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 -1/1 -3/7 -2/5 -1/3 -1/4 0/1 1/5 1/3 1/2 5/7 1/1 5/4 7/5 3/2 5/3 2/1 25/11 5/2 3/1 7/2 4/1 9/2 5/1 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 1/2 -4/9 0/1 1/1 -3/7 0/1 -8/19 1/3 2/5 -5/12 0/1 1/2 -2/5 1/2 -3/8 0/1 1/0 -4/11 0/1 1/5 -1/3 1/2 -4/13 1/0 -3/10 0/1 1/0 -2/7 0/1 1/3 -1/4 1/2 -2/9 2/3 1/1 -1/5 1/1 0/1 0/1 1/1 1/5 1/0 1/4 0/1 1/0 2/7 1/2 1/3 1/1 1/2 1/0 3/5 -1/1 8/13 1/0 13/21 -3/2 5/8 -1/1 -1/2 2/3 -1/1 0/1 5/7 0/1 8/11 0/1 1/5 3/4 0/1 1/2 1/1 1/0 6/5 -1/1 -4/5 5/4 -1/2 14/11 -1/3 -2/7 9/7 0/1 4/3 -1/3 0/1 7/5 0/1 10/7 0/1 1/7 3/2 0/1 1/2 11/7 1/2 8/5 0/1 1/1 5/3 1/1 12/7 0/1 1/1 7/4 1/2 1/1 2/1 1/0 9/4 -3/2 -1/1 25/11 -1/1 16/7 -1/1 -6/7 7/3 -1/2 5/2 0/1 1/0 18/7 1/1 2/1 13/5 1/0 8/3 -2/1 -1/1 3/1 0/1 10/3 2/3 1/1 7/2 1/0 18/5 -4/3 -1/1 47/13 -1/1 29/8 -1/1 -5/6 11/3 -1/2 4/1 0/1 1/1 9/2 1/1 1/0 5/1 1/0 11/2 -1/1 1/0 6/1 -1/1 0/1 1/0 0/1 1/0 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(21,10,2,1) (-1/2,-4/9) -> (6/1,1/0) Hyperbolic Matrix(109,48,84,37) (-4/9,-3/7) -> (9/7,4/3) Hyperbolic Matrix(269,114,210,89) (-3/7,-8/19) -> (14/11,9/7) Hyperbolic Matrix(311,130,122,51) (-8/19,-5/12) -> (5/2,18/7) Hyperbolic Matrix(63,26,-206,-85) (-5/12,-2/5) -> (-4/13,-3/10) Hyperbolic Matrix(21,8,76,29) (-2/5,-3/8) -> (1/4,2/7) Hyperbolic Matrix(113,42,78,29) (-3/8,-4/11) -> (10/7,3/2) Hyperbolic Matrix(133,48,36,13) (-4/11,-1/3) -> (11/3,4/1) Hyperbolic Matrix(193,60,312,97) (-1/3,-4/13) -> (8/13,13/21) Hyperbolic Matrix(101,30,138,41) (-3/10,-2/7) -> (8/11,3/4) Hyperbolic Matrix(15,4,-64,-17) (-2/7,-1/4) -> (-1/4,-2/9) Parabolic Matrix(105,22,62,13) (-2/9,-1/5) -> (5/3,12/7) Hyperbolic Matrix(45,8,28,5) (-1/5,0/1) -> (8/5,5/3) Hyperbolic Matrix(53,-8,20,-3) (0/1,1/5) -> (13/5,8/3) Hyperbolic Matrix(59,-14,38,-9) (1/5,1/4) -> (3/2,11/7) Hyperbolic Matrix(45,-14,74,-23) (2/7,1/3) -> (3/5,8/13) Hyperbolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(697,-432,192,-119) (13/21,5/8) -> (29/8,11/3) Hyperbolic Matrix(59,-38,14,-9) (5/8,2/3) -> (4/1,9/2) Hyperbolic Matrix(71,-50,98,-69) (2/3,5/7) -> (5/7,8/11) Parabolic Matrix(33,-26,14,-11) (3/4,1/1) -> (7/3,5/2) Hyperbolic Matrix(51,-58,22,-25) (1/1,6/5) -> (16/7,7/3) Hyperbolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(71,-98,50,-69) (4/3,7/5) -> (7/5,10/7) Parabolic Matrix(191,-302,74,-117) (11/7,8/5) -> (18/7,13/5) Hyperbolic Matrix(101,-174,18,-31) (12/7,7/4) -> (11/2,6/1) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(507,-1148,140,-317) (9/4,25/11) -> (47/13,29/8) Hyperbolic Matrix(527,-1202,146,-333) (25/11,16/7) -> (18/5,47/13) Hyperbolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(57,-196,16,-55) (10/3,7/2) -> (7/2,18/5) Parabolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(21,10,2,1) -> Matrix(1,0,-2,1) Matrix(109,48,84,37) -> Matrix(1,0,-4,1) Matrix(269,114,210,89) -> Matrix(1,0,-6,1) Matrix(311,130,122,51) -> Matrix(1,0,-2,1) Matrix(63,26,-206,-85) -> Matrix(1,0,-2,1) Matrix(21,8,76,29) -> Matrix(1,0,0,1) Matrix(113,42,78,29) -> Matrix(1,0,2,1) Matrix(133,48,36,13) -> Matrix(1,0,-4,1) Matrix(193,60,312,97) -> Matrix(1,-2,0,1) Matrix(101,30,138,41) -> Matrix(1,0,2,1) Matrix(15,4,-64,-17) -> Matrix(5,-2,8,-3) Matrix(105,22,62,13) -> Matrix(3,-2,2,-1) Matrix(45,8,28,5) -> Matrix(1,0,0,1) Matrix(53,-8,20,-3) -> Matrix(1,-2,0,1) Matrix(59,-14,38,-9) -> Matrix(1,0,2,1) Matrix(45,-14,74,-23) -> Matrix(1,0,-2,1) Matrix(9,-4,16,-7) -> Matrix(1,-2,0,1) Matrix(697,-432,192,-119) -> Matrix(3,4,-4,-5) Matrix(59,-38,14,-9) -> Matrix(1,0,2,1) Matrix(71,-50,98,-69) -> Matrix(1,0,6,1) Matrix(33,-26,14,-11) -> Matrix(1,0,-2,1) Matrix(51,-58,22,-25) -> Matrix(1,2,-2,-3) Matrix(81,-100,64,-79) -> Matrix(3,2,-8,-5) Matrix(71,-98,50,-69) -> Matrix(1,0,10,1) Matrix(191,-302,74,-117) -> Matrix(3,-2,2,-1) Matrix(101,-174,18,-31) -> Matrix(1,0,-2,1) Matrix(17,-32,8,-15) -> Matrix(1,-2,0,1) Matrix(507,-1148,140,-317) -> Matrix(7,8,-8,-9) Matrix(527,-1202,146,-333) -> Matrix(11,10,-10,-9) Matrix(19,-54,6,-17) -> Matrix(1,0,2,1) Matrix(57,-196,16,-55) -> Matrix(1,-2,0,1) Matrix(21,-100,4,-19) -> Matrix(1,-2,0,1) 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): 11 Degree of the the map X: 11 Degree of the the map Y: 32 Permutation triple for Y: ((1,6,20,11,27,32,29,30,28,14,7,2)(3,12,13,4)(5,17,22,21,26,25,16,15,23,10,9,18)(8,19,31,24); (1,4,16,29,12,5)(3,10,11)(6,18,19)(7,22,31,27,23,8)(9,26)(13,21,14)(20,28)(24,30,25); (1,2,8,25,9,3)(4,14,20,19,23,15)(5,6)(7,21)(10,27)(11,28,24,22,17,12)(13,29,32,31,18,26)(16,30)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- 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 1 1 0/1 (0/1,1/1) 0 12 1/5 1/0 2 2 1/4 (0/1,1/0) 0 12 1/3 1/1 1 3 1/2 1/0 2 4 3/5 -1/1 1 3 5/8 (-1/1,-1/2) 0 12 2/3 (-1/1,0/1) 0 12 5/7 0/1 3 1 3/4 (0/1,1/2) 0 12 1/1 1/0 1 6 6/5 (-1/1,-4/5) 0 12 5/4 -1/2 2 4 4/3 (-1/3,0/1) 0 12 7/5 0/1 5 1 3/2 (0/1,1/2) 0 12 5/3 1/1 1 3 7/4 (1/2,1/1) 0 12 2/1 1/0 1 4 9/4 (-3/2,-1/1) 0 12 25/11 -1/1 9 1 16/7 (-1/1,-6/7) 0 12 7/3 -1/2 1 6 5/2 (0/1,1/0) 0 12 13/5 1/0 2 2 8/3 (-2/1,-1/1) 0 12 3/1 0/1 1 3 10/3 (2/3,1/1) 0 12 7/2 1/0 2 4 4/1 (0/1,1/1) 0 12 9/2 (1/1,1/0) 0 12 5/1 1/0 1 2 1/0 (0/1,1/0) 0 12 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(53,-8,20,-3) (0/1,1/5) -> (13/5,8/3) Hyperbolic Matrix(9,-2,40,-9) (1/5,1/4) -> (1/5,1/4) Reflection Matrix(29,-8,18,-5) (1/4,1/3) -> (3/2,5/3) Glide Reflection Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(75,-46,44,-27) (3/5,5/8) -> (5/3,7/4) Glide Reflection Matrix(59,-38,14,-9) (5/8,2/3) -> (4/1,9/2) Hyperbolic Matrix(29,-20,42,-29) (2/3,5/7) -> (2/3,5/7) Reflection Matrix(41,-30,56,-41) (5/7,3/4) -> (5/7,3/4) Reflection Matrix(33,-26,14,-11) (3/4,1/1) -> (7/3,5/2) Hyperbolic Matrix(51,-58,22,-25) (1/1,6/5) -> (16/7,7/3) Hyperbolic Matrix(75,-92,22,-27) (6/5,5/4) -> (10/3,7/2) Glide Reflection Matrix(37,-48,10,-13) (5/4,4/3) -> (7/2,4/1) Glide Reflection Matrix(41,-56,30,-41) (4/3,7/5) -> (4/3,7/5) Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) Reflection Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(199,-450,88,-199) (9/4,25/11) -> (9/4,25/11) Reflection Matrix(351,-800,154,-351) (25/11,16/7) -> (25/11,16/7) Reflection Matrix(51,-130,20,-51) (5/2,13/5) -> (5/2,13/5) Reflection Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(19,-90,4,-19) (9/2,5/1) -> (9/2,5/1) Reflection Matrix(-1,10,0,1) (5/1,1/0) -> (5/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(-1,0,2,1) -> Matrix(1,0,2,-1) (-1/1,0/1) -> (0/1,1/1) Matrix(53,-8,20,-3) -> Matrix(1,-2,0,1) 1/0 Matrix(9,-2,40,-9) -> Matrix(1,0,0,-1) (1/5,1/4) -> (0/1,1/0) Matrix(29,-8,18,-5) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(9,-4,16,-7) -> Matrix(1,-2,0,1) 1/0 Matrix(75,-46,44,-27) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(59,-38,14,-9) -> Matrix(1,0,2,1) 0/1 Matrix(29,-20,42,-29) -> Matrix(-1,0,2,1) (2/3,5/7) -> (-1/1,0/1) Matrix(41,-30,56,-41) -> Matrix(1,0,4,-1) (5/7,3/4) -> (0/1,1/2) Matrix(33,-26,14,-11) -> Matrix(1,0,-2,1) 0/1 Matrix(51,-58,22,-25) -> Matrix(1,2,-2,-3) -1/1 Matrix(75,-92,22,-27) -> Matrix(3,2,2,1) Matrix(37,-48,10,-13) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(41,-56,30,-41) -> Matrix(-1,0,6,1) (4/3,7/5) -> (-1/3,0/1) Matrix(29,-42,20,-29) -> Matrix(1,0,4,-1) (7/5,3/2) -> (0/1,1/2) Matrix(17,-32,8,-15) -> Matrix(1,-2,0,1) 1/0 Matrix(199,-450,88,-199) -> Matrix(5,6,-4,-5) (9/4,25/11) -> (-3/2,-1/1) Matrix(351,-800,154,-351) -> Matrix(13,12,-14,-13) (25/11,16/7) -> (-1/1,-6/7) Matrix(51,-130,20,-51) -> Matrix(1,0,0,-1) (5/2,13/5) -> (0/1,1/0) Matrix(19,-54,6,-17) -> Matrix(1,0,2,1) 0/1 Matrix(19,-90,4,-19) -> Matrix(-1,2,0,1) (9/2,5/1) -> (1/1,1/0) Matrix(-1,10,0,1) -> Matrix(1,0,0,-1) (5/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.