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 -4/1 -24/7 -10/3 -8/3 -12/5 -2/1 -12/7 -4/3 0/1 1/1 4/3 3/2 12/7 2/1 16/7 12/5 8/3 3/1 4/1 16/3 6/1 32/5 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/1 -1/2 0/1 -7/2 -1/1 0/1 -24/7 -1/2 1/0 -17/5 -1/1 0/1 -10/3 -1/1 -3/1 -1/2 0/1 -8/3 -1/2 -5/2 -1/2 -2/5 -12/5 -1/2 -2/5 -1/3 -19/8 -1/2 -2/5 -26/11 -1/3 -7/3 -2/5 -1/3 -2/1 -1/3 -9/5 -1/3 0/1 -16/9 -1/3 -7/4 -1/3 -2/7 -12/7 -1/3 -2/7 -1/4 -17/10 -1/3 -2/7 -5/3 -2/7 -1/4 -8/5 -1/4 -3/2 -1/4 0/1 -4/3 -1/4 -1/5 0/1 -5/4 -1/4 0/1 -16/13 -1/4 -11/9 -1/4 -2/9 -6/5 -1/5 -19/16 -1/6 0/1 -32/27 0/1 -13/11 -1/4 0/1 -7/6 -1/5 0/1 -8/7 -1/4 -1/6 -1/1 -1/5 0/1 0/1 0/1 1/1 0/1 1/3 4/3 0/1 1/3 1/2 7/5 0/1 1/3 24/17 1/4 1/2 17/12 0/1 1/3 10/7 1/3 3/2 0/1 1/2 8/5 1/2 5/3 1/2 2/3 12/7 1/2 2/3 1/1 19/11 1/2 2/3 26/15 1/1 7/4 2/3 1/1 2/1 1/1 9/4 0/1 1/1 16/7 1/1 7/3 1/1 2/1 12/5 1/1 2/1 1/0 17/7 1/1 2/1 5/2 2/1 1/0 8/3 1/0 3/1 0/1 1/0 4/1 -1/1 0/1 1/0 5/1 0/1 1/0 16/3 1/0 11/2 -2/1 1/0 6/1 -1/1 19/3 -1/2 0/1 32/5 0/1 13/2 0/1 1/0 7/1 -1/1 0/1 8/1 -1/2 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,32,-2,-9) (-4/1,1/0) -> (-4/1,-7/2) Parabolic Matrix(167,576,118,407) (-7/2,-24/7) -> (24/17,17/12) Hyperbolic Matrix(169,576,120,409) (-24/7,-17/5) -> (7/5,24/17) Hyperbolic Matrix(151,512,-64,-217) (-17/5,-10/3) -> (-26/11,-7/3) Hyperbolic Matrix(39,128,-32,-105) (-10/3,-3/1) -> (-11/9,-6/5) Hyperbolic Matrix(23,64,14,39) (-3/1,-8/3) -> (8/5,5/3) Hyperbolic Matrix(25,64,16,41) (-8/3,-5/2) -> (3/2,8/5) Hyperbolic Matrix(119,288,-50,-121) (-5/2,-12/5) -> (-12/5,-19/8) Parabolic Matrix(257,608,-216,-511) (-19/8,-26/11) -> (-6/5,-19/16) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,62,111) (-9/5,-16/9) -> (16/7,7/3) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(167,288,-98,-169) (-7/4,-12/7) -> (-12/7,-17/10) Parabolic Matrix(151,256,-128,-217) (-17/10,-5/3) -> (-13/11,-7/6) Hyperbolic Matrix(39,64,14,23) (-5/3,-8/5) -> (8/3,3/1) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(23,32,-18,-25) (-3/2,-4/3) -> (-4/3,-5/4) Parabolic Matrix(207,256,38,47) (-5/4,-16/13) -> (16/3,11/2) Hyperbolic Matrix(209,256,40,49) (-16/13,-11/9) -> (5/1,16/3) Hyperbolic Matrix(863,1024,134,159) (-19/16,-32/27) -> (32/5,13/2) Hyperbolic Matrix(865,1024,136,161) (-32/27,-13/11) -> (19/3,32/5) Hyperbolic Matrix(55,64,6,7) (-7/6,-8/7) -> (8/1,1/0) Hyperbolic Matrix(57,64,8,9) (-8/7,-1/1) -> (7/1,8/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic Matrix(361,-512,208,-295) (17/12,10/7) -> (26/15,7/4) Hyperbolic Matrix(89,-128,16,-23) (10/7,3/2) -> (11/2,6/1) Hyperbolic Matrix(169,-288,98,-167) (5/3,12/7) -> (12/7,19/11) Parabolic Matrix(351,-608,56,-97) (19/11,26/15) -> (6/1,19/3) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(121,-288,50,-119) (7/3,12/5) -> (12/5,17/7) Parabolic Matrix(105,-256,16,-39) (17/7,5/2) -> (13/2,7/1) Hyperbolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,32,-2,-9) -> Matrix(1,0,0,1) Matrix(167,576,118,407) -> Matrix(1,0,4,1) Matrix(169,576,120,409) -> Matrix(1,0,4,1) Matrix(151,512,-64,-217) -> Matrix(3,2,-8,-5) Matrix(39,128,-32,-105) -> Matrix(3,2,-14,-9) Matrix(23,64,14,39) -> Matrix(3,2,4,3) Matrix(25,64,16,41) -> Matrix(5,2,12,5) Matrix(119,288,-50,-121) -> Matrix(1,0,0,1) Matrix(257,608,-216,-511) -> Matrix(5,2,-28,-11) Matrix(15,32,-8,-17) -> Matrix(5,2,-18,-7) Matrix(143,256,62,111) -> Matrix(5,2,2,1) Matrix(145,256,64,113) -> Matrix(7,2,10,3) Matrix(167,288,-98,-169) -> Matrix(1,0,0,1) Matrix(151,256,-128,-217) -> Matrix(7,2,-32,-9) Matrix(39,64,14,23) -> Matrix(7,2,-4,-1) Matrix(41,64,16,25) -> Matrix(9,2,4,1) Matrix(23,32,-18,-25) -> Matrix(1,0,0,1) Matrix(207,256,38,47) -> Matrix(7,2,-4,-1) Matrix(209,256,40,49) -> Matrix(9,2,4,1) Matrix(863,1024,134,159) -> Matrix(1,0,6,1) Matrix(865,1024,136,161) -> Matrix(1,0,2,1) Matrix(55,64,6,7) -> Matrix(1,0,4,1) Matrix(57,64,8,9) -> Matrix(1,0,4,1) Matrix(1,0,2,1) -> Matrix(1,0,8,1) Matrix(25,-32,18,-23) -> Matrix(1,0,0,1) Matrix(361,-512,208,-295) -> Matrix(5,-2,8,-3) Matrix(89,-128,16,-23) -> Matrix(5,-2,-2,1) Matrix(169,-288,98,-167) -> Matrix(1,0,0,1) Matrix(351,-608,56,-97) -> Matrix(3,-2,-4,3) Matrix(17,-32,8,-15) -> Matrix(3,-2,2,-1) Matrix(121,-288,50,-119) -> Matrix(1,0,0,1) Matrix(105,-256,16,-39) -> Matrix(1,-2,0,1) Matrix(9,-32,2,-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: ((2,6,17,7)(3,11,12,4)(15,27,29,19)(20,31,25,21); (1,4,14,25,30,29,18,17,23,11,22,20,26,15,5,2)(3,9,21,28,19,8,7,13,12,24,31,32,27,16,6,10); (1,2,8,19,30,25,24,12,23,17,16,27,26,20,9,3)(4,13,7,18,29,32,31,22,11,10,6,5,15,28,21,14)) ----------------------------------------------------------------------- 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 0/1 4/3 24/17 8/5 12/7 2/1 16/7 12/5 8/3 3/1 4/1 16/3 6/1 32/5 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 0/1 1/3 4/3 0/1 1/3 1/2 7/5 0/1 1/3 24/17 1/4 1/2 17/12 0/1 1/3 10/7 1/3 3/2 0/1 1/2 8/5 1/2 5/3 1/2 2/3 12/7 1/2 2/3 1/1 19/11 1/2 2/3 26/15 1/1 7/4 2/3 1/1 2/1 1/1 9/4 0/1 1/1 16/7 1/1 7/3 1/1 2/1 12/5 1/1 2/1 1/0 17/7 1/1 2/1 5/2 2/1 1/0 8/3 1/0 3/1 0/1 1/0 4/1 -1/1 0/1 1/0 5/1 0/1 1/0 16/3 1/0 11/2 -2/1 1/0 6/1 -1/1 19/3 -1/2 0/1 32/5 0/1 13/2 0/1 1/0 7/1 -1/1 0/1 8/1 -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(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic Matrix(409,-576,289,-407) (7/5,24/17) -> (24/17,17/12) Parabolic Matrix(361,-512,208,-295) (17/12,10/7) -> (26/15,7/4) Hyperbolic Matrix(89,-128,16,-23) (10/7,3/2) -> (11/2,6/1) Hyperbolic Matrix(41,-64,25,-39) (3/2,8/5) -> (8/5,5/3) Parabolic Matrix(169,-288,98,-167) (5/3,12/7) -> (12/7,19/11) Parabolic Matrix(351,-608,56,-97) (19/11,26/15) -> (6/1,19/3) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(113,-256,49,-111) (9/4,16/7) -> (16/7,7/3) Parabolic Matrix(121,-288,50,-119) (7/3,12/5) -> (12/5,17/7) Parabolic Matrix(105,-256,16,-39) (17/7,5/2) -> (13/2,7/1) Hyperbolic Matrix(25,-64,9,-23) (5/2,8/3) -> (8/3,3/1) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic Matrix(49,-256,9,-47) (5/1,16/3) -> (16/3,11/2) Parabolic Matrix(161,-1024,25,-159) (19/3,32/5) -> (32/5,13/2) Parabolic Matrix(9,-64,1,-7) (7/1,8/1) -> (8/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,4,1) Matrix(25,-32,18,-23) -> Matrix(1,0,0,1) Matrix(409,-576,289,-407) -> Matrix(1,0,0,1) Matrix(361,-512,208,-295) -> Matrix(5,-2,8,-3) Matrix(89,-128,16,-23) -> Matrix(5,-2,-2,1) Matrix(41,-64,25,-39) -> Matrix(5,-2,8,-3) Matrix(169,-288,98,-167) -> Matrix(1,0,0,1) Matrix(351,-608,56,-97) -> Matrix(3,-2,-4,3) Matrix(17,-32,8,-15) -> Matrix(3,-2,2,-1) Matrix(113,-256,49,-111) -> Matrix(3,-2,2,-1) Matrix(121,-288,50,-119) -> Matrix(1,0,0,1) Matrix(105,-256,16,-39) -> Matrix(1,-2,0,1) Matrix(25,-64,9,-23) -> Matrix(1,-2,0,1) Matrix(9,-32,2,-7) -> Matrix(1,0,0,1) Matrix(49,-256,9,-47) -> Matrix(1,-2,0,1) Matrix(161,-1024,25,-159) -> Matrix(1,0,2,1) Matrix(9,-64,1,-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 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 0/1 0/1 4 1 2/1 1/1 2 8 16/7 1/1 2 1 7/3 (1/1,2/1) 0 16 12/5 0 4 17/7 (1/1,2/1) 0 16 5/2 (2/1,1/0) 0 16 8/3 1/0 4 2 3/1 (0/1,1/0) 0 16 4/1 0 4 5/1 (0/1,1/0) 0 16 16/3 1/0 2 1 6/1 -1/1 2 8 32/5 0/1 2 1 13/2 (0/1,1/0) 0 16 7/1 (-1/1,0/1) 0 16 8/1 0 2 1/0 (-1/1,0/1) 0 16 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(15,-32,7,-15) (2/1,16/7) -> (2/1,16/7) Reflection Matrix(97,-224,42,-97) (16/7,7/3) -> (16/7,7/3) Reflection Matrix(121,-288,50,-119) (7/3,12/5) -> (12/5,17/7) Parabolic Matrix(105,-256,16,-39) (17/7,5/2) -> (13/2,7/1) Hyperbolic Matrix(25,-64,9,-23) (5/2,8/3) -> (8/3,3/1) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic Matrix(31,-160,6,-31) (5/1,16/3) -> (5/1,16/3) Reflection Matrix(17,-96,3,-17) (16/3,6/1) -> (16/3,6/1) Reflection Matrix(31,-192,5,-31) (6/1,32/5) -> (6/1,32/5) Reflection Matrix(129,-832,20,-129) (32/5,13/2) -> (32/5,13/2) Reflection Matrix(9,-64,1,-7) (7/1,8/1) -> (8/1,1/0) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,1,-1) -> Matrix(1,0,2,-1) (0/1,2/1) -> (0/1,1/1) Matrix(15,-32,7,-15) -> Matrix(-1,2,0,1) (2/1,16/7) -> (1/1,1/0) Matrix(97,-224,42,-97) -> Matrix(3,-4,2,-3) (16/7,7/3) -> (1/1,2/1) Matrix(121,-288,50,-119) -> Matrix(1,0,0,1) Matrix(105,-256,16,-39) -> Matrix(1,-2,0,1) 1/0 Matrix(25,-64,9,-23) -> Matrix(1,-2,0,1) 1/0 Matrix(9,-32,2,-7) -> Matrix(1,0,0,1) Matrix(31,-160,6,-31) -> Matrix(1,0,0,-1) (5/1,16/3) -> (0/1,1/0) Matrix(17,-96,3,-17) -> Matrix(1,2,0,-1) (16/3,6/1) -> (-1/1,1/0) Matrix(31,-192,5,-31) -> Matrix(-1,0,2,1) (6/1,32/5) -> (-1/1,0/1) Matrix(129,-832,20,-129) -> Matrix(1,0,0,-1) (32/5,13/2) -> (0/1,1/0) Matrix(9,-64,1,-7) -> Matrix(1,0,0,1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.