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 -4/3 -1/1 -2/3 -1/2 0/1 1/3 2/5 1/2 2/3 4/5 5/6 1/1 4/3 5/3 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 1/0 -5/3 -2/1 1/0 -8/5 -2/1 -3/2 -3/2 1/0 -4/3 -1/1 -5/4 -1/2 1/0 -1/1 -1/1 0/1 1/0 -2/3 0/1 -3/5 0/1 1/2 1/1 -7/12 1/0 -4/7 0/1 -1/2 1/2 1/0 -3/7 0/1 1/1 1/0 -5/12 1/2 -2/5 1/1 -1/3 1/1 1/0 0/1 1/0 1/3 -1/1 1/0 2/5 -1/1 5/12 -1/2 3/7 -1/1 0/1 1/0 1/2 -1/2 1/0 2/3 0/1 3/4 1/2 1/0 4/5 0/1 5/6 1/2 1/1 0/1 1/1 1/0 4/3 1/1 7/5 1/1 3/2 2/1 3/2 3/2 1/0 8/5 2/1 5/3 2/1 1/0 2/1 1/0 1/0 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(11,20,6,11) (-2/1,-5/3) -> (5/3,2/1) Hyperbolic Matrix(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(23,36,30,47) (-8/5,-3/2) -> (3/4,4/5) Hyperbolic Matrix(23,32,-18,-25) (-3/2,-4/3) -> (-4/3,-5/4) Parabolic Matrix(13,16,30,37) (-5/4,-1/1) -> (3/7,1/2) Hyperbolic Matrix(11,8,-18,-13) (-1/1,-2/3) -> (-2/3,-3/5) Parabolic Matrix(61,36,144,85) (-3/5,-7/12) -> (5/12,3/7) Hyperbolic Matrix(83,48,102,59) (-7/12,-4/7) -> (4/5,5/6) Hyperbolic Matrix(37,20,24,13) (-4/7,-1/2) -> (3/2,8/5) Hyperbolic Matrix(35,16,24,11) (-1/2,-3/7) -> (7/5,3/2) Hyperbolic Matrix(47,20,54,23) (-3/7,-5/12) -> (5/6,1/1) Hyperbolic Matrix(49,20,120,49) (-5/12,-2/5) -> (2/5,5/12) Hyperbolic Matrix(11,4,30,11) (-2/5,-1/3) -> (1/3,2/5) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(13,-8,18,-11) (1/2,2/3) -> (2/3,3/4) Parabolic Matrix(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,2,0,1) Matrix(11,20,6,11) -> Matrix(1,4,0,1) Matrix(49,80,30,49) -> Matrix(1,4,0,1) Matrix(23,36,30,47) -> Matrix(1,2,0,1) Matrix(23,32,-18,-25) -> Matrix(1,2,-2,-3) Matrix(13,16,30,37) -> Matrix(1,0,0,1) Matrix(11,8,-18,-13) -> Matrix(1,0,2,1) Matrix(61,36,144,85) -> Matrix(1,0,-2,1) Matrix(83,48,102,59) -> Matrix(1,0,2,1) Matrix(37,20,24,13) -> Matrix(3,-2,2,-1) Matrix(35,16,24,11) -> Matrix(3,-2,2,-1) Matrix(47,20,54,23) -> Matrix(1,0,0,1) Matrix(49,20,120,49) -> Matrix(3,-2,-4,3) Matrix(11,4,30,11) -> Matrix(1,-2,0,1) Matrix(1,0,6,1) -> Matrix(1,-2,0,1) Matrix(13,-8,18,-11) -> Matrix(1,0,2,1) Matrix(25,-32,18,-23) -> 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): 4 Degree of the the map X: 4 Degree of the the map Y: 16 Permutation triple for Y: ((1,6,2)(3,10,4)(7,14,15)(11,13,16); (1,4,12,15,13,5)(2,8,16,14,9,3)(6,7)(10,11); (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 -2/3 -1/2 0/1 1/3 2/5 1/2 2/3 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 0/1 1/0 -2/3 0/1 -3/5 0/1 1/2 1/1 -1/2 1/2 1/0 -3/7 0/1 1/1 1/0 -5/12 1/2 -2/5 1/1 -1/3 1/1 1/0 0/1 1/0 1/3 -1/1 1/0 2/5 -1/1 1/2 -1/2 1/0 2/3 0/1 3/4 1/2 1/0 4/5 0/1 5/6 1/2 1/1 0/1 1/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(11,8,-18,-13) (-1/1,-2/3) -> (-2/3,-3/5) Parabolic Matrix(11,6,-24,-13) (-3/5,-1/2) -> (-1/2,-3/7) Parabolic Matrix(47,20,54,23) (-3/7,-5/12) -> (5/6,1/1) Hyperbolic Matrix(73,30,90,37) (-5/12,-2/5) -> (4/5,5/6) Hyperbolic Matrix(11,4,30,11) (-2/5,-1/3) -> (1/3,2/5) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(23,-10,30,-13) (2/5,1/2) -> (3/4,4/5) Hyperbolic Matrix(13,-8,18,-11) (1/2,2/3) -> (2/3,3/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,1,0,1) Matrix(11,8,-18,-13) -> Matrix(1,0,2,1) Matrix(11,6,-24,-13) -> Matrix(1,-1,2,-1) Matrix(47,20,54,23) -> Matrix(1,0,0,1) Matrix(73,30,90,37) -> Matrix(1,-1,4,-3) Matrix(11,4,30,11) -> Matrix(1,-2,0,1) Matrix(1,0,6,1) -> Matrix(1,-2,0,1) Matrix(23,-10,30,-13) -> Matrix(1,1,0,1) Matrix(13,-8,18,-11) -> Matrix(1,0,2,1) 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 1/0 1 3 1/3 (-1/1,1/0) 0 6 2/5 -1/1 2 3 1/2 (-1/1,0/1).(-1/2,1/0) 0 6 2/3 0/1 3 3 3/4 (0/1,1/1).(1/2,1/0) 0 6 4/5 0/1 2 3 5/6 1/2 3 6 1/1 (1/2,1/0) 0 6 1/0 1/0 3 6 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(1,0,6,-1) (0/1,1/3) -> (0/1,1/3) Reflection Matrix(11,-4,30,-11) (1/3,2/5) -> (1/3,2/5) Reflection Matrix(23,-10,30,-13) (2/5,1/2) -> (3/4,4/5) Hyperbolic Matrix(13,-8,18,-11) (1/2,2/3) -> (2/3,3/4) Parabolic Matrix(49,-40,60,-49) (4/5,5/6) -> (4/5,5/6) Reflection Matrix(11,-10,12,-11) (5/6,1/1) -> (5/6,1/1) 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(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,6,-1) -> Matrix(1,2,0,-1) (0/1,1/3) -> (-1/1,1/0) Matrix(11,-4,30,-11) -> Matrix(1,2,0,-1) (1/3,2/5) -> (-1/1,1/0) Matrix(23,-10,30,-13) -> Matrix(1,1,0,1) 1/0 Matrix(13,-8,18,-11) -> Matrix(1,0,2,1) 0/1 Matrix(49,-40,60,-49) -> Matrix(1,0,4,-1) (4/5,5/6) -> (0/1,1/2) Matrix(11,-10,12,-11) -> Matrix(-1,1,0,1) (5/6,1/1) -> (1/2,1/0) Matrix(-1,2,0,1) -> Matrix(-1,1,0,1) (1/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.