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 -7/1 -6/1 -5/1 -14/3 -13/3 -4/1 -7/2 -3/1 -8/3 -7/3 -9/4 -2/1 -1/1 -8/13 -6/11 -1/2 -4/9 -1/3 -2/7 -4/19 0/1 1/4 3/7 1/2 3/5 2/3 4/5 9/11 1/1 8/7 11/9 4/3 3/2 8/5 19/11 16/9 2/1 7/3 12/5 5/2 8/3 48/17 3/1 4/1 16/3 17/3 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 0/1 -6/1 -1/1 -17/3 -2/3 -11/2 -1/1 -2/3 -1/2 -5/1 0/1 -14/3 -1/1 1/0 -9/2 -1/1 0/1 1/0 -22/5 -1/1 1/0 -13/3 -1/1 -4/1 -1/1 0/1 -7/2 0/1 -24/7 0/1 1/0 -41/12 -2/1 -1/1 1/0 -17/5 0/1 -10/3 -1/1 0/1 -3/1 0/1 -8/3 -1/2 1/0 -13/5 0/1 -18/7 -1/1 1/0 -5/2 -2/1 -1/1 1/0 -22/9 -4/3 -1/1 -17/7 -1/1 -12/5 -1/1 -1/2 -7/3 0/1 -16/7 -1/1 1/0 -9/4 -1/1 -11/5 0/1 -13/6 -1/1 0/1 1/0 -15/7 0/1 -2/1 -1/1 0/1 -1/1 0/1 -2/3 0/1 1/1 -9/14 1/1 -7/11 0/1 -12/19 1/2 1/1 -17/27 1/1 -5/8 1/1 2/1 1/0 -8/13 1/2 1/0 -11/18 1/1 2/1 1/0 -3/5 0/1 -10/17 0/1 1/1 -17/29 0/1 -24/41 0/1 1/0 -7/12 0/1 -4/7 0/1 1/1 -13/23 1/1 -9/16 0/1 1/1 1/0 -5/9 0/1 -11/20 1/2 2/3 1/1 -6/11 1/1 -1/2 0/1 1/1 1/0 -4/9 1/0 -7/16 -2/1 -1/1 1/0 -10/23 -2/1 -1/1 -3/7 0/1 -5/12 2/1 3/1 1/0 -2/5 -1/1 1/0 -9/23 0/1 -7/18 -1/1 0/1 1/0 -5/13 0/1 -3/8 1/0 -1/3 0/1 -6/19 0/1 1/1 -5/16 0/1 1/1 1/0 -4/13 -1/1 1/0 -7/23 0/1 -10/33 -1/1 1/0 -3/10 -1/1 0/1 1/0 -2/7 0/1 -5/18 0/1 1/2 1/1 -3/11 0/1 -1/4 -1/1 0/1 1/0 -2/9 0/1 1/1 -3/14 1/1 2/1 1/0 -4/19 1/0 -5/24 -1/1 0/1 1/0 -1/5 0/1 0/1 0/1 1/0 1/5 0/1 3/14 -2/1 -1/1 1/0 2/9 -1/1 1/0 3/13 0/1 1/4 0/1 4/15 0/1 1/1 3/11 0/1 5/18 0/1 1/1 1/0 2/7 1/1 1/0 5/17 0/1 3/10 1/1 2/1 1/0 1/3 0/1 4/11 1/0 3/8 -2/1 -1/1 1/0 5/13 0/1 2/5 -1/1 0/1 5/12 -1/2 -1/3 0/1 3/7 0/1 7/16 0/1 1/4 1/3 4/9 0/1 1/2 5/11 0/1 1/2 0/1 1/1 1/0 5/9 -2/1 4/7 -1/1 1/0 11/19 -1/1 7/12 -1/1 -2/3 -1/2 10/17 -1/1 0/1 3/5 0/1 2/3 0/1 5/7 0/1 13/18 0/1 1/2 1/1 8/11 0/1 1/1 19/26 0/1 11/15 0/1 14/19 2/3 1/1 3/4 0/1 1/1 1/0 7/9 0/1 11/14 1/1 4/5 1/1 1/0 13/16 2/1 3/1 1/0 9/11 1/0 5/6 -1/1 0/1 1/0 1/1 0/1 8/7 1/2 1/0 15/13 0/1 7/6 0/1 1/2 1/1 6/5 1/2 1/1 11/9 1/1 5/4 0/1 1/1 1/0 14/11 1/1 23/18 0/1 1/1 1/0 9/7 0/1 4/3 1/1 1/0 15/11 2/1 11/8 1/1 2/1 1/0 29/21 1/1 18/13 1/1 3/2 25/18 1/1 3/2 2/1 32/23 3/2 1/0 7/5 2/1 3/2 1/0 11/7 0/1 30/19 1/1 1/0 19/12 -2/1 -1/1 1/0 27/17 0/1 8/5 0/1 1/0 29/18 0/1 1/2 1/1 21/13 0/1 13/8 1/1 2/1 1/0 31/19 1/0 49/30 -4/1 -3/1 1/0 18/11 -1/1 1/0 5/3 0/1 17/10 0/1 1/2 1/1 12/7 0/1 1/1 31/18 0/1 1/2 1/1 19/11 1/1 45/26 1/1 4/3 3/2 26/15 1/1 2/1 33/19 2/1 7/4 0/1 1/1 1/0 16/9 1/0 9/5 0/1 11/6 0/1 1/1 1/0 13/7 1/1 2/1 1/1 1/0 9/4 4/1 5/1 1/0 7/3 1/0 19/8 -10/1 -9/1 1/0 31/13 -8/1 43/18 -8/1 -7/1 1/0 12/5 -6/1 1/0 41/17 -6/1 29/12 -5/1 -4/1 1/0 17/7 -4/1 5/2 -3/1 -2/1 1/0 13/5 -2/1 21/8 -3/2 -4/3 -1/1 29/11 -1/1 8/3 -1/1 1/0 19/7 -2/1 11/4 -1/1 14/5 -1/1 0/1 45/16 -1/1 -2/3 -1/2 31/11 0/1 48/17 -1/2 1/0 17/6 -1/1 -1/2 0/1 3/1 0/1 4/1 1/0 5/1 -2/1 16/3 -2/1 -1/1 11/2 -2/1 -3/2 -1/1 17/3 -1/1 6/1 -1/1 0/1 7/1 0/1 1/0 -1/1 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(11,92,8,67) (-7/1,1/0) -> (15/11,11/8) Hyperbolic Matrix(23,144,-4,-25) (-7/1,-6/1) -> (-6/1,-17/3) Parabolic Matrix(121,680,50,281) (-17/3,-11/2) -> (29/12,17/7) Hyperbolic Matrix(55,296,34,183) (-11/2,-5/1) -> (21/13,13/8) Hyperbolic Matrix(11,52,48,227) (-5/1,-14/3) -> (2/9,3/13) Hyperbolic Matrix(33,152,28,129) (-14/3,-9/2) -> (7/6,6/5) Hyperbolic Matrix(35,156,-116,-517) (-9/2,-22/5) -> (-10/33,-3/10) Hyperbolic Matrix(71,312,38,167) (-22/5,-13/3) -> (13/7,2/1) Hyperbolic Matrix(25,104,-44,-183) (-13/3,-4/1) -> (-4/7,-13/23) Hyperbolic Matrix(15,56,-26,-97) (-4/1,-7/2) -> (-7/12,-4/7) Hyperbolic Matrix(97,336,-166,-575) (-7/2,-24/7) -> (-24/41,-7/12) Hyperbolic Matrix(445,1524,186,637) (-24/7,-41/12) -> (43/18,12/5) Hyperbolic Matrix(431,1472,248,847) (-41/12,-17/5) -> (33/19,7/4) Hyperbolic Matrix(51,172,8,27) (-17/5,-10/3) -> (6/1,7/1) Hyperbolic Matrix(17,56,44,145) (-10/3,-3/1) -> (5/13,2/5) Hyperbolic Matrix(47,128,-18,-49) (-3/1,-8/3) -> (-8/3,-13/5) Parabolic Matrix(125,324,76,197) (-13/5,-18/7) -> (18/11,5/3) Hyperbolic Matrix(25,64,116,297) (-18/7,-5/2) -> (3/14,2/9) Hyperbolic Matrix(55,136,74,183) (-5/2,-22/9) -> (14/19,3/4) Hyperbolic Matrix(195,476,34,83) (-22/9,-17/7) -> (17/3,6/1) Hyperbolic Matrix(169,408,-268,-647) (-17/7,-12/5) -> (-12/19,-17/27) Hyperbolic Matrix(57,136,44,105) (-12/5,-7/3) -> (9/7,4/3) Hyperbolic Matrix(61,140,-200,-459) (-7/3,-16/7) -> (-4/13,-7/23) Hyperbolic Matrix(93,212,118,269) (-16/7,-9/4) -> (11/14,4/5) Hyperbolic Matrix(131,292,48,107) (-9/4,-11/5) -> (19/7,11/4) Hyperbolic Matrix(271,592,168,367) (-11/5,-13/6) -> (29/18,21/13) Hyperbolic Matrix(139,300,120,259) (-13/6,-15/7) -> (15/13,7/6) Hyperbolic Matrix(215,456,124,263) (-15/7,-2/1) -> (26/15,33/19) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(229,148,82,53) (-2/3,-9/14) -> (11/4,14/5) Hyperbolic Matrix(219,140,280,179) (-9/14,-7/11) -> (7/9,11/14) Hyperbolic Matrix(329,208,242,153) (-7/11,-12/19) -> (4/3,15/11) Hyperbolic Matrix(433,272,78,49) (-17/27,-5/8) -> (11/2,17/3) Hyperbolic Matrix(207,128,-338,-209) (-5/8,-8/13) -> (-8/13,-11/18) Parabolic Matrix(105,64,356,217) (-11/18,-3/5) -> (5/17,3/10) Hyperbolic Matrix(101,60,170,101) (-3/5,-10/17) -> (10/17,3/5) Hyperbolic Matrix(191,112,-602,-353) (-10/17,-17/29) -> (-1/3,-6/19) Hyperbolic Matrix(2029,1188,842,493) (-17/29,-24/41) -> (12/5,41/17) Hyperbolic Matrix(461,260,250,141) (-13/23,-9/16) -> (11/6,13/7) Hyperbolic Matrix(93,52,338,189) (-9/16,-5/9) -> (3/11,5/18) Hyperbolic Matrix(711,392,448,247) (-5/9,-11/20) -> (19/12,27/17) Hyperbolic Matrix(533,292,418,229) (-11/20,-6/11) -> (14/11,23/18) Hyperbolic Matrix(83,44,66,35) (-6/11,-1/2) -> (5/4,14/11) Hyperbolic Matrix(71,32,-162,-73) (-1/2,-4/9) -> (-4/9,-7/16) Parabolic Matrix(211,92,-672,-293) (-7/16,-10/23) -> (-6/19,-5/16) Hyperbolic Matrix(351,152,478,207) (-10/23,-3/7) -> (11/15,14/19) Hyperbolic Matrix(67,28,122,51) (-3/7,-5/12) -> (1/2,5/9) Hyperbolic Matrix(69,28,32,13) (-5/12,-2/5) -> (2/1,9/4) Hyperbolic Matrix(265,104,-874,-343) (-2/5,-9/23) -> (-7/23,-10/33) Hyperbolic Matrix(133,52,-642,-251) (-9/23,-7/18) -> (-5/24,-1/5) Hyperbolic Matrix(259,100,360,139) (-7/18,-5/13) -> (5/7,13/18) Hyperbolic Matrix(127,48,82,31) (-5/13,-3/8) -> (3/2,11/7) Hyperbolic Matrix(65,24,46,17) (-3/8,-1/3) -> (7/5,3/2) Hyperbolic Matrix(233,72,288,89) (-5/16,-4/13) -> (4/5,13/16) Hyperbolic Matrix(55,16,-196,-57) (-3/10,-2/7) -> (-2/7,-5/18) Parabolic Matrix(277,76,164,45) (-5/18,-3/11) -> (5/3,17/10) Hyperbolic Matrix(107,28,42,11) (-3/11,-1/4) -> (5/2,13/5) Hyperbolic Matrix(53,12,128,29) (-1/4,-2/9) -> (2/5,5/12) Hyperbolic Matrix(203,44,346,75) (-2/9,-3/14) -> (7/12,10/17) Hyperbolic Matrix(151,32,-722,-153) (-3/14,-4/19) -> (-4/19,-5/24) Parabolic Matrix(1,0,10,1) (-1/5,0/1) -> (0/1,1/5) Parabolic Matrix(529,-112,222,-47) (1/5,3/14) -> (19/8,31/13) Hyperbolic Matrix(291,-68,398,-93) (3/13,1/4) -> (19/26,11/15) Hyperbolic Matrix(317,-84,434,-115) (1/4,4/15) -> (8/11,19/26) Hyperbolic Matrix(251,-68,48,-13) (4/15,3/11) -> (5/1,16/3) Hyperbolic Matrix(499,-140,360,-101) (5/18,2/7) -> (18/13,25/18) Hyperbolic Matrix(587,-172,372,-109) (2/7,5/17) -> (11/7,30/19) Hyperbolic Matrix(185,-56,76,-23) (3/10,1/3) -> (17/7,5/2) Hyperbolic Matrix(147,-52,82,-29) (1/3,4/11) -> (16/9,9/5) Hyperbolic Matrix(205,-76,116,-43) (4/11,3/8) -> (7/4,16/9) Hyperbolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(85,-36,196,-83) (5/12,3/7) -> (3/7,7/16) Parabolic Matrix(389,-172,242,-107) (7/16,4/9) -> (8/5,29/18) Hyperbolic Matrix(331,-148,208,-93) (4/9,5/11) -> (27/17,8/5) Hyperbolic Matrix(193,-88,68,-31) (5/11,1/2) -> (17/6,3/1) Hyperbolic Matrix(205,-116,76,-43) (5/9,4/7) -> (8/3,19/7) Hyperbolic Matrix(355,-204,134,-77) (4/7,11/19) -> (29/11,8/3) Hyperbolic Matrix(557,-324,404,-235) (11/19,7/12) -> (11/8,29/21) Hyperbolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(557,-404,324,-235) (13/18,8/11) -> (12/7,31/18) Hyperbolic Matrix(135,-104,74,-57) (3/4,7/9) -> (9/5,11/6) Hyperbolic Matrix(1035,-844,634,-517) (13/16,9/11) -> (31/19,49/30) Hyperbolic Matrix(329,-272,202,-167) (9/11,5/6) -> (13/8,31/19) Hyperbolic Matrix(43,-36,6,-5) (5/6,1/1) -> (7/1,1/0) Hyperbolic Matrix(113,-128,98,-111) (1/1,8/7) -> (8/7,15/13) Parabolic Matrix(307,-372,222,-269) (6/5,11/9) -> (29/21,18/13) Hyperbolic Matrix(305,-376,116,-143) (11/9,5/4) -> (21/8,29/11) Hyperbolic Matrix(859,-1100,360,-461) (23/18,9/7) -> (31/13,43/18) Hyperbolic Matrix(1255,-1744,444,-617) (25/18,32/23) -> (48/17,17/6) Hyperbolic Matrix(953,-1328,338,-471) (32/23,7/5) -> (31/11,48/17) Hyperbolic Matrix(1147,-1812,702,-1109) (30/19,19/12) -> (49/30,18/11) Hyperbolic Matrix(237,-404,44,-75) (17/10,12/7) -> (16/3,11/2) Hyperbolic Matrix(837,-1444,484,-835) (31/18,19/11) -> (19/11,45/26) Parabolic Matrix(1039,-1800,370,-641) (45/26,26/15) -> (14/5,45/16) Hyperbolic Matrix(85,-196,36,-83) (9/4,7/3) -> (7/3,19/8) Parabolic Matrix(1137,-2744,404,-975) (41/17,29/12) -> (45/16,31/11) Hyperbolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,92,8,67) -> Matrix(1,2,0,1) Matrix(23,144,-4,-25) -> Matrix(1,2,-2,-3) Matrix(121,680,50,281) -> Matrix(11,6,-2,-1) Matrix(55,296,34,183) -> Matrix(1,0,2,1) Matrix(11,52,48,227) -> Matrix(1,0,0,1) Matrix(33,152,28,129) -> Matrix(1,0,2,1) Matrix(35,156,-116,-517) -> Matrix(1,0,0,1) Matrix(71,312,38,167) -> Matrix(1,2,0,1) Matrix(25,104,-44,-183) -> Matrix(1,0,2,1) Matrix(15,56,-26,-97) -> Matrix(1,0,2,1) Matrix(97,336,-166,-575) -> Matrix(1,0,0,1) Matrix(445,1524,186,637) -> Matrix(1,-6,0,1) Matrix(431,1472,248,847) -> Matrix(1,2,0,1) Matrix(51,172,8,27) -> Matrix(1,0,0,1) Matrix(17,56,44,145) -> Matrix(1,0,0,1) Matrix(47,128,-18,-49) -> Matrix(1,0,0,1) Matrix(125,324,76,197) -> Matrix(1,0,0,1) Matrix(25,64,116,297) -> Matrix(1,0,0,1) Matrix(55,136,74,183) -> Matrix(1,2,0,1) Matrix(195,476,34,83) -> Matrix(3,4,-4,-5) Matrix(169,408,-268,-647) -> Matrix(3,2,4,3) Matrix(57,136,44,105) -> Matrix(1,0,2,1) Matrix(61,140,-200,-459) -> Matrix(1,0,0,1) Matrix(93,212,118,269) -> Matrix(1,2,0,1) Matrix(131,292,48,107) -> Matrix(3,2,-2,-1) Matrix(271,592,168,367) -> Matrix(1,0,2,1) Matrix(139,300,120,259) -> Matrix(1,0,2,1) Matrix(215,456,124,263) -> Matrix(1,2,0,1) Matrix(3,4,-4,-5) -> Matrix(1,0,2,1) Matrix(229,148,82,53) -> Matrix(1,0,-2,1) Matrix(219,140,280,179) -> Matrix(1,0,0,1) Matrix(329,208,242,153) -> Matrix(3,-2,2,-1) Matrix(433,272,78,49) -> Matrix(3,-4,-2,3) Matrix(207,128,-338,-209) -> Matrix(1,0,0,1) Matrix(105,64,356,217) -> Matrix(1,0,0,1) Matrix(101,60,170,101) -> Matrix(1,0,-2,1) Matrix(191,112,-602,-353) -> Matrix(1,0,0,1) Matrix(2029,1188,842,493) -> Matrix(1,-6,0,1) Matrix(461,260,250,141) -> Matrix(1,0,0,1) Matrix(93,52,338,189) -> Matrix(1,0,0,1) Matrix(711,392,448,247) -> Matrix(1,0,-2,1) Matrix(533,292,418,229) -> Matrix(3,-2,2,-1) Matrix(83,44,66,35) -> Matrix(1,0,0,1) Matrix(71,32,-162,-73) -> Matrix(1,-2,0,1) Matrix(211,92,-672,-293) -> Matrix(1,2,0,1) Matrix(351,152,478,207) -> Matrix(1,0,2,1) Matrix(67,28,122,51) -> Matrix(1,-2,0,1) Matrix(69,28,32,13) -> Matrix(1,2,0,1) Matrix(265,104,-874,-343) -> Matrix(1,0,0,1) Matrix(133,52,-642,-251) -> Matrix(1,0,0,1) Matrix(259,100,360,139) -> Matrix(1,0,2,1) Matrix(127,48,82,31) -> Matrix(1,0,0,1) Matrix(65,24,46,17) -> Matrix(1,2,0,1) Matrix(233,72,288,89) -> Matrix(1,2,0,1) Matrix(55,16,-196,-57) -> Matrix(1,0,2,1) Matrix(277,76,164,45) -> Matrix(1,0,0,1) Matrix(107,28,42,11) -> Matrix(1,-2,0,1) Matrix(53,12,128,29) -> Matrix(1,0,-2,1) Matrix(203,44,346,75) -> Matrix(1,0,-2,1) Matrix(151,32,-722,-153) -> Matrix(1,-2,0,1) Matrix(1,0,10,1) -> Matrix(1,0,0,1) Matrix(529,-112,222,-47) -> Matrix(1,-8,0,1) Matrix(291,-68,398,-93) -> Matrix(1,0,2,1) Matrix(317,-84,434,-115) -> Matrix(1,0,0,1) Matrix(251,-68,48,-13) -> Matrix(1,-2,0,1) Matrix(499,-140,360,-101) -> Matrix(3,-2,2,-1) Matrix(587,-172,372,-109) -> Matrix(1,0,0,1) Matrix(185,-56,76,-23) -> Matrix(1,-4,0,1) Matrix(147,-52,82,-29) -> Matrix(1,0,0,1) Matrix(205,-76,116,-43) -> Matrix(1,2,0,1) Matrix(377,-144,144,-55) -> Matrix(3,2,-2,-1) Matrix(85,-36,196,-83) -> Matrix(1,0,6,1) Matrix(389,-172,242,-107) -> Matrix(1,0,-2,1) Matrix(331,-148,208,-93) -> Matrix(1,0,-2,1) Matrix(193,-88,68,-31) -> Matrix(1,0,-2,1) Matrix(205,-116,76,-43) -> Matrix(1,0,0,1) Matrix(355,-204,134,-77) -> Matrix(1,0,0,1) Matrix(557,-324,404,-235) -> Matrix(1,0,2,1) Matrix(25,-16,36,-23) -> Matrix(1,0,2,1) Matrix(557,-404,324,-235) -> Matrix(1,0,0,1) Matrix(135,-104,74,-57) -> Matrix(1,0,0,1) Matrix(1035,-844,634,-517) -> Matrix(1,-6,0,1) Matrix(329,-272,202,-167) -> Matrix(1,2,0,1) Matrix(43,-36,6,-5) -> Matrix(1,0,0,1) Matrix(113,-128,98,-111) -> Matrix(1,0,0,1) Matrix(307,-372,222,-269) -> Matrix(5,-4,4,-3) Matrix(305,-376,116,-143) -> Matrix(3,-4,-2,3) Matrix(859,-1100,360,-461) -> Matrix(1,-8,0,1) Matrix(1255,-1744,444,-617) -> Matrix(1,-2,0,1) Matrix(953,-1328,338,-471) -> Matrix(1,-2,0,1) Matrix(1147,-1812,702,-1109) -> Matrix(1,-2,0,1) Matrix(237,-404,44,-75) -> Matrix(1,-2,0,1) Matrix(837,-1444,484,-835) -> Matrix(5,-4,4,-3) Matrix(1039,-1800,370,-641) -> Matrix(1,-2,0,1) Matrix(85,-196,36,-83) -> Matrix(1,-14,0,1) Matrix(1137,-2744,404,-975) -> Matrix(1,6,-2,-11) Matrix(9,-32,2,-7) -> Matrix(1,-2,0,1) 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): 16 Degree of the the map X: 16 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 -6/1 -1/1 2 4 -11/2 0 20 -5/1 0/1 1 10 -9/2 0 20 -13/3 -1/1 1 2 -4/1 (-1/1,0/1) 0 20 -7/2 0/1 1 4 -24/7 (0/1,1/0) 0 20 -17/5 0/1 1 10 -10/3 (-1/1,0/1) 0 20 -3/1 0/1 1 10 -8/3 0 4 -13/5 0/1 1 10 -5/2 0 20 -17/7 -1/1 5 2 -12/5 (-1/1,-1/2) 0 20 -7/3 0/1 1 10 -9/4 -1/1 2 4 -2/1 (-1/1,0/1) 0 20 -1/1 0/1 1 2 0/1 (0/1,1/0) 0 20 1/4 0/1 1 4 4/15 (0/1,1/1) 0 20 3/11 0/1 1 10 5/18 0 20 2/7 (1/1,1/0) 0 20 5/17 0/1 1 10 3/10 0 20 1/3 0/1 1 10 4/11 1/0 4 4 3/8 0 20 5/13 0/1 1 10 2/5 (-1/1,0/1) 0 20 3/7 0/1 3 2 4/9 (0/1,1/2) 0 20 5/11 0/1 1 10 1/2 0 20 4/7 (-1/1,1/0) 0 20 11/19 -1/1 4 2 7/12 0 20 3/5 0/1 1 10 2/3 0/1 2 4 5/7 0/1 1 10 8/11 (0/1,1/1) 0 20 11/15 0/1 1 10 3/4 0 20 7/9 0/1 1 10 11/14 1/1 2 4 4/5 (1/1,1/0) 0 20 9/11 1/0 4 2 5/6 0 20 1/1 0/1 1 10 8/7 0 4 15/13 0/1 1 10 7/6 0 20 6/5 (1/2,1/1) 0 20 11/9 1/1 4 2 5/4 0 20 14/11 1/1 2 4 23/18 0 20 9/7 0/1 1 10 4/3 (1/1,1/0) 0 20 3/2 1/0 1 4 8/5 (0/1,1/0) 0 20 21/13 0/1 1 10 13/8 0 20 31/19 1/0 4 2 18/11 (-1/1,1/0) 0 20 5/3 0/1 1 10 12/7 (0/1,1/1) 0 20 19/11 1/1 2 2 26/15 (1/1,2/1) 0 20 33/19 2/1 1 10 7/4 0 20 16/9 1/0 4 4 9/5 0/1 1 10 11/6 0 20 13/7 1/1 1 2 2/1 (1/1,1/0) 0 20 7/3 1/0 7 2 12/5 (-6/1,1/0) 0 20 41/17 -6/1 1 10 29/12 0 20 17/7 -4/1 1 10 5/2 0 20 8/3 (-1/1,1/0) 0 20 11/4 -1/1 2 4 14/5 (-1/1,0/1) 0 20 31/11 0/1 1 10 17/6 0 20 3/1 0/1 1 10 4/1 1/0 4 4 5/1 -2/1 1 10 11/2 0 20 17/3 -1/1 5 2 6/1 (-1/1,0/1) 0 20 7/1 0/1 1 10 1/0 0 20 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,44,4,35) (-6/1,1/0) -> (5/4,14/11) Glide Reflection Matrix(51,292,40,229) (-6/1,-11/2) -> (14/11,23/18) Glide Reflection Matrix(55,296,34,183) (-11/2,-5/1) -> (21/13,13/8) Hyperbolic Matrix(11,52,40,189) (-5/1,-9/2) -> (3/11,5/18) Glide Reflection Matrix(59,260,32,141) (-9/2,-13/3) -> (11/6,13/7) Glide Reflection Matrix(25,104,-6,-25) (-13/3,-4/1) -> (-13/3,-4/1) Reflection Matrix(15,56,-4,-15) (-4/1,-7/2) -> (-4/1,-7/2) Reflection Matrix(97,336,-28,-97) (-7/2,-24/7) -> (-7/2,-24/7) Reflection Matrix(347,1188,144,493) (-24/7,-17/5) -> (12/5,41/17) Glide Reflection Matrix(51,172,8,27) (-17/5,-10/3) -> (6/1,7/1) Hyperbolic Matrix(17,56,44,145) (-10/3,-3/1) -> (5/13,2/5) Hyperbolic Matrix(47,128,-18,-49) (-3/1,-8/3) -> (-8/3,-13/5) Parabolic Matrix(25,64,84,215) (-13/5,-5/2) -> (5/17,3/10) Glide Reflection Matrix(111,272,20,49) (-5/2,-17/7) -> (11/2,17/3) Glide Reflection Matrix(169,408,-70,-169) (-17/7,-12/5) -> (-17/7,-12/5) Reflection Matrix(57,136,44,105) (-12/5,-7/3) -> (9/7,4/3) Hyperbolic Matrix(61,140,78,179) (-7/3,-9/4) -> (7/9,11/14) Glide Reflection Matrix(67,148,24,53) (-9/4,-2/1) -> (11/4,14/5) Glide Reflection Matrix(3,4,-2,-3) (-2/1,-1/1) -> (-2/1,-1/1) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,8,-1) (0/1,1/4) -> (0/1,1/4) Reflection Matrix(31,-8,120,-31) (1/4,4/15) -> (1/4,4/15) Reflection Matrix(253,-68,346,-93) (4/15,3/11) -> (8/11,11/15) Glide Reflection Matrix(157,-44,132,-37) (5/18,2/7) -> (7/6,6/5) Glide Reflection Matrix(341,-100,208,-61) (2/7,5/17) -> (18/11,5/3) Glide Reflection Matrix(185,-56,76,-23) (3/10,1/3) -> (17/7,5/2) Hyperbolic Matrix(147,-52,82,-29) (1/3,4/11) -> (16/9,9/5) Hyperbolic Matrix(205,-76,116,-43) (4/11,3/8) -> (7/4,16/9) Hyperbolic Matrix(115,-44,196,-75) (3/8,5/13) -> (7/12,3/5) Glide Reflection Matrix(29,-12,70,-29) (2/5,3/7) -> (2/5,3/7) Reflection Matrix(55,-24,126,-55) (3/7,4/9) -> (3/7,4/9) Reflection Matrix(277,-124,172,-77) (4/9,5/11) -> (8/5,21/13) Glide Reflection Matrix(193,-88,68,-31) (5/11,1/2) -> (17/6,3/1) Hyperbolic Matrix(51,-28,20,-11) (1/2,4/7) -> (5/2,8/3) Glide Reflection Matrix(153,-88,266,-153) (4/7,11/19) -> (4/7,11/19) Reflection Matrix(227,-132,184,-107) (11/19,7/12) -> (11/9,5/4) Glide Reflection Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(139,-100,82,-59) (5/7,8/11) -> (5/3,12/7) Glide Reflection Matrix(185,-136,34,-25) (11/15,3/4) -> (5/1,11/2) Glide Reflection Matrix(135,-104,74,-57) (3/4,7/9) -> (9/5,11/6) Hyperbolic Matrix(111,-88,140,-111) (11/14,4/5) -> (11/14,4/5) Reflection Matrix(89,-72,110,-89) (4/5,9/11) -> (4/5,9/11) Reflection Matrix(329,-272,202,-167) (9/11,5/6) -> (13/8,31/19) Hyperbolic Matrix(43,-36,6,-5) (5/6,1/1) -> (7/1,1/0) Hyperbolic Matrix(113,-128,98,-111) (1/1,8/7) -> (8/7,15/13) Parabolic Matrix(407,-472,144,-167) (15/13,7/6) -> (31/11,17/6) Glide Reflection Matrix(109,-132,90,-109) (6/5,11/9) -> (6/5,11/9) Reflection Matrix(509,-652,210,-269) (23/18,9/7) -> (29/12,17/7) Glide Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(31,-48,20,-31) (3/2,8/5) -> (3/2,8/5) Reflection Matrix(683,-1116,418,-683) (31/19,18/11) -> (31/19,18/11) Reflection Matrix(265,-456,154,-265) (12/7,19/11) -> (12/7,19/11) Reflection Matrix(571,-988,330,-571) (19/11,26/15) -> (19/11,26/15) Reflection Matrix(731,-1268,260,-451) (26/15,33/19) -> (14/5,31/11) Glide Reflection Matrix(715,-1244,296,-515) (33/19,7/4) -> (41/17,29/12) Glide Reflection Matrix(27,-52,14,-27) (13/7,2/1) -> (13/7,2/1) Reflection Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(65,-176,24,-65) (8/3,11/4) -> (8/3,11/4) Reflection Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic Matrix(35,-204,6,-35) (17/3,6/1) -> (17/3,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(5,44,4,35) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(51,292,40,229) -> Matrix(3,2,2,1) Matrix(55,296,34,183) -> Matrix(1,0,2,1) 0/1 Matrix(11,52,40,189) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(59,260,32,141) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(25,104,-6,-25) -> Matrix(-1,0,2,1) (-13/3,-4/1) -> (-1/1,0/1) Matrix(15,56,-4,-15) -> Matrix(-1,0,2,1) (-4/1,-7/2) -> (-1/1,0/1) Matrix(97,336,-28,-97) -> Matrix(1,0,0,-1) (-7/2,-24/7) -> (0/1,1/0) Matrix(347,1188,144,493) -> Matrix(1,6,0,-1) *** -> (-3/1,1/0) Matrix(51,172,8,27) -> Matrix(1,0,0,1) Matrix(17,56,44,145) -> Matrix(1,0,0,1) Matrix(47,128,-18,-49) -> Matrix(1,0,0,1) Matrix(25,64,84,215) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(111,272,20,49) -> Matrix(3,4,-2,-3) *** -> (-2/1,-1/1) Matrix(169,408,-70,-169) -> Matrix(3,2,-4,-3) (-17/7,-12/5) -> (-1/1,-1/2) Matrix(57,136,44,105) -> Matrix(1,0,2,1) 0/1 Matrix(61,140,78,179) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(67,148,24,53) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(3,4,-2,-3) -> Matrix(-1,0,2,1) (-2/1,-1/1) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,8,-1) -> Matrix(1,0,0,-1) (0/1,1/4) -> (0/1,1/0) Matrix(31,-8,120,-31) -> Matrix(1,0,2,-1) (1/4,4/15) -> (0/1,1/1) Matrix(253,-68,346,-93) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(157,-44,132,-37) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(341,-100,208,-61) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(185,-56,76,-23) -> Matrix(1,-4,0,1) 1/0 Matrix(147,-52,82,-29) -> Matrix(1,0,0,1) Matrix(205,-76,116,-43) -> Matrix(1,2,0,1) 1/0 Matrix(115,-44,196,-75) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(29,-12,70,-29) -> Matrix(-1,0,2,1) (2/5,3/7) -> (-1/1,0/1) Matrix(55,-24,126,-55) -> Matrix(1,0,4,-1) (3/7,4/9) -> (0/1,1/2) Matrix(277,-124,172,-77) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(193,-88,68,-31) -> Matrix(1,0,-2,1) 0/1 Matrix(51,-28,20,-11) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(153,-88,266,-153) -> Matrix(1,2,0,-1) (4/7,11/19) -> (-1/1,1/0) Matrix(227,-132,184,-107) -> Matrix(3,2,2,1) Matrix(25,-16,36,-23) -> Matrix(1,0,2,1) 0/1 Matrix(139,-100,82,-59) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(185,-136,34,-25) -> Matrix(3,-2,-2,1) Matrix(135,-104,74,-57) -> Matrix(1,0,0,1) Matrix(111,-88,140,-111) -> Matrix(-1,2,0,1) (11/14,4/5) -> (1/1,1/0) Matrix(89,-72,110,-89) -> Matrix(-1,2,0,1) (4/5,9/11) -> (1/1,1/0) Matrix(329,-272,202,-167) -> Matrix(1,2,0,1) 1/0 Matrix(43,-36,6,-5) -> Matrix(1,0,0,1) Matrix(113,-128,98,-111) -> Matrix(1,0,0,1) Matrix(407,-472,144,-167) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(109,-132,90,-109) -> Matrix(3,-2,4,-3) (6/5,11/9) -> (1/2,1/1) Matrix(509,-652,210,-269) -> Matrix(1,4,0,-1) *** -> (-2/1,1/0) Matrix(17,-24,12,-17) -> Matrix(-1,2,0,1) (4/3,3/2) -> (1/1,1/0) Matrix(31,-48,20,-31) -> Matrix(1,0,0,-1) (3/2,8/5) -> (0/1,1/0) Matrix(683,-1116,418,-683) -> Matrix(1,2,0,-1) (31/19,18/11) -> (-1/1,1/0) Matrix(265,-456,154,-265) -> Matrix(1,0,2,-1) (12/7,19/11) -> (0/1,1/1) Matrix(571,-988,330,-571) -> Matrix(3,-4,2,-3) (19/11,26/15) -> (1/1,2/1) Matrix(731,-1268,260,-451) -> Matrix(1,-2,-2,3) Matrix(715,-1244,296,-515) -> Matrix(1,4,0,-1) *** -> (-2/1,1/0) Matrix(27,-52,14,-27) -> Matrix(-1,2,0,1) (13/7,2/1) -> (1/1,1/0) Matrix(13,-28,6,-13) -> Matrix(-1,2,0,1) (2/1,7/3) -> (1/1,1/0) Matrix(71,-168,30,-71) -> Matrix(1,12,0,-1) (7/3,12/5) -> (-6/1,1/0) Matrix(65,-176,24,-65) -> Matrix(1,2,0,-1) (8/3,11/4) -> (-1/1,1/0) Matrix(9,-32,2,-7) -> Matrix(1,-2,0,1) 1/0 Matrix(35,-204,6,-35) -> Matrix(-1,0,2,1) (17/3,6/1) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.