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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 18 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 2/5 1/2 3/4 1/1 4/3 3/2 13/8 9/5 2/1 5/2 3/1 11/3 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/5 -4/9 1/5 1/4 -3/7 2/9 1/4 -2/5 0/1 1/4 -5/13 0/1 1/3 -3/8 1/5 1/4 -1/3 1/4 1/3 -2/7 1/3 1/2 -5/18 0/1 1/3 -3/11 1/3 2/5 -4/15 2/5 1/2 -1/4 0/1 1/2 -1/5 0/1 1/4 -2/11 1/4 1/3 -1/6 1/3 1/2 -1/7 0/1 1/1 0/1 0/1 1/3 1/4 1/4 1/3 1/3 0/1 1/2 2/5 0/1 3/7 0/1 1/6 1/2 0/1 1/4 3/5 2/7 1/3 2/3 1/4 1/3 3/4 1/3 4/5 1/3 3/8 1/1 1/3 1/2 4/3 1/2 7/5 1/2 3/5 3/2 1/2 1/1 8/5 0/1 1/1 13/8 0/1 18/11 0/1 1/5 5/3 0/1 1/3 7/4 2/5 1/2 9/5 1/2 11/6 1/2 2/3 2/1 0/1 1/2 5/2 1/2 8/3 1/2 4/7 3/1 1/2 2/3 7/2 3/4 1/1 11/3 1/1 15/4 1/1 1/0 4/1 1/2 1/1 5/1 2/3 1/1 6/1 1/1 7/1 1/1 2/1 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(49,22,-176,-79) (-1/2,-4/9) -> (-2/7,-5/18) Hyperbolic Matrix(23,10,-122,-53) (-4/9,-3/7) -> (-1/5,-2/11) Hyperbolic Matrix(71,30,26,11) (-3/7,-2/5) -> (8/3,3/1) Hyperbolic Matrix(67,26,-250,-97) (-2/5,-5/13) -> (-3/11,-4/15) Hyperbolic Matrix(21,8,-134,-51) (-5/13,-3/8) -> (-1/6,-1/7) Hyperbolic Matrix(65,24,46,17) (-3/8,-1/3) -> (7/5,3/2) Hyperbolic Matrix(19,6,22,7) (-1/3,-2/7) -> (4/5,1/1) Hyperbolic Matrix(137,38,18,5) (-5/18,-3/11) -> (7/1,1/0) Hyperbolic Matrix(173,46,94,25) (-4/15,-1/4) -> (11/6,2/1) Hyperbolic Matrix(17,4,38,9) (-1/4,-1/5) -> (3/7,1/2) Hyperbolic Matrix(189,34,50,9) (-2/11,-1/6) -> (15/4,4/1) Hyperbolic Matrix(131,18,80,11) (-1/7,0/1) -> (18/11,5/3) Hyperbolic Matrix(35,-8,22,-5) (0/1,1/4) -> (3/2,8/5) Hyperbolic Matrix(33,-10,10,-3) (1/4,1/3) -> (3/1,7/2) Hyperbolic Matrix(21,-8,50,-19) (1/3,2/5) -> (2/5,3/7) Parabolic Matrix(45,-26,26,-15) (1/2,3/5) -> (5/3,7/4) Hyperbolic Matrix(35,-22,8,-5) (3/5,2/3) -> (4/1,5/1) Hyperbolic Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic Matrix(209,-338,128,-207) (8/5,13/8) -> (13/8,18/11) Parabolic Matrix(91,-162,50,-89) (7/4,9/5) -> (9/5,11/6) Parabolic Matrix(21,-50,8,-19) (2/1,5/2) -> (5/2,8/3) Parabolic Matrix(67,-242,18,-65) (7/2,11/3) -> (11/3,15/4) Parabolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(49,22,-176,-79) -> Matrix(1,0,-2,1) Matrix(23,10,-122,-53) -> Matrix(9,-2,32,-7) Matrix(71,30,26,11) -> Matrix(17,-4,30,-7) Matrix(67,26,-250,-97) -> Matrix(7,-2,18,-5) Matrix(21,8,-134,-51) -> Matrix(1,0,-2,1) Matrix(65,24,46,17) -> Matrix(9,-2,14,-3) Matrix(19,6,22,7) -> Matrix(7,-2,18,-5) Matrix(137,38,18,5) -> Matrix(1,0,-2,1) Matrix(173,46,94,25) -> Matrix(5,-2,8,-3) Matrix(17,4,38,9) -> Matrix(1,0,2,1) Matrix(189,34,50,9) -> Matrix(1,0,-2,1) Matrix(131,18,80,11) -> Matrix(1,0,2,1) Matrix(35,-8,22,-5) -> Matrix(1,0,-2,1) Matrix(33,-10,10,-3) -> Matrix(5,-2,8,-3) Matrix(21,-8,50,-19) -> Matrix(1,0,4,1) Matrix(45,-26,26,-15) -> Matrix(7,-2,18,-5) Matrix(35,-22,8,-5) -> Matrix(1,0,-2,1) Matrix(25,-18,32,-23) -> Matrix(13,-4,36,-11) Matrix(25,-32,18,-23) -> Matrix(9,-4,16,-7) Matrix(209,-338,128,-207) -> Matrix(1,0,4,1) Matrix(91,-162,50,-89) -> Matrix(9,-4,16,-7) Matrix(21,-50,8,-19) -> Matrix(9,-4,16,-7) Matrix(67,-242,18,-65) -> Matrix(5,-4,4,-3) Matrix(13,-72,2,-11) -> Matrix(5,-4,4,-3) 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: 24 Permutation triple for Y: ((1,6,13,24,17,7,2)(3,11,8,19,23,12,4)(5,15,18,20,22,10,9); (1,4,14,8,7,15,5)(3,10,6,16,17,20,11)(9,21,18,19,24,13,12); (1,2,8,20,21,9,3)(4,13,10,22,17,19,14)(5,12,23,18,7,16,6)) ----------------------------------------------------------------------- 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): 24 Minimal number of generators: 5 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: 4 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 4/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/5 0/1 0/1 1/3 1/2 0/1 1/4 1/1 1/3 1/2 4/3 1/2 3/2 1/2 1/1 2/1 0/1 1/2 3/1 1/2 2/3 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(7,2,3,1) (-1/2,0/1) -> (2/1,3/1) Hyperbolic Matrix(7,-2,4,-1) (0/1,1/2) -> (3/2,2/1) Hyperbolic Matrix(7,-4,2,-1) (1/2,1/1) -> (3/1,1/0) Hyperbolic Matrix(13,-16,9,-11) (1/1,4/3) -> (4/3,3/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(7,2,3,1) -> Matrix(3,-1,7,-2) Matrix(7,-2,4,-1) -> Matrix(3,-1,7,-2) Matrix(7,-4,2,-1) -> Matrix(4,-1,5,-1) Matrix(13,-16,9,-11) -> Matrix(5,-2,8,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 2 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: 2 Number of equivalence classes of cusps: 1 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 -1/1 0/1 2 1 0/1 (0/1,1/3) 0 7 1/1 (1/3,1/2) 0 7 4/3 1/2 2 1 2/1 (0/1,1/2) 0 7 1/0 (0/1,1/1) 0 7 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(3,-2,1,-1) (0/1,1/1) -> (2/1,1/0) Glide Reflection Matrix(7,-8,6,-7) (1/1,4/3) -> (1/1,4/3) Reflection Matrix(5,-8,3,-5) (4/3,2/1) -> (4/3,2/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,2,-1) (-1/1,1/0) -> (0/1,1/1) Matrix(-1,0,2,1) -> Matrix(1,0,6,-1) (-1/1,0/1) -> (0/1,1/3) Matrix(3,-2,1,-1) -> Matrix(3,-1,5,-2) Matrix(7,-8,6,-7) -> Matrix(5,-2,12,-5) (1/1,4/3) -> (1/3,1/2) Matrix(5,-8,3,-5) -> Matrix(1,0,4,-1) (4/3,2/1) -> (0/1,1/2) ----------------------------------------------------------------------- The pullback map has no extra symmetries.