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): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -4/1 -3/1 -7/3 -2/1 -4/3 -1/1 -2/3 0/1 1/2 2/3 3/4 1/1 4/3 3/2 8/5 2/1 16/7 7/3 8/3 3/1 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -4/3 -1/1 -4/1 -1/1 -3/1 -1/2 -8/3 0/1 -5/2 1/0 -7/3 -1/1 0/1 -2/1 1/0 -9/5 -3/1 -2/1 -16/9 -2/1 -7/4 -3/2 -5/3 1/0 -8/5 -2/1 -3/2 -3/2 -7/5 -7/6 -4/3 -1/1 -1/1 -1/1 0/1 -4/5 -1/1 -7/9 -7/8 -3/4 -3/4 -8/11 -2/3 -5/7 -1/2 -7/10 -3/4 -2/3 -1/2 -9/14 -1/4 -16/25 0/1 -7/11 -1/1 0/1 -5/8 -1/2 -8/13 0/1 -3/5 1/0 -4/7 -1/1 -5/9 -1/1 -4/5 -1/2 -1/2 0/1 0/1 1/2 1/0 3/5 1/1 2/1 5/8 5/2 7/11 7/2 2/3 1/0 9/13 -5/2 7/10 -3/2 5/7 -1/1 0/1 3/4 1/0 1/1 1/0 5/4 1/0 4/3 -1/1 11/8 -1/2 7/5 -1/1 0/1 3/2 1/0 8/5 0/1 13/8 1/2 5/3 0/1 1/1 7/4 3/2 2/1 1/0 9/4 -9/2 16/7 -4/1 23/10 -23/6 7/3 -7/2 5/2 -5/2 8/3 -2/1 11/4 -7/4 3/1 -2/1 -1/1 7/2 -3/2 4/1 -1/1 9/2 -3/4 5/1 -1/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,40,4,23) (-5/1,1/0) -> (5/3,7/4) Hyperbolic Matrix(9,40,-16,-71) (-5/1,-4/1) -> (-4/7,-5/9) Hyperbolic Matrix(7,24,-12,-41) (-4/1,-3/1) -> (-3/5,-4/7) Hyperbolic Matrix(17,48,-28,-79) (-3/1,-8/3) -> (-8/13,-3/5) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(23,56,16,39) (-5/2,-7/3) -> (7/5,3/2) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,-224,-401) (-9/5,-16/9) -> (-16/25,-7/11) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(23,40,4,7) (-7/4,-5/3) -> (5/1,1/0) Hyperbolic Matrix(49,80,-68,-111) (-5/3,-8/5) -> (-8/11,-5/7) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(39,56,16,23) (-3/2,-7/5) -> (7/3,5/2) Hyperbolic Matrix(41,56,-52,-71) (-7/5,-4/3) -> (-4/5,-7/9) Hyperbolic Matrix(7,8,-8,-9) (-4/3,-1/1) -> (-1/1,-4/5) Parabolic Matrix(73,56,116,89) (-7/9,-3/4) -> (5/8,7/11) Hyperbolic Matrix(175,128,108,79) (-3/4,-8/11) -> (8/5,13/8) Hyperbolic Matrix(113,80,24,17) (-5/7,-7/10) -> (9/2,5/1) Hyperbolic Matrix(47,32,-72,-49) (-7/10,-2/3) -> (-2/3,-9/14) Parabolic Matrix(799,512,348,223) (-9/14,-16/25) -> (16/7,23/10) Hyperbolic Matrix(177,112,128,81) (-7/11,-5/8) -> (11/8,7/5) Hyperbolic Matrix(207,128,76,47) (-5/8,-8/13) -> (8/3,11/4) Hyperbolic Matrix(73,40,104,57) (-5/9,-1/2) -> (7/10,5/7) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(41,-24,12,-7) (1/2,3/5) -> (3/1,7/2) Hyperbolic Matrix(79,-48,28,-17) (3/5,5/8) -> (11/4,3/1) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(369,-256,160,-111) (9/13,7/10) -> (23/10,7/3) Hyperbolic Matrix(111,-80,68,-49) (5/7,3/4) -> (13/8,5/3) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,40,4,23) -> Matrix(3,4,2,3) Matrix(9,40,-16,-71) -> Matrix(7,8,-8,-9) Matrix(7,24,-12,-41) -> Matrix(3,2,-2,-1) Matrix(17,48,-28,-79) -> Matrix(1,0,2,1) Matrix(31,80,12,31) -> Matrix(5,-2,-2,1) Matrix(23,56,16,39) -> Matrix(1,0,0,1) Matrix(15,32,-8,-17) -> Matrix(1,-2,0,1) Matrix(143,256,-224,-401) -> Matrix(1,2,0,1) Matrix(145,256,64,113) -> Matrix(17,30,-4,-7) Matrix(23,40,4,7) -> Matrix(1,2,-2,-3) Matrix(49,80,-68,-111) -> Matrix(1,4,-2,-7) Matrix(31,48,20,31) -> Matrix(1,2,-2,-3) Matrix(39,56,16,23) -> Matrix(11,14,-4,-5) Matrix(41,56,-52,-71) -> Matrix(13,14,-14,-15) Matrix(7,8,-8,-9) -> Matrix(1,0,0,1) Matrix(73,56,116,89) -> Matrix(17,14,6,5) Matrix(175,128,108,79) -> Matrix(3,2,10,7) Matrix(113,80,24,17) -> Matrix(1,0,0,1) Matrix(47,32,-72,-49) -> Matrix(3,2,-8,-5) Matrix(799,512,348,223) -> Matrix(39,4,-10,-1) Matrix(177,112,128,81) -> Matrix(1,0,0,1) Matrix(207,128,76,47) -> Matrix(11,2,-6,-1) Matrix(73,40,104,57) -> Matrix(5,4,-4,-3) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(41,-24,12,-7) -> Matrix(3,-4,-2,3) Matrix(79,-48,28,-17) -> Matrix(3,-4,-2,3) Matrix(49,-32,72,-47) -> Matrix(1,-6,0,1) Matrix(369,-256,160,-111) -> Matrix(15,34,-4,-9) Matrix(111,-80,68,-49) -> Matrix(1,0,2,1) Matrix(9,-8,8,-7) -> Matrix(1,-2,0,1) Matrix(49,-64,36,-47) -> Matrix(1,2,-2,-3) Matrix(17,-32,8,-15) -> Matrix(1,-6,0,1) Matrix(17,-64,4,-15) -> Matrix(5,6,-6,-7) 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): 16 Degree of the the map X: 16 Degree of the the map Y: 32 Permutation triple for Y: ((1,2)(3,10,29,11)(4,16,17,5)(6,21,31,22)(7,25,26,8)(9,28)(12,23)(13,14)(15,24)(18,27)(19,20)(30,32); (1,5,20,6)(2,8,9,3)(4,14,22,15)(7,12,11,24)(10,27,26,13)(16,28,31,30)(17,23,21,18)(19,29,32,25); (1,3,12,17,30,29,13,4)(2,6,14,26,32,31,23,7)(5,18,10,9,16,15,11,19)(8,27,21,20,25,24,22,28)) ----------------------------------------------------------------------- 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): 96 Minimal number of generators: 17 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: 14 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -3/1 -2/1 -4/3 -1/1 0/1 1/2 1/1 2/1 7/3 3/1 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -4/3 -1/1 -4/1 -1/1 -3/1 -1/2 -2/1 1/0 -5/3 1/0 -8/5 -2/1 -3/2 -3/2 -7/5 -7/6 -4/3 -1/1 -1/1 -1/1 0/1 -4/5 -1/1 -7/9 -7/8 -3/4 -3/4 -5/7 -1/2 -2/3 -1/2 -3/5 1/0 -4/7 -1/1 -1/2 -1/2 0/1 0/1 1/2 1/0 1/1 1/0 3/2 1/0 5/3 0/1 1/1 7/4 3/2 2/1 1/0 9/4 -9/2 7/3 -7/2 5/2 -5/2 8/3 -2/1 3/1 -2/1 -1/1 7/2 -3/2 4/1 -1/1 5/1 -1/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,40,4,23) (-5/1,1/0) -> (5/3,7/4) Hyperbolic Matrix(11,52,4,19) (-5/1,-4/1) -> (8/3,3/1) Hyperbolic Matrix(7,24,-12,-41) (-4/1,-3/1) -> (-3/5,-4/7) Hyperbolic Matrix(5,12,-8,-19) (-3/1,-2/1) -> (-2/3,-3/5) Hyperbolic Matrix(11,20,-16,-29) (-2/1,-5/3) -> (-5/7,-2/3) Hyperbolic Matrix(37,60,8,13) (-5/3,-8/5) -> (4/1,5/1) Hyperbolic Matrix(43,68,12,19) (-8/5,-3/2) -> (7/2,4/1) Hyperbolic Matrix(39,56,16,23) (-3/2,-7/5) -> (7/3,5/2) Hyperbolic Matrix(41,56,-52,-71) (-7/5,-4/3) -> (-4/5,-7/9) Hyperbolic Matrix(7,8,-8,-9) (-4/3,-1/1) -> (-1/1,-4/5) Parabolic Matrix(109,84,48,37) (-7/9,-3/4) -> (9/4,7/3) Hyperbolic Matrix(27,20,4,3) (-3/4,-5/7) -> (5/1,1/0) Hyperbolic Matrix(51,28,20,11) (-4/7,-1/2) -> (5/2,8/3) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(5,-4,4,-3) (1/2,1/1) -> (1/1,3/2) Parabolic Matrix(27,-44,8,-13) (3/2,5/3) -> (3/1,7/2) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,40,4,23) -> Matrix(3,4,2,3) Matrix(11,52,4,19) -> Matrix(7,9,-4,-5) Matrix(7,24,-12,-41) -> Matrix(3,2,-2,-1) Matrix(5,12,-8,-19) -> Matrix(1,1,-2,-1) Matrix(11,20,-16,-29) -> Matrix(1,3,-2,-5) Matrix(37,60,8,13) -> Matrix(1,3,-2,-5) Matrix(43,68,12,19) -> Matrix(5,9,-4,-7) Matrix(39,56,16,23) -> Matrix(11,14,-4,-5) Matrix(41,56,-52,-71) -> Matrix(13,14,-14,-15) Matrix(7,8,-8,-9) -> Matrix(1,0,0,1) Matrix(109,84,48,37) -> Matrix(25,21,-6,-5) Matrix(27,20,4,3) -> Matrix(1,1,-4,-3) Matrix(51,28,20,11) -> Matrix(9,7,-4,-3) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(5,-4,4,-3) -> Matrix(1,-1,0,1) Matrix(27,-44,8,-13) -> Matrix(3,-1,-2,1) Matrix(17,-32,8,-15) -> Matrix(1,-6,0,1) 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): 16 ----------------------------------------------------------------------- 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 -4/1 -1/1 5 2 -3/1 -1/2 2 8 -2/1 1/0 1 4 -5/3 1/0 2 8 -3/2 -3/2 1 8 -7/5 -7/6 2 8 -4/3 -1/1 7 2 -1/1 (-1/1,0/1) 0 8 0/1 0/1 1 2 1/1 1/0 2 8 2/1 1/0 3 4 7/3 -7/2 2 8 5/2 -5/2 1 8 8/3 -2/1 5 2 3/1 (-2/1,-1/1) 0 8 4/1 -1/1 3 2 5/1 -1/2 2 8 1/0 1/0 1 8 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,28,2,11) (-4/1,1/0) -> (5/2,8/3) Glide Reflection Matrix(7,24,-2,-7) (-4/1,-3/1) -> (-4/1,-3/1) Reflection Matrix(5,12,-2,-5) (-3/1,-2/1) -> (-3/1,-2/1) Reflection Matrix(11,20,-6,-11) (-2/1,-5/3) -> (-2/1,-5/3) Reflection Matrix(13,20,2,3) (-5/3,-3/2) -> (5/1,1/0) Glide Reflection Matrix(39,56,16,23) (-3/2,-7/5) -> (7/3,5/2) Hyperbolic Matrix(41,56,-30,-41) (-7/5,-4/3) -> (-7/5,-4/3) Reflection Matrix(7,8,-6,-7) (-4/3,-1/1) -> (-4/3,-1/1) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(3,-4,2,-3) (1/1,2/1) -> (1/1,2/1) Reflection Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection Matrix(9,-40,2,-9) (4/1,5/1) -> (4/1,5/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(5,28,2,11) -> Matrix(5,7,-2,-3) Matrix(7,24,-2,-7) -> Matrix(3,2,-4,-3) (-4/1,-3/1) -> (-1/1,-1/2) Matrix(5,12,-2,-5) -> Matrix(1,1,0,-1) (-3/1,-2/1) -> (-1/2,1/0) Matrix(11,20,-6,-11) -> Matrix(1,3,0,-1) (-2/1,-5/3) -> (-3/2,1/0) Matrix(13,20,2,3) -> Matrix(1,1,-2,-3) Matrix(39,56,16,23) -> Matrix(11,14,-4,-5) Matrix(41,56,-30,-41) -> Matrix(13,14,-12,-13) (-7/5,-4/3) -> (-7/6,-1/1) Matrix(7,8,-6,-7) -> Matrix(-1,0,2,1) (-4/3,-1/1) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(1,0,0,-1) (0/1,1/1) -> (0/1,1/0) Matrix(3,-4,2,-3) -> Matrix(1,1,0,-1) (1/1,2/1) -> (-1/2,1/0) Matrix(13,-28,6,-13) -> Matrix(1,7,0,-1) (2/1,7/3) -> (-7/2,1/0) Matrix(17,-48,6,-17) -> Matrix(3,4,-2,-3) (8/3,3/1) -> (-2/1,-1/1) Matrix(7,-24,2,-7) -> Matrix(3,4,-2,-3) (3/1,4/1) -> (-2/1,-1/1) Matrix(9,-40,2,-9) -> Matrix(3,2,-4,-3) (4/1,5/1) -> (-1/1,-1/2) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.