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: 16 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -4/1 -12/5 -2/1 0/1 1/1 3/2 2/1 12/5 8/3 3/1 24/7 4/1 16/3 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/4 -5/1 -5/24 -4/1 -1/6 -7/2 -7/48 -10/3 -5/36 -3/1 -1/8 -8/3 -1/9 -5/2 -5/48 -12/5 -1/10 -7/3 -7/72 -16/7 -2/21 -9/4 -3/32 -2/1 -1/12 -9/5 -3/40 -16/9 -2/27 -7/4 -7/96 -12/7 -1/14 -5/3 -5/72 -13/8 -13/192 -8/5 -1/15 -3/2 -1/16 -10/7 -5/84 -17/12 -17/288 -24/17 -1/17 -7/5 -7/120 -4/3 -1/18 -5/4 -5/96 -16/13 -2/39 -11/9 -11/216 -6/5 -1/20 -7/6 -7/144 -1/1 -1/24 0/1 0/1 1/1 1/24 6/5 1/20 5/4 5/96 4/3 1/18 7/5 7/120 10/7 5/84 3/2 1/16 8/5 1/15 5/3 5/72 12/7 1/14 7/4 7/96 16/9 2/27 9/5 3/40 2/1 1/12 9/4 3/32 16/7 2/21 7/3 7/72 12/5 1/10 5/2 5/48 13/5 13/120 8/3 1/9 3/1 1/8 10/3 5/36 17/5 17/120 24/7 1/7 7/2 7/48 4/1 1/6 5/1 5/24 16/3 2/9 11/2 11/48 6/1 1/4 7/1 7/24 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,48,-6,-41) (-6/1,1/0) -> (-6/5,-7/6) Hyperbolic Matrix(17,96,-14,-79) (-6/1,-5/1) -> (-11/9,-6/5) 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(29,96,16,53) (-10/3,-3/1) -> (9/5,2/1) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(55,144,-34,-89) (-8/3,-5/2) -> (-13/8,-8/5) Hyperbolic Matrix(59,144,34,83) (-5/2,-12/5) -> (12/7,7/4) Hyperbolic Matrix(61,144,36,85) (-12/5,-7/3) -> (5/3,12/7) Hyperbolic Matrix(125,288,-102,-235) (-7/3,-16/7) -> (-16/13,-11/9) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(43,96,30,67) (-9/4,-2/1) -> (10/7,3/2) Hyperbolic Matrix(53,96,16,29) (-2/1,-9/5) -> (3/1,10/3) Hyperbolic Matrix(161,288,90,161) (-9/5,-16/9) -> (16/9,9/5) Hyperbolic Matrix(109,192,-88,-155) (-16/9,-7/4) -> (-5/4,-16/13) Hyperbolic 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(29,48,-26,-43) (-5/3,-13/8) -> (-7/6,-1/1) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(67,96,30,43) (-3/2,-10/7) -> (2/1,9/4) 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(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(41,-48,6,-7) (1/1,6/5) -> (6/1,7/1) Hyperbolic Matrix(79,-96,14,-17) (6/5,5/4) -> (11/2,6/1) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(163,-288,30,-53) (7/4,16/9) -> (16/3,11/2) Hyperbolic Matrix(83,-192,16,-37) (16/7,7/3) -> (5/1,16/3) Hyperbolic Matrix(19,-48,2,-5) (5/2,13/5) -> (7/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,48,-6,-41) -> Matrix(7,2,-144,-41) Matrix(17,96,-14,-79) -> Matrix(17,4,-336,-79) Matrix(11,48,8,35) -> Matrix(11,2,192,35) Matrix(13,48,10,37) -> Matrix(13,2,240,37) Matrix(71,240,-50,-169) -> Matrix(71,10,-1200,-169) Matrix(29,96,16,53) -> Matrix(29,4,384,53) Matrix(17,48,6,17) -> Matrix(17,2,144,17) Matrix(55,144,-34,-89) -> Matrix(55,6,-816,-89) Matrix(59,144,34,83) -> Matrix(59,6,816,83) Matrix(61,144,36,85) -> Matrix(61,6,864,85) Matrix(125,288,-102,-235) -> Matrix(125,12,-2448,-235) Matrix(127,288,56,127) -> Matrix(127,12,1344,127) Matrix(43,96,30,67) -> Matrix(43,4,720,67) Matrix(53,96,16,29) -> Matrix(53,4,384,29) Matrix(161,288,90,161) -> Matrix(161,12,2160,161) Matrix(109,192,-88,-155) -> Matrix(109,8,-2112,-155) Matrix(83,144,34,59) -> Matrix(83,6,816,59) Matrix(85,144,36,61) -> Matrix(85,6,864,61) Matrix(29,48,-26,-43) -> Matrix(29,2,-624,-43) Matrix(31,48,20,31) -> Matrix(31,2,480,31) Matrix(67,96,30,43) -> Matrix(67,4,720,43) Matrix(407,576,118,167) -> Matrix(407,24,2832,167) Matrix(409,576,120,169) -> Matrix(409,24,2880,169) Matrix(35,48,8,11) -> Matrix(35,2,192,11) Matrix(37,48,10,13) -> Matrix(37,2,240,13) Matrix(1,0,2,1) -> Matrix(1,0,48,1) Matrix(41,-48,6,-7) -> Matrix(41,-2,144,-7) Matrix(79,-96,14,-17) -> Matrix(79,-4,336,-17) Matrix(169,-240,50,-71) -> Matrix(169,-10,1200,-71) Matrix(89,-144,34,-55) -> Matrix(89,-6,816,-55) Matrix(163,-288,30,-53) -> Matrix(163,-12,720,-53) Matrix(83,-192,16,-37) -> Matrix(83,-8,384,-37) Matrix(19,-48,2,-5) -> Matrix(19,-2,48,-5) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 16 Genus: 9 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 32 Degree of the the map Y: 32 Permutation triple for Y: ((2,6,21,28,22,7)(3,12,18,24,13,4)(5,10,9)(8,16,15)(11,19,14)(17,25)(26,27)(29,30,31); (1,4,16,26,14,13,23,22,31,25,15,28,32,18,9,27,29,12,20,6,19,17,5,2)(3,10,21,30,24,8,7,11); (1,2,8,25,19,7,23,13,30,27,16,24,32,28,10,17,31,21,20,12,11,26,9,3)(4,14,6,5,18,29,22,15)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1, lambda2, lambda1+lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z. ----------------------------------------------------------------------- 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: 8 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 12/5 8/3 3/1 4/1 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 1/24 4/3 1/18 3/2 1/16 8/5 1/15 5/3 5/72 2/1 1/12 7/3 7/72 12/5 1/10 5/2 5/48 13/5 13/120 8/3 1/9 3/1 1/8 4/1 1/6 5/1 5/24 6/1 1/4 7/1 7/24 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(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(17,-24,5,-7) (4/3,3/2) -> (3/1,4/1) Hyperbolic Matrix(31,-48,11,-17) (3/2,8/5) -> (8/3,3/1) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic 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(19,-48,2,-5) (5/2,13/5) -> (7/1,1/0) Hyperbolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,24,1) Matrix(19,-24,4,-5) -> Matrix(19,-1,96,-5) Matrix(17,-24,5,-7) -> Matrix(17,-1,120,-7) Matrix(31,-48,11,-17) -> Matrix(31,-2,264,-17) Matrix(89,-144,34,-55) -> Matrix(89,-6,816,-55) Matrix(13,-24,6,-11) -> Matrix(13,-1,144,-11) Matrix(61,-144,25,-59) -> Matrix(61,-6,600,-59) Matrix(19,-48,2,-5) -> Matrix(19,-2,48,-5) Matrix(13,-72,2,-11) -> Matrix(13,-3,48,-11) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM 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: 8 Genus: 1 Degree of H/liftables -> H/(image of liftables): 1 ----------------------------------------------------------------------- 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 24 1 2/1 1/12 2 12 12/5 1/10 12 2 5/2 5/48 1 24 8/3 1/9 8 3 3/1 1/8 3 8 4/1 1/6 4 6 6/1 1/4 6 4 1/0 1/0 1 24 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(19,-48,2,-5) (5/2,13/5) -> (7/1,1/0) Hyperbolic Matrix(55,-144,21,-55) (18/7,8/3) -> (18/7,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection Matrix(5,-24,1,-5) (4/1,6/1) -> (4/1,6/1) Reflection Matrix(7,-48,1,-7) (6/1,8/1) -> (6/1,8/1) 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(1,0,24,-1) (0/1,2/1) -> (0/1,1/12) Matrix(11,-24,5,-11) -> Matrix(11,-1,120,-11) (2/1,12/5) -> (1/12,1/10) Matrix(49,-120,20,-49) -> Matrix(49,-5,480,-49) (12/5,5/2) -> (1/10,5/48) Matrix(19,-48,2,-5) -> Matrix(19,-2,48,-5) Matrix(55,-144,21,-55) -> Matrix(55,-6,504,-55) (18/7,8/3) -> (3/28,1/9) Matrix(17,-48,6,-17) -> Matrix(17,-2,144,-17) (8/3,3/1) -> (1/9,1/8) Matrix(7,-24,2,-7) -> Matrix(7,-1,48,-7) (3/1,4/1) -> (1/8,1/6) Matrix(5,-24,1,-5) -> Matrix(5,-1,24,-5) (4/1,6/1) -> (1/6,1/4) Matrix(7,-48,1,-7) -> Matrix(7,-2,24,-7) (6/1,8/1) -> (1/4,1/3) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.