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 0/1 1/2 2/3 3/4 1/1 6/5 5/4 3/2 2/1 9/4 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -1/1 1/1 -5/2 1/1 -12/5 1/0 -7/3 1/1 -9/4 1/1 1/0 -2/1 1/0 -3/2 -1/1 -4/3 0/1 -5/4 0/1 1/0 -6/5 1/0 -1/1 -1/1 -3/4 -1/1 0/1 -2/3 0/1 -5/8 0/1 1/0 -3/5 -1/1 1/1 -1/2 -1/1 0/1 0/1 1/2 1/1 3/5 -1/1 1/1 5/8 0/1 1/0 2/3 0/1 3/4 0/1 1/1 1/1 1/1 6/5 1/0 5/4 0/1 1/0 4/3 0/1 3/2 1/1 2/1 1/0 9/4 -1/1 1/0 7/3 -1/1 12/5 1/0 5/2 -1/1 3/1 -1/1 1/1 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(7,18,12,31) (-3/1,-5/2) -> (1/2,3/5) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(23,54,20,47) (-12/5,-7/3) -> (1/1,6/5) Hyperbolic Matrix(55,126,24,55) (-7/3,-9/4) -> (9/4,7/3) Hyperbolic Matrix(17,36,8,17) (-9/4,-2/1) -> (2/1,9/4) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(23,30,36,47) (-4/3,-5/4) -> (5/8,2/3) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(7,6,8,7) (-1/1,-3/4) -> (3/4,1/1) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(47,30,36,23) (-2/3,-5/8) -> (5/4,4/3) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(31,18,12,7) (-3/5,-1/2) -> (5/2,3/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) 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(1,0,0,1) Matrix(49,120,20,49) -> Matrix(1,-2,0,1) Matrix(23,54,20,47) -> Matrix(1,0,0,1) Matrix(55,126,24,55) -> Matrix(1,-2,0,1) Matrix(17,36,8,17) -> Matrix(1,-2,0,1) Matrix(7,12,4,7) -> Matrix(1,2,0,1) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(23,30,36,47) -> Matrix(1,0,0,1) Matrix(49,60,40,49) -> Matrix(1,0,0,1) Matrix(47,54,20,23) -> Matrix(1,0,0,1) Matrix(7,6,8,7) -> Matrix(1,0,2,1) Matrix(17,12,24,17) -> Matrix(1,0,2,1) Matrix(47,30,36,23) -> Matrix(1,0,0,1) Matrix(49,30,80,49) -> Matrix(1,0,0,1) Matrix(31,18,12,7) -> Matrix(1,0,0,1) Matrix(1,0,4,1) -> 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): 3 Degree of the the map X: 3 Degree of the the map Y: 16 Permutation triple for Y: ((1,2)(3,5,4,11,15,10)(6,8,7,12,9,13)(14,16); (1,5,12,16,15,11,14,9,3,2,8,6)(4,10,13,7); (1,3,4)(2,6,10,16,11,7)(5,9)(12,13,14)) ----------------------------------------------------------------------- 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 1/2 3/4 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 -1/1 1/1 -2/1 1/0 -3/2 -1/1 -1/1 -1/1 -3/4 -1/1 0/1 -2/3 0/1 -3/5 -1/1 1/1 -1/2 -1/1 0/1 0/1 1/2 1/1 2/3 0/1 3/4 0/1 1/1 1/1 1/1 3/2 1/1 2/1 1/0 5/2 -1/1 3/1 -1/1 1/1 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(5,12,-8,-19) (-3/1,-2/1) -> (-2/3,-3/5) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(5,6,4,5) (-3/2,-1/1) -> (1/1,3/2) Hyperbolic Matrix(7,6,8,7) (-1/1,-3/4) -> (3/4,1/1) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(31,18,12,7) (-3/5,-1/2) -> (5/2,3/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(19,-12,8,-5) (1/2,2/3) -> (2/1,5/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,0,0,1) Matrix(5,12,-8,-19) -> Matrix(0,-1,1,0) Matrix(7,12,4,7) -> Matrix(1,2,0,1) Matrix(5,6,4,5) -> Matrix(0,-1,1,0) Matrix(7,6,8,7) -> Matrix(1,0,2,1) Matrix(17,12,24,17) -> Matrix(1,0,2,1) Matrix(31,18,12,7) -> Matrix(1,0,0,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(19,-12,8,-5) -> Matrix(0,-1,1,0) 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 ----------------------------------------------------------------------- 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 12 2/3 0/1 2 6 3/4 (0/1,1/1) 0 4 1/1 1/1 1 12 3/2 1/1 2 4 2/1 1/0 2 6 5/2 -1/1 2 12 3/1 (-1/1,1/1).(0/1,1/0) 0 4 1/0 (0/1,1/0) 0 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(19,-12,8,-5) (1/2,2/3) -> (2/1,5/2) Hyperbolic Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(7,-6,8,-7) (3/4,1/1) -> (3/4,1/1) Reflection Matrix(5,-6,4,-5) (1/1,3/2) -> (1/1,3/2) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) 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(1,0,0,-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(19,-12,8,-5) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) Matrix(17,-12,24,-17) -> Matrix(1,0,2,-1) (2/3,3/4) -> (0/1,1/1) Matrix(7,-6,8,-7) -> Matrix(1,0,2,-1) (3/4,1/1) -> (0/1,1/1) Matrix(5,-6,4,-5) -> Matrix(0,1,1,0) (1/1,3/2) -> (-1/1,1/1) Matrix(7,-12,4,-7) -> Matrix(-1,2,0,1) (3/2,2/1) -> (1/1,1/0) Matrix(11,-30,4,-11) -> Matrix(0,1,1,0) (5/2,3/1) -> (-1/1,1/1) Matrix(-1,6,0,1) -> Matrix(1,0,0,-1) (3/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.