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): 36 Minimal number of generators: 7 Number of equivalence classes of cusps: 6 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 1/1 3/2 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 1/10 -1/3 2/15 0/1 1/5 1/3 4/15 1/2 3/10 2/3 1/3 1/1 2/5 3/2 1/2 5/3 8/15 2/1 3/5 3/1 4/5 4/1 1/1 1/0 1/0 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(5,2,12,5) (-1/2,-1/3) -> (1/3,1/2) Hyperbolic Matrix(9,2,4,1) (-1/3,0/1) -> (2/1,3/1) Hyperbolic Matrix(11,-2,6,-1) (0/1,1/3) -> (5/3,2/1) Hyperbolic Matrix(11,-6,2,-1) (1/2,2/3) -> (4/1,1/0) Hyperbolic Matrix(13,-10,4,-3) (2/3,1/1) -> (3/1,4/1) Hyperbolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,10,1) Matrix(5,2,12,5) -> Matrix(17,-2,60,-7) Matrix(9,2,4,1) -> Matrix(13,-2,20,-3) Matrix(11,-2,6,-1) -> Matrix(17,-4,30,-7) Matrix(11,-6,2,-1) -> Matrix(13,-4,10,-3) Matrix(13,-10,4,-3) -> Matrix(17,-6,20,-7) Matrix(13,-18,8,-11) -> Matrix(21,-10,40,-19) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 36 Minimal number of generators: 7 Number of equivalence classes of cusps: 6 Genus: 1 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 6 Degree of the the map Y: 6 Permutation triple for Y: ((1,6,4,3,2);(1,4,6,3,5);(1,2,6,5,3)) ----------------------------------------------------------------------- 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, lambda1, lambda2, lambda1+lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z+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): 6 Minimal number of generators: 3 Number of equivalence classes of elliptic points of order 2: 2 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 1/5 1/1 2/5 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(1,-1,2,-1) (0/1,1/1) -> (0/1,1/1) Elliptic Matrix(1,-2,1,-1) (1/1,1/0) -> (1/1,1/0) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,5,1) Matrix(1,-1,2,-1) -> Matrix(3,-1,10,-3) Matrix(1,-2,1,-1) -> Matrix(2,-1,5,-2) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 3 Number of equivalence classes of elliptic points of order 2: 2 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 -1/1 0/1 5 1 1/1 2/5 1 5 1/0 1/0 1 5 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(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(1,-2,1,-1) (1/1,1/0) -> (1/1,1/0) Elliptic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(0,1,1,0) -> Matrix(1,0,5,-1) (-1/1,1/1) -> (0/1,2/5) Matrix(1,-2,1,-1) -> Matrix(2,-1,5,-2) (0/1,1/2).(1/3,1/1) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.