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 -2/5 -1/3 0/1 1/5 1/3 1/2 3/5 1/1 7/5 3/2 5/3 19/11 2/1 25/11 7/3 13/5 3/1 7/2 11/3 19/5 4/1 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 1/3 -3/7 0/1 1/3 1/2 -2/5 1/3 -5/13 1/3 2/5 1/2 -3/8 2/5 1/2 -1/3 1/2 -3/10 0/1 1/2 -2/7 1/2 2/3 -3/11 1/2 2/3 1/1 -1/4 1/1 -1/5 0/1 1/2 1/1 0/1 0/1 1/1 1/5 1/2 1/0 1/4 0/1 1/0 3/11 1/2 2/7 1/1 1/3 0/1 1/1 1/0 3/8 0/1 1/1 5/13 1/2 1/0 2/5 0/1 1/0 3/7 1/2 1/2 1/1 5/9 1/0 4/7 1/1 2/1 7/12 2/1 1/0 3/5 1/1 2/1 1/0 8/13 1/1 13/21 1/0 5/8 1/1 2/1 2/3 2/1 1/0 1/1 1/0 4/3 0/1 1/0 7/5 1/0 10/7 -4/1 1/0 3/2 -2/1 1/0 11/7 -3/2 1/0 8/5 -2/1 -1/1 13/8 -1/1 5/3 -2/1 -1/1 1/0 12/7 -2/1 1/0 19/11 -3/2 1/0 7/4 -2/1 -1/1 2/1 -1/1 9/4 -1/1 0/1 25/11 -1/1 16/7 -1/1 -2/3 7/3 -1/2 12/5 -1/4 0/1 17/7 0/1 5/2 0/1 1/0 13/5 -1/2 1/0 21/8 0/1 1/0 8/3 -1/1 0/1 3/1 -1/1 0/1 1/0 7/2 -1/1 18/5 -1/1 0/1 47/13 -1/1 29/8 -1/1 -2/3 11/3 -1/2 15/4 -1/4 0/1 19/5 0/1 4/1 0/1 1/0 5/1 -1/2 1/0 6/1 0/1 1/0 1/0 -1/1 0/1 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(23,10,62,27) (-1/2,-3/7) -> (1/3,3/8) Hyperbolic Matrix(19,8,64,27) (-3/7,-2/5) -> (2/7,1/3) Hyperbolic Matrix(119,46,194,75) (-2/5,-5/13) -> (3/5,8/13) Hyperbolic Matrix(115,44,196,75) (-5/13,-3/8) -> (7/12,3/5) Hyperbolic Matrix(17,6,-54,-19) (-3/8,-1/3) -> (-1/3,-3/10) Parabolic Matrix(121,36,84,25) (-3/10,-2/7) -> (10/7,3/2) Hyperbolic Matrix(167,46,98,27) (-2/7,-3/11) -> (5/3,12/7) Hyperbolic Matrix(163,44,100,27) (-3/11,-1/4) -> (13/8,5/3) Hyperbolic Matrix(47,10,14,3) (-1/4,-1/5) -> (3/1,7/2) Hyperbolic Matrix(43,8,16,3) (-1/5,0/1) -> (8/3,3/1) Hyperbolic Matrix(51,-8,32,-5) (0/1,1/5) -> (11/7,8/5) Hyperbolic Matrix(59,-14,38,-9) (1/5,1/4) -> (3/2,11/7) Hyperbolic Matrix(209,-56,56,-15) (1/4,3/11) -> (11/3,15/4) Hyperbolic Matrix(179,-50,290,-81) (3/11,2/7) -> (8/13,13/21) Hyperbolic Matrix(243,-92,140,-53) (3/8,5/13) -> (19/11,7/4) Hyperbolic Matrix(251,-98,146,-57) (5/13,2/5) -> (12/7,19/11) Hyperbolic Matrix(119,-50,50,-21) (2/5,3/7) -> (7/3,12/5) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(193,-108,84,-47) (5/9,4/7) -> (16/7,7/3) Hyperbolic Matrix(243,-140,92,-53) (4/7,7/12) -> (21/8,8/3) Hyperbolic Matrix(697,-432,192,-119) (13/21,5/8) -> (29/8,11/3) Hyperbolic Matrix(51,-32,8,-5) (5/8,2/3) -> (6/1,1/0) Hyperbolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(71,-98,50,-69) (4/3,7/5) -> (7/5,10/7) Parabolic Matrix(179,-290,50,-81) (8/5,13/8) -> (7/2,18/5) 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(123,-296,32,-77) (12/5,17/7) -> (19/5,4/1) Hyperbolic Matrix(143,-350,38,-93) (17/7,5/2) -> (15/4,19/5) Hyperbolic Matrix(131,-338,50,-129) (5/2,13/5) -> (13/5,21/8) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(23,10,62,27) -> Matrix(1,0,-2,1) Matrix(19,8,64,27) -> Matrix(1,0,-2,1) Matrix(119,46,194,75) -> Matrix(1,0,-2,1) Matrix(115,44,196,75) -> Matrix(1,0,-2,1) Matrix(17,6,-54,-19) -> Matrix(5,-2,8,-3) Matrix(121,36,84,25) -> Matrix(5,-2,-2,1) Matrix(167,46,98,27) -> Matrix(1,0,-2,1) Matrix(163,44,100,27) -> Matrix(1,0,-2,1) Matrix(47,10,14,3) -> Matrix(1,0,-2,1) Matrix(43,8,16,3) -> Matrix(1,0,-2,1) Matrix(51,-8,32,-5) -> Matrix(1,-2,0,1) Matrix(59,-14,38,-9) -> Matrix(1,-2,0,1) Matrix(209,-56,56,-15) -> Matrix(1,0,-4,1) Matrix(179,-50,290,-81) -> Matrix(3,-2,2,-1) Matrix(243,-92,140,-53) -> Matrix(1,-2,0,1) Matrix(251,-98,146,-57) -> Matrix(1,-2,0,1) Matrix(119,-50,50,-21) -> Matrix(1,0,-4,1) Matrix(17,-8,32,-15) -> Matrix(3,-2,2,-1) Matrix(193,-108,84,-47) -> Matrix(1,0,-2,1) Matrix(243,-140,92,-53) -> Matrix(1,-2,0,1) Matrix(697,-432,192,-119) -> Matrix(1,0,-2,1) Matrix(51,-32,8,-5) -> Matrix(1,-2,0,1) Matrix(7,-6,6,-5) -> Matrix(1,-2,0,1) Matrix(71,-98,50,-69) -> Matrix(1,-4,0,1) Matrix(179,-290,50,-81) -> Matrix(1,2,-2,-3) Matrix(17,-32,8,-15) -> Matrix(1,2,-2,-3) Matrix(507,-1148,140,-317) -> Matrix(1,2,-2,-3) Matrix(527,-1202,146,-333) -> Matrix(3,2,-2,-1) Matrix(123,-296,32,-77) -> Matrix(1,0,4,1) Matrix(143,-350,38,-93) -> Matrix(1,0,-4,1) Matrix(131,-338,50,-129) -> Matrix(1,0,0,1) Matrix(11,-50,2,-9) -> 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: ((1,6,21,14,26,32,29,17,27,11,7,2)(3,12,13,4)(5,15,23,22,30,19,18,28,25,10,9,16)(8,24,31,20); (1,4,5)(3,10,11)(6,19,20)(7,23,8)(12,18,29)(13,22,14)(16,24,17)(25,31,26); (1,2,8,19,30,13,29,32,31,16,9,3)(4,14,21,20,25,28,12,11,27,24,23,15)(5,17,18,6)(7,10,26,22)) ----------------------------------------------------------------------- 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): 96 Minimal number of generators: 17 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: 16 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/5 1/3 1/2 3/5 1/1 7/5 5/3 2/1 25/11 7/3 17/7 3/1 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 0/1 1/1 1/5 1/2 1/0 1/4 0/1 1/0 1/3 0/1 1/1 1/0 2/5 0/1 1/0 3/7 1/2 1/2 1/1 5/9 1/0 4/7 1/1 2/1 3/5 1/1 2/1 1/0 5/8 1/1 2/1 2/3 2/1 1/0 1/1 1/0 4/3 0/1 1/0 7/5 1/0 3/2 -2/1 1/0 11/7 -3/2 1/0 8/5 -2/1 -1/1 5/3 -2/1 -1/1 1/0 7/4 -2/1 -1/1 2/1 -1/1 9/4 -1/1 0/1 25/11 -1/1 16/7 -1/1 -2/3 7/3 -1/2 12/5 -1/4 0/1 17/7 0/1 5/2 0/1 1/0 3/1 -1/1 0/1 1/0 4/1 0/1 1/0 5/1 -1/2 1/0 6/1 0/1 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(51,-8,32,-5) (0/1,1/5) -> (11/7,8/5) Hyperbolic Matrix(59,-14,38,-9) (1/5,1/4) -> (3/2,11/7) Hyperbolic Matrix(10,-3,27,-8) (1/4,1/3) -> (1/3,2/5) Parabolic Matrix(119,-50,50,-21) (2/5,3/7) -> (7/3,12/5) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(193,-108,84,-47) (5/9,4/7) -> (16/7,7/3) Hyperbolic Matrix(46,-27,75,-44) (4/7,3/5) -> (3/5,5/8) Parabolic Matrix(51,-32,8,-5) (5/8,2/3) -> (6/1,1/0) Hyperbolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(36,-49,25,-34) (4/3,7/5) -> (7/5,3/2) Parabolic Matrix(46,-75,27,-44) (8/5,5/3) -> (5/3,7/4) Parabolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(276,-625,121,-274) (9/4,25/11) -> (25/11,16/7) Parabolic Matrix(120,-289,49,-118) (12/5,17/7) -> (17/7,5/2) Parabolic Matrix(10,-27,3,-8) (5/2,3/1) -> (3/1,4/1) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,2,1) Matrix(51,-8,32,-5) -> Matrix(1,-2,0,1) Matrix(59,-14,38,-9) -> Matrix(1,-2,0,1) Matrix(10,-3,27,-8) -> Matrix(1,0,0,1) Matrix(119,-50,50,-21) -> Matrix(1,0,-4,1) Matrix(17,-8,32,-15) -> Matrix(3,-2,2,-1) Matrix(193,-108,84,-47) -> Matrix(1,0,-2,1) Matrix(46,-27,75,-44) -> Matrix(1,0,0,1) Matrix(51,-32,8,-5) -> Matrix(1,-2,0,1) Matrix(7,-6,6,-5) -> Matrix(1,-2,0,1) Matrix(36,-49,25,-34) -> Matrix(1,-2,0,1) Matrix(46,-75,27,-44) -> Matrix(1,0,0,1) Matrix(17,-32,8,-15) -> Matrix(1,2,-2,-3) Matrix(276,-625,121,-274) -> Matrix(1,2,-2,-3) Matrix(120,-289,49,-118) -> Matrix(1,0,4,1) Matrix(10,-27,3,-8) -> Matrix(1,0,0,1) Matrix(11,-50,2,-9) -> 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 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 -1/1 0/1 2 1 1/1 1/0 2 6 7/5 1/0 2 1 3/2 (-2/1,1/0) 0 12 8/5 (-2/1,-1/1) 0 12 5/3 0 3 7/4 (-2/1,-1/1) 0 12 2/1 -1/1 1 4 9/4 (-1/1,0/1) 0 12 25/11 -1/1 2 1 7/3 -1/2 2 6 17/7 0/1 4 1 5/2 (0/1,1/0) 0 12 3/1 0 3 4/1 (0/1,1/0) 0 12 5/1 0 2 6/1 (0/1,1/0) 0 12 1/0 (-1/1,0/1) 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(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(6,-7,5,-6) (1/1,7/5) -> (1/1,7/5) Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) Reflection Matrix(32,-51,5,-8) (3/2,8/5) -> (6/1,1/0) Glide Reflection Matrix(46,-75,27,-44) (8/5,5/3) -> (5/3,7/4) Parabolic 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(76,-175,33,-76) (25/11,7/3) -> (25/11,7/3) Reflection Matrix(50,-119,21,-50) (7/3,17/7) -> (7/3,17/7) Reflection Matrix(69,-170,28,-69) (17/7,5/2) -> (17/7,5/2) Reflection Matrix(10,-27,3,-8) (5/2,3/1) -> (3/1,4/1) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic 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) -> (-1/1,0/1) Matrix(0,1,1,0) -> Matrix(1,0,0,-1) (-1/1,1/1) -> (0/1,1/0) Matrix(6,-7,5,-6) -> Matrix(1,2,0,-1) (1/1,7/5) -> (-1/1,1/0) Matrix(29,-42,20,-29) -> Matrix(1,4,0,-1) (7/5,3/2) -> (-2/1,1/0) Matrix(32,-51,5,-8) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(46,-75,27,-44) -> Matrix(1,0,0,1) Matrix(17,-32,8,-15) -> Matrix(1,2,-2,-3) -1/1 Matrix(199,-450,88,-199) -> Matrix(-1,0,2,1) (9/4,25/11) -> (-1/1,0/1) Matrix(76,-175,33,-76) -> Matrix(3,2,-4,-3) (25/11,7/3) -> (-1/1,-1/2) Matrix(50,-119,21,-50) -> Matrix(-1,0,4,1) (7/3,17/7) -> (-1/2,0/1) Matrix(69,-170,28,-69) -> Matrix(1,0,0,-1) (17/7,5/2) -> (0/1,1/0) Matrix(10,-27,3,-8) -> Matrix(1,0,0,1) Matrix(11,-50,2,-9) -> Matrix(1,0,0,1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.