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): 48 Minimal number of generators: 9 Number of equivalence classes of cusps: 10 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/2 -1/1 -1/2 -1/4 0/1 1/2 1/1 3/2 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 -3/2 -1/1 0/1 1/0 -4/3 -1/1 -1/1 -1/1 0/1 1/0 -1/2 -1/1 0/1 1/0 -1/3 -1/1 0/1 1/0 -2/7 -1/1 -1/4 -1/2 1/0 0/1 0/1 1/2 0/1 1/1 1/0 2/3 0/1 3/4 1/2 1/0 1/1 0/1 1/1 1/0 3/2 0/1 1/1 1/0 5/3 0/1 1/1 1/0 7/4 1/2 1/0 2/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(11,18,-8,-13) (-2/1,-3/2) -> (-3/2,-4/3) Parabolic Matrix(5,6,-16,-19) (-4/3,-1/1) -> (-1/3,-2/7) Hyperbolic Matrix(3,2,-8,-5) (-1/1,-1/2) -> (-1/2,-1/3) Parabolic Matrix(57,16,32,9) (-2/7,-1/4) -> (7/4,2/1) Hyperbolic Matrix(11,2,16,3) (-1/4,0/1) -> (2/3,3/4) Hyperbolic Matrix(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic Matrix(27,-22,16,-13) (3/4,1/1) -> (5/3,7/4) Hyperbolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,2,0,1) Matrix(11,18,-8,-13) -> Matrix(1,0,0,1) Matrix(5,6,-16,-19) -> Matrix(1,0,0,1) Matrix(3,2,-8,-5) -> Matrix(1,0,0,1) Matrix(57,16,32,9) -> Matrix(1,0,2,1) Matrix(11,2,16,3) -> Matrix(1,0,2,1) Matrix(5,-2,8,-3) -> Matrix(1,0,0,1) Matrix(27,-22,16,-13) -> Matrix(1,0,0,1) Matrix(13,-18,8,-11) -> 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): 1 Degree of the the map X: 1 Degree of the the map Y: 8 Permutation triple for Y: ((1,5,6,2)(3,7,8,4); (1,4,8,5)(2,6,7,3); (1,3)(6,8)) ----------------------------------------------------------------------- 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): 24 Minimal number of generators: 5 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: 6 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/2 -1/4 0/1 1/2 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 1/0 -1/2 -1/1 0/1 1/0 -1/3 -1/1 0/1 1/0 -1/4 -1/2 1/0 0/1 0/1 1/2 0/1 1/1 1/0 2/3 0/1 3/4 1/2 1/0 1/1 0/1 1/1 1/0 1/0 1/0 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(3,2,-8,-5) (-1/1,-1/2) -> (-1/2,-1/3) Parabolic Matrix(13,4,16,5) (-1/3,-1/4) -> (3/4,1/1) Hyperbolic Matrix(11,2,16,3) (-1/4,0/1) -> (2/3,3/4) Hyperbolic Matrix(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,1,0,1) Matrix(3,2,-8,-5) -> Matrix(1,0,0,1) Matrix(13,4,16,5) -> Matrix(1,1,0,1) Matrix(11,2,16,3) -> Matrix(1,0,2,1) Matrix(5,-2,8,-3) -> Matrix(1,0,0,1) 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): 1 ----------------------------------------------------------------------- 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 1 4 1/2 0 4 2/3 0/1 1 4 3/4 (0/1,1/1).(1/2,1/0) 0 4 1/1 (1/2,1/0) 0 4 1/0 1/0 2 4 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(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(7,-6,8,-7) (3/4,1/1) -> (3/4,1/1) 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(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(5,-2,8,-3) -> Matrix(1,0,0,1) Matrix(17,-12,24,-17) -> Matrix(1,0,2,-1) (2/3,3/4) -> (0/1,1/1) Matrix(7,-6,8,-7) -> Matrix(-1,1,0,1) (3/4,1/1) -> (1/2,1/0) Matrix(-1,2,0,1) -> Matrix(-1,1,0,1) (1/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.