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 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 Since the preimage of every curve is trivial, the pure modular group virtual endomorphism is trivial. ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1 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 -1/2 1/0 1/1 -1/1 0/1 6/5 -1/2 1/0 5/4 -1/1 0/1 4/3 -1/2 1/0 3/2 -1/2 1/0 8/5 -1/2 1/0 5/3 -1/1 0/1 2/1 -1/2 1/0 7/3 -1/1 0/1 12/5 -1/2 1/0 5/2 -1/1 0/1 3/1 -1/2 1/0 7/2 -1/1 0/1 4/1 -1/2 1/0 5/1 -1/1 0/1 6/1 -1/2 1/0 1/0 -1/1 0/1 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 MULTI-ENDOMORPHISM This map is 2-valued. Matrix(1,0,1,1) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(11,-12,1,-1) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(49,-60,9,-11) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(47,-60,29,-37) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(13,-24,6,-11) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(61,-144,25,-59) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(13,-36,4,-11) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(13,-48,3,-11) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(1,0,0,1) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) ----------------------------------------------------------------------- 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 0/1 (-1/1,0/1).(-1/2,1/0) 2/1 (-1/1,0/1).(-1/2,1/0) 12/5 (-1/1,0/1).(-1/2,1/0) 5/2 (-1/1,0/1).(-1/2,1/0) 3/1 (-1/1,0/1).(-1/2,1/0) 7/2 (-1/1,0/1).(-1/2,1/0) 4/1 (-1/1,0/1).(-1/2,1/0) 5/1 (-1/1,0/1).(-1/2,1/0) 6/1 (-1/1,0/1).(-1/2,1/0) 1/0 (-1/1,0/1).(-1/2,1/0) 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 MULTI-ENDOMORPHISM FIXED POINT OF IMAGE This map is 2-valued. Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(1,0,1,-1) -> Matrix(1,1,0,-1) (0/1,2/1) -> (-1/2,1/0) -> Matrix(-1,0,2,1) -> (-1/1,0/1) Matrix(11,-24,5,-11) -> Matrix(-1,0,2,1) (2/1,12/5) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(49,-120,20,-49) -> Matrix(-1,0,2,1) (12/5,5/2) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(13,-36,4,-11) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) -> Matrix(1,0,0,1) Matrix(13,-48,3,-11) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(11,-60,2,-11) -> Matrix(1,1,0,-1) (5/1,6/1) -> (-1/2,1/0) -> Matrix(-1,0,2,1) -> (-1/1,0/1) Matrix(-1,12,0,1) -> Matrix(-1,0,2,1) (6/1,1/0) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(1,0,0,1) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0)