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: 14 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/1 0/1 1/1 4/3 3/2 8/5 2/1 16/7 5/2 8/3 3/1 7/2 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 1/0 -7/2 -1/1 1/0 -3/1 1/0 -8/3 -1/1 -5/2 -1/1 0/1 -7/3 1/0 -2/1 -1/2 1/0 -9/5 1/0 -16/9 -1/1 -7/4 -1/1 -1/2 -5/3 -1/2 -8/5 0/1 -3/2 -1/1 0/1 -4/3 -1/2 1/0 -5/4 -1/1 0/1 -1/1 -1/2 0/1 0/1 1/1 1/2 5/4 0/1 1/1 4/3 1/2 1/0 3/2 0/1 1/1 8/5 0/1 5/3 1/2 7/4 1/2 1/1 2/1 1/2 1/0 9/4 1/2 1/1 16/7 1/1 7/3 1/0 5/2 0/1 1/1 8/3 1/1 3/1 1/0 7/2 1/1 1/0 4/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,0,1) (-4/1,1/0) -> (4/1,1/0) Parabolic Matrix(15,56,4,15) (-4/1,-7/2) -> (7/2,4/1) Hyperbolic Matrix(17,56,10,33) (-7/2,-3/1) -> (5/3,7/4) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(17,40,14,33) (-5/2,-7/3) -> (1/1,5/4) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,62,111) (-9/5,-16/9) -> (16/7,7/3) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(33,56,10,17) (-7/4,-5/3) -> (3/1,7/2) Hyperbolic Matrix(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(33,40,14,17) (-5/4,-1/1) -> (7/3,5/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,0,0,1) Matrix(15,56,4,15) -> Matrix(1,2,0,1) Matrix(17,56,10,33) -> Matrix(1,0,2,1) Matrix(17,48,6,17) -> Matrix(1,2,0,1) Matrix(31,80,12,31) -> Matrix(1,0,2,1) Matrix(17,40,14,33) -> Matrix(1,0,2,1) Matrix(15,32,-8,-17) -> Matrix(1,0,0,1) Matrix(143,256,62,111) -> Matrix(1,2,0,1) Matrix(145,256,64,113) -> Matrix(3,2,4,3) Matrix(33,56,10,17) -> Matrix(1,0,2,1) Matrix(49,80,30,49) -> Matrix(1,0,4,1) Matrix(31,48,20,31) -> Matrix(1,0,2,1) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(31,40,24,31) -> Matrix(1,0,2,1) Matrix(33,40,14,17) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(17,-32,8,-15) -> Matrix(1,0,0,1) 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: ((2,6,15,7)(3,11,12,4)(5,14)(8,9); (1,4,14,15,16,11,5,2)(3,9,6,13,12,8,7,10); (1,2,8,3)(4,13,6,5)(7,14,11,10)(9,12,16,15)) ----------------------------------------------------------------------- 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 -2/1 0/1 1/1 4/3 3/2 2/1 8/3 3/1 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 1/0 -3/1 1/0 -2/1 -1/2 1/0 -5/3 -1/2 -8/5 0/1 -3/2 -1/1 0/1 -4/3 -1/2 1/0 -1/1 -1/2 0/1 0/1 1/1 1/2 4/3 1/2 1/0 3/2 0/1 1/1 2/1 1/2 1/0 5/2 0/1 1/1 8/3 1/1 3/1 1/0 4/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,0,1) (-4/1,1/0) -> (4/1,1/0) Parabolic Matrix(7,24,2,7) (-4/1,-3/1) -> (3/1,4/1) Hyperbolic Matrix(7,16,-4,-9) (-3/1,-2/1) -> (-2/1,-5/3) Parabolic Matrix(39,64,14,23) (-5/3,-8/5) -> (8/3,3/1) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(7,8,6,7) (-4/3,-1/1) -> (1/1,4/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,0,0,1) Matrix(7,24,2,7) -> Matrix(1,1,0,1) Matrix(7,16,-4,-9) -> Matrix(1,1,-2,-1) Matrix(39,64,14,23) -> Matrix(3,1,2,1) Matrix(41,64,16,25) -> Matrix(1,1,0,1) Matrix(17,24,12,17) -> Matrix(1,0,2,1) Matrix(7,8,6,7) -> Matrix(1,1,0,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(9,-16,4,-7) -> 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 8 4/3 (0/1,1/1).(1/2,1/0) 0 2 3/2 (0/1,1/1) 0 8 2/1 (0/1,1/1).(1/2,1/0) 0 4 5/2 (0/1,1/1) 0 8 8/3 1/1 1 1 3/1 1/0 1 8 4/1 1/0 1 2 1/0 (0/1,1/0) 0 8 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(7,-8,6,-7) (1/1,4/3) -> (1/1,4/3) Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection Matrix(-1,8,0,1) (4/1,1/0) -> (4/1,1/0) 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(7,-8,6,-7) -> Matrix(-1,1,0,1) (1/1,4/3) -> (1/2,1/0) Matrix(17,-24,12,-17) -> Matrix(1,0,2,-1) (4/3,3/2) -> (0/1,1/1) Matrix(9,-16,4,-7) -> Matrix(1,-1,2,-1) (0/1,1/1).(1/2,1/0) Matrix(31,-80,12,-31) -> Matrix(1,0,2,-1) (5/2,8/3) -> (0/1,1/1) Matrix(17,-48,6,-17) -> Matrix(-1,2,0,1) (8/3,3/1) -> (1/1,1/0) Matrix(7,-24,2,-7) -> Matrix(-1,1,0,1) (3/1,4/1) -> (1/2,1/0) Matrix(-1,8,0,1) -> Matrix(1,0,0,-1) (4/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.