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 -4/1 -3/1 -2/1 -3/2 0/1 1/1 6/5 3/2 2/1 12/5 5/2 3/1 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 -5/1 -1/1 -2/3 -1/2 -4/1 -1/1 -1/2 0/1 -3/1 0/1 -8/3 0/1 1/1 1/0 -5/2 1/0 -2/1 -1/1 0/1 1/0 -7/4 1/0 -12/7 -2/1 0/1 -5/3 -1/1 0/1 1/0 -3/2 -1/2 1/0 -7/5 -1/1 0/1 1/0 -4/3 -1/1 -1/2 0/1 -5/4 -1/2 -6/5 -1/1 -1/3 -1/1 -1/1 -1/2 0/1 0/1 0/1 1/1 0/1 1/3 1/2 6/5 1/3 1/1 5/4 1/2 4/3 0/1 1/3 1/2 3/2 1/2 8/5 1/2 2/3 1/1 5/3 0/1 1/2 1/1 2/1 0/1 1/2 1/1 7/3 0/1 1/2 1/1 12/5 0/1 2/3 5/2 1/2 3/1 1/1 7/2 1/0 4/1 0/1 1/1 1/0 5/1 0/1 1/2 1/1 6/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(11,60,2,11) (-6/1,-5/1) -> (5/1,6/1) Hyperbolic Matrix(13,60,8,37) (-5/1,-4/1) -> (8/5,5/3) Hyperbolic Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) 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(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(35,48,8,11) (-7/5,-4/3) -> (4/1,5/1) Hyperbolic Matrix(37,48,10,13) (-4/3,-5/4) -> (7/2,4/1) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(11,12,10,11) (-6/5,-1/1) -> (1/1,6/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,2,0,1) Matrix(11,60,2,11) -> Matrix(3,2,4,3) Matrix(13,60,8,37) -> Matrix(3,2,4,3) Matrix(11,36,-4,-13) -> Matrix(1,0,2,1) Matrix(23,60,18,47) -> Matrix(1,0,2,1) Matrix(11,24,-6,-13) -> Matrix(1,0,0,1) Matrix(83,144,34,59) -> Matrix(1,0,2,1) Matrix(85,144,36,61) -> Matrix(1,0,2,1) Matrix(23,36,-16,-25) -> Matrix(1,0,0,1) Matrix(35,48,8,11) -> Matrix(1,0,2,1) Matrix(37,48,10,13) -> Matrix(1,0,2,1) Matrix(49,60,40,49) -> Matrix(1,0,4,1) Matrix(11,12,10,11) -> Matrix(1,0,4,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(25,-36,16,-23) -> Matrix(5,-2,8,-3) Matrix(13,-24,6,-11) -> Matrix(1,0,0,1) Matrix(13,-36,4,-11) -> 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): 3 Degree of the the map X: 3 Degree of the the map Y: 16 Permutation triple for Y: ((2,6,7)(3,11,4)(5,10,9)(8,13,12); (1,4,13,14,5,2)(3,10)(6,9,16,12,11,15)(7,8); (1,2,8,16,9,3)(4,12)(5,6)(7,15,11,10,14,13)) ----------------------------------------------------------------------- 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 -2/1 -3/2 -6/5 0/1 1/1 3/2 2/1 5/2 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 0/1 -5/2 1/0 -2/1 -1/1 0/1 1/0 -3/2 -1/2 1/0 -4/3 -1/1 -1/2 0/1 -5/4 -1/2 -6/5 -1/1 -1/3 -1/1 -1/1 -1/2 0/1 0/1 0/1 1/1 0/1 1/3 1/2 3/2 1/2 5/3 0/1 1/2 1/1 2/1 0/1 1/2 1/1 7/3 0/1 1/2 1/1 12/5 0/1 2/3 5/2 1/2 3/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,6,0,1) (-3/1,1/0) -> (3/1,1/0) Parabolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(13,30,-10,-23) (-5/2,-2/1) -> (-4/3,-5/4) Hyperbolic Matrix(11,18,-8,-13) (-2/1,-3/2) -> (-3/2,-4/3) Parabolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) 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(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,1,0,1) Matrix(11,30,4,11) -> Matrix(1,-1,2,-1) Matrix(13,30,-10,-23) -> Matrix(1,1,-2,-1) Matrix(11,18,-8,-13) -> Matrix(1,1,-2,-1) Matrix(73,90,30,37) -> Matrix(1,1,0,1) Matrix(47,54,20,23) -> Matrix(1,1,0,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(13,-18,8,-11) -> Matrix(3,-1,4,-1) Matrix(13,-24,6,-11) -> Matrix(1,0,0,1) 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): 3 ----------------------------------------------------------------------- The image of the extended modular group liftables in PGL(2,Z) equals the image of the modular liftables. ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.