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: 8 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/2 -3/8 -1/4 0/1 1/4 1/2 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/8 -3/4 1/6 -2/3 3/16 -1/2 1/4 -2/5 5/16 -3/8 1/3 -1/3 3/8 -1/4 1/2 0/1 1/0 1/4 -1/2 1/3 -3/8 1/2 -1/4 3/5 -5/24 5/8 -1/5 2/3 -3/16 3/4 -1/6 1/1 -1/8 1/0 0/1 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(5,4,16,13) (-1/1,-3/4) -> (1/4,1/3) Hyperbolic Matrix(3,2,16,11) (-3/4,-2/3) -> (0/1,1/4) Hyperbolic Matrix(7,4,-16,-9) (-2/3,-1/2) -> (-1/2,-2/5) Parabolic Matrix(41,16,64,25) (-2/5,-3/8) -> (5/8,2/3) Hyperbolic Matrix(39,14,64,23) (-3/8,-1/3) -> (3/5,5/8) Hyperbolic 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(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-16,1) Matrix(5,4,16,13) -> Matrix(13,-2,-32,5) Matrix(3,2,16,11) -> Matrix(11,-2,-16,3) Matrix(7,4,-16,-9) -> Matrix(9,-2,32,-7) Matrix(41,16,64,25) -> Matrix(25,-8,-128,41) Matrix(39,14,64,23) -> Matrix(23,-8,-112,39) Matrix(13,4,16,5) -> Matrix(5,-2,-32,13) Matrix(11,2,16,3) -> Matrix(3,-2,-16,11) Matrix(9,-4,16,-7) -> Matrix(7,2,-32,-9) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 48 Minimal number of generators: 9 Number of equivalence classes of cusps: 8 Genus: 1 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 8 Degree of the the map Y: 8 Permutation triple for Y: ((1,4,7,6,8,3,5,2); (1,2,7,4,8,6,5,3); (2,6)(3,4)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1, lambda2, lambda1+lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z. ----------------------------------------------------------------------- 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): 12 Minimal number of generators: 3 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: 4 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/2 3/4 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/2 -1/4 2/3 -3/16 3/4 -1/6 1/1 -1/8 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(5,-2,8,-3) (0/1,1/2) -> (1/2,2/3) Parabolic Matrix(13,-9,16,-11) (2/3,3/4) -> (3/4,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-8,1) Matrix(5,-2,8,-3) -> Matrix(3,1,-16,-5) Matrix(13,-9,16,-11) -> Matrix(11,2,-72,-13) 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 4 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 0/1 1/0 1 8 1/4 -1/2 4 2 1/2 -1/4 2 4 1/0 0/1 8 1 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(1,0,8,-1) (0/1,1/4) -> (0/1,1/4) Reflection Matrix(3,-1,8,-3) (1/4,1/2) -> (1/4,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) 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(1,0,8,-1) -> Matrix(1,1,0,-1) (0/1,1/4) -> (-1/2,1/0) Matrix(3,-1,8,-3) -> Matrix(3,1,-8,-3) (1/4,1/2) -> (-1/2,-1/4) Matrix(-1,1,0,1) -> Matrix(-1,0,8,1) (1/2,1/0) -> (-1/4,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.