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 0/1 1/1 5/3 2/1 5/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -3/5 -5/2 -1/2 -2/1 -2/5 -5/3 -1/3 -3/2 -3/10 -1/1 -1/5 0/1 0/1 1/1 1/5 3/2 3/10 5/3 1/3 2/1 2/5 5/2 1/2 3/1 3/5 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,10,2,7) (-3/1,1/0) -> (1/1,3/2) Hyperbolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(11,20,6,11) (-2/1,-5/3) -> (5/3,2/1) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(7,10,2,3) (-3/2,-1/1) -> (3/1,1/0) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,10,2,7) -> Matrix(3,2,10,7) Matrix(11,30,4,11) -> Matrix(11,6,20,11) Matrix(9,20,4,9) -> Matrix(9,4,20,9) Matrix(11,20,6,11) -> Matrix(11,4,30,11) Matrix(19,30,12,19) -> Matrix(19,6,60,19) Matrix(7,10,2,3) -> Matrix(7,2,10,3) Matrix(1,0,2,1) -> Matrix(1,0,10,1) 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: ((2,5,4,3,6);(1,4,6,3,2);(1,2,4,5,3)) ----------------------------------------------------------------------- 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 0/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/5 2/1 2/5 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(3,-5,2,-3) (1/1,2/1) -> (1/1,2/1) Elliptic Matrix(2,-5,1,-2) (2/1,1/0) -> (2/1,1/0) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,5,1) Matrix(3,-5,2,-3) -> Matrix(3,-1,10,-3) Matrix(2,-5,1,-2) -> 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 0/1 0/1 5 1 2/1 2/5 1 5 1/0 1/0 1 5 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(2,-5,1,-2) (2/1,1/0) -> (2/1,1/0) Elliptic 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(1,0,1,-1) -> Matrix(1,0,5,-1) (0/1,2/1) -> (0/1,2/5) Matrix(2,-5,1,-2) -> 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.