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: 40 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/6 -2/3 -1/2 -4/9 -6/17 -1/3 -8/27 -4/15 -19/72 -1/4 -2/9 -1/6 -1/8 -1/9 0/1 1/8 1/7 2/13 1/6 2/11 1/5 2/9 1/4 4/15 5/18 2/7 8/27 1/3 7/18 2/5 4/9 1/2 5/9 11/18 2/3 13/18 7/9 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -5/6 -1/1 -4/5 0/1 1/0 -7/9 1/0 -10/13 -3/1 1/0 -13/17 1/0 -3/4 -3/2 1/0 -8/11 -7/6 -1/1 -13/18 -1/1 -5/7 -3/4 -2/3 -1/1 0/1 -7/11 -3/4 -12/19 -1/1 -1/2 -17/27 -1/2 -5/8 -1/2 1/0 -8/13 -5/4 -1/1 -11/18 -1/1 -3/5 -3/4 -10/17 -1/1 -1/2 -7/12 -1/1 -11/19 -3/4 -4/7 -2/3 -1/2 -5/9 -1/2 -6/11 -1/2 -1/3 -1/2 -1/2 1/0 -4/9 -1/2 -7/16 -1/2 -7/16 -10/23 -3/7 -5/12 -3/7 -3/8 -8/19 -1/2 -1/3 -5/12 -1/3 -2/5 -1/4 0/1 -7/18 0/1 -5/13 1/4 -13/34 1/2 1/0 -8/21 0/1 1/1 -3/8 1/2 1/0 -10/27 1/0 -7/19 1/0 -4/11 -3/2 -1/1 -5/14 -3/4 -1/2 -6/17 -1/1 -1/2 -1/3 -1/2 -6/19 -1/2 -1/3 -5/16 -1/2 -1/4 -4/13 -3/8 -1/3 -3/10 -3/10 -1/4 -8/27 -1/4 -5/17 -1/4 -2/7 -1/6 0/1 -5/18 0/1 -3/11 1/4 -7/26 1/2 1/0 -4/15 0/1 1/1 -9/34 1/2 1/0 -14/53 1/2 1/1 -19/72 1/1 -5/19 1/0 -1/4 -1/2 1/0 -2/9 -1/2 -3/14 -1/2 -3/8 -4/19 -1/2 -1/3 -1/5 -1/4 -3/16 -1/2 1/0 -2/11 -1/2 -1/3 -3/17 -1/4 -1/6 0/1 -3/19 1/0 -2/13 -1/1 1/0 -1/7 -1/4 -1/8 -1/2 1/0 -1/9 -1/2 0/1 -1/2 0/1 1/8 -1/2 -1/4 1/7 1/0 2/13 -1/3 -1/4 1/6 0/1 2/11 -1/1 -1/2 1/5 1/0 2/9 -1/2 3/13 -3/8 1/4 -1/2 -1/4 5/19 -1/4 4/15 -1/5 0/1 3/11 -1/8 5/18 0/1 2/7 0/1 1/2 5/17 1/0 8/27 1/0 3/10 -3/2 1/0 1/3 -1/2 4/11 -1/3 -3/10 11/30 -2/7 7/19 -1/4 10/27 -1/4 3/8 -1/4 -1/6 11/29 -1/8 8/21 -1/5 0/1 21/55 -1/4 13/34 -1/4 -1/6 5/13 -1/8 7/18 0/1 2/5 0/1 1/0 5/12 -1/1 8/19 -1/1 -1/2 3/7 -3/4 4/9 -1/2 5/11 -5/12 6/13 -3/8 -1/3 1/2 -1/2 -1/4 7/13 1/0 6/11 -1/1 -1/2 5/9 -1/2 4/7 -1/2 -2/5 7/12 -1/3 10/17 -1/2 -1/3 13/22 -1/2 -1/4 3/5 -3/8 11/18 -1/3 8/13 -1/3 -5/16 21/34 -3/10 -1/4 13/21 -3/10 5/8 -1/2 -1/4 2/3 -1/3 0/1 7/10 -1/2 -1/4 26/37 -1/2 0/1 19/27 -1/2 12/17 -1/2 -1/3 5/7 -3/8 13/18 -1/3 8/11 -1/3 -7/22 11/15 -3/10 25/34 -3/10 -1/4 39/53 -1/4 53/72 -1/3 14/19 -1/3 -3/10 3/4 -3/10 -1/4 13/17 -1/4 23/30 -2/7 10/13 -3/11 -1/4 7/9 -1/4 4/5 -1/4 0/1 5/6 -1/3 6/7 -3/10 -2/7 1/1 -1/4 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(73,62,-126,-107) (-1/1,-5/6) -> (-7/12,-11/19) Hyperbolic Matrix(37,30,90,73) (-5/6,-4/5) -> (2/5,5/12) Hyperbolic Matrix(71,56,90,71) (-4/5,-7/9) -> (7/9,4/5) Hyperbolic Matrix(181,140,234,181) (-7/9,-10/13) -> (10/13,7/9) Hyperbolic Matrix(73,56,-468,-359) (-10/13,-13/17) -> (-3/19,-2/13) Hyperbolic Matrix(37,28,144,109) (-13/17,-3/4) -> (1/4,5/19) Hyperbolic Matrix(71,52,-198,-145) (-3/4,-8/11) -> (-4/11,-5/14) Hyperbolic Matrix(287,208,396,287) (-8/11,-13/18) -> (13/18,8/11) Hyperbolic Matrix(181,130,252,181) (-13/18,-5/7) -> (5/7,13/18) Hyperbolic Matrix(35,24,-54,-37) (-5/7,-2/3) -> (-2/3,-7/11) Parabolic Matrix(145,92,342,217) (-7/11,-12/19) -> (8/19,3/7) Hyperbolic Matrix(685,432,972,613) (-12/19,-17/27) -> (19/27,12/17) Hyperbolic Matrix(35,22,-288,-181) (-17/27,-5/8) -> (-1/8,-1/9) Hyperbolic Matrix(71,44,-234,-145) (-5/8,-8/13) -> (-4/13,-3/10) Hyperbolic Matrix(287,176,468,287) (-8/13,-11/18) -> (11/18,8/13) Hyperbolic Matrix(109,66,180,109) (-11/18,-3/5) -> (3/5,11/18) 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(73,42,252,145) (-11/19,-4/7) -> (2/7,5/17) Hyperbolic Matrix(71,40,126,71) (-4/7,-5/9) -> (5/9,4/7) Hyperbolic Matrix(109,60,198,109) (-5/9,-6/11) -> (6/11,5/9) Hyperbolic Matrix(37,20,-198,-107) (-6/11,-1/2) -> (-3/16,-2/11) Hyperbolic Matrix(71,32,-162,-73) (-1/2,-4/9) -> (-4/9,-7/16) Parabolic Matrix(611,266,990,431) (-7/16,-10/23) -> (8/13,21/34) Hyperbolic Matrix(37,16,252,109) (-10/23,-3/7) -> (1/7,2/13) 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(73,30,90,37) (-5/12,-2/5) -> (4/5,5/6) Hyperbolic Matrix(71,28,180,71) (-2/5,-7/18) -> (7/18,2/5) Hyperbolic Matrix(181,70,468,181) (-7/18,-5/13) -> (5/13,7/18) Hyperbolic Matrix(251,96,468,179) (-5/13,-13/34) -> (1/2,7/13) Hyperbolic Matrix(325,124,-1224,-467) (-13/34,-8/21) -> (-4/15,-9/34) Hyperbolic Matrix(179,68,-666,-253) (-8/21,-3/8) -> (-7/26,-4/15) Hyperbolic Matrix(145,54,486,181) (-3/8,-10/27) -> (8/27,3/10) Hyperbolic Matrix(287,106,972,359) (-10/27,-7/19) -> (5/17,8/27) Hyperbolic Matrix(71,26,-396,-145) (-7/19,-4/11) -> (-2/11,-3/17) 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(395,124,-1494,-469) (-6/19,-5/16) -> (-9/34,-14/53) Hyperbolic Matrix(109,34,234,73) (-5/16,-4/13) -> (6/13,1/2) Hyperbolic Matrix(181,54,486,145) (-3/10,-8/27) -> (10/27,3/8) Hyperbolic Matrix(359,106,972,287) (-8/27,-5/17) -> (7/19,10/27) Hyperbolic Matrix(109,32,126,37) (-5/17,-2/7) -> (6/7,1/1) Hyperbolic Matrix(71,20,252,71) (-2/7,-5/18) -> (5/18,2/7) Hyperbolic Matrix(109,30,396,109) (-5/18,-3/11) -> (3/11,5/18) Hyperbolic Matrix(37,10,270,73) (-3/11,-7/26) -> (1/8,1/7) Hyperbolic Matrix(3817,1008,5184,1369) (-14/53,-19/72) -> (53/72,14/19) Hyperbolic Matrix(3815,1006,5184,1367) (-19/72,-5/19) -> (39/53,53/72) Hyperbolic Matrix(109,28,144,37) (-5/19,-1/4) -> (3/4,13/17) Hyperbolic Matrix(35,8,-162,-37) (-1/4,-2/9) -> (-2/9,-3/14) Parabolic Matrix(253,54,342,73) (-3/14,-4/19) -> (14/19,3/4) Hyperbolic Matrix(145,28,378,73) (-1/5,-3/16) -> (13/34,5/13) Hyperbolic Matrix(35,6,-216,-37) (-3/17,-1/6) -> (-1/6,-3/19) Parabolic Matrix(107,16,234,35) (-2/13,-1/7) -> (5/11,6/13) Hyperbolic Matrix(109,14,288,37) (-1/7,-1/8) -> (3/8,11/29) Hyperbolic Matrix(253,26,360,37) (-1/9,0/1) -> (26/37,19/27) Hyperbolic Matrix(215,-26,306,-37) (0/1,1/8) -> (7/10,26/37) Hyperbolic Matrix(359,-56,468,-73) (2/13,1/6) -> (23/30,10/13) Hyperbolic Matrix(145,-26,396,-71) (1/6,2/11) -> (4/11,11/30) Hyperbolic Matrix(107,-20,198,-37) (2/11,1/5) -> (7/13,6/11) Hyperbolic Matrix(37,-8,162,-35) (1/5,2/9) -> (2/9,3/13) Parabolic Matrix(181,-42,306,-71) (3/13,1/4) -> (13/22,3/5) Hyperbolic Matrix(467,-124,1224,-325) (5/19,4/15) -> (8/21,21/55) Hyperbolic Matrix(253,-68,666,-179) (4/15,3/11) -> (11/29,8/21) Hyperbolic Matrix(145,-44,234,-71) (3/10,1/3) -> (13/21,5/8) Hyperbolic Matrix(145,-52,198,-71) (1/3,4/11) -> (8/11,11/15) Hyperbolic Matrix(827,-304,1080,-397) (11/30,7/19) -> (13/17,23/30) Hyperbolic Matrix(2701,-1032,3672,-1403) (21/55,13/34) -> (25/34,39/53) Hyperbolic Matrix(73,-32,162,-71) (3/7,4/9) -> (4/9,5/11) Parabolic Matrix(107,-62,126,-73) (4/7,7/12) -> (5/6,6/7) Hyperbolic Matrix(899,-556,1224,-757) (21/34,13/21) -> (11/15,25/34) Hyperbolic Matrix(37,-24,54,-35) (5/8,2/3) -> (2/3,7/10) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(73,62,-126,-107) -> Matrix(3,4,-4,-5) Matrix(37,30,90,73) -> Matrix(1,0,0,1) Matrix(71,56,90,71) -> Matrix(1,0,-4,1) Matrix(181,140,234,181) -> Matrix(1,6,-4,-23) Matrix(73,56,-468,-359) -> Matrix(1,2,0,1) Matrix(37,28,144,109) -> Matrix(1,2,-4,-7) Matrix(71,52,-198,-145) -> Matrix(3,4,-4,-5) Matrix(287,208,396,287) -> Matrix(13,14,-40,-43) Matrix(181,130,252,181) -> Matrix(7,6,-20,-17) Matrix(35,24,-54,-37) -> Matrix(1,0,0,1) Matrix(145,92,342,217) -> Matrix(1,0,0,1) Matrix(685,432,972,613) -> Matrix(3,2,-8,-5) Matrix(35,22,-288,-181) -> Matrix(1,0,0,1) Matrix(71,44,-234,-145) -> Matrix(1,2,-4,-7) Matrix(287,176,468,287) -> Matrix(9,10,-28,-31) Matrix(109,66,180,109) -> Matrix(7,6,-20,-17) Matrix(37,22,-180,-107) -> Matrix(3,2,-8,-5) Matrix(181,106,432,253) -> Matrix(1,0,0,1) Matrix(73,42,252,145) -> Matrix(3,2,4,3) Matrix(71,40,126,71) -> Matrix(7,4,-16,-9) Matrix(109,60,198,109) -> Matrix(5,2,-8,-3) Matrix(37,20,-198,-107) -> Matrix(1,0,0,1) Matrix(71,32,-162,-73) -> Matrix(7,4,-16,-9) Matrix(611,266,990,431) -> Matrix(23,10,-76,-33) Matrix(37,16,252,109) -> Matrix(5,2,-8,-3) Matrix(179,76,252,107) -> Matrix(1,0,0,1) Matrix(253,106,432,181) -> Matrix(1,0,0,1) Matrix(73,30,90,37) -> Matrix(1,0,0,1) Matrix(71,28,180,71) -> Matrix(1,0,4,1) Matrix(181,70,468,181) -> Matrix(1,0,-12,1) Matrix(251,96,468,179) -> Matrix(1,0,-4,1) Matrix(325,124,-1224,-467) -> Matrix(1,0,0,1) Matrix(179,68,-666,-253) -> Matrix(1,0,0,1) Matrix(145,54,486,181) -> Matrix(1,-2,0,1) Matrix(287,106,972,359) -> Matrix(1,4,0,1) Matrix(71,26,-396,-145) -> Matrix(1,2,-4,-7) Matrix(361,128,612,217) -> Matrix(3,2,-8,-5) Matrix(35,12,-108,-37) -> Matrix(3,2,-8,-5) Matrix(395,124,-1494,-469) -> Matrix(1,0,4,1) Matrix(109,34,234,73) -> Matrix(1,0,0,1) Matrix(181,54,486,145) -> Matrix(7,2,-32,-9) Matrix(359,106,972,287) -> Matrix(17,4,-64,-15) Matrix(109,32,126,37) -> Matrix(9,2,-32,-7) Matrix(71,20,252,71) -> Matrix(1,0,8,1) Matrix(109,30,396,109) -> Matrix(1,0,-12,1) Matrix(37,10,270,73) -> Matrix(1,0,-4,1) Matrix(3817,1008,5184,1369) -> Matrix(5,-4,-16,13) Matrix(3815,1006,5184,1367) -> Matrix(1,0,-4,1) Matrix(109,28,144,37) -> Matrix(1,2,-4,-7) Matrix(35,8,-162,-37) -> Matrix(3,2,-8,-5) Matrix(253,54,342,73) -> Matrix(11,4,-36,-13) Matrix(145,28,378,73) -> Matrix(1,0,-4,1) Matrix(35,6,-216,-37) -> Matrix(1,0,4,1) Matrix(107,16,234,35) -> Matrix(3,2,-8,-5) Matrix(109,14,288,37) -> Matrix(1,0,-4,1) Matrix(253,26,360,37) -> Matrix(1,0,0,1) Matrix(215,-26,306,-37) -> Matrix(1,0,0,1) Matrix(359,-56,468,-73) -> Matrix(9,2,-32,-7) Matrix(145,-26,396,-71) -> Matrix(1,2,-4,-7) Matrix(107,-20,198,-37) -> Matrix(1,0,0,1) Matrix(37,-8,162,-35) -> Matrix(3,2,-8,-5) Matrix(181,-42,306,-71) -> Matrix(1,0,0,1) Matrix(467,-124,1224,-325) -> Matrix(1,0,0,1) Matrix(253,-68,666,-179) -> Matrix(1,0,0,1) Matrix(145,-44,234,-71) -> Matrix(1,2,-4,-7) Matrix(145,-52,198,-71) -> Matrix(11,4,-36,-13) Matrix(827,-304,1080,-397) -> Matrix(1,0,0,1) Matrix(2701,-1032,3672,-1403) -> Matrix(9,2,-32,-7) Matrix(73,-32,162,-71) -> Matrix(7,4,-16,-9) Matrix(107,-62,126,-73) -> Matrix(11,4,-36,-13) Matrix(899,-556,1224,-757) -> Matrix(1,0,0,1) Matrix(37,-24,54,-35) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 12 Degree of the the map X: 24 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 -2/3 -17/27 -7/18 -1/3 -8/27 -5/18 -1/6 -1/7 0/1 1/6 1/5 2/9 1/4 2/7 1/3 3/8 2/5 4/9 1/2 5/9 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 1/0 -5/6 -1/1 -4/5 0/1 1/0 -7/9 1/0 -3/4 -3/2 1/0 -5/7 -3/4 -2/3 -1/1 0/1 -7/11 -3/4 -12/19 -1/1 -1/2 -17/27 -1/2 -5/8 -1/2 1/0 -3/5 -3/4 -7/12 -1/1 -11/19 -3/4 -4/7 -2/3 -1/2 -5/9 -1/2 -1/2 -1/2 1/0 -4/9 -1/2 -3/7 -3/8 -5/12 -1/3 -2/5 -1/4 0/1 -7/18 0/1 -5/13 1/4 -3/8 1/2 1/0 -1/3 -1/2 -3/10 -3/10 -1/4 -8/27 -1/4 -5/17 -1/4 -2/7 -1/6 0/1 -5/18 0/1 -3/11 1/4 -1/4 -1/2 1/0 -2/9 -1/2 -1/5 -1/4 -1/6 0/1 -1/7 -1/4 0/1 -1/2 0/1 1/6 0/1 1/5 1/0 2/9 -1/2 1/4 -1/2 -1/4 2/7 0/1 1/2 1/3 -1/2 4/11 -1/3 -3/10 7/19 -1/4 10/27 -1/4 3/8 -1/4 -1/6 2/5 0/1 1/0 5/12 -1/1 8/19 -1/1 -1/2 3/7 -3/4 4/9 -1/2 1/2 -1/2 -1/4 5/9 -1/2 4/7 -1/2 -2/5 7/12 -1/3 3/5 -3/8 11/18 -1/3 8/13 -1/3 -5/16 5/8 -1/2 -1/4 2/3 -1/3 0/1 7/10 -1/2 -1/4 19/27 -1/2 12/17 -1/2 -1/3 5/7 -3/8 13/18 -1/3 8/11 -1/3 -7/22 3/4 -3/10 -1/4 7/9 -1/4 4/5 -1/4 0/1 5/6 -1/3 6/7 -3/10 -2/7 1/1 -1/4 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(73,62,-126,-107) (-1/1,-5/6) -> (-7/12,-11/19) Hyperbolic Matrix(37,30,90,73) (-5/6,-4/5) -> (2/5,5/12) Hyperbolic Matrix(71,56,90,71) (-4/5,-7/9) -> (7/9,4/5) 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(35,24,-54,-37) (-5/7,-2/3) -> (-2/3,-7/11) Parabolic Matrix(145,92,342,217) (-7/11,-12/19) -> (8/19,3/7) Hyperbolic Matrix(685,432,972,613) (-12/19,-17/27) -> (19/27,12/17) Hyperbolic Matrix(341,214,486,305) (-17/27,-5/8) -> (7/10,19/27) Hyperbolic Matrix(55,34,-144,-89) (-5/8,-3/5) -> (-5/13,-3/8) Hyperbolic Matrix(17,10,90,53) (-3/5,-7/12) -> (1/6,1/5) Hyperbolic Matrix(125,72,342,197) (-11/19,-4/7) -> (4/11,7/19) Hyperbolic Matrix(71,40,126,71) (-4/7,-5/9) -> (5/9,4/7) 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(55,24,126,55) (-4/9,-3/7) -> (3/7,4/9) Hyperbolic Matrix(19,8,-126,-53) (-3/7,-5/12) -> (-1/6,-1/7) Hyperbolic Matrix(73,30,90,37) (-5/12,-2/5) -> (4/5,5/6) 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(17,6,-54,-19) (-3/8,-1/3) -> (-1/3,-3/10) Parabolic Matrix(181,54,486,145) (-3/10,-8/27) -> (10/27,3/8) Hyperbolic Matrix(359,106,972,287) (-8/27,-5/17) -> (7/19,10/27) Hyperbolic Matrix(109,32,126,37) (-5/17,-2/7) -> (6/7,1/1) 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(19,4,90,19) (-2/9,-1/5) -> (1/5,2/9) Hyperbolic Matrix(53,10,90,17) (-1/5,-1/6) -> (7/12,3/5) Hyperbolic Matrix(89,12,126,17) (-1/7,0/1) -> (12/17,5/7) Hyperbolic Matrix(53,-8,126,-19) (0/1,1/6) -> (5/12,8/19) Hyperbolic Matrix(53,-14,72,-19) (1/4,2/7) -> (8/11,3/4) Hyperbolic Matrix(19,-6,54,-17) (2/7,1/3) -> (1/3,4/11) Parabolic Matrix(89,-34,144,-55) (3/8,2/5) -> (8/13,5/8) Hyperbolic Matrix(107,-62,126,-73) (4/7,7/12) -> (5/6,6/7) Hyperbolic Matrix(37,-24,54,-35) (5/8,2/3) -> (2/3,7/10) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(73,62,-126,-107) -> Matrix(3,4,-4,-5) Matrix(37,30,90,73) -> Matrix(1,0,0,1) Matrix(71,56,90,71) -> Matrix(1,0,-4,1) Matrix(55,42,72,55) -> Matrix(1,3,-4,-11) Matrix(19,14,-72,-53) -> Matrix(1,1,0,1) Matrix(35,24,-54,-37) -> Matrix(1,0,0,1) Matrix(145,92,342,217) -> Matrix(1,0,0,1) Matrix(685,432,972,613) -> Matrix(3,2,-8,-5) Matrix(341,214,486,305) -> Matrix(1,1,-4,-3) Matrix(55,34,-144,-89) -> Matrix(1,1,0,1) Matrix(17,10,90,53) -> Matrix(1,1,-4,-3) Matrix(125,72,342,197) -> Matrix(7,5,-24,-17) Matrix(71,40,126,71) -> Matrix(7,4,-16,-9) Matrix(19,10,36,19) -> Matrix(1,1,-4,-3) Matrix(17,8,36,17) -> Matrix(1,1,-4,-3) Matrix(55,24,126,55) -> Matrix(7,3,-12,-5) Matrix(19,8,-126,-53) -> Matrix(3,1,-4,-1) Matrix(73,30,90,37) -> Matrix(1,0,0,1) Matrix(199,78,324,127) -> Matrix(9,1,-28,-3) Matrix(197,76,324,125) -> Matrix(7,-1,-20,3) Matrix(17,6,-54,-19) -> Matrix(1,1,-4,-3) Matrix(181,54,486,145) -> Matrix(7,2,-32,-9) Matrix(359,106,972,287) -> Matrix(17,4,-64,-15) Matrix(109,32,126,37) -> Matrix(9,2,-32,-7) Matrix(235,66,324,91) -> Matrix(13,1,-40,-3) Matrix(233,64,324,89) -> Matrix(7,-1,-20,3) Matrix(17,4,72,17) -> Matrix(1,1,-4,-3) Matrix(19,4,90,19) -> Matrix(3,1,-4,-1) Matrix(53,10,90,17) -> Matrix(1,1,-4,-3) Matrix(89,12,126,17) -> Matrix(1,1,-4,-3) Matrix(53,-8,126,-19) -> Matrix(3,1,-4,-1) Matrix(53,-14,72,-19) -> Matrix(5,1,-16,-3) Matrix(19,-6,54,-17) -> Matrix(1,1,-4,-3) Matrix(89,-34,144,-55) -> Matrix(5,1,-16,-3) Matrix(107,-62,126,-73) -> Matrix(11,4,-36,-13) Matrix(37,-24,54,-35) -> 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): 12 ----------------------------------------------------------------------- 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 (-1/2,0/1) 0 18 1/6 0/1 2 3 1/5 1/0 1 18 2/9 -1/2 2 2 1/4 (-1/2,-1/4) 0 9 2/7 (0/1,1/2) 0 18 1/3 -1/2 1 6 4/11 (-1/3,-3/10) 0 18 7/19 -1/4 1 18 10/27 -1/4 6 2 3/8 (-1/4,-1/6) 0 9 2/5 (0/1,1/0) 0 18 5/12 -1/1 2 3 8/19 (-1/1,-1/2) 0 18 3/7 -3/4 1 18 4/9 -1/2 4 2 1/2 (-1/2,-1/4) 0 9 5/9 -1/2 3 2 4/7 (-1/2,-2/5) 0 18 7/12 -1/3 2 3 3/5 -3/8 1 18 11/18 -1/3 8 1 8/13 (-1/3,-5/16) 0 18 5/8 (-1/2,-1/4) 0 9 2/3 0 6 7/10 (-1/2,-1/4) 0 9 19/27 -1/2 1 2 12/17 (-1/2,-1/3) 0 18 5/7 -3/8 1 18 13/18 -1/3 10 1 8/11 (-1/3,-7/22) 0 18 3/4 (-3/10,-1/4) 0 9 7/9 -1/4 3 2 4/5 (-1/4,0/1) 0 18 5/6 -1/3 2 3 6/7 (-3/10,-2/7) 0 18 1/1 -1/4 1 18 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,-8,126,-19) (0/1,1/6) -> (5/12,8/19) Hyperbolic Matrix(53,-10,90,-17) (1/6,1/5) -> (7/12,3/5) Glide Reflection Matrix(19,-4,90,-19) (1/5,2/9) -> (1/5,2/9) Reflection Matrix(17,-4,72,-17) (2/9,1/4) -> (2/9,1/4) Reflection Matrix(53,-14,72,-19) (1/4,2/7) -> (8/11,3/4) Hyperbolic Matrix(19,-6,54,-17) (2/7,1/3) -> (1/3,4/11) Parabolic Matrix(125,-46,144,-53) (4/11,7/19) -> (6/7,1/1) Glide Reflection Matrix(379,-140,1026,-379) (7/19,10/27) -> (7/19,10/27) Reflection Matrix(161,-60,432,-161) (10/27,3/8) -> (10/27,3/8) Reflection Matrix(89,-34,144,-55) (3/8,2/5) -> (8/13,5/8) Hyperbolic Matrix(73,-30,90,-37) (2/5,5/12) -> (4/5,5/6) Glide Reflection Matrix(179,-76,252,-107) (8/19,3/7) -> (12/17,5/7) Glide Reflection Matrix(55,-24,126,-55) (3/7,4/9) -> (3/7,4/9) Reflection Matrix(17,-8,36,-17) (4/9,1/2) -> (4/9,1/2) Reflection Matrix(19,-10,36,-19) (1/2,5/9) -> (1/2,5/9) Reflection Matrix(71,-40,126,-71) (5/9,4/7) -> (5/9,4/7) Reflection Matrix(107,-62,126,-73) (4/7,7/12) -> (5/6,6/7) Hyperbolic Matrix(109,-66,180,-109) (3/5,11/18) -> (3/5,11/18) Reflection Matrix(287,-176,468,-287) (11/18,8/13) -> (11/18,8/13) Reflection Matrix(37,-24,54,-35) (5/8,2/3) -> (2/3,7/10) Parabolic Matrix(379,-266,540,-379) (7/10,19/27) -> (7/10,19/27) Reflection Matrix(647,-456,918,-647) (19/27,12/17) -> (19/27,12/17) Reflection Matrix(181,-130,252,-181) (5/7,13/18) -> (5/7,13/18) Reflection Matrix(287,-208,396,-287) (13/18,8/11) -> (13/18,8/11) Reflection Matrix(55,-42,72,-55) (3/4,7/9) -> (3/4,7/9) Reflection Matrix(71,-56,90,-71) (7/9,4/5) -> (7/9,4/5) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,4,1) (0/1,1/0) -> (-1/2,0/1) Matrix(53,-8,126,-19) -> Matrix(3,1,-4,-1) -1/2 Matrix(53,-10,90,-17) -> Matrix(3,1,-8,-3) *** -> (-1/2,-1/4) Matrix(19,-4,90,-19) -> Matrix(1,1,0,-1) (1/5,2/9) -> (-1/2,1/0) Matrix(17,-4,72,-17) -> Matrix(3,1,-8,-3) (2/9,1/4) -> (-1/2,-1/4) Matrix(53,-14,72,-19) -> Matrix(5,1,-16,-3) -1/4 Matrix(19,-6,54,-17) -> Matrix(1,1,-4,-3) -1/2 Matrix(125,-46,144,-53) -> Matrix(11,3,-40,-11) *** -> (-3/10,-1/4) Matrix(379,-140,1026,-379) -> Matrix(19,5,-72,-19) (7/19,10/27) -> (-5/18,-1/4) Matrix(161,-60,432,-161) -> Matrix(5,1,-24,-5) (10/27,3/8) -> (-1/4,-1/6) Matrix(89,-34,144,-55) -> Matrix(5,1,-16,-3) -1/4 Matrix(73,-30,90,-37) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(179,-76,252,-107) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(55,-24,126,-55) -> Matrix(5,3,-8,-5) (3/7,4/9) -> (-3/4,-1/2) Matrix(17,-8,36,-17) -> Matrix(3,1,-8,-3) (4/9,1/2) -> (-1/2,-1/4) Matrix(19,-10,36,-19) -> Matrix(3,1,-8,-3) (1/2,5/9) -> (-1/2,-1/4) Matrix(71,-40,126,-71) -> Matrix(9,4,-20,-9) (5/9,4/7) -> (-1/2,-2/5) Matrix(107,-62,126,-73) -> Matrix(11,4,-36,-13) -1/3 Matrix(109,-66,180,-109) -> Matrix(17,6,-48,-17) (3/5,11/18) -> (-3/8,-1/3) Matrix(287,-176,468,-287) -> Matrix(31,10,-96,-31) (11/18,8/13) -> (-1/3,-5/16) Matrix(37,-24,54,-35) -> Matrix(1,0,0,1) Matrix(379,-266,540,-379) -> Matrix(3,1,-8,-3) (7/10,19/27) -> (-1/2,-1/4) Matrix(647,-456,918,-647) -> Matrix(5,2,-12,-5) (19/27,12/17) -> (-1/2,-1/3) Matrix(181,-130,252,-181) -> Matrix(17,6,-48,-17) (5/7,13/18) -> (-3/8,-1/3) Matrix(287,-208,396,-287) -> Matrix(43,14,-132,-43) (13/18,8/11) -> (-1/3,-7/22) Matrix(55,-42,72,-55) -> Matrix(11,3,-40,-11) (3/4,7/9) -> (-3/10,-1/4) Matrix(71,-56,90,-71) -> Matrix(-1,0,8,1) (7/9,4/5) -> (-1/4,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,8,1) (1/1,1/0) -> (-1/4,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.