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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 20 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/2 -2/1 -3/2 -1/1 -2/3 -1/2 -2/5 0/1 1/3 2/5 1/2 2/3 5/6 1/1 6/5 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 -1/2 1/0 -5/2 -1/1 1/0 -2/1 -1/1 -7/4 -1/1 -3/4 -12/7 -1/1 -5/3 -3/4 -3/2 -1/1 -2/3 -1/2 -4/3 -1/2 -5/4 -1/1 -1/2 -6/5 -1/1 -1/1 -1/2 -6/7 -1/3 -5/6 -1/3 0/1 -4/5 0/1 -3/4 -1/1 -1/2 0/1 -2/3 -1/2 -5/8 -1/2 -1/3 -3/5 -1/2 -1/4 -1/2 -1/2 0/1 -3/7 -1/2 -1/4 -5/12 -1/3 0/1 -2/5 0/1 -3/8 -1/1 0/1 1/0 -1/3 -1/2 0/1 0/1 1/3 1/0 2/5 0/1 1/2 0/1 1/0 4/7 0/1 7/12 0/1 1/1 3/5 1/2 1/0 2/3 1/0 3/4 -1/1 0/1 1/0 4/5 0/1 5/6 0/1 1/1 1/1 1/0 7/6 -2/1 -1/1 6/5 -1/1 5/4 -1/1 1/0 4/3 1/0 3/2 -2/1 -1/1 1/0 8/5 -2/1 5/3 -3/2 2/1 -1/1 7/3 -1/2 12/5 -1/1 5/2 -1/1 -1/2 8/3 -1/2 3/1 -1/2 1/0 1/0 -1/1 0/1 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,-18,-49) (-3/1,-5/2) -> (-5/8,-3/5) Hyperbolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic Matrix(59,102,48,83) (-7/4,-12/7) -> (6/5,5/4) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(11,18,-30,-49) (-5/3,-3/2) -> (-3/8,-1/3) Hyperbolic Matrix(13,18,18,25) (-3/2,-4/3) -> (2/3,3/4) Hyperbolic Matrix(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(11,12,-12,-13) (-6/5,-1/1) -> (-1/1,-6/7) Parabolic Matrix(85,72,72,61) (-6/7,-5/6) -> (7/6,6/5) Hyperbolic Matrix(59,48,102,83) (-5/6,-4/5) -> (4/7,7/12) Hyperbolic Matrix(47,36,30,23) (-4/5,-3/4) -> (3/2,8/5) Hyperbolic Matrix(25,18,18,13) (-3/4,-2/3) -> (4/3,3/2) Hyperbolic Matrix(47,30,36,23) (-2/3,-5/8) -> (5/4,4/3) Hyperbolic Matrix(11,6,-24,-13) (-3/5,-1/2) -> (-1/2,-3/7) Parabolic Matrix(85,36,144,61) (-3/7,-5/12) -> (7/12,3/5) Hyperbolic Matrix(73,30,90,37) (-5/12,-2/5) -> (4/5,5/6) Hyperbolic Matrix(47,18,60,23) (-2/5,-3/8) -> (3/4,4/5) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(49,-18,30,-11) (1/3,2/5) -> (8/5,5/3) Hyperbolic Matrix(13,-6,24,-11) (2/5,1/2) -> (1/2,4/7) Parabolic Matrix(49,-30,18,-11) (3/5,2/3) -> (8/3,3/1) Hyperbolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) 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(1,0,0,1) Matrix(11,30,-18,-49) -> Matrix(1,0,-2,1) Matrix(11,24,-6,-13) -> Matrix(3,4,-4,-5) Matrix(59,102,48,83) -> Matrix(5,4,-4,-3) Matrix(85,144,36,61) -> Matrix(3,2,-2,-1) Matrix(11,18,-30,-49) -> Matrix(3,2,-2,-1) Matrix(13,18,18,25) -> Matrix(3,2,-2,-1) Matrix(47,60,18,23) -> Matrix(1,0,0,1) Matrix(73,90,30,37) -> Matrix(1,0,0,1) Matrix(11,12,-12,-13) -> Matrix(3,2,-8,-5) Matrix(85,72,72,61) -> Matrix(7,2,-4,-1) Matrix(59,48,102,83) -> Matrix(1,0,4,1) Matrix(47,36,30,23) -> Matrix(3,2,-2,-1) Matrix(25,18,18,13) -> Matrix(3,2,-2,-1) Matrix(47,30,36,23) -> Matrix(1,0,2,1) Matrix(11,6,-24,-13) -> Matrix(1,0,0,1) Matrix(85,36,144,61) -> Matrix(1,0,4,1) Matrix(73,30,90,37) -> Matrix(1,0,4,1) Matrix(47,18,60,23) -> Matrix(1,0,0,1) Matrix(1,0,6,1) -> Matrix(1,0,2,1) Matrix(49,-18,30,-11) -> Matrix(3,2,-2,-1) Matrix(13,-6,24,-11) -> Matrix(1,0,0,1) Matrix(49,-30,18,-11) -> Matrix(1,0,-2,1) Matrix(13,-12,12,-11) -> Matrix(1,-2,0,1) Matrix(13,-24,6,-11) -> Matrix(3,4,-4,-5) 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): 5 Degree of the the map X: 5 Degree of the the map Y: 24 Permutation triple for Y: ((1,7,2)(3,12,13)(4,15,5)(6,11,14)(8,18,20)(9,17,10)(16,19,21)(22,24,23); (1,5,17,24,18,6)(2,10,21,22,11,3)(4,13,9,20,19,14)(7,16,15,23,12,8); (1,3,4)(2,8,9)(5,16,10)(6,19,7)(11,18,12)(13,23,17)(14,22,15)(20,24,21)) ----------------------------------------------------------------------- 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): 72 Minimal number of generators: 13 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: 12 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/1 -1/1 -1/2 0/1 1/3 1/2 2/3 1/1 3/2 2/1 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -1/2 1/0 -5/2 -1/1 1/0 -2/1 -1/1 -5/3 -3/4 -3/2 -1/1 -2/3 -1/2 -1/1 -1/2 -3/4 -1/1 -1/2 0/1 -2/3 -1/2 -3/5 -1/2 -1/4 -1/2 -1/2 0/1 -2/5 0/1 -1/3 -1/2 0/1 0/1 1/3 1/0 2/5 0/1 1/2 0/1 1/0 3/5 1/2 1/0 2/3 1/0 1/1 1/0 4/3 1/0 3/2 -2/1 -1/1 1/0 5/3 -3/2 2/1 -1/1 5/2 -1/1 -1/2 3/1 -1/2 1/0 1/0 -1/1 0/1 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(7,18,12,31) (-3/1,-5/2) -> (1/2,3/5) Hyperbolic Matrix(5,12,12,29) (-5/2,-2/1) -> (2/5,1/2) Hyperbolic Matrix(7,12,18,31) (-2/1,-5/3) -> (1/3,2/5) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(5,6,-6,-7) (-3/2,-1/1) -> (-1/1,-3/4) Parabolic Matrix(25,18,18,13) (-3/4,-2/3) -> (4/3,3/2) Hyperbolic Matrix(19,12,30,19) (-2/3,-3/5) -> (3/5,2/3) Hyperbolic Matrix(31,18,12,7) (-3/5,-1/2) -> (5/2,3/1) Hyperbolic Matrix(29,12,12,5) (-1/2,-2/5) -> (2/1,5/2) Hyperbolic Matrix(31,12,18,7) (-2/5,-1/3) -> (5/3,2/1) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,0,0,1) Matrix(7,18,12,31) -> Matrix(1,1,0,1) Matrix(5,12,12,29) -> Matrix(1,1,0,1) Matrix(7,12,18,31) -> Matrix(1,1,-4,-3) Matrix(19,30,12,19) -> Matrix(5,3,-2,-1) Matrix(5,6,-6,-7) -> Matrix(1,1,-4,-3) Matrix(25,18,18,13) -> Matrix(3,2,-2,-1) Matrix(19,12,30,19) -> Matrix(3,1,2,1) Matrix(31,18,12,7) -> Matrix(3,1,-4,-1) Matrix(29,12,12,5) -> Matrix(3,1,-4,-1) Matrix(31,12,18,7) -> Matrix(1,-1,0,1) Matrix(1,0,6,1) -> Matrix(1,0,2,1) Matrix(7,-6,6,-5) -> Matrix(1,-1,0,1) 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): 5 ----------------------------------------------------------------------- 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 1 3 1/3 1/0 1 6 2/5 0/1 2 3 1/2 0 6 3/5 (0/1,1/1).(1/2,1/0) 0 6 2/3 1/0 1 3 1/1 1/0 1 6 4/3 1/0 1 3 3/2 (-3/2,1/0) 0 6 5/3 -3/2 1 6 2/1 -1/1 2 3 5/2 0 6 3/1 (-1/1,0/1).(-1/2,1/0) 0 6 1/0 (-1/1,0/1) 0 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(31,-12,18,-7) (1/3,2/5) -> (5/3,2/1) Glide Reflection Matrix(29,-12,12,-5) (2/5,1/2) -> (2/1,5/2) Glide Reflection Matrix(31,-18,12,-7) (1/2,3/5) -> (5/2,3/1) Glide Reflection Matrix(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(-1,6,0,1) (3/1,1/0) -> (3/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,6,-1) -> Matrix(1,0,0,-1) (0/1,1/3) -> (0/1,1/0) Matrix(31,-12,18,-7) -> Matrix(3,1,-2,-1) Matrix(29,-12,12,-5) -> Matrix(1,-1,-2,1) Matrix(31,-18,12,-7) -> Matrix(1,-1,-2,1) Matrix(19,-12,30,-19) -> Matrix(-1,1,0,1) (3/5,2/3) -> (1/2,1/0) Matrix(7,-6,6,-5) -> Matrix(1,-1,0,1) 1/0 Matrix(17,-24,12,-17) -> Matrix(1,3,0,-1) (4/3,3/2) -> (-3/2,1/0) Matrix(19,-30,12,-19) -> Matrix(1,3,0,-1) (3/2,5/3) -> (-3/2,1/0) Matrix(-1,6,0,1) -> Matrix(-1,0,2,1) (3/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.