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 2/1 -4/1 2/1 -3/1 1/0 -8/3 -2/1 -5/2 -2/1 1/0 -2/1 -2/1 0/1 -7/4 -2/1 1/0 -12/7 -2/1 -5/3 -2/1 -1/1 -3/2 -1/1 -7/5 -1/1 -2/3 -4/3 -2/3 -5/4 -2/3 -1/2 -6/5 -1/2 -1/1 -1/2 0/1 0/1 0/1 1/1 0/1 1/4 6/5 1/4 5/4 1/4 2/7 4/3 2/7 3/2 1/3 8/5 2/5 5/3 1/3 2/5 2/1 0/1 2/5 7/3 1/3 2/5 12/5 2/5 5/2 2/5 1/2 3/1 1/2 7/2 1/2 2/3 4/1 2/3 5/1 2/3 1/1 6/1 1/1 1/0 0/1 1/1 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(1,0,0,1) Matrix(11,60,2,11) -> Matrix(3,-4,4,-5) Matrix(13,60,8,37) -> Matrix(1,0,2,1) Matrix(11,36,-4,-13) -> Matrix(1,-4,0,1) Matrix(23,60,18,47) -> Matrix(1,0,4,1) Matrix(11,24,-6,-13) -> Matrix(1,0,0,1) Matrix(83,144,34,59) -> Matrix(1,4,2,9) Matrix(85,144,36,61) -> Matrix(3,4,8,11) Matrix(23,36,-16,-25) -> Matrix(3,4,-4,-5) Matrix(35,48,8,11) -> Matrix(5,4,6,5) Matrix(37,48,10,13) -> Matrix(7,4,12,7) Matrix(49,60,40,49) -> Matrix(7,4,26,15) Matrix(11,12,10,11) -> Matrix(1,0,6,1) Matrix(1,0,2,1) -> Matrix(1,0,6,1) Matrix(25,-36,16,-23) -> Matrix(13,-4,36,-11) Matrix(13,-24,6,-11) -> Matrix(1,0,0,1) 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): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 3 Degree of the the map X: 6 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 0/1 4/3 3/2 2/1 12/5 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 0/1 0/1 1/1 0/1 1/4 6/5 1/4 5/4 1/4 2/7 4/3 2/7 3/2 1/3 8/5 2/5 5/3 1/3 2/5 2/1 0/1 2/5 7/3 1/3 2/5 12/5 2/5 5/2 2/5 1/2 3/1 1/2 7/2 1/2 2/3 4/1 2/3 5/1 2/3 1/1 6/1 1/1 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(11,-12,1,-1) (1/1,6/5) -> (6/1,1/0) Hyperbolic Matrix(49,-60,9,-11) (6/5,5/4) -> (5/1,6/1) Hyperbolic Matrix(47,-60,29,-37) (5/4,4/3) -> (8/5,5/3) Hyperbolic 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(61,-144,25,-59) (7/3,12/5) -> (12/5,5/2) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(13,-48,3,-11) (7/2,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,3,1) Matrix(11,-12,1,-1) -> Matrix(1,0,-3,1) Matrix(49,-60,9,-11) -> Matrix(15,-4,19,-5) Matrix(47,-60,29,-37) -> Matrix(1,0,-1,1) Matrix(25,-36,16,-23) -> Matrix(13,-4,36,-11) Matrix(13,-24,6,-11) -> Matrix(1,0,0,1) Matrix(61,-144,25,-59) -> Matrix(11,-4,25,-9) Matrix(13,-36,4,-11) -> Matrix(9,-4,16,-7) Matrix(13,-48,3,-11) -> Matrix(7,-4,9,-5) 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): 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 3 1 2/1 (0/1,2/5) 0 3 12/5 2/5 1 1 5/2 (2/5,1/2) 0 6 3/1 1/2 2 2 7/2 (1/2,2/3) 0 6 4/1 2/3 1 3 5/1 (2/3,1/1) 0 6 6/1 1/1 2 1 1/0 (0/1,1/1) 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(11,-24,5,-11) (2/1,12/5) -> (2/1,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(13,-48,3,-11) (7/2,4/1) -> (4/1,5/1) Parabolic 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(1,0,2,-1) (0/1,1/0) -> (0/1,1/1) Matrix(1,0,1,-1) -> Matrix(1,0,5,-1) (0/1,2/1) -> (0/1,2/5) Matrix(11,-24,5,-11) -> Matrix(1,0,5,-1) (2/1,12/5) -> (0/1,2/5) Matrix(49,-120,20,-49) -> Matrix(9,-4,20,-9) (12/5,5/2) -> (2/5,1/2) Matrix(13,-36,4,-11) -> Matrix(9,-4,16,-7) 1/2 Matrix(13,-48,3,-11) -> Matrix(7,-4,9,-5) 2/3 Matrix(11,-60,2,-11) -> Matrix(5,-4,6,-5) (5/1,6/1) -> (2/3,1/1) Matrix(-1,12,0,1) -> Matrix(1,0,2,-1) (6/1,1/0) -> (0/1,1/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.