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 1/1 2/1 -4/1 2/1 -3/1 2/1 3/1 -8/3 5/2 1/0 -5/2 2/1 5/2 3/1 -7/3 2/1 3/1 -2/1 3/1 -9/5 3/1 4/1 -16/9 7/2 1/0 -7/4 3/1 4/1 1/0 -5/3 3/1 4/1 -8/5 7/2 1/0 -3/2 3/1 4/1 1/0 -7/5 3/1 4/1 -4/3 4/1 -1/1 4/1 1/0 -4/5 4/1 -7/9 4/1 5/1 -3/4 4/1 5/1 1/0 -8/11 9/2 1/0 -5/7 4/1 5/1 -7/10 4/1 5/1 1/0 -2/3 5/1 -9/14 5/1 11/2 6/1 -16/25 11/2 1/0 -7/11 5/1 6/1 -5/8 5/1 11/2 6/1 -8/13 11/2 1/0 -3/5 5/1 6/1 -4/7 6/1 -5/9 6/1 7/1 -1/2 6/1 7/1 1/0 0/1 1/0 1/2 -4/1 -3/1 1/0 3/5 -3/1 -2/1 5/8 -3/1 -5/2 -2/1 7/11 -3/1 -2/1 2/3 -2/1 9/13 -2/1 -1/1 7/10 -2/1 -1/1 1/0 5/7 -2/1 -1/1 3/4 -2/1 -1/1 1/0 1/1 -1/1 1/0 5/4 -2/1 -1/1 1/0 4/3 -1/1 11/8 -1/1 -1/2 0/1 7/5 -1/1 0/1 3/2 -1/1 0/1 1/0 8/5 -1/2 1/0 13/8 -1/1 -1/2 0/1 5/3 -1/1 0/1 7/4 -1/1 0/1 1/0 2/1 0/1 9/4 0/1 1/2 1/1 16/7 1/2 1/0 23/10 0/1 1/1 1/0 7/3 0/1 1/1 5/2 0/1 1/2 1/1 8/3 1/2 1/0 11/4 0/1 1/1 1/0 3/1 0/1 1/1 7/2 1/2 2/3 1/1 4/1 1/1 9/2 1/1 3/2 2/1 5/1 1/1 2/1 1/0 1/1 2/1 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(1,-2,0,1) Matrix(9,40,-16,-71) -> Matrix(13,-20,2,-3) Matrix(7,24,-12,-41) -> Matrix(11,-28,2,-5) Matrix(17,48,-28,-79) -> Matrix(11,-28,2,-5) Matrix(31,80,12,31) -> Matrix(1,-2,0,1) Matrix(23,56,16,39) -> Matrix(1,-2,-2,5) Matrix(15,32,-8,-17) -> Matrix(7,-18,2,-5) Matrix(143,256,-224,-401) -> Matrix(1,2,0,1) Matrix(145,256,64,113) -> Matrix(1,-4,2,-7) Matrix(23,40,4,7) -> Matrix(1,-2,0,1) Matrix(49,80,-68,-111) -> Matrix(9,-32,2,-7) Matrix(31,48,20,31) -> Matrix(1,-4,0,1) Matrix(39,56,16,23) -> Matrix(1,-4,2,-7) Matrix(41,56,-52,-71) -> Matrix(9,-32,2,-7) Matrix(7,8,-8,-9) -> Matrix(1,0,0,1) Matrix(73,56,116,89) -> Matrix(5,-22,-2,9) Matrix(175,128,108,79) -> Matrix(1,-4,-2,9) Matrix(113,80,24,17) -> Matrix(3,-14,2,-9) Matrix(47,32,-72,-49) -> Matrix(11,-50,2,-9) Matrix(799,512,348,223) -> Matrix(1,-6,2,-11) Matrix(177,112,128,81) -> Matrix(1,-6,0,1) Matrix(207,128,76,47) -> Matrix(1,-6,2,-11) Matrix(73,40,104,57) -> Matrix(1,-8,0,1) Matrix(1,0,4,1) -> Matrix(1,-10,0,1) Matrix(41,-24,12,-7) -> Matrix(1,2,2,5) Matrix(79,-48,28,-17) -> Matrix(1,2,2,5) Matrix(49,-32,72,-47) -> Matrix(3,8,-2,-5) Matrix(369,-256,160,-111) -> Matrix(1,2,0,1) Matrix(111,-80,68,-49) -> Matrix(1,2,-2,-3) Matrix(9,-8,8,-7) -> Matrix(1,0,0,1) Matrix(49,-64,36,-47) -> Matrix(1,2,-2,-3) Matrix(17,-32,8,-15) -> Matrix(1,0,2,1) Matrix(17,-64,4,-15) -> Matrix(5,-4,4,-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): 5 Degree of the the map X: 5 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 -2/1 0/1 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 1/1 2/1 -4/1 2/1 -3/1 2/1 3/1 -8/3 5/2 1/0 -5/2 2/1 5/2 3/1 -7/3 2/1 3/1 -2/1 3/1 -9/5 3/1 4/1 -16/9 7/2 1/0 -7/4 3/1 4/1 1/0 -5/3 3/1 4/1 -8/5 7/2 1/0 -3/2 3/1 4/1 1/0 -7/5 3/1 4/1 -4/3 4/1 -1/1 4/1 1/0 0/1 1/0 1/1 -1/1 1/0 4/3 -1/1 7/5 -1/1 0/1 3/2 -1/1 0/1 1/0 8/5 -1/2 1/0 5/3 -1/1 0/1 7/4 -1/1 0/1 1/0 2/1 0/1 9/4 0/1 1/2 1/1 16/7 1/2 1/0 7/3 0/1 1/1 5/2 0/1 1/2 1/1 8/3 1/2 1/0 3/1 0/1 1/1 4/1 1/1 5/1 1/1 2/1 1/0 1/1 2/1 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(9,40,2,9) (-5/1,-4/1) -> (4/1,5/1) Hyperbolic Matrix(7,24,2,7) (-4/1,-3/1) -> (3/1,4/1) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) 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(143,256,62,111) (-9/5,-16/9) -> (16/7,7/3) 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(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) 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(41,56,30,41) (-7/5,-4/3) -> (4/3,7/5) Hyperbolic Matrix(7,8,6,7) (-4/3,-1/1) -> (1/1,4/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic 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(1,-2,0,1) Matrix(9,40,2,9) -> Matrix(3,-5,2,-3) Matrix(7,24,2,7) -> Matrix(1,-3,2,-5) Matrix(17,48,6,17) -> Matrix(1,-3,2,-5) Matrix(31,80,12,31) -> Matrix(1,-2,0,1) Matrix(23,56,16,39) -> Matrix(1,-2,-2,5) Matrix(15,32,-8,-17) -> Matrix(7,-18,2,-5) Matrix(143,256,62,111) -> Matrix(1,-3,0,1) Matrix(145,256,64,113) -> Matrix(1,-4,2,-7) Matrix(23,40,4,7) -> Matrix(1,-2,0,1) Matrix(49,80,30,49) -> Matrix(1,-3,-2,7) Matrix(31,48,20,31) -> Matrix(1,-4,0,1) Matrix(39,56,16,23) -> Matrix(1,-4,2,-7) Matrix(41,56,30,41) -> Matrix(1,-3,-2,7) Matrix(7,8,6,7) -> Matrix(1,-5,0,1) Matrix(1,0,2,1) -> Matrix(1,-5,0,1) Matrix(17,-32,8,-15) -> Matrix(1,0,2,1) 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): 5 ----------------------------------------------------------------------- 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 5 2 1/1 (-1/1,1/0) 0 8 4/3 -1/1 1 2 7/5 (-1/1,0/1) 0 8 3/2 (-1/2,1/0) 0 8 8/5 (-1/1,0/1).(-1/2,1/0) 0 2 5/3 (-1/1,0/1) 0 8 7/4 (-1/2,1/0) 0 8 2/1 0/1 1 4 9/4 (1/2,1/0) 0 8 16/7 (0/1,1/1).(1/2,1/0) 0 2 7/3 (0/1,1/1) 0 8 5/2 (1/2,1/0) 0 8 8/3 (0/1,1/1).(1/2,1/0) 0 2 3/1 (0/1,1/1) 0 8 4/1 1/1 2 2 5/1 (1/1,2/1) 0 8 1/0 (3/2,1/0) 0 8 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(7,-8,6,-7) (1/1,4/3) -> (1/1,4/3) Reflection Matrix(41,-56,30,-41) (4/3,7/5) -> (4/3,7/5) Reflection Matrix(39,-56,16,-23) (7/5,3/2) -> (7/3,5/2) Glide Reflection Matrix(31,-48,20,-31) (3/2,8/5) -> (3/2,8/5) Reflection Matrix(49,-80,30,-49) (8/5,5/3) -> (8/5,5/3) Reflection Matrix(23,-40,4,-7) (5/3,7/4) -> (5/1,1/0) Glide Reflection Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(97,-224,42,-97) (16/7,7/3) -> (16/7,7/3) Reflection Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) 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(1,0,0,-1) -> Matrix(-1,3,0,1) (0/1,1/0) -> (3/2,1/0) Matrix(1,0,2,-1) -> Matrix(1,2,0,-1) (0/1,1/1) -> (-1/1,1/0) Matrix(7,-8,6,-7) -> Matrix(1,2,0,-1) (1/1,4/3) -> (-1/1,1/0) Matrix(41,-56,30,-41) -> Matrix(-1,0,2,1) (4/3,7/5) -> (-1/1,0/1) Matrix(39,-56,16,-23) -> Matrix(1,1,2,1) Matrix(31,-48,20,-31) -> Matrix(1,1,0,-1) (3/2,8/5) -> (-1/2,1/0) Matrix(49,-80,30,-49) -> Matrix(-1,0,2,1) (8/5,5/3) -> (-1/1,0/1) Matrix(23,-40,4,-7) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(17,-32,8,-15) -> Matrix(1,0,2,1) 0/1 Matrix(127,-288,56,-127) -> Matrix(-1,1,0,1) (9/4,16/7) -> (1/2,1/0) Matrix(97,-224,42,-97) -> Matrix(1,0,2,-1) (16/7,7/3) -> (0/1,1/1) Matrix(31,-80,12,-31) -> Matrix(-1,1,0,1) (5/2,8/3) -> (1/2,1/0) Matrix(17,-48,6,-17) -> Matrix(1,0,2,-1) (8/3,3/1) -> (0/1,1/1) Matrix(7,-24,2,-7) -> Matrix(1,0,2,-1) (3/1,4/1) -> (0/1,1/1) Matrix(9,-40,2,-9) -> Matrix(3,-4,2,-3) (4/1,5/1) -> (1/1,2/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.