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 -2/1 -3/2 -5/4 -6/5 -1/1 -3/4 0/1 1/2 3/5 3/4 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 1/1 -12/5 1/1 -7/3 1/1 3/1 -2/1 -1/1 1/1 -3/2 0/1 -4/3 1/3 1/1 -5/4 0/1 1/1 -6/5 1/1 -1/1 -1/1 1/1 -3/4 1/0 -5/7 -5/1 -3/1 -2/3 -3/1 -1/1 -5/8 -2/1 -1/1 -3/5 -1/1 -1/2 -1/1 0/1 0/1 1/2 1/1 3/5 1/1 5/8 1/1 2/1 2/3 1/1 3/1 3/4 1/0 4/5 -3/1 -1/1 1/1 -1/1 1/1 3/2 0/1 5/3 1/3 1/1 12/7 1/1 7/4 0/1 1/1 2/1 -1/1 1/1 7/3 -3/1 -1/1 12/5 -1/1 5/2 -1/1 3/1 0/1 1/0 0/1 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(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(61,144,36,85) (-12/5,-7/3) -> (5/3,12/7) Hyperbolic Matrix(13,30,16,37) (-7/3,-2/1) -> (4/5,1/1) Hyperbolic Matrix(11,18,-8,-13) (-2/1,-3/2) -> (-3/2,-4/3) Parabolic Matrix(23,30,36,47) (-4/3,-5/4) -> (5/8,2/3) Hyperbolic Matrix(83,102,48,59) (-5/4,-6/5) -> (12/7,7/4) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(23,18,-32,-25) (-1/1,-3/4) -> (-3/4,-5/7) Parabolic Matrix(35,24,16,11) (-5/7,-2/3) -> (2/1,7/3) Hyperbolic Matrix(37,24,20,13) (-2/3,-5/8) -> (7/4,2/1) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(11,6,20,11) (-3/5,-1/2) -> (1/2,3/5) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,0,0,1) Matrix(11,30,4,11) -> Matrix(1,0,-2,1) Matrix(49,120,20,49) -> Matrix(1,-2,0,1) Matrix(61,144,36,85) -> Matrix(1,-2,2,-3) Matrix(13,30,16,37) -> Matrix(1,-2,0,1) Matrix(11,18,-8,-13) -> Matrix(1,0,2,1) Matrix(23,30,36,47) -> Matrix(3,-2,2,-1) Matrix(83,102,48,59) -> Matrix(1,0,0,1) Matrix(47,54,20,23) -> Matrix(1,-2,0,1) Matrix(23,18,-32,-25) -> Matrix(1,-4,0,1) Matrix(35,24,16,11) -> Matrix(1,2,0,1) Matrix(37,24,20,13) -> Matrix(1,2,0,1) Matrix(49,30,80,49) -> Matrix(3,4,2,3) Matrix(11,6,20,11) -> Matrix(1,2,0,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(25,-18,32,-23) -> Matrix(1,-4,0,1) Matrix(13,-18,8,-11) -> 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: 16 Permutation triple for Y: ((1,2)(3,9,14,6,13,10)(4,11,8,7,12,5)(15,16); (1,5,12,16,13,6)(2,8,11,15,9,3)(4,10)(7,14); (1,3,4)(2,6,7)(10,16,11)(12,14,15)) ----------------------------------------------------------------------- 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 -1/1 -3/4 0/1 1/2 3/5 3/4 1/1 3/2 7/4 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 1/1 -3/4 1/0 -2/3 -3/1 -1/1 -5/8 -2/1 -1/1 -3/5 -1/1 -1/2 -1/1 0/1 0/1 1/2 1/1 3/5 1/1 5/8 1/1 2/1 2/3 1/1 3/1 3/4 1/0 1/1 -1/1 1/1 3/2 0/1 5/3 1/3 1/1 7/4 0/1 1/1 2/1 -1/1 1/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,3,0,1) (-1/1,1/0) -> (2/1,1/0) Parabolic Matrix(11,9,-16,-13) (-1/1,-3/4) -> (-3/4,-2/3) Parabolic Matrix(37,24,20,13) (-2/3,-5/8) -> (7/4,2/1) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(11,6,20,11) (-3/5,-1/2) -> (1/2,3/5) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(61,-39,36,-23) (5/8,2/3) -> (5/3,7/4) Hyperbolic Matrix(13,-9,16,-11) (2/3,3/4) -> (3/4,1/1) Parabolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,3,0,1) -> Matrix(1,0,0,1) Matrix(11,9,-16,-13) -> Matrix(1,-2,0,1) Matrix(37,24,20,13) -> Matrix(1,2,0,1) Matrix(49,30,80,49) -> Matrix(3,4,2,3) Matrix(11,6,20,11) -> Matrix(1,2,0,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(61,-39,36,-23) -> Matrix(1,-2,2,-3) Matrix(13,-9,16,-11) -> Matrix(1,-2,0,1) Matrix(13,-18,8,-11) -> 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 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): 2 ----------------------------------------------------------------------- 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 2 1/2 1/1 2 6 3/5 1/1 1 2 5/8 (1/1,2/1) 0 6 2/3 0 6 3/4 1/0 4 2 1/1 0 6 5/4 (-1/1,0/1) 0 6 3/2 0/1 2 2 1/0 (0/1,1/0) 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(11,-6,20,-11) (1/2,3/5) -> (1/2,3/5) Reflection Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(23,-15,20,-13) (5/8,2/3) -> (1/1,5/4) Hyperbolic Matrix(13,-9,16,-11) (2/3,3/4) -> (3/4,1/1) Parabolic Matrix(11,-15,8,-11) (5/4,3/2) -> (5/4,3/2) Reflection Matrix(-1,3,0,1) (3/2,1/0) -> (3/2,1/0) 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(1,0,4,-1) -> Matrix(1,0,2,-1) (0/1,1/2) -> (0/1,1/1) Matrix(11,-6,20,-11) -> Matrix(-1,2,0,1) (1/2,3/5) -> (1/1,1/0) Matrix(49,-30,80,-49) -> Matrix(3,-4,2,-3) (3/5,5/8) -> (1/1,2/1) Matrix(23,-15,20,-13) -> Matrix(1,-2,0,1) 1/0 Matrix(13,-9,16,-11) -> Matrix(1,-2,0,1) 1/0 Matrix(11,-15,8,-11) -> Matrix(-1,0,2,1) (5/4,3/2) -> (-1/1,0/1) Matrix(-1,3,0,1) -> Matrix(1,0,0,-1) (3/2,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.