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 -4/1 -7/2 -3/1 -8/3 -9/4 -2/1 -1/1 -9/14 -8/13 -7/12 -6/11 -1/2 -4/9 -3/8 -2/7 -1/4 -1/6 0/1 1/4 3/7 1/2 11/19 2/3 3/4 1/1 6/5 11/9 5/4 4/3 3/2 5/3 19/11 13/7 2/1 7/3 5/2 8/3 3/1 7/2 4/1 5/1 17/3 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -3/5 -6/1 -1/2 -17/3 -7/15 -11/2 -1/2 -9/20 -5/1 -2/5 -9/2 -1/2 -3/8 -4/1 -1/3 0/1 -19/5 -1/1 -15/4 -1/2 -3/8 -26/7 -3/10 -11/3 0/1 -7/2 0/1 -17/5 0/1 -10/3 1/0 -3/1 -1/1 -14/5 -5/8 -11/4 -7/12 -1/2 -8/3 -1/2 -21/8 -1/2 -13/28 -34/13 -11/24 -13/5 -4/9 -18/7 -1/2 -5/2 -1/2 -3/8 -12/5 -1/3 -2/7 -7/3 0/1 -9/4 0/1 -11/5 1/1 -13/6 1/2 1/0 -2/1 -1/2 -1/1 0/1 -2/3 1/0 -13/20 -1/2 -1/4 -11/17 -1/3 -9/14 0/1 -7/11 0/1 -12/19 2/3 1/1 -5/8 3/2 1/0 -18/29 1/0 -13/21 4/1 -8/13 1/0 -19/31 -6/1 -30/49 -9/2 -11/18 -7/2 1/0 -3/5 -1/1 -10/17 -1/2 -7/12 0/1 -18/31 1/2 -11/19 0/1 -26/45 3/4 -15/26 3/2 1/0 -19/33 -1/1 -4/7 0/1 1/1 -9/16 3/2 1/0 -5/9 2/1 -6/11 1/0 -7/13 -3/1 -1/2 -1/2 1/0 -6/13 1/0 -5/11 1/1 -4/9 1/0 -3/7 -2/1 -5/12 -3/2 1/0 -2/5 1/0 -7/18 -3/2 -5/4 -5/13 -2/1 -3/8 -1/1 -7/19 -2/3 -18/49 -1/2 -11/30 -3/4 -1/2 -15/41 -1/3 -19/52 -1/2 -1/4 -4/11 -1/1 0/1 -9/25 -2/3 -5/14 -1/2 -1/4 -6/17 1/2 -1/3 -1/1 -3/10 1/2 1/0 -2/7 1/0 -5/18 -7/2 1/0 -3/11 -2/1 -10/37 1/0 -7/26 -5/2 1/0 -4/15 -3/1 -2/1 -5/19 -1/1 -1/4 -3/2 1/0 -2/9 -1/2 -1/5 -2/1 -3/16 -3/2 1/0 -2/11 -5/4 -1/6 -1/1 -2/13 -3/4 -3/20 -3/4 -7/10 -4/27 -2/3 -3/5 -1/7 -1/1 0/1 -1/1 0/1 1/6 -1/2 1/0 1/5 -1/1 1/4 -1/1 3/11 -1/1 8/29 -13/15 -6/7 5/18 -17/20 -5/6 2/7 -5/6 3/10 -3/4 -1/2 1/3 -2/3 4/11 -1/2 3/8 -1/2 -1/4 5/13 -1/1 2/5 1/0 5/12 -3/2 -5/4 3/7 -1/1 7/16 -11/12 -9/10 4/9 -1/1 -6/7 1/2 -3/4 -1/2 5/9 -1/1 4/7 -2/3 -3/5 15/26 -5/8 -1/2 11/19 -3/5 7/12 -5/8 -1/2 3/5 -2/3 2/3 -1/2 5/7 -2/5 13/18 -3/8 -5/14 8/11 -1/3 0/1 11/15 -1/3 14/19 -1/4 3/4 -1/2 -1/4 7/9 0/1 11/14 0/1 15/19 -1/1 4/5 -1/3 0/1 9/11 0/1 5/6 1/2 1/0 1/1 -1/1 7/6 -1/2 -3/8 6/5 -1/4 17/14 -1/8 -1/10 11/9 0/1 5/4 1/2 1/0 19/15 3/1 14/11 1/0 9/7 -2/1 4/3 -1/1 0/1 3/2 -1/1 8/5 -1/1 -4/5 29/18 -11/14 -3/4 21/13 -7/9 13/8 -3/4 -1/2 31/19 -1/1 49/30 -5/6 -3/4 18/11 -3/4 5/3 -2/3 17/10 -1/2 1/0 12/7 -1/1 0/1 31/18 -3/2 -5/4 19/11 -1/1 45/26 -15/16 -13/14 26/15 -9/10 7/4 -5/6 -3/4 16/9 -3/4 9/5 -5/7 11/6 -3/4 -7/10 24/13 -13/19 -2/3 13/7 -2/3 2/1 -1/2 9/4 -1/2 1/0 7/3 -1/1 19/8 -7/8 -5/6 12/5 -1/1 -4/5 5/2 -3/4 -1/2 13/5 -5/7 21/8 -7/10 -11/16 29/11 -2/3 8/3 -1/1 -2/3 19/7 -5/7 11/4 -2/3 3/1 -2/3 10/3 -7/12 17/5 -3/5 7/2 -9/16 -1/2 4/1 -1/2 9/2 -1/2 -7/16 14/3 -5/12 33/7 -2/5 52/11 -2/5 -9/23 19/4 -1/2 -3/8 5/1 -1/3 11/2 -1/2 -1/4 28/5 -1/5 0/1 17/3 0/1 40/7 0/1 1/1 23/4 -1/2 1/0 6/1 -1/2 7/1 0/1 1/0 -1/2 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(13,112,8,69) (-7/1,1/0) -> (21/13,13/8) Hyperbolic Matrix(23,144,-4,-25) (-7/1,-6/1) -> (-6/1,-17/3) Parabolic Matrix(83,464,-144,-805) (-17/3,-11/2) -> (-15/26,-19/33) Hyperbolic Matrix(23,124,-64,-345) (-11/2,-5/1) -> (-9/25,-5/14) Hyperbolic Matrix(5,24,16,77) (-5/1,-9/2) -> (3/10,1/3) Hyperbolic Matrix(29,128,12,53) (-9/2,-4/1) -> (12/5,5/2) Hyperbolic Matrix(43,164,156,595) (-4/1,-19/5) -> (3/11,8/29) Hyperbolic Matrix(101,380,80,301) (-19/5,-15/4) -> (5/4,19/15) Hyperbolic Matrix(153,572,88,329) (-15/4,-26/7) -> (26/15,7/4) Hyperbolic Matrix(223,824,-364,-1345) (-26/7,-11/3) -> (-19/31,-30/49) Hyperbolic Matrix(55,196,-16,-57) (-11/3,-7/2) -> (-7/2,-17/5) Parabolic 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(11,32,-32,-93) (-3/1,-14/5) -> (-6/17,-1/3) Hyperbolic Matrix(23,64,-124,-345) (-14/5,-11/4) -> (-3/16,-2/11) Hyperbolic Matrix(95,256,-36,-97) (-11/4,-8/3) -> (-8/3,-21/8) Parabolic Matrix(185,484,-284,-743) (-21/8,-34/13) -> (-2/3,-13/20) Hyperbolic Matrix(273,712,-472,-1231) (-34/13,-13/5) -> (-11/19,-26/45) Hyperbolic Matrix(125,324,76,197) (-13/5,-18/7) -> (18/11,5/3) Hyperbolic Matrix(39,100,140,359) (-18/7,-5/2) -> (5/18,2/7) Hyperbolic Matrix(13,32,28,69) (-5/2,-12/5) -> (4/9,1/2) Hyperbolic Matrix(57,136,44,105) (-12/5,-7/3) -> (9/7,4/3) Hyperbolic Matrix(45,104,16,37) (-7/3,-9/4) -> (11/4,3/1) 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(191,412,-312,-673) (-13/6,-2/1) -> (-30/49,-11/18) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(623,404,-1704,-1105) (-13/20,-11/17) -> (-15/41,-19/52) Hyperbolic Matrix(397,256,504,325) (-11/17,-9/14) -> (11/14,15/19) Hyperbolic Matrix(219,140,280,179) (-9/14,-7/11) -> (7/9,11/14) Hyperbolic Matrix(215,136,-596,-377) (-7/11,-12/19) -> (-4/11,-9/25) Hyperbolic Matrix(165,104,376,237) (-12/19,-5/8) -> (7/16,4/9) Hyperbolic Matrix(103,64,-684,-425) (-5/8,-18/29) -> (-2/13,-3/20) Hyperbolic Matrix(581,360,-1580,-979) (-18/29,-13/21) -> (-7/19,-18/49) Hyperbolic Matrix(415,256,-676,-417) (-13/21,-8/13) -> (-8/13,-19/31) Parabolic Matrix(53,32,48,29) (-11/18,-3/5) -> (1/1,7/6) Hyperbolic Matrix(339,200,100,59) (-3/5,-10/17) -> (10/3,17/5) Hyperbolic Matrix(335,196,-576,-337) (-10/17,-7/12) -> (-7/12,-18/31) Parabolic Matrix(283,164,-1044,-605) (-18/31,-11/19) -> (-3/11,-10/37) Hyperbolic Matrix(2701,1560,1560,901) (-26/45,-15/26) -> (45/26,26/15) Hyperbolic Matrix(139,80,-940,-541) (-19/33,-4/7) -> (-4/27,-1/7) Hyperbolic Matrix(325,184,136,77) (-4/7,-9/16) -> (19/8,12/5) Hyperbolic Matrix(43,24,-224,-125) (-9/16,-5/9) -> (-1/5,-3/16) Hyperbolic Matrix(225,124,176,97) (-5/9,-6/11) -> (14/11,9/7) Hyperbolic Matrix(391,212,308,167) (-6/11,-7/13) -> (19/15,14/11) Hyperbolic Matrix(173,92,-472,-251) (-7/13,-1/2) -> (-11/30,-15/41) Hyperbolic Matrix(111,52,-412,-193) (-1/2,-6/13) -> (-10/37,-7/26) Hyperbolic Matrix(297,136,404,185) (-6/13,-5/11) -> (11/15,14/19) Hyperbolic Matrix(257,116,144,65) (-5/11,-4/9) -> (16/9,9/5) Hyperbolic Matrix(37,16,104,45) (-4/9,-3/7) -> (1/3,4/11) Hyperbolic Matrix(105,44,136,57) (-3/7,-5/12) -> (3/4,7/9) Hyperbolic Matrix(69,28,32,13) (-5/12,-2/5) -> (2/1,9/4) Hyperbolic Matrix(297,116,64,25) (-2/5,-7/18) -> (9/2,14/3) Hyperbolic Matrix(259,100,360,139) (-7/18,-5/13) -> (5/7,13/18) Hyperbolic Matrix(95,36,-256,-97) (-5/13,-3/8) -> (-3/8,-7/19) Parabolic Matrix(2941,1080,1800,661) (-18/49,-11/30) -> (49/30,18/11) Hyperbolic Matrix(2333,852,408,149) (-19/52,-4/11) -> (40/7,23/4) Hyperbolic Matrix(373,132,308,109) (-5/14,-6/17) -> (6/5,17/14) Hyperbolic Matrix(145,44,56,17) (-1/3,-3/10) -> (5/2,13/5) 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(329,88,572,153) (-7/26,-4/15) -> (4/7,15/26) Hyperbolic Matrix(301,80,380,101) (-4/15,-5/19) -> (15/19,4/5) Hyperbolic Matrix(381,100,80,21) (-5/19,-1/4) -> (19/4,5/1) Hyperbolic Matrix(53,12,128,29) (-1/4,-2/9) -> (2/5,5/12) Hyperbolic Matrix(77,16,24,5) (-2/9,-1/5) -> (3/1,10/3) Hyperbolic Matrix(23,4,-144,-25) (-2/11,-1/6) -> (-1/6,-2/13) Parabolic Matrix(1553,232,328,49) (-3/20,-4/27) -> (52/11,19/4) Hyperbolic Matrix(67,8,92,11) (-1/7,0/1) -> (8/11,11/15) Hyperbolic Matrix(89,-12,52,-7) (0/1,1/6) -> (17/10,12/7) Hyperbolic Matrix(85,-16,16,-3) (1/6,1/5) -> (5/1,11/2) Hyperbolic Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(751,-208,408,-113) (8/29,5/18) -> (11/6,24/13) Hyperbolic Matrix(109,-32,92,-27) (2/7,3/10) -> (7/6,6/5) 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(117,-64,64,-35) (1/2,5/9) -> (9/5,11/6) Hyperbolic Matrix(205,-116,76,-43) (5/9,4/7) -> (8/3,19/7) Hyperbolic Matrix(1737,-1004,1064,-615) (15/26,11/19) -> (31/19,49/30) Hyperbolic Matrix(619,-360,380,-221) (11/19,7/12) -> (13/8,31/19) Hyperbolic Matrix(89,-52,12,-7) (7/12,3/5) -> (7/1,1/0) 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(461,-340,80,-59) (14/19,3/4) -> (23/4,6/1) Hyperbolic Matrix(233,-188,88,-71) (4/5,9/11) -> (29/11,8/3) Hyperbolic Matrix(253,-208,208,-171) (9/11,5/6) -> (17/14,11/9) Hyperbolic Matrix(109,-92,32,-27) (5/6,1/1) -> (17/5,7/2) Hyperbolic Matrix(305,-376,116,-143) (11/9,5/4) -> (21/8,29/11) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(559,-900,100,-161) (8/5,29/18) -> (11/2,28/5) Hyperbolic Matrix(837,-1444,484,-835) (31/18,19/11) -> (19/11,45/26) Parabolic Matrix(793,-1468,168,-311) (24/13,13/7) -> (33/7,52/11) Hyperbolic Matrix(131,-248,28,-53) (13/7,2/1) -> (14/3,33/7) Hyperbolic Matrix(85,-196,36,-83) (9/4,7/3) -> (7/3,19/8) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic Matrix(205,-1156,36,-203) (28/5,17/3) -> (17/3,40/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(13,112,8,69) -> Matrix(1,2,-2,-3) Matrix(23,144,-4,-25) -> Matrix(19,10,-40,-21) Matrix(83,464,-144,-805) -> Matrix(13,6,2,1) Matrix(23,124,-64,-345) -> Matrix(9,4,-16,-7) Matrix(5,24,16,77) -> Matrix(11,4,-14,-5) Matrix(29,128,12,53) -> Matrix(11,4,-14,-5) Matrix(43,164,156,595) -> Matrix(5,6,-6,-7) Matrix(101,380,80,301) -> Matrix(5,2,2,1) Matrix(153,572,88,329) -> Matrix(17,6,-20,-7) Matrix(223,824,-364,-1345) -> Matrix(23,6,-4,-1) Matrix(55,196,-16,-57) -> Matrix(1,0,4,1) Matrix(51,172,8,27) -> Matrix(1,0,-2,1) Matrix(17,56,44,145) -> Matrix(1,0,0,1) Matrix(11,32,-32,-93) -> Matrix(3,2,-2,-1) Matrix(23,64,-124,-345) -> Matrix(17,10,-12,-7) Matrix(95,256,-36,-97) -> Matrix(19,10,-40,-21) Matrix(185,484,-284,-743) -> Matrix(13,6,-24,-11) Matrix(273,712,-472,-1231) -> Matrix(9,4,20,9) Matrix(125,324,76,197) -> Matrix(23,10,-30,-13) Matrix(39,100,140,359) -> Matrix(37,16,-44,-19) Matrix(13,32,28,69) -> Matrix(11,4,-14,-5) Matrix(57,136,44,105) -> Matrix(7,2,-4,-1) Matrix(45,104,16,37) -> Matrix(9,2,-14,-3) Matrix(131,292,48,107) -> Matrix(7,-2,-10,3) Matrix(271,592,168,367) -> Matrix(3,4,-4,-5) Matrix(191,412,-312,-673) -> Matrix(1,-4,0,1) Matrix(3,4,-4,-5) -> Matrix(1,0,2,1) Matrix(623,404,-1704,-1105) -> Matrix(1,0,0,1) Matrix(397,256,504,325) -> Matrix(1,0,2,1) Matrix(219,140,280,179) -> Matrix(1,0,-6,1) Matrix(215,136,-596,-377) -> Matrix(3,-2,-4,3) Matrix(165,104,376,237) -> Matrix(9,-8,-10,9) Matrix(103,64,-684,-425) -> Matrix(3,-8,-4,11) Matrix(581,360,-1580,-979) -> Matrix(1,-2,-2,5) Matrix(415,256,-676,-417) -> Matrix(1,-10,0,1) Matrix(53,32,48,29) -> Matrix(1,2,-2,-3) Matrix(339,200,100,59) -> Matrix(1,4,-2,-7) Matrix(335,196,-576,-337) -> Matrix(1,0,4,1) Matrix(283,164,-1044,-605) -> Matrix(5,-2,-2,1) Matrix(2701,1560,1560,901) -> Matrix(13,-12,-14,13) Matrix(139,80,-940,-541) -> Matrix(1,2,-2,-3) Matrix(325,184,136,77) -> Matrix(5,-4,-6,5) Matrix(43,24,-224,-125) -> Matrix(3,-4,-2,3) Matrix(225,124,176,97) -> Matrix(1,-4,0,1) Matrix(391,212,308,167) -> Matrix(1,6,0,1) Matrix(173,92,-472,-251) -> Matrix(1,2,-2,-3) Matrix(111,52,-412,-193) -> Matrix(1,-2,0,1) Matrix(297,136,404,185) -> Matrix(1,0,-4,1) Matrix(257,116,144,65) -> Matrix(3,-8,-4,11) Matrix(37,16,104,45) -> Matrix(1,4,-2,-7) Matrix(105,44,136,57) -> Matrix(1,2,-4,-7) Matrix(69,28,32,13) -> Matrix(1,2,-2,-3) Matrix(297,116,64,25) -> Matrix(5,8,-12,-19) Matrix(259,100,360,139) -> Matrix(1,0,-2,1) Matrix(95,36,-256,-97) -> Matrix(3,4,-4,-5) Matrix(2941,1080,1800,661) -> Matrix(1,2,-2,-3) Matrix(2333,852,408,149) -> Matrix(1,0,2,1) Matrix(373,132,308,109) -> Matrix(1,0,-6,1) Matrix(145,44,56,17) -> Matrix(3,-2,-4,3) Matrix(55,16,-196,-57) -> Matrix(1,-4,0,1) Matrix(277,76,164,45) -> Matrix(1,4,-2,-7) Matrix(329,88,572,153) -> Matrix(5,12,-8,-19) Matrix(301,80,380,101) -> Matrix(1,2,-2,-3) Matrix(381,100,80,21) -> Matrix(1,0,-2,1) Matrix(53,12,128,29) -> Matrix(3,2,-2,-1) Matrix(77,16,24,5) -> Matrix(1,4,-2,-7) Matrix(23,4,-144,-25) -> Matrix(7,8,-8,-9) Matrix(1553,232,328,49) -> Matrix(17,12,-44,-31) Matrix(67,8,92,11) -> Matrix(1,0,-2,1) Matrix(89,-12,52,-7) -> Matrix(1,0,0,1) Matrix(85,-16,16,-3) -> Matrix(1,0,-2,1) Matrix(17,-4,64,-15) -> Matrix(7,8,-8,-9) Matrix(751,-208,408,-113) -> Matrix(61,52,-88,-75) Matrix(109,-32,92,-27) -> Matrix(5,4,-14,-11) Matrix(205,-76,116,-43) -> Matrix(11,4,-14,-5) Matrix(377,-144,144,-55) -> Matrix(3,-2,-4,3) Matrix(85,-36,196,-83) -> Matrix(13,14,-14,-15) Matrix(117,-64,64,-35) -> Matrix(13,8,-18,-11) Matrix(205,-116,76,-43) -> Matrix(13,8,-18,-11) Matrix(1737,-1004,1064,-615) -> Matrix(23,14,-28,-17) Matrix(619,-360,380,-221) -> Matrix(13,8,-18,-11) Matrix(89,-52,12,-7) -> Matrix(3,2,-8,-5) Matrix(25,-16,36,-23) -> Matrix(7,4,-16,-9) Matrix(557,-404,324,-235) -> Matrix(1,0,2,1) Matrix(461,-340,80,-59) -> Matrix(1,0,2,1) Matrix(233,-188,88,-71) -> Matrix(5,2,-8,-3) Matrix(253,-208,208,-171) -> Matrix(1,0,-10,1) Matrix(109,-92,32,-27) -> Matrix(1,4,-2,-7) Matrix(305,-376,116,-143) -> Matrix(11,-2,-16,3) Matrix(25,-36,16,-23) -> Matrix(3,4,-4,-5) Matrix(559,-900,100,-161) -> Matrix(5,4,-24,-19) Matrix(837,-1444,484,-835) -> Matrix(17,18,-18,-19) Matrix(793,-1468,168,-311) -> Matrix(65,44,-164,-111) Matrix(131,-248,28,-53) -> Matrix(19,12,-46,-29) Matrix(85,-196,36,-83) -> Matrix(5,6,-6,-7) Matrix(17,-64,4,-15) -> Matrix(15,8,-32,-17) Matrix(205,-1156,36,-203) -> Matrix(1,0,6,1) 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): 36 Degree of the the map X: 36 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: 24 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -4/1 -7/2 -3/1 -8/3 -9/4 -2/1 -1/1 0/1 1/4 3/7 1/2 2/3 1/1 11/9 3/2 2/1 7/3 5/2 3/1 7/2 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/2 -11/2 -1/2 -9/20 -5/1 -2/5 -9/2 -1/2 -3/8 -4/1 -1/3 0/1 -7/2 0/1 -3/1 -1/1 -14/5 -5/8 -11/4 -7/12 -1/2 -8/3 -1/2 -5/2 -1/2 -3/8 -12/5 -1/3 -2/7 -7/3 0/1 -9/4 0/1 -2/1 -1/2 -1/1 0/1 -2/3 1/0 -7/11 0/1 -12/19 2/3 1/1 -5/8 3/2 1/0 -8/13 1/0 -11/18 -7/2 1/0 -3/5 -1/1 -4/7 0/1 1/1 -9/16 3/2 1/0 -5/9 2/1 -6/11 1/0 -1/2 -1/2 1/0 -3/7 -2/1 -5/12 -3/2 1/0 -2/5 1/0 -5/13 -2/1 -3/8 -1/1 -4/11 -1/1 0/1 -9/25 -2/3 -5/14 -1/2 -1/4 -6/17 1/2 -1/3 -1/1 -1/4 -3/2 1/0 -2/9 -1/2 -1/5 -2/1 -3/16 -3/2 1/0 -2/11 -5/4 -1/6 -1/1 0/1 -1/1 0/1 1/5 -1/1 1/4 -1/1 3/11 -1/1 2/7 -5/6 3/10 -3/4 -1/2 1/3 -2/3 3/8 -1/2 -1/4 5/13 -1/1 2/5 1/0 5/12 -3/2 -5/4 3/7 -1/1 7/16 -11/12 -9/10 4/9 -1/1 -6/7 1/2 -3/4 -1/2 3/5 -2/3 2/3 -1/2 5/7 -2/5 3/4 -1/2 -1/4 7/9 0/1 4/5 -1/3 0/1 1/1 -1/1 7/6 -1/2 -3/8 6/5 -1/4 11/9 0/1 5/4 1/2 1/0 14/11 1/0 9/7 -2/1 4/3 -1/1 0/1 3/2 -1/1 8/5 -1/1 -4/5 5/3 -2/3 12/7 -1/1 0/1 19/11 -1/1 26/15 -9/10 7/4 -5/6 -3/4 2/1 -1/2 9/4 -1/2 1/0 7/3 -1/1 19/8 -7/8 -5/6 12/5 -1/1 -4/5 5/2 -3/4 -1/2 13/5 -5/7 21/8 -7/10 -11/16 29/11 -2/3 8/3 -1/1 -2/3 11/4 -2/3 3/1 -2/3 10/3 -7/12 7/2 -9/16 -1/2 4/1 -1/2 9/2 -1/2 -7/16 5/1 -1/3 1/0 -1/2 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,-2,-23) (-6/1,1/0) -> (-6/11,-1/2) Hyperbolic Matrix(33,184,26,145) (-6/1,-11/2) -> (5/4,14/11) Hyperbolic Matrix(23,124,-64,-345) (-11/2,-5/1) -> (-9/25,-5/14) Hyperbolic Matrix(5,24,16,77) (-5/1,-9/2) -> (3/10,1/3) Hyperbolic Matrix(29,128,12,53) (-9/2,-4/1) -> (12/5,5/2) Hyperbolic Matrix(11,40,-30,-109) (-4/1,-7/2) -> (-3/8,-4/11) Hyperbolic Matrix(13,44,-34,-115) (-7/2,-3/1) -> (-5/13,-3/8) Hyperbolic Matrix(11,32,-32,-93) (-3/1,-14/5) -> (-6/17,-1/3) Hyperbolic Matrix(23,64,-124,-345) (-14/5,-11/4) -> (-3/16,-2/11) Hyperbolic Matrix(65,176,-106,-287) (-11/4,-8/3) -> (-8/13,-11/18) Hyperbolic Matrix(31,80,-50,-129) (-8/3,-5/2) -> (-5/8,-8/13) Hyperbolic Matrix(13,32,28,69) (-5/2,-12/5) -> (4/9,1/2) Hyperbolic Matrix(57,136,44,105) (-12/5,-7/3) -> (9/7,4/3) Hyperbolic Matrix(45,104,16,37) (-7/3,-9/4) -> (11/4,3/1) Hyperbolic Matrix(9,20,-50,-111) (-9/4,-2/1) -> (-2/11,-1/6) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(31,20,110,71) (-2/3,-7/11) -> (3/11,2/7) Hyperbolic Matrix(215,136,-596,-377) (-7/11,-12/19) -> (-4/11,-9/25) Hyperbolic Matrix(165,104,376,237) (-12/19,-5/8) -> (7/16,4/9) Hyperbolic Matrix(53,32,48,29) (-11/18,-3/5) -> (1/1,7/6) Hyperbolic Matrix(75,44,46,27) (-3/5,-4/7) -> (8/5,5/3) Hyperbolic Matrix(325,184,136,77) (-4/7,-9/16) -> (19/8,12/5) Hyperbolic Matrix(43,24,-224,-125) (-9/16,-5/9) -> (-1/5,-3/16) Hyperbolic Matrix(225,124,176,97) (-5/9,-6/11) -> (14/11,9/7) Hyperbolic Matrix(17,8,2,1) (-1/2,-3/7) -> (5/1,1/0) Hyperbolic Matrix(105,44,136,57) (-3/7,-5/12) -> (3/4,7/9) Hyperbolic Matrix(69,28,32,13) (-5/12,-2/5) -> (2/1,9/4) Hyperbolic Matrix(51,20,130,51) (-2/5,-5/13) -> (5/13,2/5) Hyperbolic Matrix(483,172,278,99) (-5/14,-6/17) -> (26/15,7/4) Hyperbolic Matrix(15,4,26,7) (-1/3,-1/4) -> (1/2,3/5) Hyperbolic Matrix(53,12,128,29) (-1/4,-2/9) -> (2/5,5/12) Hyperbolic Matrix(77,16,24,5) (-2/9,-1/5) -> (3/1,10/3) Hyperbolic Matrix(59,8,22,3) (-1/6,0/1) -> (8/3,11/4) Hyperbolic Matrix(27,-4,34,-5) (0/1,1/5) -> (7/9,4/5) Hyperbolic Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(109,-32,92,-27) (2/7,3/10) -> (7/6,6/5) Hyperbolic Matrix(67,-24,14,-5) (1/3,3/8) -> (9/2,5/1) 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(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(87,-64,34,-25) (5/7,3/4) -> (5/2,13/5) Hyperbolic Matrix(37,-32,22,-19) (4/5,1/1) -> (5/3,12/7) Hyperbolic Matrix(329,-400,190,-231) (6/5,11/9) -> (19/11,26/15) Hyperbolic Matrix(305,-376,116,-143) (11/9,5/4) -> (21/8,29/11) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(291,-500,110,-189) (12/7,19/11) -> (29/11,8/3) Hyperbolic Matrix(47,-84,14,-25) (7/4,2/1) -> (10/3,7/2) Hyperbolic Matrix(85,-196,36,-83) (9/4,7/3) -> (7/3,19/8) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,-2,-23) -> Matrix(1,1,-2,-1) Matrix(33,184,26,145) -> Matrix(11,5,2,1) Matrix(23,124,-64,-345) -> Matrix(9,4,-16,-7) Matrix(5,24,16,77) -> Matrix(11,4,-14,-5) Matrix(29,128,12,53) -> Matrix(11,4,-14,-5) Matrix(11,40,-30,-109) -> Matrix(3,1,-4,-1) Matrix(13,44,-34,-115) -> Matrix(1,-1,0,1) Matrix(11,32,-32,-93) -> Matrix(3,2,-2,-1) Matrix(23,64,-124,-345) -> Matrix(17,10,-12,-7) Matrix(65,176,-106,-287) -> Matrix(13,7,-2,-1) Matrix(31,80,-50,-129) -> Matrix(7,3,2,1) Matrix(13,32,28,69) -> Matrix(11,4,-14,-5) Matrix(57,136,44,105) -> Matrix(7,2,-4,-1) Matrix(45,104,16,37) -> Matrix(9,2,-14,-3) Matrix(9,20,-50,-111) -> Matrix(3,-1,-2,1) Matrix(3,4,-4,-5) -> Matrix(1,0,2,1) Matrix(31,20,110,71) -> Matrix(5,1,-6,-1) Matrix(215,136,-596,-377) -> Matrix(3,-2,-4,3) Matrix(165,104,376,237) -> Matrix(9,-8,-10,9) Matrix(53,32,48,29) -> Matrix(1,2,-2,-3) Matrix(75,44,46,27) -> Matrix(3,1,-4,-1) Matrix(325,184,136,77) -> Matrix(5,-4,-6,5) Matrix(43,24,-224,-125) -> Matrix(3,-4,-2,3) Matrix(225,124,176,97) -> Matrix(1,-4,0,1) Matrix(17,8,2,1) -> Matrix(1,1,-2,-1) Matrix(105,44,136,57) -> Matrix(1,2,-4,-7) Matrix(69,28,32,13) -> Matrix(1,2,-2,-3) Matrix(51,20,130,51) -> Matrix(1,1,0,1) Matrix(483,172,278,99) -> Matrix(7,1,-8,-1) Matrix(15,4,26,7) -> Matrix(1,3,-2,-5) Matrix(53,12,128,29) -> Matrix(3,2,-2,-1) Matrix(77,16,24,5) -> Matrix(1,4,-2,-7) Matrix(59,8,22,3) -> Matrix(3,1,-4,-1) Matrix(27,-4,34,-5) -> Matrix(1,1,-4,-3) Matrix(17,-4,64,-15) -> Matrix(7,8,-8,-9) Matrix(109,-32,92,-27) -> Matrix(5,4,-14,-11) Matrix(67,-24,14,-5) -> Matrix(5,3,-12,-7) Matrix(377,-144,144,-55) -> Matrix(3,-2,-4,3) Matrix(85,-36,196,-83) -> Matrix(13,14,-14,-15) Matrix(25,-16,36,-23) -> Matrix(7,4,-16,-9) Matrix(87,-64,34,-25) -> Matrix(5,1,-6,-1) Matrix(37,-32,22,-19) -> Matrix(3,1,-4,-1) Matrix(329,-400,190,-231) -> Matrix(13,1,-14,-1) Matrix(305,-376,116,-143) -> Matrix(11,-2,-16,3) Matrix(25,-36,16,-23) -> Matrix(3,4,-4,-5) Matrix(291,-500,110,-189) -> Matrix(3,1,-4,-1) Matrix(47,-84,14,-25) -> Matrix(15,11,-26,-19) Matrix(85,-196,36,-83) -> Matrix(5,6,-6,-7) Matrix(17,-64,4,-15) -> Matrix(15,8,-32,-17) 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): 36 ----------------------------------------------------------------------- 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 -1/2 5 2 -5/1 -2/5 1 10 -9/2 (-1/2,-3/8) 0 10 -4/1 (-1/3,0/1) 0 10 -3/1 -1/1 1 10 -11/4 (-7/12,-1/2) 0 10 -8/3 -1/2 5 2 -5/2 (-1/2,-3/8) 0 10 -12/5 (-1/3,-2/7) 0 10 -7/3 0/1 1 10 -2/1 -1/2 1 10 -1/1 0/1 1 2 0/1 (-1/1,0/1) 0 10 1/5 -1/1 1 10 1/4 -1/1 4 2 3/11 -1/1 1 10 2/7 -5/6 1 10 3/10 (-3/4,-1/2) 0 10 1/3 -2/3 1 10 3/8 (-1/2,-1/4) 0 10 2/5 1/0 1 10 3/7 -1/1 7 2 4/9 (-1/1,-6/7) 0 10 1/2 (-3/4,-1/2) 0 10 2/3 -1/2 2 2 3/4 (-1/2,-1/4) 0 10 4/5 (-1/3,0/1) 0 10 1/1 -1/1 1 10 7/6 (-1/2,-3/8) 0 10 6/5 -1/4 1 10 11/9 0/1 9 2 5/4 (1/2,1/0) 0 10 14/11 1/0 5 2 9/7 -2/1 1 10 4/3 (-1/1,0/1) 0 10 3/2 -1/1 2 2 8/5 (-1/1,-4/5) 0 10 5/3 -2/3 1 10 12/7 (-1/1,0/1) 0 10 19/11 -1/1 9 2 7/4 (-5/6,-3/4) 0 10 2/1 -1/2 1 10 7/3 -1/1 3 2 12/5 (-1/1,-4/5) 0 10 5/2 (-3/4,-1/2) 0 10 8/3 (-1/1,-2/3) 0 10 3/1 -2/3 1 10 7/2 (-9/16,-1/2) 0 10 4/1 -1/2 4 2 1/0 (-1/2,1/0) 0 10 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,-1) (-6/1,1/0) -> (-6/1,1/0) Reflection Matrix(23,124,18,97) (-6/1,-5/1) -> (14/11,9/7) Glide Reflection Matrix(5,24,16,77) (-5/1,-9/2) -> (3/10,1/3) Hyperbolic Matrix(29,128,12,53) (-9/2,-4/1) -> (12/5,5/2) Hyperbolic Matrix(13,44,8,27) (-4/1,-3/1) -> (8/5,5/3) Glide Reflection Matrix(11,32,10,29) (-3/1,-11/4) -> (1/1,7/6) Glide Reflection Matrix(65,176,-24,-65) (-11/4,-8/3) -> (-11/4,-8/3) Reflection Matrix(31,80,-12,-31) (-8/3,-5/2) -> (-8/3,-5/2) Reflection Matrix(13,32,28,69) (-5/2,-12/5) -> (4/9,1/2) Hyperbolic Matrix(57,136,44,105) (-12/5,-7/3) -> (9/7,4/3) Hyperbolic Matrix(9,20,32,71) (-7/3,-2/1) -> (3/11,2/7) 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(43,-8,16,-3) (0/1,1/5) -> (8/3,3/1) Glide Reflection Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(109,-32,92,-27) (2/7,3/10) -> (7/6,6/5) Hyperbolic Matrix(45,-16,14,-5) (1/3,3/8) -> (3/1,7/2) Glide Reflection Matrix(51,-20,28,-11) (3/8,2/5) -> (7/4,2/1) 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(7,-4,12,-7) (1/2,2/3) -> (1/2,2/3) Reflection Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(57,-44,22,-17) (3/4,4/5) -> (5/2,8/3) Glide Reflection Matrix(37,-32,22,-19) (4/5,1/1) -> (5/3,12/7) Hyperbolic Matrix(109,-132,90,-109) (6/5,11/9) -> (6/5,11/9) Reflection Matrix(139,-172,80,-99) (11/9,5/4) -> (19/11,7/4) Glide Reflection Matrix(111,-140,88,-111) (5/4,14/11) -> (5/4,14/11) Reflection Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(265,-456,154,-265) (12/7,19/11) -> (12/7,19/11) 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(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) Reflection Matrix(-1,8,0,1) (4/1,1/0) -> (4/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,12,0,-1) -> Matrix(1,1,0,-1) (-6/1,1/0) -> (-1/2,1/0) Matrix(23,124,18,97) -> Matrix(9,4,-2,-1) Matrix(5,24,16,77) -> Matrix(11,4,-14,-5) Matrix(29,128,12,53) -> Matrix(11,4,-14,-5) Matrix(13,44,8,27) -> Matrix(1,-1,-2,1) Matrix(11,32,10,29) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(65,176,-24,-65) -> Matrix(13,7,-24,-13) (-11/4,-8/3) -> (-7/12,-1/2) Matrix(31,80,-12,-31) -> Matrix(7,3,-16,-7) (-8/3,-5/2) -> (-1/2,-3/8) Matrix(13,32,28,69) -> Matrix(11,4,-14,-5) Matrix(57,136,44,105) -> Matrix(7,2,-4,-1) Matrix(9,20,32,71) -> Matrix(3,-1,-4,1) Matrix(3,4,-2,-3) -> Matrix(-1,0,4,1) (-2/1,-1/1) -> (-1/2,0/1) Matrix(-1,0,2,1) -> Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Matrix(43,-8,16,-3) -> Matrix(1,-1,-2,1) Matrix(17,-4,64,-15) -> Matrix(7,8,-8,-9) -1/1 Matrix(109,-32,92,-27) -> Matrix(5,4,-14,-11) Matrix(45,-16,14,-5) -> Matrix(7,4,-12,-7) *** -> (-2/3,-1/2) Matrix(51,-20,28,-11) -> Matrix(1,-1,-2,1) Matrix(29,-12,70,-29) -> Matrix(1,2,0,-1) (2/5,3/7) -> (-1/1,1/0) Matrix(55,-24,126,-55) -> Matrix(13,12,-14,-13) (3/7,4/9) -> (-1/1,-6/7) Matrix(7,-4,12,-7) -> Matrix(5,3,-8,-5) (1/2,2/3) -> (-3/4,-1/2) Matrix(17,-12,24,-17) -> Matrix(3,1,-8,-3) (2/3,3/4) -> (-1/2,-1/4) Matrix(57,-44,22,-17) -> Matrix(7,2,-10,-3) Matrix(37,-32,22,-19) -> Matrix(3,1,-4,-1) -1/2 Matrix(109,-132,90,-109) -> Matrix(-1,0,8,1) (6/5,11/9) -> (-1/4,0/1) Matrix(139,-172,80,-99) -> Matrix(5,-1,-6,1) Matrix(111,-140,88,-111) -> Matrix(-1,1,0,1) (5/4,14/11) -> (1/2,1/0) Matrix(25,-36,16,-23) -> Matrix(3,4,-4,-5) -1/1 Matrix(265,-456,154,-265) -> Matrix(-1,0,2,1) (12/7,19/11) -> (-1/1,0/1) Matrix(13,-28,6,-13) -> Matrix(3,2,-4,-3) (2/1,7/3) -> (-1/1,-1/2) Matrix(71,-168,30,-71) -> Matrix(9,8,-10,-9) (7/3,12/5) -> (-1/1,-4/5) Matrix(15,-56,4,-15) -> Matrix(17,9,-32,-17) (7/2,4/1) -> (-9/16,-1/2) Matrix(-1,8,0,1) -> Matrix(1,1,0,-1) (4/1,1/0) -> (-1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.