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): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -4/1 -10/3 -3/1 -2/1 -4/3 -1/1 -2/3 0/1 1/2 2/3 3/4 1/1 4/3 3/2 2/1 8/3 3/1 10/3 4/1 16/3 6/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/2 1/0 -5/1 -1/1 -1/3 -4/1 0/1 -7/2 1/1 -10/3 1/0 -3/1 -1/1 -2/1 0/1 -5/3 1/1 -8/5 0/1 -3/2 1/1 -16/11 1/0 -13/9 -1/1 1/1 -10/7 1/2 1/0 -17/12 0/1 1/1 -24/17 1/1 -7/5 1/1 -4/3 1/0 -5/4 0/1 1/0 -16/13 1/0 -11/9 -1/1 -6/5 1/0 -7/6 -1/1 -8/7 -1/1 -1/1 -1/1 1/1 -4/5 0/1 -3/4 0/1 1/0 -2/3 1/0 -5/8 -2/1 1/0 -8/13 -2/1 -3/5 -1/1 -4/7 -1/1 -1/2 -1/1 0/1 -1/2 1/0 1/2 -1/1 4/7 0/1 3/5 -1/1 1/1 2/3 -2/1 0/1 5/7 -1/1 1/1 3/4 0/1 1/0 4/5 1/0 1/1 -1/1 6/5 -2/3 0/1 5/4 -1/2 0/1 4/3 0/1 7/5 -1/1 1/1 10/7 0/1 3/2 -1/1 2/1 -1/2 1/0 5/2 -1/1 8/3 0/1 11/4 0/1 1/0 3/1 -1/1 1/1 16/5 1/0 13/4 -2/1 1/0 10/3 -2/1 0/1 17/5 -1/1 24/7 -1/1 7/2 -1/1 4/1 -1/1 5/1 -1/1 16/3 -1/2 11/2 -1/3 6/1 0/1 7/1 -1/1 1/1 8/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(17,112,-12,-79) (-6/1,1/0) -> (-10/7,-17/12) Hyperbolic Matrix(23,128,-16,-89) (-6/1,-5/1) -> (-13/9,-10/7) Hyperbolic Matrix(7,32,12,55) (-5/1,-4/1) -> (4/7,3/5) Hyperbolic Matrix(9,32,16,57) (-4/1,-7/2) -> (1/2,4/7) Hyperbolic Matrix(33,112,-28,-95) (-7/2,-10/3) -> (-6/5,-7/6) Hyperbolic Matrix(39,128,-32,-105) (-10/3,-3/1) -> (-11/9,-6/5) Hyperbolic Matrix(7,16,-4,-9) (-3/1,-2/1) -> (-2/1,-5/3) Parabolic Matrix(39,64,-64,-105) (-5/3,-8/5) -> (-8/13,-3/5) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(153,224,28,41) (-3/2,-16/11) -> (16/3,11/2) Hyperbolic Matrix(177,256,56,81) (-16/11,-13/9) -> (3/1,16/5) Hyperbolic Matrix(113,160,12,17) (-17/12,-24/17) -> (8/1,1/0) Hyperbolic Matrix(409,576,120,169) (-24/17,-7/5) -> (17/5,24/7) Hyperbolic Matrix(23,32,28,39) (-7/5,-4/3) -> (4/5,1/1) Hyperbolic Matrix(25,32,32,41) (-4/3,-5/4) -> (3/4,4/5) Hyperbolic Matrix(233,288,72,89) (-5/4,-16/13) -> (16/5,13/4) Hyperbolic Matrix(209,256,40,49) (-16/13,-11/9) -> (5/1,16/3) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(57,64,8,9) (-8/7,-1/1) -> (7/1,8/1) Hyperbolic Matrix(39,32,28,23) (-1/1,-4/5) -> (4/3,7/5) Hyperbolic Matrix(41,32,32,25) (-4/5,-3/4) -> (5/4,4/3) Hyperbolic Matrix(23,16,-36,-25) (-3/4,-2/3) -> (-2/3,-5/8) Parabolic Matrix(207,128,76,47) (-5/8,-8/13) -> (8/3,11/4) Hyperbolic Matrix(55,32,12,7) (-3/5,-4/7) -> (4/1,5/1) Hyperbolic Matrix(57,32,16,9) (-4/7,-1/2) -> (7/2,4/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(89,-64,32,-23) (5/7,3/4) -> (11/4,3/1) Hyperbolic Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(105,-128,32,-39) (6/5,5/4) -> (13/4,10/3) Hyperbolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(89,-128,16,-23) (10/7,3/2) -> (11/2,6/1) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(17,112,-12,-79) -> Matrix(1,0,2,1) Matrix(23,128,-16,-89) -> Matrix(1,0,2,1) Matrix(7,32,12,55) -> Matrix(1,0,2,1) Matrix(9,32,16,57) -> Matrix(1,0,-2,1) Matrix(33,112,-28,-95) -> Matrix(1,-2,0,1) Matrix(39,128,-32,-105) -> Matrix(1,0,0,1) Matrix(7,16,-4,-9) -> Matrix(1,0,2,1) Matrix(39,64,-64,-105) -> Matrix(1,-2,0,1) Matrix(41,64,16,25) -> Matrix(1,0,-2,1) Matrix(153,224,28,41) -> Matrix(1,-2,-2,5) Matrix(177,256,56,81) -> Matrix(1,0,0,1) Matrix(113,160,12,17) -> Matrix(1,0,-2,1) Matrix(409,576,120,169) -> Matrix(1,-2,0,1) Matrix(23,32,28,39) -> Matrix(1,-2,0,1) Matrix(25,32,32,41) -> Matrix(1,0,0,1) Matrix(233,288,72,89) -> Matrix(1,-2,0,1) Matrix(209,256,40,49) -> Matrix(1,2,-2,-3) Matrix(193,224,56,65) -> Matrix(1,0,0,1) Matrix(57,64,8,9) -> Matrix(1,0,0,1) Matrix(39,32,28,23) -> Matrix(1,0,0,1) Matrix(41,32,32,25) -> Matrix(1,0,-2,1) Matrix(23,16,-36,-25) -> Matrix(1,-2,0,1) Matrix(207,128,76,47) -> Matrix(1,2,0,1) Matrix(55,32,12,7) -> Matrix(1,2,-2,-3) Matrix(57,32,16,9) -> Matrix(1,0,0,1) Matrix(1,0,4,1) -> Matrix(1,0,0,1) Matrix(25,-16,36,-23) -> Matrix(1,0,0,1) Matrix(89,-64,32,-23) -> Matrix(1,0,0,1) Matrix(95,-112,28,-33) -> Matrix(3,2,-2,-1) Matrix(105,-128,32,-39) -> Matrix(3,2,-2,-1) Matrix(79,-112,12,-17) -> Matrix(1,0,0,1) Matrix(89,-128,16,-23) -> Matrix(1,0,-2,1) Matrix(9,-16,4,-7) -> 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): 6 Degree of the the map X: 6 Degree of the the map Y: 32 Permutation triple for Y: ((1,2)(3,10)(4,5)(6,18)(7,8)(9,17)(11,19)(12,13)(14,20)(15,16)(21,23)(22,24)(25,26)(27,28)(29,31)(30,32); (1,5,16,27,31,22,17,6)(2,8,15,25,29,21,9,3)(4,13,28,32,24,19,18,14)(7,12,26,30,23,11,10,20); (1,3,11,24,31,25,12,4)(2,6,19,23,29,27,13,7)(5,14,10,9,22,32,26,15)(8,20,18,17,21,30,28,16)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- The image of the extended modular group liftables in PGL(2,Z) equals the image of the modular liftables. ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.