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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 6 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 2/1 23/10 23/5 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -5/23 -4/1 -4/23 -11/3 -11/69 -7/2 -7/46 -10/3 -10/69 -3/1 -3/23 -5/2 -5/46 -2/1 -2/23 -9/5 -9/115 -16/9 -16/207 -23/13 -1/13 -7/4 -7/92 -5/3 -5/69 -8/5 -8/115 -3/2 -3/46 -10/7 -10/161 -7/5 -7/115 -4/3 -4/69 -9/7 -9/161 -23/18 -1/18 -14/11 -14/253 -5/4 -5/92 -6/5 -6/115 -1/1 -1/23 0/1 0/1 1/1 1/23 5/4 5/92 4/3 4/69 11/8 11/184 7/5 7/115 10/7 10/161 3/2 3/46 5/3 5/69 2/1 2/23 9/4 9/92 16/7 16/161 23/10 1/10 7/3 7/69 5/2 5/46 8/3 8/69 3/1 3/23 10/3 10/69 7/2 7/46 4/1 4/23 9/2 9/46 23/5 1/5 14/3 14/69 5/1 5/23 6/1 6/23 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,46,-2,-13) (-5/1,1/0) -> (-11/3,-7/2) Hyperbolic Matrix(11,46,-6,-25) (-5/1,-4/1) -> (-2/1,-9/5) Hyperbolic Matrix(37,138,26,97) (-4/1,-11/3) -> (7/5,10/7) Hyperbolic Matrix(41,138,30,101) (-7/2,-10/3) -> (4/3,11/8) Hyperbolic Matrix(43,138,-24,-77) (-10/3,-3/1) -> (-9/5,-16/9) Hyperbolic Matrix(17,46,-10,-27) (-3/1,-5/2) -> (-7/4,-5/3) Hyperbolic Matrix(19,46,-12,-29) (-5/2,-2/1) -> (-8/5,-3/2) Hyperbolic Matrix(389,690,84,149) (-16/9,-23/13) -> (23/5,14/3) Hyperbolic Matrix(209,368,46,81) (-23/13,-7/4) -> (9/2,23/5) Hyperbolic Matrix(57,92,-44,-71) (-5/3,-8/5) -> (-4/3,-9/7) Hyperbolic Matrix(63,92,-50,-73) (-3/2,-10/7) -> (-14/11,-5/4) Hyperbolic Matrix(65,92,12,17) (-10/7,-7/5) -> (5/1,6/1) Hyperbolic Matrix(33,46,-28,-39) (-7/5,-4/3) -> (-6/5,-1/1) Hyperbolic Matrix(287,368,124,159) (-9/7,-23/18) -> (23/10,7/3) Hyperbolic Matrix(541,690,236,301) (-23/18,-14/11) -> (16/7,23/10) Hyperbolic Matrix(75,92,22,27) (-5/4,-6/5) -> (10/3,7/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(39,-46,28,-33) (1/1,5/4) -> (11/8,7/5) Hyperbolic Matrix(35,-46,16,-21) (5/4,4/3) -> (2/1,9/4) Hyperbolic Matrix(95,-138,42,-61) (10/7,3/2) -> (9/4,16/7) Hyperbolic Matrix(29,-46,12,-19) (3/2,5/3) -> (7/3,5/2) Hyperbolic Matrix(27,-46,10,-17) (5/3,2/1) -> (8/3,3/1) Hyperbolic Matrix(35,-92,8,-21) (5/2,8/3) -> (4/1,9/2) Hyperbolic Matrix(29,-92,6,-19) (3/1,10/3) -> (14/3,5/1) Hyperbolic Matrix(13,-46,2,-7) (7/2,4/1) -> (6/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,46,-2,-13) -> Matrix(7,2,-46,-13) Matrix(11,46,-6,-25) -> Matrix(11,2,-138,-25) Matrix(37,138,26,97) -> Matrix(37,6,598,97) Matrix(41,138,30,101) -> Matrix(41,6,690,101) Matrix(43,138,-24,-77) -> Matrix(43,6,-552,-77) Matrix(17,46,-10,-27) -> Matrix(17,2,-230,-27) Matrix(19,46,-12,-29) -> Matrix(19,2,-276,-29) Matrix(389,690,84,149) -> Matrix(389,30,1932,149) Matrix(209,368,46,81) -> Matrix(209,16,1058,81) Matrix(57,92,-44,-71) -> Matrix(57,4,-1012,-71) Matrix(63,92,-50,-73) -> Matrix(63,4,-1150,-73) Matrix(65,92,12,17) -> Matrix(65,4,276,17) Matrix(33,46,-28,-39) -> Matrix(33,2,-644,-39) Matrix(287,368,124,159) -> Matrix(287,16,2852,159) Matrix(541,690,236,301) -> Matrix(541,30,5428,301) Matrix(75,92,22,27) -> Matrix(75,4,506,27) Matrix(1,0,2,1) -> Matrix(1,0,46,1) Matrix(39,-46,28,-33) -> Matrix(39,-2,644,-33) Matrix(35,-46,16,-21) -> Matrix(35,-2,368,-21) Matrix(95,-138,42,-61) -> Matrix(95,-6,966,-61) Matrix(29,-46,12,-19) -> Matrix(29,-2,276,-19) Matrix(27,-46,10,-17) -> Matrix(27,-2,230,-17) Matrix(35,-92,8,-21) -> Matrix(35,-4,184,-21) Matrix(29,-92,6,-19) -> Matrix(29,-4,138,-19) Matrix(13,-46,2,-7) -> Matrix(13,-2,46,-7) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 6 Genus: 10 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 24 Degree of the the map Y: 24 Permutation triple for Y: ((2,6,18,8,20,22,23,17,10,9,11,4,3,12,14,13,19,24,21,15,5,16,7); (1,4,14,13,20,18,19,8,7,6,16,23,22,12,11,3,10,21,9,17,15,5,2); (1,2,8,20,19,14,22,13,4,12,11,21,24,18,7,6,5,17,16,15,10,9,3)) ----------------------------------------------------------------------- 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, lambda1, lambda2, lambda1+lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z+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): 24 Minimal number of generators: 5 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: 2 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/23 4/3 4/69 3/2 3/46 5/3 5/69 2/1 2/23 5/2 5/46 3/1 3/23 4/1 4/23 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(18,-23,11,-14) (1/1,4/3) -> (3/2,5/3) Hyperbolic Matrix(16,-23,7,-10) (4/3,3/2) -> (2/1,5/2) Hyperbolic Matrix(13,-23,4,-7) (5/3,2/1) -> (3/1,4/1) Hyperbolic Matrix(9,-23,2,-5) (5/2,3/1) -> (4/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,23,1) Matrix(18,-23,11,-14) -> Matrix(18,-1,253,-14) Matrix(16,-23,7,-10) -> Matrix(16,-1,161,-10) Matrix(13,-23,4,-7) -> Matrix(13,-1,92,-7) Matrix(9,-23,2,-5) -> Matrix(9,-1,46,-5) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 24 Minimal number of generators: 5 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: 2 Genus: 2 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 23 1 2/1 2/23 1 23 5/2 5/46 1 23 3/1 3/23 1 23 4/1 4/23 1 23 1/0 1/0 1 23 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(10,-23,3,-7) (2/1,5/2) -> (3/1,4/1) Glide Reflection Matrix(9,-23,2,-5) (5/2,3/1) -> (4/1,1/0) Hyperbolic 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(1,0,23,-1) (0/1,2/1) -> (0/1,2/23) Matrix(10,-23,3,-7) -> Matrix(10,-1,69,-7) Matrix(9,-23,2,-5) -> Matrix(9,-1,46,-5) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.