1721) (34/53,9/14) -> (19/26,68/93) Hyperbolic Matrix(281,-196,400,-279) (9/13,7/10) -> (7/10,19/27) Parabolic Matrix(121,-90,160,-119) (14/19,3/4) -> (3/4,16/21) Parabolic Matrix(801,-626,920,-719) (25/32,18/23) -> (20/23,7/8) Hyperbolic Matrix(161,-128,200,-159) (15/19,4/5) -> (4/5,17/21) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,0,1) Matrix(161,146,-440,-399) -> Matrix(3,2,-8,-5) Matrix(79,70,360,319) -> Matrix(1,0,0,1) Matrix(79,68,-280,-241) -> Matrix(1,2,-4,-7) Matrix(239,204,280,239) -> Matrix(7,8,-8,-9) Matrix(441,374,520,441) -> Matrix(21,20,-20,-19) Matrix(199,168,520,439) -> Matrix(7,6,8,7) Matrix(41,34,-240,-199) -> Matrix(3,2,-8,-5) Matrix(399,328,680,559) -> Matrix(3,2,4,3) Matrix(199,162,-640,-521) -> Matrix(1,0,0,1) Matrix(159,128,-200,-161) -> Matrix(3,2,-8,-5) Matrix(521,412,760,601) -> Matrix(1,0,4,1) Matrix(439,346,760,599) -> Matrix(1,0,0,1) Matrix(199,156,560,439) -> Matrix(1,0,4,1) Matrix(719,562,-2560,-2001) -> Matrix(1,2,-4,-7) Matrix(41,32,360,281) -> Matrix(1,0,0,1) Matrix(319,246,-520,-401) -> Matrix(5,2,-8,-3) Matrix(159,122,520,399) -> Matrix(1,-2,0,1) Matrix(119,90,-160,-121) -> Matrix(3,4,-4,-5) Matrix(401,296,760,561) -> Matrix(5,4,-4,-3) Matrix(759,560,-3200,-2361) -> Matrix(3,2,4,3) Matrix(519,382,-1360,-1001) -> Matrix(3,2,-8,-5) Matrix(159,116,-440,-321) -> Matrix(3,2,-8,-5) Matrix(199,144,-720,-521) -> Matrix(3,2,-8,-5) Matrix(319,230,-1000,-721) -> Matrix(5,4,-4,-3) Matrix(2001,1442,-3440,-2479) -> Matrix(5,4,-4,-3) Matrix(39,28,-280,-201) -> Matrix(3,2,4,3) Matrix(121,86,-280,-199) -> Matrix(3,2,-8,-5) Matrix(279,196,-400,-281) -> Matrix(11,6,-24,-13) Matrix(121,84,520,361) -> Matrix(5,2,-8,-3) Matrix(119,82,-640,-441) -> Matrix(1,0,0,1) Matrix(519,356,640,439) -> Matrix(7,2,-4,-1) Matrix(279,190,-1000,-681) -> Matrix(1,0,-4,1) Matrix(79,52,120,79) -> Matrix(1,0,0,1) Matrix(441,286,680,441) -> Matrix(13,12,-12,-11) Matrix(121,78,560,361) -> Matrix(5,4,-4,-3) Matrix(119,76,-440,-281) -> Matrix(3,2,-8,-5) Matrix(961,610,1640,1041) -> Matrix(3,2,4,3) Matrix(41,26,-440,-279) -> Matrix(3,2,-8,-5) Matrix(639,404,-1520,-961) -> Matrix(1,0,0,1) Matrix(121,76,320,201) -> Matrix(3,2,4,3) Matrix(119,74,320,199) -> Matrix(3,2,4,3) Matrix(359,222,-1360,-841) -> Matrix(3,2,-8,-5) Matrix(81,50,520,321) -> Matrix(3,2,16,11) Matrix(281,172,-1240,-759) -> Matrix(3,2,-8,-5) Matrix(119,72,-200,-121) -> Matrix(7,4,-16,-9) Matrix(1761,1040,2440,1441) -> Matrix(5,2,-8,-3) Matrix(7321,4320,12000,7081) -> Matrix(201,68,-68,-23) Matrix(4879,2878,8000,4719) -> Matrix(123,40,-40,-13) Matrix(961,566,1360,801) -> Matrix(15,4,-4,-1) Matrix(559,328,680,399) -> Matrix(7,2,-4,-1) Matrix(1041,610,1640,961) -> Matrix(7,2,-4,-1) Matrix(3599,2106,4920,2879) -> Matrix(7,2,-4,-1) Matrix(879,514,1640,959) -> Matrix(7,2,-4,-1) Matrix(921,536,1720,1001) -> Matrix(1,-2,0,1) Matrix(761,442,-1720,-999) -> Matrix(1,0,0,1) Matrix(559,324,-1520,-881) -> Matrix(1,0,0,1) Matrix(599,346,760,439) -> Matrix(1,0,0,1) Matrix(441,254,-1040,-599) -> Matrix(1,0,0,1) Matrix(81,46,-280,-159) -> Matrix(3,2,-8,-5) Matrix(1161,652,1640,921) -> Matrix(1,-2,0,1) Matrix(1281,718,2000,1121) -> Matrix(5,4,-4,-3) Matrix(1681,940,2720,1521) -> Matrix(7,2,-4,-1) Matrix(721,402,-1720,-959) -> Matrix(1,0,0,1) Matrix(199,110,360,199) -> Matrix(1,0,0,1) Matrix(241,132,440,241) -> Matrix(5,4,-4,-3) Matrix(81,44,-440,-239) -> Matrix(3,2,-8,-5) Matrix(959,514,1640,879) -> Matrix(3,2,4,3) Matrix(1239,662,1720,919) -> Matrix(17,10,-12,-7) Matrix(921,490,1280,681) -> Matrix(7,2,-4,-1) Matrix(519,274,680,359) -> Matrix(1,0,0,1) Matrix(199,94,760,359) -> Matrix(1,0,4,1) Matrix(521,244,600,281) -> Matrix(1,0,0,1) Matrix(719,334,1720,799) -> Matrix(5,2,-8,-3) Matrix(681,316,1640,761) -> Matrix(7,2,-4,-1) Matrix(161,74,-520,-239) -> Matrix(5,2,-8,-3) Matrix(199,90,440,199) -> Matrix(1,0,4,1) Matrix(161,72,360,161) -> Matrix(1,0,0,1) Matrix(2999,1324,-7160,-3161) -> Matrix(1,0,0,1) Matrix(999,440,1360,599) -> Matrix(5,4,-4,-3) Matrix(1201,528,1640,721) -> Matrix(7,2,-4,-1) Matrix(721,316,1280,561) -> Matrix(1,0,0,1) Matrix(719,314,1280,559) -> Matrix(1,0,0,1) Matrix(1121,488,1840,801) -> Matrix(13,10,-4,-3) Matrix(921,392,1600,681) -> Matrix(1,0,0,1) Matrix(919,390,1600,679) -> Matrix(1,0,0,1) Matrix(161,68,760,321) -> Matrix(1,0,0,1) Matrix(81,34,-760,-319) -> Matrix(1,0,0,1) Matrix(14881,6232,25600,10721) -> Matrix(1,0,0,1) Matrix(14879,6230,25600,10719) -> Matrix(1,0,0,1) Matrix(961,402,-3440,-1439) -> Matrix(1,0,-4,1) Matrix(761,316,1640,681) -> Matrix(3,2,4,3) Matrix(2719,1128,4240,1759) -> Matrix(5,4,-4,-3) Matrix(599,248,1640,679) -> Matrix(3,2,4,3) Matrix(121,50,680,281) -> Matrix(3,2,4,3) Matrix(79,32,-200,-81) -> Matrix(7,4,-16,-9) Matrix(1039,406,1840,719) -> Matrix(5,2,12,5) Matrix(3281,1280,8000,3121) -> Matrix(139,56,-72,-29) Matrix(4919,1918,12000,4679) -> Matrix(221,88,-108,-43) Matrix(1119,436,1640,639) -> Matrix(5,2,-28,-11) Matrix(119,46,-520,-201) -> Matrix(5,2,-8,-3) Matrix(439,168,520,199) -> Matrix(19,6,-16,-5) Matrix(1199,458,2720,1039) -> Matrix(7,2,-4,-1) Matrix(201,76,320,121) -> Matrix(7,2,-4,-1) Matrix(199,74,320,119) -> Matrix(7,2,-4,-1) Matrix(1319,484,-3600,-1321) -> Matrix(3,2,-8,-5) Matrix(679,248,1640,599) -> Matrix(7,2,-4,-1) Matrix(879,316,2000,719) -> Matrix(1,0,4,1) Matrix(2481,890,4240,1521) -> Matrix(1,0,4,1) Matrix(721,258,-2680,-959) -> Matrix(1,0,0,1) Matrix(439,156,560,199) -> Matrix(1,0,4,1) Matrix(239,84,680,239) -> Matrix(1,0,12,1) Matrix(41,14,120,41) -> Matrix(1,0,0,1) Matrix(1001,318,1640,521) -> Matrix(11,14,-4,-5) Matrix(159,50,760,239) -> Matrix(5,4,-4,-3) Matrix(119,36,-400,-121) -> Matrix(11,6,-24,-13) Matrix(521,154,680,201) -> Matrix(9,4,-16,-7) Matrix(559,164,1360,399) -> Matrix(19,8,-12,-5) Matrix(719,210,1640,479) -> Matrix(5,2,-8,-3) Matrix(599,168,1280,359) -> Matrix(7,2,-4,-1) Matrix(801,224,1720,481) -> Matrix(9,2,4,1) Matrix(999,278,2440,679) -> Matrix(1,-2,0,1) Matrix(1161,320,1600,441) -> Matrix(7,2,-4,-1) Matrix(1159,318,1600,439) -> Matrix(7,2,-4,-1) Matrix(18721,5032,25600,6881) -> Matrix(7,2,-4,-1) Matrix(18719,5030,25600,6879) -> Matrix(7,2,-4,-1) Matrix(2041,548,4920,1321) -> Matrix(7,2,-4,-1) Matrix(919,246,1640,439) -> Matrix(7,2,-4,-1) Matrix(761,202,1360,361) -> Matrix(1,0,4,1) Matrix(39,10,-160,-41) -> Matrix(1,0,4,1) Matrix(4881,1160,6400,1521) -> Matrix(3,-2,-4,3) Matrix(4879,1158,6400,1519) -> Matrix(3,-2,-4,3) Matrix(321,76,680,161) -> Matrix(1,0,0,1) Matrix(479,112,680,159) -> Matrix(1,-4,0,1) Matrix(1241,280,1600,361) -> Matrix(1,0,0,1) Matrix(1239,278,1600,359) -> Matrix(1,0,0,1) Matrix(319,70,360,79) -> Matrix(1,0,0,1) Matrix(119,26,-920,-201) -> Matrix(1,0,0,1) Matrix(361,78,560,121) -> Matrix(5,4,-4,-3) Matrix(321,68,760,161) -> Matrix(1,0,0,1) Matrix(39,8,-200,-41) -> Matrix(3,2,-8,-5) Matrix(201,38,640,121) -> Matrix(7,2,-4,-1) Matrix(281,50,680,121) -> Matrix(7,2,-4,-1) Matrix(1321,232,1600,281) -> Matrix(7,2,-4,-1) Matrix(1319,230,1600,279) -> Matrix(7,2,-4,-1) Matrix(321,50,520,81) -> Matrix(15,2,-8,-1) Matrix(79,12,520,79) -> Matrix(1,0,20,1) Matrix(41,6,280,41) -> Matrix(1,0,-8,1) Matrix(319,42,600,79) -> Matrix(1,0,0,1) Matrix(281,32,360,41) -> Matrix(1,0,0,1) Matrix(279,-26,440,-41) -> Matrix(1,-2,0,1) Matrix(201,-28,280,-39) -> Matrix(7,2,-4,-1) Matrix(199,-34,240,-41) -> Matrix(1,-2,0,1) Matrix(441,-82,640,-119) -> Matrix(1,0,0,1) Matrix(41,-8,200,-39) -> Matrix(1,-2,0,1) Matrix(1841,-402,2560,-559) -> Matrix(7,2,-4,-1) Matrix(201,-46,520,-119) -> Matrix(1,2,0,1) Matrix(41,-10,160,-39) -> Matrix(1,0,4,1) Matrix(2441,-640,3200,-839) -> Matrix(3,-2,-4,3) Matrix(841,-222,1360,-359) -> Matrix(1,-2,0,1) Matrix(281,-76,440,-119) -> Matrix(1,-2,0,1) Matrix(521,-144,720,-199) -> Matrix(1,-2,0,1) Matrix(681,-190,1000,-279) -> Matrix(1,0,-4,1) Matrix(1439,-402,3440,-961) -> Matrix(1,0,-4,1) Matrix(241,-68,280,-79) -> Matrix(3,-2,-4,3) Matrix(159,-46,280,-81) -> Matrix(1,-2,0,1) Matrix(121,-36,400,-119) -> Matrix(1,-6,0,1) Matrix(521,-162,640,-199) -> Matrix(1,0,0,1) Matrix(721,-230,1000,-319) -> Matrix(5,4,-4,-3) Matrix(321,-116,440,-159) -> Matrix(1,-2,0,1) Matrix(399,-146,440,-161) -> Matrix(1,-2,0,1) Matrix(881,-324,1520,-559) -> Matrix(1,0,0,1) Matrix(1001,-382,1360,-519) -> Matrix(1,-2,0,1) Matrix(959,-372,1240,-481) -> Matrix(1,-2,0,1) Matrix(81,-32,200,-79) -> Matrix(1,-4,0,1) Matrix(959,-402,1720,-721) -> Matrix(1,0,0,1) Matrix(961,-404,1520,-639) -> Matrix(1,0,0,1) Matrix(599,-254,1040,-441) -> Matrix(1,0,0,1) Matrix(199,-86,280,-121) -> Matrix(1,-2,0,1) Matrix(999,-442,1720,-761) -> Matrix(1,0,0,1) Matrix(359,-164,440,-201) -> Matrix(1,-2,0,1) Matrix(359,-194,520,-281) -> Matrix(1,2,0,1) Matrix(4161,-2324,7160,-3999) -> Matrix(1,0,0,1) Matrix(679,-394,760,-441) -> Matrix(1,0,0,1) Matrix(2479,-1442,3440,-2001) -> Matrix(5,4,-4,-3) Matrix(121,-72,200,-119) -> Matrix(1,-4,0,1) Matrix(401,-246,520,-319) -> Matrix(1,2,0,1) Matrix(2281,-1444,3600,-2279) -> Matrix(1,-2,0,1) Matrix(1959,-1258,2680,-1721) -> Matrix(1,0,0,1) Matrix(281,-196,400,-279) -> Matrix(1,-6,0,1) Matrix(121,-90,160,-119) -> Matrix(3,4,-4,-5) Matrix(801,-626,920,-719) -> Matrix(1,0,0,1) Matrix(161,-128,200,-159) -> Matrix(1,-2,0,1) 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): 24 Degree of the the map X: 48 Degree of the the map Y: 192 ----------------------------------------------------------------------- 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 0/1 1/7 1/6 1/5 3/14 2/9 1/4 5/18 3/10 1/3 11/30 3/8 2/5 5/12 7/16 1/2 11/20 9/16 23/40 7/12 3/5 61/100 5/8 19/30 13/20 7/10 29/40 3/4 4/5 33/40 17/20 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/1 0/1 1/7 -1/5 0/1 1/6 1/2 3/17 0/1 1/1 2/11 0/1 1/1 1/5 1/0 4/19 -1/1 -2/3 3/14 1/0 2/9 -1/1 0/1 1/4 0/1 3/11 0/1 1/1 5/18 1/0 2/7 1/1 2/1 3/10 1/0 4/13 -2/1 -1/1 1/3 -1/1 0/1 5/14 1/2 4/11 0/1 1/1 11/30 1/0 7/19 0/1 1/1 3/8 1/2 1/0 8/21 0/1 1/1 5/13 1/1 4/3 7/18 3/2 2/5 1/0 9/22 -5/2 16/39 -2/1 -13/7 7/17 -2/1 -1/1 12/29 -2/1 -1/1 5/12 -1/1 13/31 -1/1 0/1 8/19 -2/1 -1/1 11/26 -1/2 3/7 -1/1 0/1 7/16 -1/2 1/0 18/41 0/1 1/3 11/25 1/0 4/9 -1/1 0/1 1/2 1/0 6/11 -2/1 -1/1 11/20 -1/1 5/9 -1/1 0/1 9/16 -1/2 1/0 22/39 -1/5 0/1 13/23 0/1 1/1 4/7 -1/1 0/1 23/40 -1/2 1/0 19/33 -1/1 0/1 15/26 -1/2 11/19 0/1 1/1 29/50 1/0 18/31 -1/1 0/1 7/12 0/1 17/29 0/1 1/1 10/17 0/1 1/1 3/5 1/0 14/23 -4/1 -3/1 25/41 -22/7 -3/1 61/100 -3/1 36/59 -3/1 -32/11 11/18 -5/2 8/13 -7/3 -2/1 13/21 -2/1 -1/1 5/8 -3/2 1/0 12/19 -2/1 -1/1 19/30 1/0 26/41 -3/1 -2/1 7/11 -2/1 -1/1 9/14 -3/2 11/17 -6/5 -1/1 13/20 -1/1 2/3 -1/1 0/1 7/10 1/0 12/17 -4/1 -3/1 29/41 -3/1 -14/5 17/24 -5/2 1/0 5/7 -3/1 -2/1 13/18 1/0 21/29 -2/1 -1/1 29/40 -3/2 1/0 8/11 -2/1 -1/1 3/4 -1/1 7/9 -1/1 0/1 11/14 1/0 15/19 -1/3 0/1 4/5 1/0 17/21 -7/3 -2/1 13/16 -3/2 1/0 9/11 -2/1 -1/1 14/17 -2/1 -1/1 33/40 -3/2 1/0 19/23 -2/1 -1/1 5/6 -3/2 11/13 -10/9 -1/1 17/20 -1/1 6/7 -1/1 -4/5 1/1 -1/1 0/1 1/0 -1/2 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(99,-13,160,-21) (0/1,1/7) -> (8/13,13/21) Hyperbolic Matrix(101,-16,120,-19) (1/7,1/6) -> (5/6,11/13) Hyperbolic Matrix(199,-34,240,-41) (1/6,3/17) -> (19/23,5/6) Hyperbolic Matrix(399,-71,680,-121) (3/17,2/11) -> (17/29,10/17) Hyperbolic Matrix(141,-26,320,-59) (2/11,1/5) -> (11/25,4/9) Hyperbolic Matrix(299,-62,680,-141) (1/5,4/19) -> (18/41,11/25) Hyperbolic Matrix(439,-93,760,-161) (4/19,3/14) -> (15/26,11/19) Hyperbolic Matrix(179,-39,280,-61) (3/14,2/9) -> (7/11,9/14) Hyperbolic Matrix(21,-5,80,-19) (2/9,1/4) -> (1/4,3/11) Parabolic Matrix(521,-144,720,-199) (3/11,5/18) -> (13/18,21/29) Hyperbolic Matrix(139,-39,360,-101) (5/18,2/7) -> (5/13,7/18) Hyperbolic Matrix(61,-18,200,-59) (2/7,3/10) -> (3/10,4/13) Parabolic Matrix(181,-56,320,-99) (4/13,1/3) -> (13/23,4/7) Hyperbolic Matrix(181,-64,280,-99) (1/3,5/14) -> (9/14,11/17) Hyperbolic Matrix(219,-79,280,-101) (5/14,4/11) -> (7/9,11/14) Hyperbolic Matrix(859,-314,1480,-541) (4/11,11/30) -> (29/50,18/31) Hyperbolic Matrix(881,-324,1520,-559) (11/30,7/19) -> (11/19,29/50) Hyperbolic Matrix(121,-45,320,-119) (7/19,3/8) -> (3/8,8/21) Parabolic Matrix(139,-53,160,-61) (8/21,5/13) -> (6/7,1/1) Hyperbolic Matrix(81,-32,200,-79) (7/18,2/5) -> (2/5,9/22) Parabolic Matrix(1221,-500,2000,-819) (9/22,16/39) -> (36/59,11/18) Hyperbolic Matrix(961,-395,1360,-559) (16/39,7/17) -> (12/17,29/41) Hyperbolic Matrix(559,-231,680,-281) (7/17,12/29) -> (9/11,14/17) Hyperbolic Matrix(301,-125,720,-299) (12/29,5/12) -> (5/12,13/31) Parabolic Matrix(939,-394,1480,-621) (13/31,8/19) -> (26/41,7/11) Hyperbolic Matrix(599,-253,760,-321) (8/19,11/26) -> (11/14,15/19) Hyperbolic Matrix(599,-254,1040,-441) (11/26,3/7) -> (19/33,15/26) Hyperbolic Matrix(199,-86,280,-121) (3/7,7/16) -> (17/24,5/7) Hyperbolic Matrix(1161,-509,1640,-719) (7/16,18/41) -> (29/41,17/24) Hyperbolic Matrix(21,-10,40,-19) (4/9,1/2) -> (1/2,6/11) Parabolic Matrix(221,-121,400,-219) (6/11,11/20) -> (11/20,5/9) Parabolic Matrix(261,-146,320,-179) (5/9,9/16) -> (13/16,9/11) Hyperbolic Matrix(779,-439,960,-541) (9/16,22/39) -> (17/21,13/16) Hyperbolic Matrix(1121,-633,1840,-1039) (22/39,13/23) -> (14/23,25/41) Hyperbolic Matrix(921,-529,1600,-919) (4/7,23/40) -> (23/40,19/33) Parabolic Matrix(421,-245,720,-419) (18/31,7/12) -> (7/12,17/29) Parabolic Matrix(121,-72,200,-119) (10/17,3/5) -> (3/5,14/23) Parabolic Matrix(6101,-3721,10000,-6099) (25/41,61/100) -> (61/100,36/59) Parabolic Matrix(259,-159,360,-221) (11/18,8/13) -> (5/7,13/18) Hyperbolic Matrix(201,-125,320,-199) (13/21,5/8) -> (5/8,12/19) Parabolic Matrix(1141,-722,1800,-1139) (12/19,19/30) -> (19/30,26/41) Parabolic Matrix(261,-169,400,-259) (11/17,13/20) -> (13/20,2/3) Parabolic Matrix(141,-98,200,-139) (2/3,7/10) -> (7/10,12/17) Parabolic Matrix(1161,-841,1600,-1159) (21/29,29/40) -> (29/40,8/11) Parabolic Matrix(61,-45,80,-59) (8/11,3/4) -> (3/4,7/9) Parabolic Matrix(161,-128,200,-159) (15/19,4/5) -> (4/5,17/21) Parabolic Matrix(1321,-1089,1600,-1319) (14/17,33/40) -> (33/40,19/23) Parabolic Matrix(341,-289,400,-339) (11/13,17/20) -> (17/20,6/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,1,-2,-1) Matrix(99,-13,160,-21) -> Matrix(3,2,-2,-1) Matrix(101,-16,120,-19) -> Matrix(5,-1,-4,1) Matrix(199,-34,240,-41) -> Matrix(1,-2,0,1) Matrix(399,-71,680,-121) -> Matrix(1,-1,2,-1) Matrix(141,-26,320,-59) -> Matrix(1,-1,0,1) Matrix(299,-62,680,-141) -> Matrix(1,1,0,1) Matrix(439,-93,760,-161) -> Matrix(1,1,-2,-1) Matrix(179,-39,280,-61) -> Matrix(3,2,-2,-1) Matrix(21,-5,80,-19) -> Matrix(1,0,2,1) Matrix(521,-144,720,-199) -> Matrix(1,-2,0,1) Matrix(139,-39,360,-101) -> Matrix(3,-2,2,-1) Matrix(61,-18,200,-59) -> Matrix(1,-3,0,1) Matrix(181,-56,320,-99) -> Matrix(1,1,0,1) Matrix(181,-64,280,-99) -> Matrix(5,-1,-4,1) Matrix(219,-79,280,-101) -> Matrix(1,0,-2,1) Matrix(859,-314,1480,-541) -> Matrix(1,-1,0,1) Matrix(881,-324,1520,-559) -> Matrix(1,0,0,1) Matrix(121,-45,320,-119) -> Matrix(1,-1,2,-1) Matrix(139,-53,160,-61) -> Matrix(1,0,-2,1) Matrix(81,-32,200,-79) -> Matrix(1,-4,0,1) Matrix(1221,-500,2000,-819) -> Matrix(11,25,-4,-9) Matrix(961,-395,1360,-559) -> Matrix(7,11,-2,-3) Matrix(559,-231,680,-281) -> Matrix(3,5,-2,-3) Matrix(301,-125,720,-299) -> Matrix(1,2,-2,-3) Matrix(939,-394,1480,-621) -> Matrix(1,-1,0,1) Matrix(599,-253,760,-321) -> Matrix(1,1,-2,-1) Matrix(599,-254,1040,-441) -> Matrix(1,0,0,1) Matrix(199,-86,280,-121) -> Matrix(1,-2,0,1) Matrix(1161,-509,1640,-719) -> Matrix(5,3,-2,-1) Matrix(21,-10,40,-19) -> Matrix(1,-1,0,1) Matrix(221,-121,400,-219) -> Matrix(1,2,-2,-3) Matrix(261,-146,320,-179) -> Matrix(1,-1,0,1) Matrix(779,-439,960,-541) -> Matrix(3,2,-2,-1) Matrix(1121,-633,1840,-1039) -> Matrix(7,-3,-2,1) Matrix(921,-529,1600,-919) -> Matrix(1,1,-2,-1) Matrix(421,-245,720,-419) -> Matrix(1,0,2,1) Matrix(121,-72,200,-119) -> Matrix(1,-4,0,1) Matrix(6101,-3721,10000,-6099) -> Matrix(53,162,-18,-55) Matrix(259,-159,360,-221) -> Matrix(3,8,-2,-5) Matrix(201,-125,320,-199) -> Matrix(3,5,-2,-3) Matrix(1141,-722,1800,-1139) -> Matrix(1,-1,0,1) Matrix(261,-169,400,-259) -> Matrix(5,6,-6,-7) Matrix(141,-98,200,-139) -> Matrix(1,-3,0,1) Matrix(1161,-841,1600,-1159) -> Matrix(3,5,-2,-3) Matrix(61,-45,80,-59) -> Matrix(1,2,-2,-3) Matrix(161,-128,200,-159) -> Matrix(1,-2,0,1) Matrix(1321,-1089,1600,-1319) -> Matrix(3,5,-2,-3) Matrix(341,-289,400,-339) -> Matrix(13,14,-14,-15) 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): 24 ----------------------------------------------------------------------- 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 20 1/7 (-1/5,0/1) 0 20 3/20 0/1 14 1 1/6 1/2 1 10 7/40 (0/1,1/1).(1/2,1/0) 0 1 3/17 (0/1,1/1) 0 20 2/11 (0/1,1/1) 0 20 1/5 1/0 1 4 4/19 (-1/1,-2/3) 0 20 3/14 1/0 1 10 2/9 (-1/1,0/1) 0 20 1/4 0/1 2 5 3/11 (0/1,1/1) 0 20 11/40 (0/1,1/1).(1/2,1/0) 0 1 5/18 1/0 1 10 2/7 (1/1,2/1) 0 20 3/10 1/0 3 2 4/13 (-2/1,-1/1) 0 20 1/3 (-1/1,0/1) 0 20 7/20 0/1 6 1 5/14 1/2 1 10 4/11 (0/1,1/1) 0 20 11/30 1/0 1 2 7/19 (0/1,1/1) 0 20 3/8 (0/1,1/1).(1/2,1/0) 0 5 8/21 (0/1,1/1) 0 20 5/13 (1/1,4/3) 0 20 7/18 3/2 1 10 2/5 1/0 2 4 9/22 -5/2 1 10 41/100 -2/1 18 1 16/39 (-2/1,-13/7) 0 20 7/17 (-2/1,-1/1) 0 20 12/29 (-2/1,-1/1) 0 20 5/12 -1/1 2 5 13/31 (-1/1,0/1) 0 20 21/50 1/0 1 2 8/19 (-2/1,-1/1) 0 20 11/26 -1/2 1 10 17/40 (-1/1,0/1).(-1/2,1/0) 0 1 3/7 (-1/1,0/1) 0 20 10/23 (-2/1,-1/1) 0 20 17/39 (-1/1,-4/5) 0 20 7/16 (-1/1,0/1).(-1/2,1/0) 0 5 18/41 (0/1,1/3) 0 20 11/25 1/0 1 4 4/9 (-1/1,0/1) 0 20 9/20 0/1 2 1 1/2 1/0 1 10 1/0 (-1/1,0/1).(-1/2,1/0) 0 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(61,-8,160,-21) (0/1,1/7) -> (8/21,5/13) Glide Reflection Matrix(41,-6,280,-41) (1/7,3/20) -> (1/7,3/20) Reflection Matrix(19,-3,120,-19) (3/20,1/6) -> (3/20,1/6) Reflection Matrix(41,-7,240,-41) (1/6,7/40) -> (1/6,7/40) Reflection Matrix(239,-42,1360,-239) (7/40,3/17) -> (7/40,3/17) Reflection Matrix(281,-50,680,-121) (3/17,2/11) -> (7/17,12/29) Glide Reflection Matrix(141,-26,320,-59) (2/11,1/5) -> (11/25,4/9) Hyperbolic Matrix(299,-62,680,-141) (1/5,4/19) -> (18/41,11/25) Hyperbolic Matrix(321,-68,760,-161) (4/19,3/14) -> (8/19,11/26) Glide Reflection Matrix(101,-22,280,-61) (3/14,2/9) -> (5/14,4/11) Glide Reflection Matrix(21,-5,80,-19) (2/9,1/4) -> (1/4,3/11) Parabolic Matrix(241,-66,880,-241) (3/11,11/40) -> (3/11,11/40) Reflection Matrix(199,-55,720,-199) (11/40,5/18) -> (11/40,5/18) Reflection Matrix(139,-39,360,-101) (5/18,2/7) -> (5/13,7/18) Hyperbolic Matrix(61,-18,200,-59) (2/7,3/10) -> (3/10,4/13) Parabolic Matrix(139,-43,320,-99) (4/13,1/3) -> (3/7,10/23) Glide Reflection Matrix(41,-14,120,-41) (1/3,7/20) -> (1/3,7/20) Reflection Matrix(99,-35,280,-99) (7/20,5/14) -> (7/20,5/14) Reflection Matrix(621,-227,1480,-541) (4/11,11/30) -> (13/31,21/50) Glide Reflection Matrix(639,-235,1520,-559) (11/30,7/19) -> (21/50,8/19) Glide Reflection Matrix(121,-45,320,-119) (7/19,3/8) -> (3/8,8/21) Parabolic Matrix(81,-32,200,-79) (7/18,2/5) -> (2/5,9/22) Parabolic Matrix(901,-369,2200,-901) (9/22,41/100) -> (9/22,41/100) Reflection Matrix(3199,-1312,7800,-3199) (41/100,16/39) -> (41/100,16/39) Reflection Matrix(679,-279,1560,-641) (16/39,7/17) -> (10/23,17/39) Hyperbolic Matrix(301,-125,720,-299) (12/29,5/12) -> (5/12,13/31) Parabolic Matrix(441,-187,1040,-441) (11/26,17/40) -> (11/26,17/40) Reflection Matrix(239,-102,560,-239) (17/40,3/7) -> (17/40,3/7) Reflection Matrix(561,-245,1280,-559) (17/39,7/16) -> (7/16,18/41) Parabolic Matrix(161,-72,360,-161) (4/9,9/20) -> (4/9,9/20) Reflection Matrix(19,-9,40,-19) (9/20,1/2) -> (9/20,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) 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(61,-8,160,-21) -> Matrix(1,1,2,1) Matrix(41,-6,280,-41) -> Matrix(-1,0,10,1) (1/7,3/20) -> (-1/5,0/1) Matrix(19,-3,120,-19) -> Matrix(1,0,4,-1) (3/20,1/6) -> (0/1,1/2) Matrix(41,-7,240,-41) -> Matrix(-1,1,0,1) (1/6,7/40) -> (1/2,1/0) Matrix(239,-42,1360,-239) -> Matrix(1,0,2,-1) (7/40,3/17) -> (0/1,1/1) Matrix(281,-50,680,-121) -> Matrix(3,-2,-2,1) Matrix(141,-26,320,-59) -> Matrix(1,-1,0,1) 1/0 Matrix(299,-62,680,-141) -> Matrix(1,1,0,1) 1/0 Matrix(321,-68,760,-161) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(101,-22,280,-61) -> Matrix(1,1,2,1) Matrix(21,-5,80,-19) -> Matrix(1,0,2,1) 0/1 Matrix(241,-66,880,-241) -> Matrix(1,0,2,-1) (3/11,11/40) -> (0/1,1/1) Matrix(199,-55,720,-199) -> Matrix(-1,1,0,1) (11/40,5/18) -> (1/2,1/0) Matrix(139,-39,360,-101) -> Matrix(3,-2,2,-1) 1/1 Matrix(61,-18,200,-59) -> Matrix(1,-3,0,1) 1/0 Matrix(139,-43,320,-99) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(41,-14,120,-41) -> Matrix(-1,0,2,1) (1/3,7/20) -> (-1/1,0/1) Matrix(99,-35,280,-99) -> Matrix(1,0,4,-1) (7/20,5/14) -> (0/1,1/2) Matrix(621,-227,1480,-541) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(639,-235,1520,-559) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(121,-45,320,-119) -> Matrix(1,-1,2,-1) (0/1,1/1).(1/2,1/0) Matrix(81,-32,200,-79) -> Matrix(1,-4,0,1) 1/0 Matrix(901,-369,2200,-901) -> Matrix(9,20,-4,-9) (9/22,41/100) -> (-5/2,-2/1) Matrix(3199,-1312,7800,-3199) -> Matrix(27,52,-14,-27) (41/100,16/39) -> (-2/1,-13/7) Matrix(679,-279,1560,-641) -> Matrix(3,5,-2,-3) (-2/1,-1/1).(-3/2,1/0) Matrix(301,-125,720,-299) -> Matrix(1,2,-2,-3) -1/1 Matrix(441,-187,1040,-441) -> Matrix(1,1,0,-1) (11/26,17/40) -> (-1/2,1/0) Matrix(239,-102,560,-239) -> Matrix(-1,0,2,1) (17/40,3/7) -> (-1/1,0/1) Matrix(561,-245,1280,-559) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(161,-72,360,-161) -> Matrix(-1,0,2,1) (4/9,9/20) -> (-1/1,0/1) Matrix(19,-9,40,-19) -> Matrix(1,0,0,-1) (9/20,1/2) -> (0/1,1/0) Matrix(-1,1,0,1) -> Matrix(1,1,0,-1) (1/2,1/0) -> (-1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.