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: 12 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/1 -6/5 0/1 1/1 3/2 2/1 12/5 8/3 3/1 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/1 0/1 -3/1 0/1 -8/3 0/1 -5/2 1/0 -2/1 -1/1 -7/4 -1/2 -12/7 -1/2 -5/3 -1/2 -13/8 -1/4 -8/5 0/1 -3/2 0/1 -4/3 -1/1 0/1 -5/4 1/0 -6/5 -1/1 -7/6 -3/4 -1/1 -1/2 0/1 0/1 1/1 1/2 4/3 0/1 1/1 3/2 0/1 8/5 0/1 5/3 1/2 2/1 1/1 7/3 1/0 12/5 1/0 5/2 1/0 13/5 -1/2 8/3 0/1 3/1 0/1 4/1 0/1 1/1 5/1 1/2 6/1 1/1 7/1 3/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,24,-4,-19) (-4/1,1/0) -> (-4/3,-5/4) Hyperbolic Matrix(7,24,2,7) (-4/1,-3/1) -> (3/1,4/1) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(55,144,-34,-89) (-8/3,-5/2) -> (-13/8,-8/5) Hyperbolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic Matrix(83,144,34,59) (-7/4,-12/7) -> (12/5,5/2) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(29,48,-26,-43) (-5/3,-13/8) -> (-7/6,-1/1) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(59,72,-50,-61) (-5/4,-6/5) -> (-6/5,-7/6) Parabolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(19,-48,2,-5) (5/2,13/5) -> (7/1,1/0) Hyperbolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,24,-4,-19) -> Matrix(1,0,0,1) Matrix(7,24,2,7) -> Matrix(1,0,2,1) Matrix(17,48,6,17) -> Matrix(1,0,0,1) Matrix(55,144,-34,-89) -> Matrix(1,0,-4,1) Matrix(11,24,-6,-13) -> Matrix(1,2,-2,-3) Matrix(83,144,34,59) -> Matrix(7,4,-2,-1) Matrix(85,144,36,61) -> Matrix(9,4,2,1) Matrix(29,48,-26,-43) -> Matrix(5,2,-8,-3) Matrix(31,48,20,31) -> Matrix(1,0,4,1) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(59,72,-50,-61) -> Matrix(3,4,-4,-5) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(19,-24,4,-5) -> Matrix(1,0,0,1) Matrix(89,-144,34,-55) -> Matrix(1,0,-4,1) Matrix(13,-24,6,-11) -> Matrix(3,-2,2,-1) Matrix(19,-48,2,-5) -> Matrix(1,2,0,1) Matrix(13,-72,2,-11) -> Matrix(5,-4,4,-3) 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): 6 Degree of the the map X: 6 Degree of the the map Y: 16 Permutation triple for Y: ((2,6,7)(3,11,4)(5,8,9)(10,14,12); (1,4,9,16,12,11,15,6,14,13,5,2)(3,8,7,10); (1,2,8,13,14,7,15,11,10,16,9,3)(4,12,6,5)) ----------------------------------------------------------------------- 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: 8 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 12/5 8/3 3/1 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/2 4/3 0/1 1/1 3/2 0/1 8/5 0/1 5/3 1/2 2/1 1/1 7/3 1/0 12/5 1/0 5/2 1/0 13/5 -1/2 8/3 0/1 3/1 0/1 4/1 0/1 1/1 5/1 1/2 6/1 1/1 7/1 3/2 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(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(17,-24,5,-7) (4/3,3/2) -> (3/1,4/1) Hyperbolic Matrix(31,-48,11,-17) (3/2,8/5) -> (8/3,3/1) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(61,-144,25,-59) (7/3,12/5) -> (12/5,5/2) Parabolic Matrix(19,-48,2,-5) (5/2,13/5) -> (7/1,1/0) Hyperbolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,2,1) Matrix(19,-24,4,-5) -> Matrix(1,0,0,1) Matrix(17,-24,5,-7) -> Matrix(1,0,0,1) Matrix(31,-48,11,-17) -> Matrix(1,0,-2,1) Matrix(89,-144,34,-55) -> Matrix(1,0,-4,1) Matrix(13,-24,6,-11) -> Matrix(3,-2,2,-1) Matrix(61,-144,25,-59) -> Matrix(1,-4,0,1) Matrix(19,-48,2,-5) -> Matrix(1,2,0,1) Matrix(13,-72,2,-11) -> Matrix(5,-4,4,-3) 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): 3 ----------------------------------------------------------------------- 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 4 2 2/1 1/1 2 6 12/5 1/0 8 2 5/2 1/0 1 12 8/3 0/1 4 6 3/1 0/1 1 4 4/1 (0/1,1/1) 0 6 6/1 1/1 4 2 1/0 1/0 1 12 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(11,-24,5,-11) (2/1,12/5) -> (2/1,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(19,-48,2,-5) (5/2,13/5) -> (7/1,1/0) Hyperbolic Matrix(55,-144,21,-55) (18/7,8/3) -> (18/7,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection Matrix(5,-24,1,-5) (4/1,6/1) -> (4/1,6/1) Reflection Matrix(7,-48,1,-7) (6/1,8/1) -> (6/1,8/1) 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(1,0,1,-1) -> Matrix(1,0,2,-1) (0/1,2/1) -> (0/1,1/1) Matrix(11,-24,5,-11) -> Matrix(-1,2,0,1) (2/1,12/5) -> (1/1,1/0) Matrix(49,-120,20,-49) -> Matrix(1,2,0,-1) (12/5,5/2) -> (-1/1,1/0) Matrix(19,-48,2,-5) -> Matrix(1,2,0,1) 1/0 Matrix(55,-144,21,-55) -> Matrix(-1,0,2,1) (18/7,8/3) -> (-1/1,0/1) Matrix(17,-48,6,-17) -> Matrix(1,0,0,-1) (8/3,3/1) -> (0/1,1/0) Matrix(7,-24,2,-7) -> Matrix(1,0,2,-1) (3/1,4/1) -> (0/1,1/1) Matrix(5,-24,1,-5) -> Matrix(1,0,2,-1) (4/1,6/1) -> (0/1,1/1) Matrix(7,-48,1,-7) -> Matrix(3,-4,2,-3) (6/1,8/1) -> (1/1,2/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.