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): 96 Minimal number of generators: 17 Number of equivalence classes of cusps: 16 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/3 -1/1 -2/3 -3/5 -1/2 -1/3 -1/5 -1/6 0/1 1/3 1/2 2/3 1/1 5/3 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 -5/3 0/1 -8/5 1/1 -3/2 1/0 -1/1 -1/1 0/1 -3/4 1/0 -2/3 -1/1 -5/8 -3/4 -3/5 -2/3 -7/12 -1/2 -4/7 -1/1 -1/2 -1/2 -1/3 0/1 -1/4 1/0 -1/5 0/1 1/0 -1/6 1/0 0/1 -1/1 1/3 -1/1 -1/2 0/1 2/5 -1/1 5/12 -1/2 3/7 0/1 1/2 -1/2 4/7 -1/1 7/12 -1/2 3/5 -1/2 0/1 2/3 -1/1 -1/3 1/1 0/1 4/3 -1/1 1/1 7/5 0/1 1/0 3/2 1/0 5/3 -1/1 0/1 1/0 7/4 1/0 2/1 -1/1 1/0 -1/2 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(29,50,-18,-31) (-2/1,-5/3) -> (-5/3,-8/5) Parabolic Matrix(13,20,24,37) (-8/5,-3/2) -> (1/2,4/7) Hyperbolic Matrix(5,6,-6,-7) (-3/2,-1/1) -> (-1/1,-3/4) Parabolic Matrix(23,16,-36,-25) (-3/4,-2/3) -> (-2/3,-5/8) Parabolic Matrix(29,18,66,41) (-5/8,-3/5) -> (3/7,1/2) Hyperbolic Matrix(61,36,144,85) (-3/5,-7/12) -> (5/12,3/7) Hyperbolic Matrix(97,56,168,97) (-7/12,-4/7) -> (4/7,7/12) Hyperbolic Matrix(53,30,30,17) (-4/7,-1/2) -> (7/4,2/1) Hyperbolic Matrix(5,2,-18,-7) (-1/2,-1/3) -> (-1/3,-1/4) Parabolic Matrix(43,10,30,7) (-1/4,-1/5) -> (7/5,3/2) Hyperbolic Matrix(53,10,90,17) (-1/5,-1/6) -> (7/12,3/5) Hyperbolic Matrix(17,2,42,5) (-1/6,0/1) -> (2/5,5/12) Hyperbolic Matrix(7,-2,18,-5) (0/1,1/3) -> (1/3,2/5) Parabolic Matrix(41,-26,30,-19) (3/5,2/3) -> (4/3,7/5) Hyperbolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,0,1) Matrix(29,50,-18,-31) -> Matrix(1,0,2,1) Matrix(13,20,24,37) -> Matrix(1,0,-2,1) Matrix(5,6,-6,-7) -> Matrix(1,0,0,1) Matrix(23,16,-36,-25) -> Matrix(3,4,-4,-5) Matrix(29,18,66,41) -> Matrix(3,2,-2,-1) Matrix(61,36,144,85) -> Matrix(3,2,-8,-5) Matrix(97,56,168,97) -> Matrix(1,0,0,1) Matrix(53,30,30,17) -> Matrix(3,2,-2,-1) Matrix(5,2,-18,-7) -> Matrix(1,0,2,1) Matrix(43,10,30,7) -> Matrix(1,0,0,1) Matrix(53,10,90,17) -> Matrix(1,0,-2,1) Matrix(17,2,42,5) -> Matrix(1,2,-2,-3) Matrix(7,-2,18,-5) -> Matrix(1,0,0,1) Matrix(41,-26,30,-19) -> Matrix(1,0,2,1) Matrix(7,-6,6,-5) -> Matrix(1,0,2,1) Matrix(31,-50,18,-29) -> 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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 4 Degree of the the map X: 4 Degree of the the map Y: 16 Permutation triple for Y: ((1,6,14,15,7,2)(3,10,13,16,11,4)(5,12)(8,9); (1,4,5)(2,9,3)(8,16,14)(12,15,13); (1,3)(2,7,12,4,11,8)(5,13,10,9,14,6)(15,16)) ----------------------------------------------------------------------- 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): 48 Minimal number of generators: 9 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: 10 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/3 -1/1 -2/3 -3/5 -1/3 0/1 1/3 1/1 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 -5/3 0/1 -3/2 1/0 -1/1 -1/1 0/1 -3/4 1/0 -2/3 -1/1 -5/8 -3/4 -3/5 -2/3 -1/2 -1/2 -1/3 0/1 0/1 -1/1 1/3 -1/1 -1/2 0/1 2/5 -1/1 3/7 0/1 1/2 -1/2 1/1 0/1 2/1 -1/1 1/0 -1/2 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(14,25,-9,-16) (-2/1,-5/3) -> (-5/3,-3/2) Parabolic Matrix(5,6,-6,-7) (-3/2,-1/1) -> (-1/1,-3/4) Parabolic Matrix(23,16,-36,-25) (-3/4,-2/3) -> (-2/3,-5/8) Parabolic Matrix(29,18,66,41) (-5/8,-3/5) -> (3/7,1/2) Hyperbolic Matrix(16,9,39,22) (-3/5,-1/2) -> (2/5,3/7) Hyperbolic Matrix(2,1,-9,-4) (-1/2,-1/3) -> (-1/3,0/1) Parabolic Matrix(7,-2,18,-5) (0/1,1/3) -> (1/3,2/5) Parabolic Matrix(4,-3,3,-2) (1/2,1/1) -> (1/1,2/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,0,1) Matrix(14,25,-9,-16) -> Matrix(1,0,1,1) Matrix(5,6,-6,-7) -> Matrix(1,0,0,1) Matrix(23,16,-36,-25) -> Matrix(3,4,-4,-5) Matrix(29,18,66,41) -> Matrix(3,2,-2,-1) Matrix(16,9,39,22) -> Matrix(3,2,-5,-3) Matrix(2,1,-9,-4) -> Matrix(1,0,1,1) Matrix(7,-2,18,-5) -> Matrix(1,0,0,1) Matrix(4,-3,3,-2) -> Matrix(1,0,1,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 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): 4 ----------------------------------------------------------------------- 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 (-1/1,0/1) 0 6 -2/3 -1/1 2 2 -3/5 -2/3 2 6 -1/2 -1/2 1 6 -1/3 0/1 2 2 0/1 -1/1 1 6 1/5 0/1 2 6 1/3 (-2/3,0/1) 0 2 1/1 0/1 2 6 1/0 (-1/1,0/1) 0 2 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(5,4,-6,-5) (-1/1,-2/3) -> (-1/1,-2/3) Reflection Matrix(19,12,-30,-19) (-2/3,-3/5) -> (-2/3,-3/5) Reflection Matrix(2,1,15,8) (-3/5,-1/2) -> (0/1,1/5) Hyperbolic Matrix(2,1,-9,-4) (-1/2,-1/3) -> (-1/3,0/1) Parabolic Matrix(4,-1,15,-4) (1/5,1/3) -> (1/5,1/3) Reflection Matrix(2,-1,3,-2) (1/3,1/1) -> (1/3,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) 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(5,4,-6,-5) -> Matrix(-1,0,2,1) (-1/1,-2/3) -> (-1/1,0/1) Matrix(19,12,-30,-19) -> Matrix(5,4,-6,-5) (-2/3,-3/5) -> (-1/1,-2/3) Matrix(2,1,15,8) -> Matrix(3,2,-5,-3) (-1/1,-1/2).(-2/3,0/1) Matrix(2,1,-9,-4) -> Matrix(1,0,1,1) 0/1 Matrix(4,-1,15,-4) -> Matrix(-1,0,3,1) (1/5,1/3) -> (-2/3,0/1) Matrix(2,-1,3,-2) -> Matrix(-1,0,3,1) (1/3,1/1) -> (-2/3,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,2,1) (1/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.