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): 432 Minimal number of generators: 73 Number of equivalence classes of cusps: 32 Genus: 21 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/3 -5/12 -7/18 -13/36 -1/3 -17/54 -11/36 -5/18 -3/16 -1/6 0/1 1/7 1/6 1/5 3/14 2/9 1/4 2/7 8/27 3/10 1/3 3/8 2/5 5/12 4/9 1/2 5/9 17/27 2/3 7/9 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/2 -6/7 3/4 -5/6 1/1 -9/11 1/2 -4/5 1/0 -11/14 -1/1 0/1 -7/9 0/1 -3/4 0/1 1/2 -5/7 1/2 -12/17 1/2 -19/27 1/2 -7/10 1/2 1/1 -9/13 1/2 -11/16 0/1 1/1 -2/3 1/2 1/0 -13/20 0/1 1/1 -11/17 1/0 -9/14 0/1 1/1 -7/11 1/4 -12/19 1/2 -17/27 1/2 -5/8 0/1 1/2 -3/5 1/2 -10/17 1/2 -7/12 2/3 -11/19 3/4 -4/7 3/4 -9/16 6/7 1/1 -5/9 1/1 -1/2 0/1 1/1 -4/9 1/1 -7/16 1/1 6/5 -10/23 5/4 -3/7 3/2 -8/19 3/2 -5/12 2/1 -7/17 1/0 -9/22 1/1 2/1 -2/5 1/0 -7/18 1/0 -5/13 1/0 -3/8 0/1 1/0 -10/27 1/0 -7/19 1/0 -4/11 -1/2 -13/36 0/1 -9/25 1/4 -5/14 0/1 1/1 -6/17 1/2 -1/3 1/2 1/0 -6/19 1/2 -17/54 1/1 -11/35 1/0 -5/16 0/1 1/1 -4/13 1/0 -11/36 0/1 2/1 -7/23 1/0 -3/10 1/1 1/0 -8/27 1/0 -5/17 1/0 -2/7 1/0 -5/18 1/0 -3/11 1/0 -1/4 0/1 1/0 -2/9 0/1 -3/14 0/1 1/3 -4/19 1/2 -1/5 1/2 -3/16 2/3 1/1 -2/11 1/0 -1/6 1/1 -1/7 3/2 0/1 1/0 1/7 -3/2 1/6 -1/1 2/11 1/0 1/5 -1/2 3/14 -1/3 0/1 2/9 0/1 1/4 0/1 1/0 2/7 1/0 5/17 1/0 8/27 1/0 3/10 -1/1 1/0 4/13 1/0 5/16 -1/1 0/1 1/3 -1/2 1/0 7/20 -1/1 0/1 6/17 -1/2 5/14 -1/1 0/1 4/11 1/2 7/19 1/0 10/27 1/0 3/8 0/1 1/0 2/5 1/0 7/17 1/0 5/12 -2/1 8/19 -3/2 3/7 -3/2 7/16 -6/5 -1/1 4/9 -1/1 1/2 -1/1 0/1 5/9 -1/1 9/16 -1/1 -6/7 13/23 -5/6 4/7 -3/4 11/19 -3/4 7/12 -2/3 10/17 -1/2 13/22 -1/1 -2/3 3/5 -1/2 11/18 -1/2 8/13 -1/2 5/8 -1/2 0/1 17/27 -1/2 12/19 -1/2 7/11 -1/4 23/36 0/1 16/25 1/2 9/14 -1/1 0/1 11/17 1/0 2/3 -1/2 1/0 13/19 1/0 37/54 -1/1 24/35 -1/2 11/16 -1/1 0/1 9/13 -1/2 25/36 -2/3 0/1 16/23 -1/2 7/10 -1/1 -1/2 19/27 -1/2 12/17 -1/2 5/7 -1/2 13/18 -1/2 8/11 -1/2 3/4 -1/2 0/1 7/9 0/1 11/14 0/1 1/1 15/19 1/0 4/5 1/0 13/16 -2/1 -1/1 9/11 -1/2 5/6 -1/1 6/7 -3/4 1/1 -1/2 1/0 0/1 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(53,46,144,125) (-1/1,-6/7) -> (4/11,7/19) Hyperbolic Matrix(19,16,108,91) (-6/7,-5/6) -> (1/6,2/11) Hyperbolic Matrix(17,14,108,89) (-5/6,-9/11) -> (1/7,1/6) Hyperbolic Matrix(125,102,-288,-235) (-9/11,-4/5) -> (-10/23,-3/7) Hyperbolic Matrix(73,58,-180,-143) (-4/5,-11/14) -> (-9/22,-2/5) Hyperbolic Matrix(197,154,252,197) (-11/14,-7/9) -> (7/9,11/14) Hyperbolic Matrix(55,42,72,55) (-7/9,-3/4) -> (3/4,7/9) Hyperbolic Matrix(19,14,-72,-53) (-3/4,-5/7) -> (-3/11,-1/4) Hyperbolic Matrix(107,76,252,179) (-5/7,-12/17) -> (8/19,3/7) Hyperbolic Matrix(613,432,972,685) (-12/17,-19/27) -> (17/27,12/19) Hyperbolic Matrix(379,266,540,379) (-19/27,-7/10) -> (7/10,19/27) Hyperbolic Matrix(109,76,-360,-251) (-7/10,-9/13) -> (-7/23,-3/10) Hyperbolic Matrix(325,224,576,397) (-9/13,-11/16) -> (9/16,13/23) Hyperbolic Matrix(71,48,-108,-73) (-11/16,-2/3) -> (-2/3,-13/20) Parabolic Matrix(305,198,-972,-631) (-13/20,-11/17) -> (-11/35,-5/16) Hyperbolic Matrix(397,256,504,325) (-11/17,-9/14) -> (11/14,15/19) Hyperbolic Matrix(181,116,-504,-323) (-9/14,-7/11) -> (-9/25,-5/14) Hyperbolic Matrix(19,12,144,91) (-7/11,-12/19) -> (0/1,1/7) Hyperbolic Matrix(685,432,972,613) (-12/19,-17/27) -> (19/27,12/17) Hyperbolic Matrix(271,170,432,271) (-17/27,-5/8) -> (5/8,17/27) Hyperbolic Matrix(55,34,-144,-89) (-5/8,-3/5) -> (-5/13,-3/8) Hyperbolic Matrix(37,22,-180,-107) (-3/5,-10/17) -> (-4/19,-1/5) Hyperbolic Matrix(181,106,432,253) (-10/17,-7/12) -> (5/12,8/19) Hyperbolic Matrix(179,104,432,251) (-7/12,-11/19) -> (7/17,5/12) Hyperbolic Matrix(73,42,252,145) (-11/19,-4/7) -> (2/7,5/17) Hyperbolic Matrix(53,30,-288,-163) (-4/7,-9/16) -> (-3/16,-2/11) Hyperbolic Matrix(161,90,288,161) (-9/16,-5/9) -> (5/9,9/16) Hyperbolic Matrix(19,10,36,19) (-5/9,-1/2) -> (1/2,5/9) Hyperbolic Matrix(17,8,36,17) (-1/2,-4/9) -> (4/9,1/2) Hyperbolic Matrix(127,56,288,127) (-4/9,-7/16) -> (7/16,4/9) Hyperbolic Matrix(179,78,576,251) (-7/16,-10/23) -> (4/13,5/16) Hyperbolic Matrix(179,76,252,107) (-3/7,-8/19) -> (12/17,5/7) Hyperbolic Matrix(253,106,432,181) (-8/19,-5/12) -> (7/12,10/17) Hyperbolic Matrix(251,104,432,179) (-5/12,-7/17) -> (11/19,7/12) Hyperbolic Matrix(395,162,612,251) (-7/17,-9/22) -> (9/14,11/17) Hyperbolic Matrix(199,78,324,127) (-2/5,-7/18) -> (11/18,8/13) Hyperbolic Matrix(197,76,324,125) (-7/18,-5/13) -> (3/5,11/18) Hyperbolic Matrix(161,60,432,161) (-3/8,-10/27) -> (10/27,3/8) Hyperbolic Matrix(287,106,972,359) (-10/27,-7/19) -> (5/17,8/27) Hyperbolic Matrix(125,46,144,53) (-7/19,-4/11) -> (6/7,1/1) Hyperbolic Matrix(829,300,1296,469) (-4/11,-13/36) -> (23/36,16/25) Hyperbolic Matrix(827,298,1296,467) (-13/36,-9/25) -> (7/11,23/36) Hyperbolic Matrix(361,128,612,217) (-5/14,-6/17) -> (10/17,13/22) Hyperbolic Matrix(35,12,-108,-37) (-6/17,-1/3) -> (-1/3,-6/19) Parabolic Matrix(1999,630,2916,919) (-6/19,-17/54) -> (37/54,24/35) Hyperbolic Matrix(1997,628,2916,917) (-17/54,-11/35) -> (13/19,37/54) Hyperbolic Matrix(233,72,288,89) (-5/16,-4/13) -> (4/5,13/16) Hyperbolic Matrix(901,276,1296,397) (-4/13,-11/36) -> (25/36,16/23) Hyperbolic Matrix(899,274,1296,395) (-11/36,-7/23) -> (9/13,25/36) Hyperbolic Matrix(161,48,540,161) (-3/10,-8/27) -> (8/27,3/10) Hyperbolic Matrix(359,106,972,287) (-8/27,-5/17) -> (7/19,10/27) Hyperbolic Matrix(145,42,252,73) (-5/17,-2/7) -> (4/7,11/19) Hyperbolic Matrix(235,66,324,91) (-2/7,-5/18) -> (13/18,8/11) Hyperbolic Matrix(233,64,324,89) (-5/18,-3/11) -> (5/7,13/18) Hyperbolic Matrix(17,4,72,17) (-1/4,-2/9) -> (2/9,1/4) Hyperbolic Matrix(55,12,252,55) (-2/9,-3/14) -> (3/14,2/9) Hyperbolic Matrix(179,38,504,107) (-3/14,-4/19) -> (6/17,5/14) Hyperbolic Matrix(199,38,288,55) (-1/5,-3/16) -> (11/16,9/13) Hyperbolic Matrix(91,16,108,19) (-2/11,-1/6) -> (5/6,6/7) Hyperbolic Matrix(89,14,108,17) (-1/6,-1/7) -> (9/11,5/6) Hyperbolic Matrix(91,12,144,19) (-1/7,0/1) -> (12/19,7/11) Hyperbolic Matrix(163,-30,288,-53) (2/11,1/5) -> (13/23,4/7) Hyperbolic Matrix(107,-22,180,-37) (1/5,3/14) -> (13/22,3/5) Hyperbolic Matrix(53,-14,72,-19) (1/4,2/7) -> (8/11,3/4) Hyperbolic Matrix(251,-76,360,-109) (3/10,4/13) -> (16/23,7/10) Hyperbolic Matrix(37,-12,108,-35) (5/16,1/3) -> (1/3,7/20) Parabolic Matrix(667,-234,972,-341) (7/20,6/17) -> (24/35,11/16) Hyperbolic Matrix(323,-116,504,-181) (5/14,4/11) -> (16/25,9/14) Hyperbolic Matrix(89,-34,144,-55) (3/8,2/5) -> (8/13,5/8) Hyperbolic Matrix(143,-58,180,-73) (2/5,7/17) -> (15/19,4/5) Hyperbolic Matrix(235,-102,288,-125) (3/7,7/16) -> (13/16,9/11) Hyperbolic Matrix(73,-48,108,-71) (11/17,2/3) -> (2/3,13/19) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(53,46,144,125) -> Matrix(3,-2,2,-1) Matrix(19,16,108,91) -> Matrix(3,-2,-4,3) Matrix(17,14,108,89) -> Matrix(1,-2,0,1) Matrix(125,102,-288,-235) -> Matrix(5,-4,4,-3) Matrix(73,58,-180,-143) -> Matrix(1,2,0,1) Matrix(197,154,252,197) -> Matrix(1,0,2,1) Matrix(55,42,72,55) -> Matrix(1,0,-4,1) Matrix(19,14,-72,-53) -> Matrix(1,0,-2,1) Matrix(107,76,252,179) -> Matrix(1,-2,0,1) Matrix(613,432,972,685) -> Matrix(5,-2,-12,5) Matrix(379,266,540,379) -> Matrix(3,-2,-4,3) Matrix(109,76,-360,-251) -> Matrix(3,-2,2,-1) Matrix(325,224,576,397) -> Matrix(7,-6,-8,7) Matrix(71,48,-108,-73) -> Matrix(1,0,0,1) Matrix(305,198,-972,-631) -> Matrix(1,0,0,1) Matrix(397,256,504,325) -> Matrix(1,0,0,1) Matrix(181,116,-504,-323) -> Matrix(1,0,0,1) Matrix(19,12,144,91) -> Matrix(5,-2,-2,1) Matrix(685,432,972,613) -> Matrix(5,-2,-12,5) Matrix(271,170,432,271) -> Matrix(1,0,-4,1) Matrix(55,34,-144,-89) -> Matrix(1,0,-2,1) Matrix(37,22,-180,-107) -> Matrix(3,-2,8,-5) Matrix(181,106,432,253) -> Matrix(13,-8,-8,5) Matrix(179,104,432,251) -> Matrix(11,-8,-4,3) Matrix(73,42,252,145) -> Matrix(3,-2,-4,3) Matrix(53,30,-288,-163) -> Matrix(5,-4,4,-3) Matrix(161,90,288,161) -> Matrix(13,-12,-14,13) Matrix(19,10,36,19) -> Matrix(1,0,-2,1) Matrix(17,8,36,17) -> Matrix(1,0,-2,1) Matrix(127,56,288,127) -> Matrix(11,-12,-10,11) Matrix(179,78,576,251) -> Matrix(5,-6,-4,5) Matrix(179,76,252,107) -> Matrix(1,-2,0,1) Matrix(253,106,432,181) -> Matrix(5,-8,-8,13) Matrix(251,104,432,179) -> Matrix(3,-8,-4,11) Matrix(395,162,612,251) -> Matrix(1,-2,0,1) Matrix(199,78,324,127) -> Matrix(1,-4,-2,9) Matrix(197,76,324,125) -> Matrix(1,4,-2,-7) Matrix(161,60,432,161) -> Matrix(1,0,0,1) Matrix(287,106,972,359) -> Matrix(1,2,0,1) Matrix(125,46,144,53) -> Matrix(1,2,-2,-3) Matrix(829,300,1296,469) -> Matrix(1,0,4,1) Matrix(827,298,1296,467) -> Matrix(1,0,-8,1) Matrix(361,128,612,217) -> Matrix(3,-2,-4,3) Matrix(35,12,-108,-37) -> Matrix(1,0,0,1) Matrix(1999,630,2916,919) -> Matrix(3,-2,-4,3) Matrix(1997,628,2916,917) -> Matrix(1,-2,0,1) Matrix(233,72,288,89) -> Matrix(1,-2,0,1) Matrix(901,276,1296,397) -> Matrix(1,0,-2,1) Matrix(899,274,1296,395) -> Matrix(1,0,-2,1) Matrix(161,48,540,161) -> Matrix(1,-2,0,1) Matrix(359,106,972,287) -> Matrix(1,2,0,1) Matrix(145,42,252,73) -> Matrix(3,-2,-4,3) Matrix(235,66,324,91) -> Matrix(1,-2,-2,5) Matrix(233,64,324,89) -> Matrix(1,2,-2,-3) Matrix(17,4,72,17) -> Matrix(1,0,0,1) Matrix(55,12,252,55) -> Matrix(1,0,-6,1) Matrix(179,38,504,107) -> Matrix(1,0,-4,1) Matrix(199,38,288,55) -> Matrix(3,-2,-4,3) Matrix(91,16,108,19) -> Matrix(3,-2,-4,3) Matrix(89,14,108,17) -> Matrix(1,-2,0,1) Matrix(91,12,144,19) -> Matrix(1,-2,-2,5) Matrix(163,-30,288,-53) -> Matrix(3,4,-4,-5) Matrix(107,-22,180,-37) -> Matrix(5,2,-8,-3) Matrix(53,-14,72,-19) -> Matrix(1,0,-2,1) Matrix(251,-76,360,-109) -> Matrix(1,2,-2,-3) Matrix(37,-12,108,-35) -> Matrix(1,0,0,1) Matrix(667,-234,972,-341) -> Matrix(1,0,0,1) Matrix(323,-116,504,-181) -> Matrix(1,0,0,1) Matrix(89,-34,144,-55) -> Matrix(1,0,-2,1) Matrix(143,-58,180,-73) -> Matrix(1,2,0,1) Matrix(235,-102,288,-125) -> Matrix(3,4,-4,-5) Matrix(73,-48,108,-71) -> Matrix(1,0,0,1) 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): 18 Degree of the the map X: 18 Degree of the the map Y: 72 ----------------------------------------------------------------------- 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): 216 Minimal number of generators: 37 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: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/7 1/6 1/5 3/14 2/9 1/4 8/27 3/10 5/16 1/3 3/8 5/12 4/9 1/2 7/12 11/18 23/36 2/3 37/54 25/36 13/18 5/6 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/7 -3/2 1/6 -1/1 2/11 1/0 1/5 -1/2 3/14 -1/3 0/1 2/9 0/1 1/4 0/1 1/0 2/7 1/0 5/17 1/0 8/27 1/0 3/10 -1/1 1/0 4/13 1/0 5/16 -1/1 0/1 1/3 -1/2 1/0 7/20 -1/1 0/1 6/17 -1/2 5/14 -1/1 0/1 4/11 1/2 7/19 1/0 10/27 1/0 3/8 0/1 1/0 2/5 1/0 7/17 1/0 5/12 -2/1 8/19 -3/2 3/7 -3/2 7/16 -6/5 -1/1 4/9 -1/1 1/2 -1/1 0/1 5/9 -1/1 9/16 -1/1 -6/7 13/23 -5/6 4/7 -3/4 11/19 -3/4 7/12 -2/3 10/17 -1/2 13/22 -1/1 -2/3 3/5 -1/2 11/18 -1/2 8/13 -1/2 5/8 -1/2 0/1 17/27 -1/2 12/19 -1/2 7/11 -1/4 23/36 0/1 16/25 1/2 9/14 -1/1 0/1 11/17 1/0 2/3 -1/2 1/0 13/19 1/0 37/54 -1/1 24/35 -1/2 11/16 -1/1 0/1 9/13 -1/2 25/36 -2/3 0/1 16/23 -1/2 7/10 -1/1 -1/2 19/27 -1/2 12/17 -1/2 5/7 -1/2 13/18 -1/2 8/11 -1/2 3/4 -1/2 0/1 7/9 0/1 11/14 0/1 1/1 15/19 1/0 4/5 1/0 13/16 -2/1 -1/1 9/11 -1/2 5/6 -1/1 6/7 -3/4 1/1 -1/2 1/0 0/1 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(53,-7,144,-19) (0/1,1/7) -> (4/11,7/19) Hyperbolic Matrix(19,-3,108,-17) (1/7,1/6) -> (1/6,2/11) Parabolic Matrix(163,-30,288,-53) (2/11,1/5) -> (13/23,4/7) Hyperbolic Matrix(107,-22,180,-37) (1/5,3/14) -> (13/22,3/5) Hyperbolic Matrix(197,-43,252,-55) (3/14,2/9) -> (7/9,11/14) Hyperbolic Matrix(55,-13,72,-17) (2/9,1/4) -> (3/4,7/9) Hyperbolic Matrix(53,-14,72,-19) (1/4,2/7) -> (8/11,3/4) Hyperbolic Matrix(107,-31,252,-73) (2/7,5/17) -> (8/19,3/7) Hyperbolic Matrix(613,-181,972,-287) (5/17,8/27) -> (17/27,12/19) Hyperbolic Matrix(379,-113,540,-161) (8/27,3/10) -> (7/10,19/27) Hyperbolic Matrix(251,-76,360,-109) (3/10,4/13) -> (16/23,7/10) Hyperbolic Matrix(325,-101,576,-179) (4/13,5/16) -> (9/16,13/23) Hyperbolic Matrix(37,-12,108,-35) (5/16,1/3) -> (1/3,7/20) Parabolic Matrix(667,-234,972,-341) (7/20,6/17) -> (24/35,11/16) Hyperbolic Matrix(397,-141,504,-179) (6/17,5/14) -> (11/14,15/19) Hyperbolic Matrix(323,-116,504,-181) (5/14,4/11) -> (16/25,9/14) Hyperbolic Matrix(685,-253,972,-359) (7/19,10/27) -> (19/27,12/17) Hyperbolic Matrix(271,-101,432,-161) (10/27,3/8) -> (5/8,17/27) Hyperbolic Matrix(89,-34,144,-55) (3/8,2/5) -> (8/13,5/8) Hyperbolic Matrix(143,-58,180,-73) (2/5,7/17) -> (15/19,4/5) Hyperbolic Matrix(181,-75,432,-179) (7/17,5/12) -> (5/12,8/19) Parabolic Matrix(235,-102,288,-125) (3/7,7/16) -> (13/16,9/11) Hyperbolic Matrix(161,-71,288,-127) (7/16,4/9) -> (5/9,9/16) Hyperbolic Matrix(19,-9,36,-17) (4/9,1/2) -> (1/2,5/9) Parabolic Matrix(179,-103,252,-145) (4/7,11/19) -> (12/17,5/7) Hyperbolic Matrix(253,-147,432,-251) (11/19,7/12) -> (7/12,10/17) Parabolic Matrix(395,-233,612,-361) (10/17,13/22) -> (9/14,11/17) Hyperbolic Matrix(199,-121,324,-197) (3/5,11/18) -> (11/18,8/13) Parabolic Matrix(125,-79,144,-91) (12/19,7/11) -> (6/7,1/1) Hyperbolic Matrix(829,-529,1296,-827) (7/11,23/36) -> (23/36,16/25) Parabolic Matrix(73,-48,108,-71) (11/17,2/3) -> (2/3,13/19) Parabolic Matrix(1999,-1369,2916,-1997) (13/19,37/54) -> (37/54,24/35) Parabolic Matrix(233,-161,288,-199) (11/16,9/13) -> (4/5,13/16) Hyperbolic Matrix(901,-625,1296,-899) (9/13,25/36) -> (25/36,16/23) Parabolic Matrix(235,-169,324,-233) (5/7,13/18) -> (13/18,8/11) Parabolic Matrix(91,-75,108,-89) (9/11,5/6) -> (5/6,6/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-2,1) Matrix(53,-7,144,-19) -> Matrix(1,2,0,1) Matrix(19,-3,108,-17) -> Matrix(1,2,-2,-3) Matrix(163,-30,288,-53) -> Matrix(3,4,-4,-5) Matrix(107,-22,180,-37) -> Matrix(5,2,-8,-3) Matrix(197,-43,252,-55) -> Matrix(1,0,4,1) Matrix(55,-13,72,-17) -> Matrix(1,0,-2,1) Matrix(53,-14,72,-19) -> Matrix(1,0,-2,1) Matrix(107,-31,252,-73) -> Matrix(3,2,-2,-1) Matrix(613,-181,972,-287) -> Matrix(1,-2,-2,5) Matrix(379,-113,540,-161) -> Matrix(1,2,-2,-3) Matrix(251,-76,360,-109) -> Matrix(1,2,-2,-3) Matrix(325,-101,576,-179) -> Matrix(5,6,-6,-7) Matrix(37,-12,108,-35) -> Matrix(1,0,0,1) Matrix(667,-234,972,-341) -> Matrix(1,0,0,1) Matrix(397,-141,504,-179) -> Matrix(1,0,2,1) Matrix(323,-116,504,-181) -> Matrix(1,0,0,1) Matrix(685,-253,972,-359) -> Matrix(1,-2,-2,5) Matrix(271,-101,432,-161) -> Matrix(1,0,-2,1) Matrix(89,-34,144,-55) -> Matrix(1,0,-2,1) Matrix(143,-58,180,-73) -> Matrix(1,2,0,1) Matrix(181,-75,432,-179) -> Matrix(3,8,-2,-5) Matrix(235,-102,288,-125) -> Matrix(3,4,-4,-5) Matrix(161,-71,288,-127) -> Matrix(11,12,-12,-13) Matrix(19,-9,36,-17) -> Matrix(1,0,0,1) Matrix(179,-103,252,-145) -> Matrix(3,2,-2,-1) Matrix(253,-147,432,-251) -> Matrix(11,8,-18,-13) Matrix(395,-233,612,-361) -> Matrix(3,2,-2,-1) Matrix(199,-121,324,-197) -> Matrix(7,4,-16,-9) Matrix(125,-79,144,-91) -> Matrix(5,2,-8,-3) Matrix(829,-529,1296,-827) -> Matrix(1,0,6,1) Matrix(73,-48,108,-71) -> Matrix(1,0,0,1) Matrix(1999,-1369,2916,-1997) -> Matrix(1,2,-2,-3) Matrix(233,-161,288,-199) -> Matrix(3,2,-2,-1) Matrix(901,-625,1296,-899) -> Matrix(1,0,0,1) Matrix(235,-169,324,-233) -> Matrix(3,2,-8,-5) Matrix(91,-75,108,-89) -> Matrix(1,2,-2,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 9 ----------------------------------------------------------------------- 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 1/0 1 18 1/7 -3/2 1 18 1/6 -1/1 2 3 2/11 1/0 1 18 1/5 -1/2 1 18 3/14 (-1/3,0/1) 0 9 2/9 0/1 3 2 1/4 (0/1,1/0) 0 9 5/18 1/0 2 1 2/7 1/0 1 18 5/17 1/0 1 18 8/27 1/0 3 2 3/10 (-1/1,1/0) 0 9 11/36 (-1/1,1/0) 0 1 4/13 1/0 1 18 5/16 (-1/1,0/1) 0 9 1/3 0 6 7/20 (-1/1,0/1) 0 9 19/54 -1/1 2 1 6/17 -1/2 1 18 5/14 (-1/1,0/1) 0 9 13/36 0/1 6 1 4/11 1/2 1 18 7/19 1/0 1 18 10/27 1/0 3 2 3/8 (0/1,1/0) 0 9 7/18 1/0 4 1 2/5 1/0 1 18 9/22 (-2/1,-1/1) 0 9 7/17 1/0 1 18 5/12 -2/1 2 3 8/19 -3/2 1 18 3/7 -3/2 1 18 10/23 -5/4 1 18 7/16 (-6/5,-1/1) 0 9 4/9 -1/1 6 2 1/2 (-1/1,0/1) 0 9 1/0 0/1 2 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(53,-7,144,-19) (0/1,1/7) -> (4/11,7/19) Hyperbolic Matrix(19,-3,108,-17) (1/7,1/6) -> (1/6,2/11) Parabolic Matrix(125,-23,288,-53) (2/11,1/5) -> (3/7,10/23) Glide Reflection Matrix(73,-15,180,-37) (1/5,3/14) -> (2/5,9/22) Glide Reflection Matrix(55,-12,252,-55) (3/14,2/9) -> (3/14,2/9) Reflection Matrix(17,-4,72,-17) (2/9,1/4) -> (2/9,1/4) Reflection Matrix(19,-5,72,-19) (1/4,5/18) -> (1/4,5/18) Reflection Matrix(71,-20,252,-71) (5/18,2/7) -> (5/18,2/7) Reflection Matrix(107,-31,252,-73) (2/7,5/17) -> (8/19,3/7) Hyperbolic Matrix(359,-106,972,-287) (5/17,8/27) -> (7/19,10/27) Glide Reflection Matrix(161,-48,540,-161) (8/27,3/10) -> (8/27,3/10) Reflection Matrix(109,-33,360,-109) (3/10,11/36) -> (3/10,11/36) Reflection Matrix(287,-88,936,-287) (11/36,4/13) -> (11/36,4/13) Reflection Matrix(251,-78,576,-179) (4/13,5/16) -> (10/23,7/16) Glide Reflection Matrix(37,-12,108,-35) (5/16,1/3) -> (1/3,7/20) Parabolic Matrix(379,-133,1080,-379) (7/20,19/54) -> (7/20,19/54) Reflection Matrix(647,-228,1836,-647) (19/54,6/17) -> (19/54,6/17) Reflection Matrix(251,-89,612,-217) (6/17,5/14) -> (9/22,7/17) Hyperbolic Matrix(181,-65,504,-181) (5/14,13/36) -> (5/14,13/36) Reflection Matrix(287,-104,792,-287) (13/36,4/11) -> (13/36,4/11) Reflection Matrix(161,-60,432,-161) (10/27,3/8) -> (10/27,3/8) Reflection Matrix(55,-21,144,-55) (3/8,7/18) -> (3/8,7/18) Reflection Matrix(71,-28,180,-71) (7/18,2/5) -> (7/18,2/5) Reflection Matrix(181,-75,432,-179) (7/17,5/12) -> (5/12,8/19) Parabolic Matrix(127,-56,288,-127) (7/16,4/9) -> (7/16,4/9) Reflection Matrix(17,-8,36,-17) (4/9,1/2) -> (4/9,1/2) 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(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(53,-7,144,-19) -> Matrix(1,2,0,1) 1/0 Matrix(19,-3,108,-17) -> Matrix(1,2,-2,-3) -1/1 Matrix(125,-23,288,-53) -> Matrix(3,4,-2,-3) *** -> (-2/1,-1/1) Matrix(73,-15,180,-37) -> Matrix(5,2,-2,-1) Matrix(55,-12,252,-55) -> Matrix(-1,0,6,1) (3/14,2/9) -> (-1/3,0/1) Matrix(17,-4,72,-17) -> Matrix(1,0,0,-1) (2/9,1/4) -> (0/1,1/0) Matrix(19,-5,72,-19) -> Matrix(1,0,0,-1) (1/4,5/18) -> (0/1,1/0) Matrix(71,-20,252,-71) -> Matrix(1,2,0,-1) (5/18,2/7) -> (-1/1,1/0) Matrix(107,-31,252,-73) -> Matrix(3,2,-2,-1) -1/1 Matrix(359,-106,972,-287) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(161,-48,540,-161) -> Matrix(1,2,0,-1) (8/27,3/10) -> (-1/1,1/0) Matrix(109,-33,360,-109) -> Matrix(1,2,0,-1) (3/10,11/36) -> (-1/1,1/0) Matrix(287,-88,936,-287) -> Matrix(1,2,0,-1) (11/36,4/13) -> (-1/1,1/0) Matrix(251,-78,576,-179) -> Matrix(5,6,-4,-5) *** -> (-3/2,-1/1) Matrix(37,-12,108,-35) -> Matrix(1,0,0,1) Matrix(379,-133,1080,-379) -> Matrix(-1,0,2,1) (7/20,19/54) -> (-1/1,0/1) Matrix(647,-228,1836,-647) -> Matrix(3,2,-4,-3) (19/54,6/17) -> (-1/1,-1/2) Matrix(251,-89,612,-217) -> Matrix(3,2,-2,-1) -1/1 Matrix(181,-65,504,-181) -> Matrix(-1,0,2,1) (5/14,13/36) -> (-1/1,0/1) Matrix(287,-104,792,-287) -> Matrix(1,0,4,-1) (13/36,4/11) -> (0/1,1/2) Matrix(161,-60,432,-161) -> Matrix(1,0,0,-1) (10/27,3/8) -> (0/1,1/0) Matrix(55,-21,144,-55) -> Matrix(1,0,0,-1) (3/8,7/18) -> (0/1,1/0) Matrix(71,-28,180,-71) -> Matrix(1,4,0,-1) (7/18,2/5) -> (-2/1,1/0) Matrix(181,-75,432,-179) -> Matrix(3,8,-2,-5) -2/1 Matrix(127,-56,288,-127) -> Matrix(11,12,-10,-11) (7/16,4/9) -> (-6/5,-1/1) Matrix(17,-8,36,-17) -> Matrix(-1,0,2,1) (4/9,1/2) -> (-1/1,0/1) Matrix(-1,1,0,1) -> Matrix(-1,0,2,1) (1/2,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.