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 -6/1 -3/1 -12/5 -3/2 0/1 1/1 9/7 3/2 27/17 9/5 2/1 9/4 12/5 5/2 18/7 8/3 36/13 3/1 54/17 36/11 10/3 27/8 7/2 18/5 4/1 9/2 14/3 5/1 16/3 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -1/1 0/1 -6/1 -1/2 1/0 -11/2 -1/1 0/1 -5/1 -1/1 -1/2 -14/3 -1/2 -9/2 -1/2 -4/1 -1/2 0/1 -7/2 0/1 1/1 -17/5 1/1 2/1 -27/8 1/0 -10/3 1/0 -13/4 -1/1 1/0 -16/5 -2/1 -1/1 -3/1 -1/1 -20/7 -1/1 0/1 -17/6 -1/1 -6/7 -14/5 -3/4 -11/4 -2/3 -3/5 -19/7 -2/3 -3/5 -27/10 -1/2 -8/3 -2/3 -1/2 -5/2 -1/2 -1/3 -17/7 -1/3 -2/7 -12/5 0/1 -19/8 -1/1 -2/3 -7/3 -1/1 0/1 -16/7 -1/1 0/1 -9/4 -1/1 -2/1 -1/2 -9/5 -1/3 -16/9 -1/3 0/1 -23/13 -1/3 -1/4 -7/4 -1/3 0/1 -19/11 -2/5 -1/3 -12/7 0/1 -17/10 -2/1 -1/1 -22/13 -1/2 -5/3 -1/1 -1/2 -18/11 -1/2 -13/8 -1/2 -5/11 -8/5 -1/2 -2/5 -27/17 -1/2 -19/12 -3/7 -2/5 -11/7 -3/7 -2/5 -36/23 -2/5 -25/16 -2/5 -9/23 -14/9 -3/8 -17/11 -6/17 -1/3 -3/2 -1/3 -19/13 -4/11 -1/3 -54/37 -1/3 -35/24 -1/3 -8/25 -16/11 -1/3 -2/7 -13/9 -1/3 -1/4 -36/25 -2/7 0/1 -23/16 -1/3 -1/4 -10/7 -1/4 -27/19 -1/4 -17/12 -2/9 -1/5 -7/5 -1/5 0/1 -18/13 0/1 -11/8 0/1 1/1 -4/3 -1/2 0/1 -9/7 -1/2 -14/11 -1/2 -19/15 -1/3 -2/7 -5/4 -1/2 -1/3 -16/13 -1/3 0/1 -11/9 -1/3 0/1 -6/5 -1/2 -1/4 -7/6 -1/3 0/1 -1/1 -1/3 0/1 0/1 0/1 1/1 0/1 1/1 7/6 0/1 1/1 6/5 1/2 1/0 11/9 0/1 1/1 5/4 1/1 1/0 14/11 1/0 9/7 1/0 4/3 0/1 1/0 7/5 0/1 1/3 17/12 1/3 2/5 27/19 1/2 10/7 1/2 13/9 1/2 1/1 16/11 2/3 1/1 3/2 1/1 20/13 0/1 1/1 17/11 1/1 6/5 14/9 3/2 11/7 2/1 3/1 19/12 2/1 3/1 27/17 1/0 8/5 2/1 1/0 5/3 -1/1 1/0 17/10 -1/1 -2/3 12/7 0/1 19/11 1/1 2/1 7/4 0/1 1/1 16/9 0/1 1/1 9/5 1/1 2/1 1/0 9/4 -1/1 16/7 -1/1 0/1 23/10 -1/1 -1/2 7/3 -1/1 0/1 19/8 -2/1 -1/1 12/5 0/1 17/7 2/3 1/1 22/9 1/0 5/2 1/1 1/0 18/7 1/0 13/5 -5/1 1/0 8/3 -2/1 1/0 27/10 1/0 19/7 -3/1 -2/1 11/4 -3/1 -2/1 36/13 -2/1 25/9 -2/1 -9/5 14/5 -3/2 17/6 -6/5 -1/1 3/1 -1/1 19/6 -4/3 -1/1 54/17 -1/1 35/11 -1/1 -8/9 16/5 -1/1 -2/3 13/4 -1/1 -1/2 36/11 -2/3 0/1 23/7 -1/1 -1/2 10/3 -1/2 27/8 -1/2 17/5 -2/5 -1/3 7/2 -1/3 0/1 18/5 0/1 11/3 0/1 1/3 4/1 0/1 1/0 9/2 1/0 14/3 1/0 19/4 -1/1 -2/3 5/1 -1/1 1/0 16/3 -1/1 0/1 11/2 -1/1 0/1 6/1 -1/2 1/0 7/1 -1/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(19,144,12,91) (-7/1,1/0) -> (11/7,19/12) Hyperbolic Matrix(17,108,14,89) (-7/1,-6/1) -> (6/5,11/9) Hyperbolic Matrix(19,108,16,91) (-6/1,-11/2) -> (7/6,6/5) Hyperbolic Matrix(53,288,-30,-163) (-11/2,-5/1) -> (-23/13,-7/4) Hyperbolic Matrix(37,180,-22,-107) (-5/1,-14/3) -> (-22/13,-5/3) Hyperbolic Matrix(55,252,12,55) (-14/3,-9/2) -> (9/2,14/3) Hyperbolic Matrix(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(19,72,-14,-53) (-4/1,-7/2) -> (-11/8,-4/3) Hyperbolic Matrix(73,252,42,145) (-7/2,-17/5) -> (19/11,7/4) Hyperbolic Matrix(287,972,106,359) (-17/5,-27/8) -> (27/10,19/7) Hyperbolic Matrix(161,540,48,161) (-27/8,-10/3) -> (10/3,27/8) Hyperbolic Matrix(109,360,-76,-251) (-10/3,-13/4) -> (-23/16,-10/7) Hyperbolic Matrix(179,576,78,251) (-13/4,-16/5) -> (16/7,23/10) Hyperbolic Matrix(35,108,-12,-37) (-16/5,-3/1) -> (-3/1,-20/7) Parabolic Matrix(341,972,-234,-667) (-20/7,-17/6) -> (-35/24,-16/11) Hyperbolic Matrix(179,504,38,107) (-17/6,-14/5) -> (14/3,19/4) Hyperbolic Matrix(181,504,-116,-323) (-14/5,-11/4) -> (-25/16,-14/9) Hyperbolic Matrix(53,144,46,125) (-11/4,-19/7) -> (1/1,7/6) Hyperbolic Matrix(359,972,106,287) (-19/7,-27/10) -> (27/8,17/5) Hyperbolic Matrix(161,432,60,161) (-27/10,-8/3) -> (8/3,27/10) Hyperbolic Matrix(55,144,-34,-89) (-8/3,-5/2) -> (-13/8,-8/5) Hyperbolic Matrix(73,180,-58,-143) (-5/2,-17/7) -> (-19/15,-5/4) Hyperbolic Matrix(179,432,104,251) (-17/7,-12/5) -> (12/7,19/11) Hyperbolic Matrix(181,432,106,253) (-12/5,-19/8) -> (17/10,12/7) Hyperbolic Matrix(107,252,76,179) (-19/8,-7/3) -> (7/5,17/12) Hyperbolic Matrix(125,288,-102,-235) (-7/3,-16/7) -> (-16/13,-11/9) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(17,36,8,17) (-9/4,-2/1) -> (2/1,9/4) Hyperbolic Matrix(19,36,10,19) (-2/1,-9/5) -> (9/5,2/1) Hyperbolic Matrix(161,288,90,161) (-9/5,-16/9) -> (16/9,9/5) Hyperbolic Matrix(325,576,224,397) (-16/9,-23/13) -> (13/9,16/11) Hyperbolic Matrix(145,252,42,73) (-7/4,-19/11) -> (17/5,7/2) Hyperbolic Matrix(251,432,104,179) (-19/11,-12/7) -> (12/5,17/7) Hyperbolic Matrix(253,432,106,181) (-12/7,-17/10) -> (19/8,12/5) Hyperbolic Matrix(361,612,128,217) (-17/10,-22/13) -> (14/5,17/6) Hyperbolic Matrix(197,324,76,125) (-5/3,-18/11) -> (18/7,13/5) Hyperbolic Matrix(199,324,78,127) (-18/11,-13/8) -> (5/2,18/7) Hyperbolic Matrix(271,432,170,271) (-8/5,-27/17) -> (27/17,8/5) Hyperbolic Matrix(613,972,432,685) (-27/17,-19/12) -> (17/12,27/19) Hyperbolic Matrix(91,144,12,19) (-19/12,-11/7) -> (7/1,1/0) Hyperbolic Matrix(827,1296,298,467) (-11/7,-36/23) -> (36/13,25/9) Hyperbolic Matrix(829,1296,300,469) (-36/23,-25/16) -> (11/4,36/13) Hyperbolic Matrix(395,612,162,251) (-14/9,-17/11) -> (17/7,22/9) Hyperbolic Matrix(71,108,-48,-73) (-17/11,-3/2) -> (-3/2,-19/13) Parabolic Matrix(1997,2916,628,917) (-19/13,-54/37) -> (54/17,35/11) Hyperbolic Matrix(1999,2916,630,919) (-54/37,-35/24) -> (19/6,54/17) Hyperbolic Matrix(199,288,38,55) (-16/11,-13/9) -> (5/1,16/3) Hyperbolic Matrix(899,1296,274,395) (-13/9,-36/25) -> (36/11,23/7) Hyperbolic Matrix(901,1296,276,397) (-36/25,-23/16) -> (13/4,36/11) Hyperbolic Matrix(379,540,266,379) (-10/7,-27/19) -> (27/19,10/7) Hyperbolic Matrix(685,972,432,613) (-27/19,-17/12) -> (19/12,27/17) Hyperbolic Matrix(179,252,76,107) (-17/12,-7/5) -> (7/3,19/8) Hyperbolic Matrix(233,324,64,89) (-7/5,-18/13) -> (18/5,11/3) Hyperbolic Matrix(235,324,66,91) (-18/13,-11/8) -> (7/2,18/5) Hyperbolic Matrix(55,72,42,55) (-4/3,-9/7) -> (9/7,4/3) Hyperbolic Matrix(197,252,154,197) (-9/7,-14/11) -> (14/11,9/7) Hyperbolic Matrix(397,504,256,325) (-14/11,-19/15) -> (17/11,14/9) Hyperbolic Matrix(233,288,72,89) (-5/4,-16/13) -> (16/5,13/4) Hyperbolic Matrix(89,108,14,17) (-11/9,-6/5) -> (6/1,7/1) Hyperbolic Matrix(91,108,16,19) (-6/5,-7/6) -> (11/2,6/1) Hyperbolic Matrix(125,144,46,53) (-7/6,-1/1) -> (19/7,11/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(235,-288,102,-125) (11/9,5/4) -> (23/10,7/3) Hyperbolic Matrix(143,-180,58,-73) (5/4,14/11) -> (22/9,5/2) Hyperbolic Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(251,-360,76,-109) (10/7,13/9) -> (23/7,10/3) Hyperbolic Matrix(73,-108,48,-71) (16/11,3/2) -> (3/2,20/13) Parabolic Matrix(631,-972,198,-305) (20/13,17/11) -> (35/11,16/5) Hyperbolic Matrix(323,-504,116,-181) (14/9,11/7) -> (25/9,14/5) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(107,-180,22,-37) (5/3,17/10) -> (19/4,5/1) Hyperbolic Matrix(163,-288,30,-53) (7/4,16/9) -> (16/3,11/2) Hyperbolic Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(19,144,12,91) -> Matrix(5,2,2,1) Matrix(17,108,14,89) -> Matrix(1,0,2,1) Matrix(19,108,16,91) -> Matrix(1,0,2,1) Matrix(53,288,-30,-163) -> Matrix(1,0,-2,1) Matrix(37,180,-22,-107) -> Matrix(1,0,0,1) Matrix(55,252,12,55) -> Matrix(3,2,-2,-1) Matrix(17,72,4,17) -> Matrix(1,0,2,1) Matrix(19,72,-14,-53) -> Matrix(1,0,0,1) Matrix(73,252,42,145) -> Matrix(1,0,0,1) Matrix(287,972,106,359) -> Matrix(1,-4,0,1) Matrix(161,540,48,161) -> Matrix(1,0,-2,1) Matrix(109,360,-76,-251) -> Matrix(1,2,-4,-7) Matrix(179,576,78,251) -> Matrix(1,2,-2,-3) Matrix(35,108,-12,-37) -> Matrix(1,2,-2,-3) Matrix(341,972,-234,-667) -> Matrix(1,2,-4,-7) Matrix(179,504,38,107) -> Matrix(5,4,-4,-3) Matrix(181,504,-116,-323) -> Matrix(17,12,-44,-31) Matrix(53,144,46,125) -> Matrix(3,2,-2,-1) Matrix(359,972,106,287) -> Matrix(7,4,-16,-9) Matrix(161,432,60,161) -> Matrix(7,4,-2,-1) Matrix(55,144,-34,-89) -> Matrix(7,4,-16,-9) Matrix(73,180,-58,-143) -> Matrix(1,0,0,1) Matrix(179,432,104,251) -> Matrix(1,0,4,1) Matrix(181,432,106,253) -> Matrix(1,0,0,1) Matrix(107,252,76,179) -> Matrix(1,0,4,1) Matrix(125,288,-102,-235) -> Matrix(1,0,-2,1) Matrix(127,288,56,127) -> Matrix(1,0,0,1) Matrix(17,36,8,17) -> Matrix(3,2,-2,-1) Matrix(19,36,10,19) -> Matrix(5,2,2,1) Matrix(161,288,90,161) -> Matrix(1,0,4,1) Matrix(325,576,224,397) -> Matrix(7,2,10,3) Matrix(145,252,42,73) -> Matrix(1,0,0,1) Matrix(251,432,104,179) -> Matrix(1,0,4,1) Matrix(253,432,106,181) -> Matrix(1,0,0,1) Matrix(361,612,128,217) -> Matrix(5,4,-4,-3) Matrix(197,324,76,125) -> Matrix(11,6,-2,-1) Matrix(199,324,78,127) -> Matrix(13,6,2,1) Matrix(271,432,170,271) -> Matrix(9,4,2,1) Matrix(613,972,432,685) -> Matrix(9,4,20,9) Matrix(91,144,12,19) -> Matrix(5,2,2,1) Matrix(827,1296,298,467) -> Matrix(59,24,-32,-13) Matrix(829,1296,300,469) -> Matrix(61,24,-28,-11) Matrix(395,612,162,251) -> Matrix(11,4,8,3) Matrix(71,108,-48,-73) -> Matrix(5,2,-18,-7) Matrix(1997,2916,628,917) -> Matrix(35,12,-38,-13) Matrix(1999,2916,630,919) -> Matrix(37,12,-34,-11) Matrix(199,288,38,55) -> Matrix(7,2,-4,-1) Matrix(899,1296,274,395) -> Matrix(1,0,2,1) Matrix(901,1296,276,397) -> Matrix(1,0,2,1) Matrix(379,540,266,379) -> Matrix(1,0,6,1) Matrix(685,972,432,613) -> Matrix(17,4,4,1) Matrix(179,252,76,107) -> Matrix(1,0,4,1) Matrix(233,324,64,89) -> Matrix(1,0,8,1) Matrix(235,324,66,91) -> Matrix(1,0,-4,1) Matrix(55,72,42,55) -> Matrix(1,0,2,1) Matrix(197,252,154,197) -> Matrix(5,2,2,1) Matrix(397,504,256,325) -> Matrix(11,4,8,3) Matrix(233,288,72,89) -> Matrix(5,2,-8,-3) Matrix(89,108,14,17) -> Matrix(1,0,2,1) Matrix(91,108,16,19) -> Matrix(1,0,2,1) Matrix(125,144,46,53) -> Matrix(3,2,-2,-1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(235,-288,102,-125) -> Matrix(1,0,-2,1) Matrix(143,-180,58,-73) -> Matrix(1,0,0,1) Matrix(53,-72,14,-19) -> Matrix(1,0,0,1) Matrix(251,-360,76,-109) -> Matrix(3,-2,-4,3) Matrix(73,-108,48,-71) -> Matrix(3,-2,2,-1) Matrix(631,-972,198,-305) -> Matrix(3,-2,-4,3) Matrix(323,-504,116,-181) -> Matrix(7,-12,-4,7) Matrix(89,-144,34,-55) -> Matrix(1,-4,0,1) Matrix(107,-180,22,-37) -> Matrix(1,0,0,1) Matrix(163,-288,30,-53) -> Matrix(1,0,-2,1) Matrix(37,-108,12,-35) -> 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 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 6/5 3/2 12/7 2/1 9/4 12/5 18/7 8/3 36/13 3/1 54/17 36/11 10/3 27/8 18/5 4/1 9/2 14/3 5/1 16/3 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 0/1 1/1 7/6 0/1 1/1 6/5 1/2 1/0 11/9 0/1 1/1 5/4 1/1 1/0 14/11 1/0 9/7 1/0 4/3 0/1 1/0 7/5 0/1 1/3 17/12 1/3 2/5 27/19 1/2 10/7 1/2 13/9 1/2 1/1 16/11 2/3 1/1 3/2 1/1 20/13 0/1 1/1 17/11 1/1 6/5 14/9 3/2 11/7 2/1 3/1 19/12 2/1 3/1 27/17 1/0 8/5 2/1 1/0 5/3 -1/1 1/0 17/10 -1/1 -2/3 12/7 0/1 19/11 1/1 2/1 7/4 0/1 1/1 16/9 0/1 1/1 9/5 1/1 2/1 1/0 9/4 -1/1 16/7 -1/1 0/1 23/10 -1/1 -1/2 7/3 -1/1 0/1 19/8 -2/1 -1/1 12/5 0/1 17/7 2/3 1/1 22/9 1/0 5/2 1/1 1/0 18/7 1/0 13/5 -5/1 1/0 8/3 -2/1 1/0 27/10 1/0 19/7 -3/1 -2/1 11/4 -3/1 -2/1 36/13 -2/1 25/9 -2/1 -9/5 14/5 -3/2 17/6 -6/5 -1/1 3/1 -1/1 19/6 -4/3 -1/1 54/17 -1/1 35/11 -1/1 -8/9 16/5 -1/1 -2/3 13/4 -1/1 -1/2 36/11 -2/3 0/1 23/7 -1/1 -1/2 10/3 -1/2 27/8 -1/2 17/5 -2/5 -1/3 7/2 -1/3 0/1 18/5 0/1 11/3 0/1 1/3 4/1 0/1 1/0 9/2 1/0 14/3 1/0 19/4 -1/1 -2/3 5/1 -1/1 1/0 16/3 -1/1 0/1 11/2 -1/1 0/1 6/1 -1/2 1/0 7/1 -1/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(125,-144,79,-91) (1/1,7/6) -> (11/7,19/12) Hyperbolic Matrix(91,-108,75,-89) (7/6,6/5) -> (6/5,11/9) Parabolic Matrix(235,-288,102,-125) (11/9,5/4) -> (23/10,7/3) Hyperbolic Matrix(143,-180,58,-73) (5/4,14/11) -> (22/9,5/2) Hyperbolic Matrix(197,-252,43,-55) (14/11,9/7) -> (9/2,14/3) Hyperbolic Matrix(55,-72,13,-17) (9/7,4/3) -> (4/1,9/2) Hyperbolic Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(179,-252,103,-145) (7/5,17/12) -> (19/11,7/4) Hyperbolic Matrix(685,-972,253,-359) (17/12,27/19) -> (27/10,19/7) Hyperbolic Matrix(379,-540,113,-161) (27/19,10/7) -> (10/3,27/8) Hyperbolic Matrix(251,-360,76,-109) (10/7,13/9) -> (23/7,10/3) Hyperbolic Matrix(397,-576,173,-251) (13/9,16/11) -> (16/7,23/10) Hyperbolic Matrix(73,-108,48,-71) (16/11,3/2) -> (3/2,20/13) Parabolic Matrix(631,-972,198,-305) (20/13,17/11) -> (35/11,16/5) Hyperbolic Matrix(325,-504,69,-107) (17/11,14/9) -> (14/3,19/4) Hyperbolic Matrix(323,-504,116,-181) (14/9,11/7) -> (25/9,14/5) Hyperbolic Matrix(613,-972,181,-287) (19/12,27/17) -> (27/8,17/5) Hyperbolic Matrix(271,-432,101,-161) (27/17,8/5) -> (8/3,27/10) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(107,-180,22,-37) (5/3,17/10) -> (19/4,5/1) Hyperbolic Matrix(253,-432,147,-251) (17/10,12/7) -> (12/7,19/11) Parabolic Matrix(163,-288,30,-53) (7/4,16/9) -> (16/3,11/2) Hyperbolic Matrix(161,-288,71,-127) (16/9,9/5) -> (9/4,16/7) Hyperbolic Matrix(19,-36,9,-17) (9/5,2/1) -> (2/1,9/4) Parabolic Matrix(107,-252,31,-73) (7/3,19/8) -> (17/5,7/2) Hyperbolic Matrix(181,-432,75,-179) (19/8,12/5) -> (12/5,17/7) Parabolic Matrix(251,-612,89,-217) (17/7,22/9) -> (14/5,17/6) Hyperbolic Matrix(127,-324,49,-125) (5/2,18/7) -> (18/7,13/5) Parabolic Matrix(53,-144,7,-19) (19/7,11/4) -> (7/1,1/0) Hyperbolic Matrix(469,-1296,169,-467) (11/4,36/13) -> (36/13,25/9) Parabolic Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic Matrix(919,-2916,289,-917) (19/6,54/17) -> (54/17,35/11) Parabolic Matrix(89,-288,17,-55) (16/5,13/4) -> (5/1,16/3) Hyperbolic Matrix(397,-1296,121,-395) (13/4,36/11) -> (36/11,23/7) Parabolic Matrix(91,-324,25,-89) (7/2,18/5) -> (18/5,11/3) Parabolic Matrix(19,-108,3,-17) (11/2,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,2,1) Matrix(125,-144,79,-91) -> Matrix(1,2,0,1) Matrix(91,-108,75,-89) -> Matrix(1,0,0,1) Matrix(235,-288,102,-125) -> Matrix(1,0,-2,1) Matrix(143,-180,58,-73) -> Matrix(1,0,0,1) Matrix(197,-252,43,-55) -> Matrix(1,-2,0,1) Matrix(55,-72,13,-17) -> Matrix(1,0,0,1) Matrix(53,-72,14,-19) -> Matrix(1,0,0,1) Matrix(179,-252,103,-145) -> Matrix(1,0,-2,1) Matrix(685,-972,253,-359) -> Matrix(9,-4,-2,1) Matrix(379,-540,113,-161) -> Matrix(1,0,-4,1) Matrix(251,-360,76,-109) -> Matrix(3,-2,-4,3) Matrix(397,-576,173,-251) -> Matrix(3,-2,-4,3) Matrix(73,-108,48,-71) -> Matrix(3,-2,2,-1) Matrix(631,-972,198,-305) -> Matrix(3,-2,-4,3) Matrix(325,-504,69,-107) -> Matrix(3,-4,-2,3) Matrix(323,-504,116,-181) -> Matrix(7,-12,-4,7) Matrix(613,-972,181,-287) -> Matrix(1,-4,-2,9) Matrix(271,-432,101,-161) -> Matrix(1,-4,0,1) Matrix(89,-144,34,-55) -> Matrix(1,-4,0,1) Matrix(107,-180,22,-37) -> Matrix(1,0,0,1) Matrix(253,-432,147,-251) -> Matrix(1,0,2,1) Matrix(163,-288,30,-53) -> Matrix(1,0,-2,1) Matrix(161,-288,71,-127) -> Matrix(1,0,-2,1) Matrix(19,-36,9,-17) -> Matrix(1,-2,0,1) Matrix(107,-252,31,-73) -> Matrix(1,0,-2,1) Matrix(181,-432,75,-179) -> Matrix(1,0,2,1) Matrix(251,-612,89,-217) -> Matrix(3,-4,-2,3) Matrix(127,-324,49,-125) -> Matrix(1,-6,0,1) Matrix(53,-144,7,-19) -> Matrix(1,2,0,1) Matrix(469,-1296,169,-467) -> Matrix(11,24,-6,-13) Matrix(37,-108,12,-35) -> Matrix(1,2,-2,-3) Matrix(919,-2916,289,-917) -> Matrix(11,12,-12,-13) Matrix(89,-288,17,-55) -> Matrix(3,2,-2,-1) Matrix(397,-1296,121,-395) -> Matrix(1,0,0,1) Matrix(91,-324,25,-89) -> Matrix(1,0,6,1) Matrix(19,-108,3,-17) -> 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 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 0/1 2 1 2/1 1/0 2 9 9/4 -1/1 1 2 16/7 (-1/1,0/1) 0 9 7/3 (-1/1,0/1) 0 18 19/8 (-2/1,-1/1) 0 18 12/5 0/1 2 3 17/7 (2/3,1/1) 0 18 5/2 (1/1,1/0) 0 18 18/7 1/0 6 1 8/3 (-2/1,1/0) 0 9 27/10 1/0 2 2 19/7 (-3/1,-2/1) 0 18 11/4 (-3/1,-2/1) 0 18 36/13 -2/1 6 1 14/5 -3/2 2 9 17/6 (-6/5,-1/1) 0 18 3/1 -1/1 1 6 19/6 (-4/3,-1/1) 0 18 54/17 -1/1 12 1 16/5 (-1/1,-2/3) 0 9 13/4 (-1/1,-1/2) 0 18 36/11 (-1/1,-1/2) 0 1 10/3 -1/2 2 9 27/8 -1/2 2 2 17/5 (-2/5,-1/3) 0 18 7/2 (-1/3,0/1) 0 18 18/5 0/1 6 1 4/1 (0/1,1/0) 0 9 9/2 1/0 1 2 14/3 1/0 2 9 19/4 (-1/1,-2/3) 0 18 5/1 (-1/1,1/0) 0 18 16/3 (-1/1,0/1) 0 9 11/2 (-1/1,0/1) 0 18 6/1 0 3 7/1 (-1/1,0/1) 0 18 1/0 (-1/1,0/1) 0 18 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(17,-36,8,-17) (2/1,9/4) -> (2/1,9/4) Reflection Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(125,-288,23,-53) (16/7,7/3) -> (16/3,11/2) Glide Reflection Matrix(107,-252,31,-73) (7/3,19/8) -> (17/5,7/2) Hyperbolic Matrix(181,-432,75,-179) (19/8,12/5) -> (12/5,17/7) Parabolic Matrix(73,-180,15,-37) (17/7,5/2) -> (19/4,5/1) Glide Reflection Matrix(71,-180,28,-71) (5/2,18/7) -> (5/2,18/7) Reflection Matrix(55,-144,21,-55) (18/7,8/3) -> (18/7,8/3) Reflection Matrix(161,-432,60,-161) (8/3,27/10) -> (8/3,27/10) Reflection Matrix(359,-972,106,-287) (27/10,19/7) -> (27/8,17/5) Glide Reflection Matrix(53,-144,7,-19) (19/7,11/4) -> (7/1,1/0) Hyperbolic Matrix(287,-792,104,-287) (11/4,36/13) -> (11/4,36/13) Reflection Matrix(181,-504,65,-181) (36/13,14/5) -> (36/13,14/5) Reflection Matrix(179,-504,38,-107) (14/5,17/6) -> (14/3,19/4) Glide Reflection Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic Matrix(647,-2052,204,-647) (19/6,54/17) -> (19/6,54/17) Reflection Matrix(271,-864,85,-271) (54/17,16/5) -> (54/17,16/5) Reflection Matrix(89,-288,17,-55) (16/5,13/4) -> (5/1,16/3) Hyperbolic Matrix(287,-936,88,-287) (13/4,36/11) -> (13/4,36/11) Reflection Matrix(109,-360,33,-109) (36/11,10/3) -> (36/11,10/3) Reflection Matrix(161,-540,48,-161) (10/3,27/8) -> (10/3,27/8) Reflection Matrix(71,-252,20,-71) (7/2,18/5) -> (7/2,18/5) Reflection Matrix(19,-72,5,-19) (18/5,4/1) -> (18/5,4/1) Reflection Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) Reflection Matrix(19,-108,3,-17) (11/2,6/1) -> (6/1,7/1) Parabolic 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(1,0,1,-1) -> Matrix(1,0,0,-1) (0/1,2/1) -> (0/1,1/0) Matrix(17,-36,8,-17) -> Matrix(1,2,0,-1) (2/1,9/4) -> (-1/1,1/0) Matrix(127,-288,56,-127) -> Matrix(-1,0,2,1) (9/4,16/7) -> (-1/1,0/1) Matrix(125,-288,23,-53) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(107,-252,31,-73) -> Matrix(1,0,-2,1) 0/1 Matrix(181,-432,75,-179) -> Matrix(1,0,2,1) 0/1 Matrix(73,-180,15,-37) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(71,-180,28,-71) -> Matrix(-1,2,0,1) (5/2,18/7) -> (1/1,1/0) Matrix(55,-144,21,-55) -> Matrix(1,4,0,-1) (18/7,8/3) -> (-2/1,1/0) Matrix(161,-432,60,-161) -> Matrix(1,4,0,-1) (8/3,27/10) -> (-2/1,1/0) Matrix(359,-972,106,-287) -> Matrix(1,4,-2,-9) Matrix(53,-144,7,-19) -> Matrix(1,2,0,1) 1/0 Matrix(287,-792,104,-287) -> Matrix(5,12,-2,-5) (11/4,36/13) -> (-3/1,-2/1) Matrix(181,-504,65,-181) -> Matrix(7,12,-4,-7) (36/13,14/5) -> (-2/1,-3/2) Matrix(179,-504,38,-107) -> Matrix(3,4,-2,-3) *** -> (-2/1,-1/1) Matrix(37,-108,12,-35) -> Matrix(1,2,-2,-3) -1/1 Matrix(647,-2052,204,-647) -> Matrix(7,8,-6,-7) (19/6,54/17) -> (-4/3,-1/1) Matrix(271,-864,85,-271) -> Matrix(5,4,-6,-5) (54/17,16/5) -> (-1/1,-2/3) Matrix(89,-288,17,-55) -> Matrix(3,2,-2,-1) -1/1 Matrix(287,-936,88,-287) -> Matrix(3,2,-4,-3) (13/4,36/11) -> (-1/1,-1/2) Matrix(109,-360,33,-109) -> Matrix(3,2,-4,-3) (36/11,10/3) -> (-1/1,-1/2) Matrix(161,-540,48,-161) -> Matrix(-1,0,4,1) (10/3,27/8) -> (-1/2,0/1) Matrix(71,-252,20,-71) -> Matrix(-1,0,6,1) (7/2,18/5) -> (-1/3,0/1) Matrix(19,-72,5,-19) -> Matrix(1,0,0,-1) (18/5,4/1) -> (0/1,1/0) Matrix(17,-72,4,-17) -> Matrix(1,0,0,-1) (4/1,9/2) -> (0/1,1/0) Matrix(55,-252,12,-55) -> Matrix(1,2,0,-1) (9/2,14/3) -> (-1/1,1/0) Matrix(19,-108,3,-17) -> Matrix(1,0,0,1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.