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 -1/1 0/1 -5/6 -1/1 -4/5 -1/1 0/1 -7/9 -1/1 -10/13 -1/1 -3/4 -13/17 -1/1 -2/3 -3/4 -1/1 -2/3 -1/2 -8/11 -2/3 -1/2 -13/18 -1/2 -5/7 -1/2 0/1 -2/3 -1/2 1/0 -7/11 -1/2 0/1 -12/19 -1/1 -1/2 -17/27 -1/2 -5/8 -1/1 -1/2 0/1 -8/13 -3/5 -1/2 -11/18 -1/2 -3/5 -1/2 -1/3 -10/17 -1/2 -1/3 -7/12 -1/3 -11/19 -1/3 0/1 -4/7 -1/3 0/1 -5/9 0/1 -6/11 0/1 1/0 -1/2 -1/1 -1/2 0/1 -4/9 0/1 -7/16 0/1 1/1 1/0 -10/23 -1/1 1/0 -3/7 0/1 1/0 -8/19 -1/1 1/0 -5/12 -1/1 -2/5 -1/1 0/1 -7/18 -1/1 -5/13 -1/1 -3/4 -13/34 -1/1 -2/3 -1/2 -8/21 -3/4 -1/2 -3/8 -1/1 -2/3 -1/2 -10/27 -1/1 -7/19 -1/1 -2/3 -4/11 -2/3 -1/2 -5/14 -1/1 -2/3 -1/2 -6/17 -1/1 -1/2 -1/3 -1/2 -6/19 -1/2 -1/3 -5/16 -1/2 -1/3 0/1 -4/13 -1/1 -1/2 -3/10 -1/2 -2/5 -1/3 -8/27 -1/3 -5/17 -1/3 0/1 -2/7 -1/3 0/1 -5/18 0/1 -3/11 0/1 1/0 -7/26 -1/1 0/1 1/0 -4/15 -1/2 1/0 -9/34 -1/1 0/1 1/0 -14/53 -1/1 1/0 -19/72 -1/1 -5/19 -1/1 0/1 -1/4 -1/1 -1/2 0/1 -2/9 -1/2 -3/14 -1/2 -2/5 -1/3 -4/19 -1/2 -1/3 -1/5 -1/2 -1/3 -3/16 -1/3 -1/4 0/1 -2/11 -1/2 0/1 -3/17 -1/3 0/1 -1/6 0/1 -3/19 -1/1 0/1 -2/13 -1/1 -1/2 -1/7 -1/2 0/1 -1/8 -1/1 -1/2 0/1 -1/9 -1/2 0/1 -1/2 0/1 1/8 -1/2 -1/3 0/1 1/7 -1/2 0/1 2/13 -1/2 -1/3 1/6 0/1 2/11 -1/2 0/1 1/5 -1/1 -1/2 2/9 -1/2 3/13 -1/2 -1/3 1/4 -1/2 -1/3 0/1 5/19 -1/3 0/1 4/15 -1/2 -1/4 3/11 -1/4 0/1 5/18 0/1 2/7 -1/1 0/1 5/17 -1/1 0/1 8/27 -1/1 3/10 -1/1 -2/3 -1/2 1/3 -1/2 4/11 -1/2 -2/5 11/30 -2/5 7/19 -2/5 -1/3 10/27 -1/3 3/8 -1/2 -2/5 -1/3 11/29 -1/2 -2/5 8/21 -1/2 -3/8 21/55 -2/5 -1/3 13/34 -1/2 -2/5 -1/3 5/13 -3/8 -1/3 7/18 -1/3 2/5 -1/3 0/1 5/12 -1/3 8/19 -1/3 -1/4 3/7 -1/4 0/1 4/9 0/1 5/11 -1/2 0/1 6/13 -1/1 -1/2 1/2 -1/2 -1/3 0/1 7/13 -1/3 -1/4 6/11 -1/4 0/1 5/9 0/1 4/7 -1/1 0/1 7/12 -1/1 10/17 -1/1 -1/2 13/22 -1/1 -2/3 -1/2 3/5 -1/1 -1/2 11/18 -1/2 8/13 -1/2 -3/7 21/34 -1/2 -2/5 -1/3 13/21 -1/2 5/8 -1/2 -1/3 0/1 2/3 -1/2 -1/4 7/10 -1/2 -1/3 0/1 26/37 -1/2 0/1 19/27 -1/2 12/17 -1/2 -1/3 5/7 -1/2 0/1 13/18 -1/2 8/11 -1/2 -2/5 11/15 -1/2 25/34 -1/2 -2/5 -1/3 39/53 -2/5 -1/3 53/72 -1/3 14/19 -1/2 -1/3 3/4 -1/2 -2/5 -1/3 13/17 -2/5 -1/3 23/30 -2/5 10/13 -3/8 -1/3 7/9 -1/3 4/5 -1/3 0/1 5/6 -1/3 6/7 -1/3 0/1 1/1 -1/3 0/1 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,-2,1) Matrix(73,62,-126,-107) -> Matrix(1,0,-2,1) Matrix(37,30,90,73) -> Matrix(1,0,-2,1) Matrix(71,56,90,71) -> Matrix(1,0,-2,1) Matrix(181,140,234,181) -> Matrix(7,6,-20,-17) Matrix(73,56,-468,-359) -> Matrix(3,2,-2,-1) Matrix(37,28,144,109) -> Matrix(3,2,-8,-5) Matrix(71,52,-198,-145) -> Matrix(1,0,0,1) Matrix(287,208,396,287) -> Matrix(7,4,-16,-9) Matrix(181,130,252,181) -> Matrix(1,0,0,1) Matrix(35,24,-54,-37) -> Matrix(1,0,0,1) Matrix(145,92,342,217) -> Matrix(1,0,-2,1) 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(3,2,-8,-5) Matrix(287,176,468,287) -> Matrix(11,6,-24,-13) Matrix(109,66,180,109) -> Matrix(5,2,-8,-3) Matrix(37,22,-180,-107) -> Matrix(1,0,0,1) Matrix(181,106,432,253) -> Matrix(5,2,-18,-7) Matrix(73,42,252,145) -> Matrix(1,0,2,1) Matrix(71,40,126,71) -> Matrix(1,0,2,1) Matrix(109,60,198,109) -> Matrix(1,0,-4,1) Matrix(37,20,-198,-107) -> Matrix(1,0,-2,1) Matrix(71,32,-162,-73) -> Matrix(1,0,2,1) Matrix(611,266,990,431) -> Matrix(1,-2,-2,5) Matrix(37,16,252,109) -> Matrix(1,0,-2,1) Matrix(179,76,252,107) -> Matrix(1,0,-2,1) Matrix(253,106,432,181) -> Matrix(1,2,-2,-3) Matrix(73,30,90,37) -> Matrix(1,0,-2,1) Matrix(71,28,180,71) -> Matrix(1,0,-2,1) Matrix(181,70,468,181) -> Matrix(7,6,-20,-17) Matrix(251,96,468,179) -> Matrix(3,2,-8,-5) Matrix(325,124,-1224,-467) -> Matrix(3,2,-2,-1) Matrix(179,68,-666,-253) -> Matrix(3,2,-2,-1) Matrix(145,54,486,181) -> Matrix(1,0,0,1) Matrix(287,106,972,359) -> Matrix(3,2,-2,-1) Matrix(71,26,-396,-145) -> Matrix(3,2,-8,-5) Matrix(361,128,612,217) -> Matrix(1,0,0,1) Matrix(35,12,-108,-37) -> Matrix(3,2,-8,-5) Matrix(395,124,-1494,-469) -> Matrix(1,0,2,1) Matrix(109,34,234,73) -> Matrix(1,0,0,1) Matrix(181,54,486,145) -> Matrix(1,0,0,1) Matrix(359,106,972,287) -> Matrix(7,2,-18,-5) Matrix(109,32,126,37) -> Matrix(1,0,0,1) Matrix(71,20,252,71) -> Matrix(1,0,2,1) Matrix(109,30,396,109) -> Matrix(1,0,-4,1) Matrix(37,10,270,73) -> Matrix(1,0,-2,1) Matrix(3817,1008,5184,1369) -> Matrix(1,0,-2,1) Matrix(3815,1006,5184,1367) -> Matrix(3,2,-8,-5) Matrix(109,28,144,37) -> Matrix(3,2,-8,-5) Matrix(35,8,-162,-37) -> Matrix(3,2,-8,-5) Matrix(253,54,342,73) -> Matrix(1,0,0,1) Matrix(145,28,378,73) -> Matrix(7,2,-18,-5) Matrix(35,6,-216,-37) -> Matrix(1,0,2,1) Matrix(107,16,234,35) -> Matrix(1,0,0,1) Matrix(109,14,288,37) -> Matrix(3,2,-8,-5) 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(7,2,-18,-5) Matrix(145,-26,396,-71) -> Matrix(3,2,-8,-5) Matrix(107,-20,198,-37) -> Matrix(1,0,-2,1) Matrix(37,-8,162,-35) -> Matrix(3,2,-8,-5) Matrix(181,-42,306,-71) -> Matrix(5,2,-8,-3) Matrix(467,-124,1224,-325) -> Matrix(7,2,-18,-5) Matrix(253,-68,666,-179) -> Matrix(7,2,-18,-5) Matrix(145,-44,234,-71) -> Matrix(3,2,-8,-5) Matrix(145,-52,198,-71) -> Matrix(1,0,0,1) Matrix(827,-304,1080,-397) -> Matrix(1,0,0,1) Matrix(2701,-1032,3672,-1403) -> Matrix(1,0,0,1) Matrix(73,-32,162,-71) -> Matrix(1,0,2,1) Matrix(107,-62,126,-73) -> Matrix(1,0,-2,1) 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): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 11 Degree of the the map X: 11 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. ----------------------------------------------------------------------- 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 0/1 (-1/2,0/1) 0 18 1/8 0 18 1/7 (-1/2,0/1) 0 18 2/13 (-1/2,-1/3) 0 18 1/6 0/1 2 6 2/11 (-1/2,0/1) 0 18 1/5 (-1/1,-1/2) 0 18 2/9 -1/2 2 2 3/13 (-1/2,-1/3) 0 18 1/4 0 18 5/19 (-1/3,0/1) 0 18 4/15 0 6 3/11 (-1/4,0/1) 0 18 5/18 0/1 6 2 2/7 (-1/1,0/1) 0 18 5/17 (-1/1,0/1) 0 18 8/27 -1/1 2 2 3/10 0 18 1/3 -1/2 1 6 4/11 (-1/2,-2/5) 0 18 11/30 -2/5 2 6 7/19 (-2/5,-1/3) 0 18 10/27 -1/3 2 2 3/8 0 18 11/29 (-1/2,-2/5) 0 18 8/21 0 6 21/55 (-2/5,-1/3) 0 18 13/34 0 18 5/13 (-3/8,-1/3) 0 18 7/18 -1/3 6 2 2/5 (-1/3,0/1) 0 18 5/12 -1/3 2 6 8/19 (-1/3,-1/4) 0 18 3/7 (-1/4,0/1) 0 18 4/9 0/1 2 2 5/11 (-1/2,0/1) 0 18 6/13 (-1/1,-1/2) 0 18 1/2 0 18 7/13 (-1/3,-1/4) 0 18 6/11 (-1/4,0/1) 0 18 5/9 0/1 3 2 4/7 (-1/1,0/1) 0 18 7/12 -1/1 2 6 10/17 (-1/1,-1/2) 0 18 13/22 0 18 3/5 (-1/1,-1/2) 0 18 11/18 -1/2 8 2 8/13 (-1/2,-3/7) 0 18 21/34 0 18 13/21 -1/2 1 6 5/8 0 18 2/3 0 6 7/10 0 18 26/37 (-1/2,0/1) 0 18 19/27 -1/2 1 2 12/17 (-1/2,-1/3) 0 18 5/7 (-1/2,0/1) 0 18 13/18 -1/2 4 2 8/11 (-1/2,-2/5) 0 18 11/15 -1/2 1 6 25/34 0 18 39/53 (-2/5,-1/3) 0 18 53/72 -1/3 2 2 14/19 (-1/2,-1/3) 0 18 3/4 0 18 13/17 (-2/5,-1/3) 0 18 23/30 -2/5 2 6 10/13 (-3/8,-1/3) 0 18 7/9 -1/3 3 2 4/5 (-1/3,0/1) 0 18 5/6 -1/3 2 6 6/7 (-1/3,0/1) 0 18 1/1 (-1/3,0/1) 0 18 1/0 0/1 2 2 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(215,-26,306,-37) (0/1,1/8) -> (7/10,26/37) Hyperbolic Matrix(109,-14,288,-37) (1/8,1/7) -> (3/8,11/29) Glide Reflection Matrix(107,-16,234,-35) (1/7,2/13) -> (5/11,6/13) Glide Reflection 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(109,-28,144,-37) (1/4,5/19) -> (3/4,13/17) Glide Reflection 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(109,-30,396,-109) (3/11,5/18) -> (3/11,5/18) Reflection Matrix(71,-20,252,-71) (5/18,2/7) -> (5/18,2/7) Reflection Matrix(109,-32,126,-37) (2/7,5/17) -> (6/7,1/1) Glide Reflection Matrix(359,-106,972,-287) (5/17,8/27) -> (7/19,10/27) Glide Reflection Matrix(181,-54,486,-145) (8/27,3/10) -> (10/27,3/8) Glide Reflection 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(251,-96,468,-179) (13/34,5/13) -> (1/2,7/13) Glide Reflection Matrix(181,-70,468,-181) (5/13,7/18) -> (5/13,7/18) Reflection Matrix(71,-28,180,-71) (7/18,2/5) -> (7/18,2/5) Reflection Matrix(73,-30,90,-37) (2/5,5/12) -> (4/5,5/6) Glide Reflection Matrix(253,-106,432,-181) (5/12,8/19) -> (7/12,10/17) Glide Reflection Matrix(179,-76,252,-107) (8/19,3/7) -> (12/17,5/7) Glide Reflection Matrix(73,-32,162,-71) (3/7,4/9) -> (4/9,5/11) Parabolic Matrix(289,-134,468,-217) (6/13,1/2) -> (8/13,21/34) Glide Reflection Matrix(109,-60,198,-109) (6/11,5/9) -> (6/11,5/9) 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(359,-212,486,-287) (10/17,13/22) -> (14/19,3/4) Glide Reflection 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(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 Matrix(1405,-988,1998,-1405) (26/37,19/27) -> (26/37,19/27) 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(5617,-4134,7632,-5617) (39/53,53/72) -> (39/53,53/72) Reflection Matrix(2015,-1484,2736,-2015) (53/72,14/19) -> (53/72,14/19) Reflection Matrix(181,-140,234,-181) (10/13,7/9) -> (10/13,7/9) 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(215,-26,306,-37) -> Matrix(1,0,0,1) Matrix(109,-14,288,-37) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(107,-16,234,-35) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(359,-56,468,-73) -> Matrix(7,2,-18,-5) -1/3 Matrix(145,-26,396,-71) -> Matrix(3,2,-8,-5) -1/2 Matrix(107,-20,198,-37) -> Matrix(1,0,-2,1) 0/1 Matrix(37,-8,162,-35) -> Matrix(3,2,-8,-5) -1/2 Matrix(181,-42,306,-71) -> Matrix(5,2,-8,-3) -1/2 Matrix(109,-28,144,-37) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(467,-124,1224,-325) -> Matrix(7,2,-18,-5) -1/3 Matrix(253,-68,666,-179) -> Matrix(7,2,-18,-5) -1/3 Matrix(109,-30,396,-109) -> Matrix(-1,0,8,1) (3/11,5/18) -> (-1/4,0/1) Matrix(71,-20,252,-71) -> Matrix(-1,0,2,1) (5/18,2/7) -> (-1/1,0/1) Matrix(109,-32,126,-37) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(359,-106,972,-287) -> Matrix(1,2,-2,-5) Matrix(181,-54,486,-145) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(145,-44,234,-71) -> Matrix(3,2,-8,-5) -1/2 Matrix(145,-52,198,-71) -> Matrix(1,0,0,1) Matrix(827,-304,1080,-397) -> Matrix(1,0,0,1) Matrix(2701,-1032,3672,-1403) -> Matrix(1,0,0,1) Matrix(251,-96,468,-179) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(181,-70,468,-181) -> Matrix(17,6,-48,-17) (5/13,7/18) -> (-3/8,-1/3) Matrix(71,-28,180,-71) -> Matrix(-1,0,6,1) (7/18,2/5) -> (-1/3,0/1) Matrix(73,-30,90,-37) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(253,-106,432,-181) -> Matrix(7,2,-10,-3) Matrix(179,-76,252,-107) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(73,-32,162,-71) -> Matrix(1,0,2,1) 0/1 Matrix(289,-134,468,-217) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(109,-60,198,-109) -> Matrix(-1,0,8,1) (6/11,5/9) -> (-1/4,0/1) Matrix(71,-40,126,-71) -> Matrix(-1,0,2,1) (5/9,4/7) -> (-1/1,0/1) Matrix(107,-62,126,-73) -> Matrix(1,0,-2,1) 0/1 Matrix(359,-212,486,-287) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(109,-66,180,-109) -> Matrix(3,2,-4,-3) (3/5,11/18) -> (-1/1,-1/2) Matrix(287,-176,468,-287) -> Matrix(13,6,-28,-13) (11/18,8/13) -> (-1/2,-3/7) Matrix(899,-556,1224,-757) -> Matrix(1,0,0,1) Matrix(37,-24,54,-35) -> Matrix(1,0,0,1) Matrix(1405,-988,1998,-1405) -> Matrix(-1,0,4,1) (26/37,19/27) -> (-1/2,0/1) Matrix(647,-456,918,-647) -> Matrix(5,2,-12,-5) (19/27,12/17) -> (-1/2,-1/3) Matrix(181,-130,252,-181) -> Matrix(-1,0,4,1) (5/7,13/18) -> (-1/2,0/1) Matrix(287,-208,396,-287) -> Matrix(9,4,-20,-9) (13/18,8/11) -> (-1/2,-2/5) Matrix(5617,-4134,7632,-5617) -> Matrix(11,4,-30,-11) (39/53,53/72) -> (-2/5,-1/3) Matrix(2015,-1484,2736,-2015) -> Matrix(5,2,-12,-5) (53/72,14/19) -> (-1/2,-1/3) Matrix(181,-140,234,-181) -> Matrix(17,6,-48,-17) (10/13,7/9) -> (-3/8,-1/3) Matrix(71,-56,90,-71) -> Matrix(-1,0,6,1) (7/9,4/5) -> (-1/3,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,6,1) (1/1,1/0) -> (-1/3,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.