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 -2/1 0/1 1/1 4/3 3/2 2/1 8/3 3/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 1/1 -3/1 1/1 3/2 2/1 -2/1 2/1 -5/3 2/1 3/1 1/0 -8/5 2/1 4/1 -3/2 2/1 3/1 1/0 -4/3 1/0 -1/1 1/1 2/1 1/0 0/1 0/1 2/1 1/1 1/1 2/1 1/0 4/3 1/0 3/2 0/1 1/1 1/0 2/1 0/1 2/1 5/2 0/1 1/1 1/0 8/3 0/1 2/1 3/1 0/1 1/1 1/0 4/1 1/1 1/0 1/1 2/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,0,1) (-4/1,1/0) -> (4/1,1/0) Parabolic Matrix(7,24,2,7) (-4/1,-3/1) -> (3/1,4/1) Hyperbolic Matrix(7,16,-4,-9) (-3/1,-2/1) -> (-2/1,-5/3) Parabolic Matrix(39,64,14,23) (-5/3,-8/5) -> (8/3,3/1) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) 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(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,0,0,1) Matrix(7,24,2,7) -> Matrix(1,-2,2,-3) Matrix(7,16,-4,-9) -> Matrix(5,-8,2,-3) Matrix(39,64,14,23) -> Matrix(1,-2,0,1) Matrix(41,64,16,25) -> Matrix(1,-2,0,1) Matrix(17,24,12,17) -> Matrix(1,-2,0,1) Matrix(7,8,6,7) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(1,0,0,1) Matrix(9,-16,4,-7) -> 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: ((2,6)(3,4); (1,4,5,2)(3,7,6,8); (1,2,7,3)(4,8,6,5)) ----------------------------------------------------------------------- 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 -2/1 0/1 1/1 2/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 2/1 -3/2 2/1 3/1 1/0 -4/3 1/0 -1/1 1/1 2/1 1/0 0/1 0/1 2/1 1/1 1/1 2/1 1/0 2/1 0/1 2/1 3/1 0/1 1/1 1/0 4/1 1/1 1/0 1/1 2/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,8,-2,-5) (-2/1,1/0) -> (-2/1,-3/2) Parabolic Matrix(11,16,2,3) (-3/2,-4/3) -> (4/1,1/0) Hyperbolic Matrix(13,16,4,5) (-4/3,-1/1) -> (3/1,4/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(5,-8,2,-3) (1/1,2/1) -> (2/1,3/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,8,-2,-5) -> Matrix(3,-4,1,-1) Matrix(11,16,2,3) -> Matrix(1,-4,1,-3) Matrix(13,16,4,5) -> Matrix(1,-2,1,-1) Matrix(1,0,2,1) -> Matrix(1,0,0,1) Matrix(5,-8,2,-3) -> Matrix(1,-2,1,-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 -4/1 1/1 2 2 -2/1 2/1 1 2 0/1 (0/1,2/1) 0 2 2/1 (0/1,2/1).(1/1,1/0) 0 2 4/1 1/1 2 2 1/0 0 4 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,0,1) (-4/1,1/0) -> (4/1,1/0) Parabolic Matrix(3,8,-1,-3) (-4/1,-2/1) -> (-4/1,-2/1) Reflection Matrix(-1,0,1,1) (-2/1,0/1) -> (-2/1,0/1) Reflection Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(3,-8,1,-3) (2/1,4/1) -> (2/1,4/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,8,0,1) -> Matrix(1,0,0,1) Matrix(3,8,-1,-3) -> Matrix(3,-4,2,-3) (-4/1,-2/1) -> (1/1,2/1) Matrix(-1,0,1,1) -> Matrix(1,0,1,-1) (-2/1,0/1) -> (0/1,2/1) Matrix(1,0,1,-1) -> Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Matrix(3,-8,1,-3) -> Matrix(-1,2,0,1) (2/1,4/1) -> (1/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.