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 1/1 4/3 3/2 2/1 12/5 5/2 3/1 4/1 9/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 1/0 -5/1 1/0 -9/2 -1/1 0/1 -4/1 0/1 -3/1 1/0 -8/3 -1/1 -5/2 -1/1 1/0 -12/5 -1/1 -7/3 -1/2 -9/4 -1/1 0/1 -2/1 -1/2 1/0 -3/2 -1/1 0/1 -4/3 -1/1 -5/4 -1/1 -1/2 -6/5 -1/2 -1/1 -1/2 0/1 0/1 1/1 1/2 6/5 1/2 5/4 1/2 1/1 4/3 1/1 3/2 0/1 1/1 2/1 1/2 1/0 9/4 0/1 1/1 7/3 1/2 12/5 1/1 5/2 1/1 1/0 8/3 1/1 3/1 1/0 4/1 0/1 9/2 0/1 1/1 5/1 1/0 6/1 1/0 1/0 0/1 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(7,36,6,31) (-6/1,-5/1) -> (1/1,6/5) Hyperbolic Matrix(23,108,10,47) (-5/1,-9/2) -> (9/4,7/3) Hyperbolic Matrix(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(7,24,2,7) (-4/1,-3/1) -> (3/1,4/1) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(47,108,10,23) (-7/3,-9/4) -> (9/2,5/1) 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(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(31,36,6,7) (-6/5,-1/1) -> (5/1,6/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,0,0,1) Matrix(7,36,6,31) -> Matrix(1,0,2,1) Matrix(23,108,10,47) -> Matrix(1,0,2,1) Matrix(17,72,4,17) -> Matrix(1,0,2,1) Matrix(7,24,2,7) -> Matrix(1,0,0,1) Matrix(17,48,6,17) -> Matrix(1,2,0,1) Matrix(23,60,18,47) -> Matrix(1,0,2,1) Matrix(49,120,20,49) -> Matrix(1,2,0,1) Matrix(71,168,30,71) -> Matrix(3,2,4,3) Matrix(47,108,10,23) -> Matrix(1,0,2,1) Matrix(17,36,8,17) -> Matrix(1,0,2,1) Matrix(7,12,4,7) -> Matrix(1,0,2,1) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(47,60,18,23) -> Matrix(1,0,2,1) Matrix(49,60,40,49) -> Matrix(3,2,4,3) Matrix(31,36,6,7) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) 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: ((2,4,3,10,13,6)(5,9,11)(7,8,15)(12,16); (1,4,11,16,15,10,14,13,8,12,5,2)(3,7,6,9); (1,2,7,16,8,3)(4,5)(6,14,10,9,12,11)(13,15)) ----------------------------------------------------------------------- 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: 8 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 3/2 2/1 3/1 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 0/1 -3/1 1/0 -2/1 -1/2 1/0 -3/2 -1/1 0/1 -4/3 -1/1 -5/4 -1/1 -1/2 -6/5 -1/2 -1/1 -1/2 0/1 0/1 1/1 1/2 4/3 1/1 3/2 0/1 1/1 2/1 1/2 1/0 3/1 1/0 4/1 0/1 5/1 1/0 6/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,24,-4,-19) (-4/1,1/0) -> (-4/3,-5/4) Hyperbolic Matrix(7,24,2,7) (-4/1,-3/1) -> (3/1,4/1) Hyperbolic Matrix(5,12,2,5) (-3/1,-2/1) -> (2/1,3/1) 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(29,36,4,5) (-5/4,-6/5) -> (6/1,1/0) Hyperbolic Matrix(31,36,6,7) (-6/5,-1/1) -> (5/1,6/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,24,-4,-19) -> Matrix(1,1,-2,-1) Matrix(7,24,2,7) -> Matrix(1,0,0,1) Matrix(5,12,2,5) -> Matrix(1,1,0,1) Matrix(7,12,4,7) -> Matrix(1,0,2,1) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(29,36,4,5) -> Matrix(1,1,-2,-1) Matrix(31,36,6,7) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(19,-24,4,-5) -> Matrix(1,-1,2,-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): 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 2 1 1/1 1/2 1 12 4/3 1/1 1 3 3/2 (0/1,1/1) 0 4 2/1 (0/1,1/1).(1/2,1/0) 0 6 3/1 1/0 1 4 4/1 0/1 1 3 5/1 1/0 1 12 6/1 1/0 1 2 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(5,-12,2,-5) (2/1,3/1) -> (2/1,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) 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(1,0,2,-1) -> Matrix(1,0,4,-1) (0/1,1/1) -> (0/1,1/2) Matrix(19,-24,4,-5) -> Matrix(1,-1,2,-1) (0/1,1/1).(1/2,1/0) Matrix(17,-24,12,-17) -> Matrix(1,0,2,-1) (4/3,3/2) -> (0/1,1/1) Matrix(7,-12,4,-7) -> Matrix(1,0,2,-1) (3/2,2/1) -> (0/1,1/1) Matrix(5,-12,2,-5) -> Matrix(-1,1,0,1) (2/1,3/1) -> (1/2,1/0) Matrix(7,-24,2,-7) -> Matrix(1,0,0,-1) (3/1,4/1) -> (0/1,1/0) Matrix(11,-60,2,-11) -> Matrix(-1,1,0,1) (5/1,6/1) -> (1/2,1/0) Matrix(-1,12,0,1) -> Matrix(1,0,0,-1) (6/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.