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 -1/1 -1/3 0/1 1/3 1/2 1/1 5/3 2/1 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/3 -2/5 1/3 -1/3 0/1 1/3 1/2 -2/7 1/3 -1/4 1/3 1/2 0/1 1/1 1/3 1/0 2/5 -3/1 1/2 -1/1 1/0 1/1 -1/1 0/1 1/0 3/2 -1/1 1/0 5/3 -2/1 0/1 7/4 -1/1 1/0 2/1 -1/1 3/1 -1/2 1/0 4/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(13,6,2,1) (-1/2,-2/5) -> (4/1,1/0) Hyperbolic Matrix(11,4,-36,-13) (-2/5,-1/3) -> (-1/3,-2/7) Parabolic Matrix(57,16,32,9) (-2/7,-1/4) -> (7/4,2/1) Hyperbolic Matrix(9,2,22,5) (-1/4,0/1) -> (2/5,1/2) Hyperbolic Matrix(7,-2,18,-5) (0/1,1/3) -> (1/3,2/5) Parabolic Matrix(5,-4,4,-3) (1/2,1/1) -> (1/1,3/2) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(13,6,2,1) -> Matrix(1,0,-4,1) Matrix(11,4,-36,-13) -> Matrix(1,0,0,1) Matrix(57,16,32,9) -> Matrix(5,-2,-2,1) Matrix(9,2,22,5) -> Matrix(5,-2,-2,1) Matrix(7,-2,18,-5) -> Matrix(1,-4,0,1) Matrix(5,-4,4,-3) -> Matrix(1,0,0,1) Matrix(31,-50,18,-29) -> Matrix(1,0,0,1) Matrix(7,-18,2,-5) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 2 Degree of the the map Y: 8 Permutation triple for Y: ((1,5,6,2)(3,7,8,4); (1,4)(6,7); (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 -1/1 0/1 1/1 2/1 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/3 0/1 1/1 1/2 -1/1 1/0 1/1 -1/1 0/1 1/0 3/2 -1/1 1/0 2/1 -1/1 3/1 -1/2 1/0 4/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic 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(9,-14,2,-3) (3/2,2/1) -> (4/1,1/0) Hyperbolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(1,0,4,1) -> Matrix(2,-1,1,0) Matrix(5,-4,4,-3) -> Matrix(1,0,0,1) Matrix(9,-14,2,-3) -> Matrix(0,-1,1,2) Matrix(7,-18,2,-5) -> 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 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 This is a reflection group. CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d -1/1 0/1 2 1 0/1 1/1 1 4 1/1 (-1/1,1/1) 0 2 2/1 -1/1 1 4 3/1 (-1/1,0/1) 0 1 1/0 (-1/1,0/1) 0 4 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) 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(5,-12,2,-5) (2/1,3/1) -> (2/1,3/1) Reflection Matrix(-1,6,0,1) (3/1,1/0) -> (3/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,2,-1) (-1/1,0/1) -> (0/1,1/1) Matrix(1,0,2,-1) -> Matrix(0,1,1,0) (0/1,1/1) -> (-1/1,1/1) Matrix(3,-4,2,-3) -> Matrix(0,1,1,0) (1/1,2/1) -> (-1/1,1/1) Matrix(5,-12,2,-5) -> Matrix(-1,0,2,1) (2/1,3/1) -> (-1/1,0/1) Matrix(-1,6,0,1) -> Matrix(-1,0,2,1) (3/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.