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 -8/3 -2/1 -3/2 -6/5 0/1 1/1 6/5 3/2 2/1 12/5 5/2 8/3 3/1 36/11 10/3 24/7 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/2 -5/1 -1/2 -4/1 -1/3 -7/2 -1/4 -10/3 -1/3 0/1 -3/1 -1/4 -14/5 -2/9 -1/5 -11/4 -3/16 -8/3 0/1 -13/5 -3/10 -5/2 -1/4 -2/1 -1/5 0/1 -7/4 -1/4 -12/7 -1/5 -5/3 -1/6 -18/11 -1/6 -13/8 -3/20 -8/5 0/1 -11/7 -3/14 -3/2 -1/6 -13/9 -5/34 -36/25 -1/7 -23/16 -5/36 -10/7 -1/7 0/1 -17/12 -1/8 -24/17 0/1 -7/5 -1/6 -4/3 -1/7 -5/4 -1/8 -6/5 -1/8 -7/6 -1/8 -1/1 -1/10 0/1 0/1 1/1 1/10 6/5 1/8 5/4 1/8 4/3 1/7 7/5 1/6 10/7 0/1 1/7 3/2 1/6 14/9 2/11 1/5 11/7 3/14 8/5 0/1 13/8 3/20 5/3 1/6 2/1 0/1 1/5 7/3 1/6 12/5 1/5 5/2 1/4 18/7 1/4 13/5 3/10 8/3 0/1 11/4 3/16 3/1 1/4 13/4 5/16 36/11 1/3 23/7 5/14 10/3 0/1 1/3 17/5 1/2 24/7 0/1 7/2 1/4 4/1 1/3 5/1 1/2 6/1 1/2 7/1 1/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(13,96,-8,-59) (-6/1,1/0) -> (-18/11,-13/8) Hyperbolic Matrix(23,120,-14,-73) (-6/1,-5/1) -> (-5/3,-18/11) Hyperbolic Matrix(11,48,8,35) (-5/1,-4/1) -> (4/3,7/5) Hyperbolic Matrix(13,48,10,37) (-4/1,-7/2) -> (5/4,4/3) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(23,72,-8,-25) (-10/3,-3/1) -> (-3/1,-14/5) Parabolic Matrix(155,432,-108,-301) (-14/5,-11/4) -> (-23/16,-10/7) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(73,192,46,121) (-8/3,-13/5) -> (11/7,8/5) Hyperbolic Matrix(37,96,-32,-83) (-13/5,-5/2) -> (-7/6,-1/1) Hyperbolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic Matrix(83,144,34,59) (-7/4,-12/7) -> (12/5,5/2) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(119,192,44,71) (-13/8,-8/5) -> (8/3,11/4) Hyperbolic Matrix(121,192,46,73) (-8/5,-11/7) -> (13/5,8/3) Hyperbolic Matrix(47,72,-32,-49) (-11/7,-3/2) -> (-3/2,-13/9) Parabolic Matrix(899,1296,274,395) (-13/9,-36/25) -> (36/11,23/7) Hyperbolic Matrix(901,1296,276,397) (-36/25,-23/16) -> (13/4,36/11) Hyperbolic Matrix(407,576,118,167) (-17/12,-24/17) -> (24/7,7/2) Hyperbolic Matrix(409,576,120,169) (-24/17,-7/5) -> (17/5,24/7) Hyperbolic Matrix(35,48,8,11) (-7/5,-4/3) -> (4/1,5/1) Hyperbolic Matrix(37,48,10,13) (-4/3,-5/4) -> (7/2,4/1) Hyperbolic Matrix(59,72,-50,-61) (-5/4,-6/5) -> (-6/5,-7/6) Parabolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(83,-96,32,-37) (1/1,6/5) -> (18/7,13/5) Hyperbolic Matrix(97,-120,38,-47) (6/5,5/4) -> (5/2,18/7) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(277,-432,84,-131) (14/9,11/7) -> (23/7,10/3) Hyperbolic Matrix(59,-96,8,-13) (13/8,5/3) -> (7/1,1/0) Hyperbolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(13,96,-8,-59) -> Matrix(3,2,-20,-13) Matrix(23,120,-14,-73) -> Matrix(5,2,-28,-11) Matrix(11,48,8,35) -> Matrix(5,2,32,13) Matrix(13,48,10,37) -> Matrix(7,2,52,15) Matrix(71,240,-50,-169) -> Matrix(1,0,-4,1) Matrix(23,72,-8,-25) -> Matrix(7,2,-32,-9) Matrix(155,432,-108,-301) -> Matrix(9,2,-68,-15) Matrix(71,192,44,119) -> Matrix(1,0,12,1) Matrix(73,192,46,121) -> Matrix(1,0,8,1) Matrix(37,96,-32,-83) -> Matrix(7,2,-60,-17) Matrix(11,24,-6,-13) -> Matrix(1,0,0,1) Matrix(83,144,34,59) -> Matrix(9,2,40,9) Matrix(85,144,36,61) -> Matrix(11,2,60,11) Matrix(119,192,44,71) -> Matrix(1,0,12,1) Matrix(121,192,46,73) -> Matrix(1,0,8,1) Matrix(47,72,-32,-49) -> Matrix(11,2,-72,-13) Matrix(899,1296,274,395) -> Matrix(69,10,200,29) Matrix(901,1296,276,397) -> Matrix(71,10,220,31) Matrix(407,576,118,167) -> Matrix(1,0,12,1) Matrix(409,576,120,169) -> Matrix(1,0,8,1) Matrix(35,48,8,11) -> Matrix(13,2,32,5) Matrix(37,48,10,13) -> Matrix(15,2,52,7) Matrix(59,72,-50,-61) -> Matrix(31,4,-256,-33) Matrix(1,0,2,1) -> Matrix(1,0,20,1) Matrix(83,-96,32,-37) -> Matrix(17,-2,60,-7) Matrix(97,-120,38,-47) -> Matrix(15,-2,68,-9) Matrix(169,-240,50,-71) -> Matrix(1,0,-4,1) Matrix(49,-72,32,-47) -> Matrix(13,-2,72,-11) Matrix(277,-432,84,-131) -> Matrix(11,-2,28,-5) Matrix(59,-96,8,-13) -> Matrix(13,-2,20,-3) Matrix(13,-24,6,-11) -> Matrix(1,0,0,1) Matrix(25,-72,8,-23) -> Matrix(9,-2,32,-7) Matrix(13,-72,2,-11) -> Matrix(9,-4,16,-7) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 8 Degree of the the map X: 16 Degree of the the map Y: 32 Permutation triple for Y: ((2,6,7)(3,12,4)(5,10,9)(8,15,14)(11,19,18)(13,22,21)(17,23,31)(26,29,27); (1,4,15,28,18,31,32,29,21,16,5,2)(3,10,27,11)(6,19,24,14,26,30,17,9,25,13,12,20)(7,22,23,8); (1,2,8,24,19,27,32,31,22,25,9,3)(4,13,29,14)(5,17,18,6)(7,20,12,11,28,15,23,30,26,10,16,21)) ----------------------------------------------------------------------- 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): 48 Minimal number of generators: 9 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: 10 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 4/3 3/2 2/1 12/5 3/1 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/10 6/5 1/8 5/4 1/8 4/3 1/7 3/2 1/6 8/5 0/1 5/3 1/6 2/1 0/1 1/5 7/3 1/6 12/5 1/5 5/2 1/4 3/1 1/4 7/2 1/4 4/1 1/3 5/1 1/2 6/1 1/2 1/0 1/0 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(11,-12,1,-1) (1/1,6/5) -> (6/1,1/0) Hyperbolic Matrix(49,-60,9,-11) (6/5,5/4) -> (5/1,6/1) Hyperbolic Matrix(47,-60,29,-37) (5/4,4/3) -> (8/5,5/3) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(61,-144,25,-59) (7/3,12/5) -> (12/5,5/2) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(13,-48,3,-11) (7/2,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,10,1) Matrix(11,-12,1,-1) -> Matrix(9,-1,10,-1) Matrix(49,-60,9,-11) -> Matrix(23,-3,54,-7) Matrix(47,-60,29,-37) -> Matrix(7,-1,50,-7) Matrix(25,-36,16,-23) -> Matrix(7,-1,36,-5) Matrix(13,-24,6,-11) -> Matrix(1,0,0,1) Matrix(61,-144,25,-59) -> Matrix(11,-2,50,-9) Matrix(13,-36,4,-11) -> Matrix(5,-1,16,-3) Matrix(13,-48,3,-11) -> Matrix(7,-2,18,-5) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 8 ----------------------------------------------------------------------- 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 10 1 2/1 (0/1,1/5) 0 6 12/5 1/5 2 1 5/2 1/4 1 12 3/1 1/4 1 4 7/2 1/4 1 12 4/1 1/3 2 3 5/1 1/2 1 12 6/1 1/2 4 2 1/0 1/0 1 12 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(11,-24,5,-11) (2/1,12/5) -> (2/1,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(13,-48,3,-11) (7/2,4/1) -> (4/1,5/1) Parabolic Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,10,-1) (0/1,2/1) -> (0/1,1/5) Matrix(11,-24,5,-11) -> Matrix(1,0,10,-1) (2/1,12/5) -> (0/1,1/5) Matrix(49,-120,20,-49) -> Matrix(9,-2,40,-9) (12/5,5/2) -> (1/5,1/4) Matrix(13,-36,4,-11) -> Matrix(5,-1,16,-3) 1/4 Matrix(13,-48,3,-11) -> Matrix(7,-2,18,-5) 1/3 Matrix(11,-60,2,-11) -> Matrix(7,-3,16,-7) (5/1,6/1) -> (3/8,1/2) Matrix(-1,12,0,1) -> Matrix(-1,1,0,1) (6/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.