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): 96 Minimal number of generators: 17 Number of equivalence classes of cusps: 16 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -3/7 -1/3 -1/5 -1/7 0/1 1/3 1/2 1/1 5/3 2/1 3/1 13/3 5/1 7/1 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 -5/12 0/1 1/5 -7/17 1/4 -2/5 1/4 1/3 -1/3 1/2 -2/7 3/4 1/1 -1/4 1/1 1/0 -1/5 0/1 -1/6 1/3 1/2 -2/13 1/2 1/1 -1/7 1/2 -1/8 2/3 1/1 0/1 0/1 1/1 1/3 0/1 2/5 0/1 1/3 3/7 1/2 1/2 1/2 1/1 1/1 1/0 3/2 -3/2 -1/1 5/3 -1/1 7/4 -1/1 -5/6 2/1 -1/1 -1/2 3/1 0/1 4/1 1/1 1/0 13/3 1/0 22/5 -3/1 1/0 9/2 -1/1 1/0 5/1 1/0 6/1 -2/1 -1/1 7/1 -1/1 8/1 -1/1 -2/3 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(41,18,-98,-43) (-1/2,-3/7) -> (-3/7,-5/12) Parabolic Matrix(29,12,-220,-91) (-5/12,-7/17) -> (-1/7,-1/8) Hyperbolic Matrix(39,16,-256,-105) (-7/17,-2/5) -> (-2/13,-1/7) Hyperbolic Matrix(11,4,-36,-13) (-2/5,-1/3) -> (-1/3,-2/7) Parabolic Matrix(57,16,32,9) (-2/7,-1/4) -> (7/4,2/1) Hyperbolic Matrix(9,2,-50,-11) (-1/4,-1/5) -> (-1/5,-1/6) Parabolic Matrix(249,40,56,9) (-1/6,-2/13) -> (22/5,9/2) Hyperbolic Matrix(65,8,8,1) (-1/8,0/1) -> (8/1,1/0) Hyperbolic Matrix(7,-2,18,-5) (0/1,1/3) -> (1/3,2/5) Parabolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(73,-32,16,-7) (3/7,1/2) -> (9/2,5/1) Hyperbolic Matrix(5,-4,4,-3) (1/2,1/1) -> (1/1,3/2) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic Matrix(79,-338,18,-77) (4/1,13/3) -> (13/3,22/5) Parabolic Matrix(15,-98,2,-13) (6/1,7/1) -> (7/1,8/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(41,18,-98,-43) -> Matrix(1,0,2,1) Matrix(29,12,-220,-91) -> Matrix(9,-2,14,-3) Matrix(39,16,-256,-105) -> Matrix(1,0,-2,1) Matrix(11,4,-36,-13) -> Matrix(5,-2,8,-3) Matrix(57,16,32,9) -> Matrix(5,-4,-6,5) Matrix(9,2,-50,-11) -> Matrix(1,0,2,1) Matrix(249,40,56,9) -> Matrix(5,-2,-2,1) Matrix(65,8,8,1) -> Matrix(3,-2,-4,3) Matrix(7,-2,18,-5) -> Matrix(1,0,2,1) Matrix(67,-28,12,-5) -> Matrix(5,-2,-2,1) Matrix(73,-32,16,-7) -> Matrix(1,0,-2,1) Matrix(5,-4,4,-3) -> Matrix(1,-2,0,1) Matrix(31,-50,18,-29) -> Matrix(7,8,-8,-9) Matrix(7,-18,2,-5) -> Matrix(1,0,2,1) Matrix(79,-338,18,-77) -> Matrix(1,-4,0,1) Matrix(15,-98,2,-13) -> Matrix(3,4,-4,-5) 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: 16 Permutation triple for Y: ((1,5,10,13,15,11,6,2)(3,8,14,16,12,7,9,4); (1,4)(6,7)(8,10)(15,16); (1,2,7,12,15,13,8,3)(4,9,6,11,16,14,10,5)) ----------------------------------------------------------------------- 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 -1/1 1/3 1/1 5/3 2/1 3/1 13/3 5/1 7/1 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 1/3 0/1 2/5 0/1 1/3 3/7 1/2 1/2 1/2 1/1 1/1 1/0 3/2 -3/2 -1/1 5/3 -1/1 2/1 -1/1 -1/2 3/1 0/1 4/1 1/1 1/0 13/3 1/0 9/2 -1/1 1/0 5/1 1/0 6/1 -2/1 -1/1 7/1 -1/1 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(7,-2,18,-5) (0/1,1/3) -> (1/3,2/5) Parabolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(73,-32,16,-7) (3/7,1/2) -> (9/2,5/1) Hyperbolic Matrix(5,-4,4,-3) (1/2,1/1) -> (1/1,3/2) Parabolic Matrix(16,-25,9,-14) (3/2,5/3) -> (5/3,2/1) Parabolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic Matrix(40,-169,9,-38) (4/1,13/3) -> (13/3,9/2) Parabolic Matrix(8,-49,1,-6) (6/1,7/1) -> (7/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,2,1) Matrix(7,-2,18,-5) -> Matrix(1,0,2,1) Matrix(67,-28,12,-5) -> Matrix(5,-2,-2,1) Matrix(73,-32,16,-7) -> Matrix(1,0,-2,1) Matrix(5,-4,4,-3) -> Matrix(1,-2,0,1) Matrix(16,-25,9,-14) -> Matrix(3,4,-4,-5) Matrix(7,-18,2,-5) -> Matrix(1,0,2,1) Matrix(40,-169,9,-38) -> Matrix(1,-2,0,1) Matrix(8,-49,1,-6) -> Matrix(1,2,-2,-3) 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 4 5/3 -1/1 4 1 2/1 (-1/1,-1/2) 0 8 3/1 0/1 2 2 4/1 (1/1,1/0) 0 8 13/3 1/0 2 1 5/1 1/0 2 4 7/1 -1/1 2 1 1/0 (-1/1,0/1) 0 8 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(4,-5,3,-4) (1/1,5/3) -> (1/1,5/3) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) Reflection Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic Matrix(25,-104,6,-25) (4/1,13/3) -> (4/1,13/3) Reflection Matrix(14,-65,3,-14) (13/3,5/1) -> (13/3,5/1) Reflection Matrix(6,-35,1,-6) (5/1,7/1) -> (5/1,7/1) Reflection Matrix(-1,14,0,1) (7/1,1/0) -> (7/1,1/0) Reflection 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(4,-5,3,-4) -> Matrix(1,2,0,-1) (1/1,5/3) -> (-1/1,1/0) Matrix(11,-20,6,-11) -> Matrix(3,2,-4,-3) (5/3,2/1) -> (-1/1,-1/2) Matrix(7,-18,2,-5) -> Matrix(1,0,2,1) 0/1 Matrix(25,-104,6,-25) -> Matrix(-1,2,0,1) (4/1,13/3) -> (1/1,1/0) Matrix(14,-65,3,-14) -> Matrix(1,0,0,-1) (13/3,5/1) -> (0/1,1/0) Matrix(6,-35,1,-6) -> Matrix(1,2,0,-1) (5/1,7/1) -> (-1/1,1/0) Matrix(-1,14,0,1) -> Matrix(-1,0,2,1) (7/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.