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 -4/1 -3/1 -2/1 -3/2 0/1 1/1 6/5 3/2 2/1 12/5 5/2 3/1 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 -5/1 -1/1 -4/5 -4/1 -2/3 -3/1 -1/2 -8/3 -2/5 -5/2 -1/2 -2/1 -1/3 -7/4 -1/4 -12/7 0/1 -5/3 -1/3 0/1 -3/2 -1/4 -7/5 -2/9 -1/5 -4/3 0/1 -5/4 -1/4 -6/5 -1/5 -1/1 -1/5 0/1 0/1 0/1 1/1 0/1 1/5 6/5 1/5 5/4 1/4 4/3 0/1 3/2 1/4 8/5 2/7 5/3 0/1 1/3 2/1 1/3 7/3 1/3 2/5 12/5 2/5 5/2 1/2 3/1 1/2 7/2 1/2 4/1 2/3 5/1 4/5 1/1 6/1 1/1 1/0 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(11,60,2,11) (-6/1,-5/1) -> (5/1,6/1) Hyperbolic Matrix(13,60,8,37) (-5/1,-4/1) -> (8/5,5/3) Hyperbolic Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) Hyperbolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic Matrix(83,144,34,59) (-7/4,-12/7) -> (12/5,5/2) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(35,48,8,11) (-7/5,-4/3) -> (4/1,5/1) Hyperbolic Matrix(37,48,10,13) (-4/3,-5/4) -> (7/2,4/1) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(11,12,10,11) (-6/5,-1/1) -> (1/1,6/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,2,0,1) Matrix(11,60,2,11) -> Matrix(9,8,10,9) Matrix(13,60,8,37) -> Matrix(5,4,16,13) Matrix(11,36,-4,-13) -> Matrix(7,4,-16,-9) Matrix(23,60,18,47) -> Matrix(5,2,22,9) Matrix(11,24,-6,-13) -> Matrix(5,2,-18,-7) Matrix(83,144,34,59) -> Matrix(9,2,22,5) Matrix(85,144,36,61) -> Matrix(5,2,12,5) Matrix(23,36,-16,-25) -> Matrix(7,2,-32,-9) Matrix(35,48,8,11) -> Matrix(11,2,16,3) Matrix(37,48,10,13) -> Matrix(7,2,10,3) Matrix(49,60,40,49) -> Matrix(9,2,40,9) Matrix(11,12,10,11) -> Matrix(1,0,10,1) Matrix(1,0,2,1) -> Matrix(1,0,10,1) Matrix(25,-36,16,-23) -> Matrix(9,-2,32,-7) Matrix(13,-24,6,-11) -> Matrix(7,-2,18,-5) Matrix(13,-36,4,-11) -> Matrix(9,-4,16,-7) 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): 8 Degree of the the map X: 8 Degree of the the map Y: 16 Permutation triple for Y: ((2,6,7)(3,11,4)(5,10,9)(8,13,12); (1,4,13,14,5,2)(3,10)(6,9,16,12,11,15)(7,8); (1,2,8,16,9,3)(4,12)(5,6)(7,15,11,10,14,13)) ----------------------------------------------------------------------- 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 -3/2 -6/5 0/1 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 -1/2 -5/2 -1/2 -2/1 -1/3 -3/2 -1/4 -4/3 0/1 -5/4 -1/4 -6/5 -1/5 -1/1 -1/5 0/1 0/1 0/1 1/1 0/1 1/5 3/2 1/4 5/3 0/1 1/3 2/1 1/3 7/3 1/3 2/5 12/5 2/5 5/2 1/2 3/1 1/2 1/0 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(13,30,-10,-23) (-5/2,-2/1) -> (-4/3,-5/4) Hyperbolic Matrix(11,18,-8,-13) (-2/1,-3/2) -> (-3/2,-4/3) Parabolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) 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(1,1,0,1) Matrix(11,30,4,11) -> Matrix(7,3,16,7) Matrix(13,30,-10,-23) -> Matrix(3,1,-10,-3) Matrix(11,18,-8,-13) -> Matrix(3,1,-16,-5) Matrix(73,90,30,37) -> Matrix(13,3,30,7) Matrix(47,54,20,23) -> Matrix(7,1,20,3) Matrix(1,0,2,1) -> Matrix(1,0,10,1) Matrix(13,-18,8,-11) -> Matrix(5,-1,16,-3) Matrix(13,-24,6,-11) -> Matrix(7,-2,18,-5) 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): 8 ----------------------------------------------------------------------- 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 10 2 1/1 (0/1,1/5) 0 12 3/2 1/4 2 4 5/3 (0/1,1/3) 0 12 2/1 1/3 2 6 7/3 (1/3,2/5) 0 12 12/5 2/5 2 2 5/2 1/2 2 12 3/1 1/2 4 4 1/0 1/0 2 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(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) 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(1,0,2,-1) -> Matrix(1,0,10,-1) (0/1,1/1) -> (0/1,1/5) Matrix(13,-18,8,-11) -> Matrix(5,-1,16,-3) 1/4 Matrix(13,-24,6,-11) -> Matrix(7,-2,18,-5) 1/3 Matrix(71,-168,30,-71) -> Matrix(11,-4,30,-11) (7/3,12/5) -> (1/3,2/5) Matrix(49,-120,20,-49) -> Matrix(9,-4,20,-9) (12/5,5/2) -> (2/5,1/2) Matrix(11,-30,4,-11) -> Matrix(7,-3,16,-7) (5/2,3/1) -> (3/8,1/2) Matrix(-1,6,0,1) -> Matrix(-1,1,0,1) (3/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.