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 -3/1 -12/5 -3/2 0/1 1/1 3/2 9/5 2/1 3/1 9/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 0/1 -5/2 1/2 -12/5 1/1 -7/3 1/0 -9/4 1/0 -2/1 1/0 -3/2 -1/2 1/0 -4/3 1/0 -9/7 -1/1 -5/4 -1/2 -6/5 -1/1 -1/1 -1/2 0/1 0/1 1/1 1/2 3/2 1/2 1/0 5/3 1/2 12/7 1/1 7/4 1/2 9/5 1/1 2/1 1/0 3/1 0/1 4/1 1/2 9/2 1/2 5/1 1/2 6/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,18,-2,-7) (-3/1,1/0) -> (-3/1,-5/2) Parabolic Matrix(59,144,34,83) (-5/2,-12/5) -> (12/7,7/4) Hyperbolic Matrix(61,144,36,85) (-12/5,-7/3) -> (5/3,12/7) Hyperbolic Matrix(47,108,10,23) (-7/3,-9/4) -> (9/2,5/1) Hyperbolic Matrix(25,54,6,13) (-9/4,-2/1) -> (4/1,9/2) Hyperbolic Matrix(11,18,-8,-13) (-2/1,-3/2) -> (-3/2,-4/3) Parabolic Matrix(41,54,22,29) (-4/3,-9/7) -> (9/5,2/1) Hyperbolic Matrix(85,108,48,61) (-9/7,-5/4) -> (7/4,9/5) Hyperbolic Matrix(29,36,4,5) (-5/4,-6/5) -> (6/1,1/0) Hyperbolic Matrix(31,36,6,7) (-6/5,-1/1) -> (5/1,6/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,18,-2,-7) -> Matrix(1,0,2,1) Matrix(59,144,34,83) -> Matrix(1,0,0,1) Matrix(61,144,36,85) -> Matrix(1,-2,2,-3) Matrix(47,108,10,23) -> Matrix(1,-2,2,-3) Matrix(25,54,6,13) -> Matrix(1,0,2,1) Matrix(11,18,-8,-13) -> Matrix(1,0,0,1) Matrix(41,54,22,29) -> Matrix(1,2,0,1) Matrix(85,108,48,61) -> Matrix(3,2,4,3) Matrix(29,36,4,5) -> Matrix(1,0,2,1) Matrix(31,36,6,7) -> Matrix(3,2,4,3) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(13,-18,8,-11) -> Matrix(1,0,0,1) Matrix(7,-18,2,-5) -> Matrix(1,0,2,1) 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): 4 Degree of the the map X: 4 Degree of the the map Y: 12 Permutation triple for Y: ((2,5,10,8,4,3,7,11,6); (1,4,8,12,11,7,9,5,2); (1,2,6,12,8,10,9,7,3)) ----------------------------------------------------------------------- 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 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: 7 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: 8 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/1 -3/2 0/1 1/1 3/2 3/1 9/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 0/1 -5/2 1/2 -2/1 1/0 -3/2 -1/2 1/0 -1/1 -1/2 0/1 0/1 1/1 1/2 3/2 1/2 1/0 2/1 1/0 3/1 0/1 4/1 1/2 9/2 1/2 5/1 1/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,18,-2,-7) (-3/1,1/0) -> (-3/1,-5/2) Parabolic Matrix(11,27,2,5) (-5/2,-2/1) -> (5/1,1/0) Hyperbolic Matrix(5,9,-4,-7) (-2/1,-3/2) -> (-3/2,-1/1) Parabolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(7,-9,4,-5) (1/1,3/2) -> (3/2,2/1) Parabolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic Matrix(19,-81,4,-17) (4/1,9/2) -> (9/2,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,18,-2,-7) -> Matrix(1,0,2,1) Matrix(11,27,2,5) -> Matrix(1,-1,2,-1) Matrix(5,9,-4,-7) -> Matrix(1,1,-2,-1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(7,-9,4,-5) -> Matrix(1,-1,2,-1) Matrix(7,-18,2,-5) -> Matrix(1,0,2,1) Matrix(19,-81,4,-17) -> Matrix(3,-1,4,-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 0/1 0/1 6 3 1/1 1/2 1 9 3/2 (0/1,1/1).(1/2,1/0) 0 3 2/1 1/0 1 9 3/1 0/1 3 3 4/1 1/2 1 9 9/2 1/2 3 3 1/0 1/0 1 9 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(7,-9,4,-5) (1/1,3/2) -> (3/2,2/1) Parabolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(-1,9,0,1) (9/2,1/0) -> (9/2,1/0) 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(1,0,2,-1) -> Matrix(1,0,4,-1) (0/1,1/1) -> (0/1,1/2) Matrix(7,-9,4,-5) -> Matrix(1,-1,2,-1) (0/1,1/1).(1/2,1/0) Matrix(7,-18,2,-5) -> Matrix(1,0,2,1) 0/1 Matrix(17,-72,4,-17) -> Matrix(1,0,4,-1) (4/1,9/2) -> (0/1,1/2) Matrix(-1,9,0,1) -> Matrix(-1,1,0,1) (9/2,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.