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): 72 Minimal number of generators: 13 Number of equivalence classes of cusps: 12 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 1/4 1/2 2/3 1/1 3/2 2/1 7/3 3/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 1/1 -1/2 1/0 -3/7 -1/1 0/1 -2/5 -1/1 0/1 -1/3 0/1 1/0 -1/4 1/0 -2/9 -1/1 0/1 -1/5 -1/1 0/1 0/1 0/1 1/0 1/4 0/1 2/7 0/1 1/2 1/3 0/1 1/1 1/2 1/0 2/3 -1/1 1/1 3/4 1/0 1/1 0/1 1/0 3/2 0/1 5/3 0/1 1/2 2/1 0/1 1/1 7/3 1/1 12/5 1/1 2/1 5/2 1/0 3/1 0/1 1/1 4/1 1/1 5/1 1/1 2/1 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(17,8,2,1) (-1/2,-3/7) -> (5/1,1/0) Hyperbolic Matrix(19,8,-88,-37) (-3/7,-2/5) -> (-2/9,-1/5) Hyperbolic Matrix(31,12,18,7) (-2/5,-1/3) -> (5/3,2/1) Hyperbolic Matrix(13,4,16,5) (-1/3,-1/4) -> (3/4,1/1) Hyperbolic Matrix(93,22,38,9) (-1/4,-2/9) -> (12/5,5/2) Hyperbolic Matrix(11,2,38,7) (-1/5,0/1) -> (2/7,1/3) Hyperbolic Matrix(9,-2,32,-7) (0/1,1/4) -> (1/4,2/7) Parabolic Matrix(21,-8,8,-3) (1/3,1/2) -> (5/2,3/1) Hyperbolic Matrix(13,-8,18,-11) (1/2,2/3) -> (2/3,3/4) Parabolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,0,1) Matrix(17,8,2,1) -> Matrix(1,2,0,1) Matrix(19,8,-88,-37) -> Matrix(1,0,0,1) Matrix(31,12,18,7) -> Matrix(1,0,2,1) Matrix(13,4,16,5) -> Matrix(1,0,0,1) Matrix(93,22,38,9) -> Matrix(1,2,0,1) Matrix(11,2,38,7) -> Matrix(1,0,2,1) Matrix(9,-2,32,-7) -> Matrix(1,0,2,1) Matrix(21,-8,8,-3) -> Matrix(1,0,0,1) Matrix(13,-8,18,-11) -> Matrix(1,0,0,1) Matrix(13,-18,8,-11) -> Matrix(1,0,2,1) Matrix(43,-98,18,-41) -> Matrix(3,-2,2,-1) Matrix(9,-32,2,-7) -> Matrix(3,-2,2,-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): 2 Degree of the the map X: 2 Degree of the the map Y: 12 Permutation triple for Y: ((1,6,11,7,2)(3,10,12,8,4); (1,4,8,7,5)(3,9,12,11,6); (1,2,8,9,3)(5,7,12,10,6)) ----------------------------------------------------------------------- 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): 36 Minimal number of generators: 9 Number of equivalence classes of elliptic points of order 2: 4 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 3/2 2/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 1/1 -1/2 1/0 -3/7 -1/1 0/1 -2/5 -1/1 0/1 -1/3 0/1 1/0 0/1 0/1 1/0 1/1 0/1 1/0 3/2 0/1 5/3 0/1 1/2 2/1 0/1 1/1 3/1 0/1 1/1 4/1 1/1 5/1 1/1 2/1 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(17,8,2,1) (-1/2,-3/7) -> (5/1,1/0) Hyperbolic Matrix(31,13,-74,-31) (-3/7,-2/5) -> (-3/7,-2/5) Elliptic Matrix(31,12,18,7) (-2/5,-1/3) -> (5/3,2/1) Hyperbolic Matrix(3,1,-10,-3) (-1/3,0/1) -> (-1/3,0/1) Elliptic Matrix(1,-1,2,-1) (0/1,1/1) -> (0/1,1/1) Elliptic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(5,-13,2,-5) (2/1,3/1) -> (2/1,3/1) Elliptic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,0,1) Matrix(17,8,2,1) -> Matrix(1,2,0,1) Matrix(31,13,-74,-31) -> Matrix(1,0,0,1) Matrix(31,12,18,7) -> Matrix(1,0,2,1) Matrix(3,1,-10,-3) -> Matrix(1,0,0,1) Matrix(1,-1,2,-1) -> Matrix(1,0,0,1) Matrix(13,-18,8,-11) -> Matrix(1,0,2,1) Matrix(5,-13,2,-5) -> Matrix(1,0,0,1) Matrix(9,-32,2,-7) -> Matrix(3,-2,2,-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 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d -1/1 (0/1,1/0) 0 1 0/1 (0/1,1/0) 0 5 1/1 (0/1,1/0) 0 5 3/2 0/1 1 1 2/1 (0/1,1/1) 0 5 3/1 (0/1,1/1) 0 5 4/1 1/1 1 1 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(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,-1,2,-1) (0/1,1/1) -> (0/1,1/1) Elliptic Matrix(5,-6,4,-5) (1/1,3/2) -> (1/1,3/2) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(5,-13,2,-5) (2/1,3/1) -> (2/1,3/1) Elliptic Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection Matrix(-1,8,0,1) (4/1,1/0) -> (4/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,-1,2,-1) -> Matrix(1,0,0,1) Matrix(5,-6,4,-5) -> Matrix(1,0,0,-1) (1/1,3/2) -> (0/1,1/0) Matrix(7,-12,4,-7) -> Matrix(1,0,2,-1) (3/2,2/1) -> (0/1,1/1) Matrix(5,-13,2,-5) -> Matrix(1,0,0,1) Matrix(7,-24,2,-7) -> Matrix(1,0,2,-1) (3/1,4/1) -> (0/1,1/1) Matrix(-1,8,0,1) -> Matrix(-1,2,0,1) (4/1,1/0) -> (1/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.