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: 18 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -7/6 0/1 1/1 7/5 3/2 7/4 2/1 7/3 5/2 14/5 3/1 7/2 4/1 14/3 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -1/2 0/1 -4/1 -1/2 0/1 -7/2 0/1 -3/1 0/1 1/0 -8/3 -1/1 0/1 -5/2 0/1 1/0 -7/3 -1/1 1/1 -2/1 0/1 1/0 -7/4 1/0 -5/3 -4/1 1/0 -13/8 -4/1 -3/1 -21/13 -3/1 -8/5 -3/1 -2/1 -11/7 -2/1 1/0 -14/9 -3/1 -1/1 -3/2 -2/1 1/0 -7/5 -3/1 -1/1 -4/3 -2/1 1/0 -9/7 -2/1 1/0 -14/11 -3/1 -1/1 -5/4 -2/1 1/0 -6/5 -3/1 -2/1 -7/6 -2/1 -1/1 -2/1 -1/1 0/1 -1/1 1/1 -1/1 -2/3 5/4 -2/3 -1/2 4/3 -2/3 -1/2 7/5 -1/1 -3/5 3/2 -2/3 -1/2 8/5 -2/3 -3/5 5/3 -4/7 -1/2 7/4 -1/2 2/1 -1/2 0/1 7/3 -1/1 -1/3 5/2 -1/2 0/1 13/5 -1/3 0/1 21/8 0/1 8/3 -1/1 0/1 11/4 -1/2 0/1 14/5 -1/1 -1/3 3/1 -1/2 0/1 7/2 0/1 4/1 0/1 1/0 9/2 0/1 1/0 14/3 -1/1 1/1 5/1 0/1 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(13,70,-8,-43) (-5/1,1/0) -> (-5/3,-13/8) Hyperbolic Matrix(13,56,-10,-43) (-5/1,-4/1) -> (-4/3,-9/7) Hyperbolic Matrix(15,56,4,15) (-4/1,-7/2) -> (7/2,4/1) Hyperbolic Matrix(13,42,4,13) (-7/2,-3/1) -> (3/1,7/2) Hyperbolic Matrix(41,112,-26,-71) (-3/1,-8/3) -> (-8/5,-11/7) Hyperbolic Matrix(27,70,-22,-57) (-8/3,-5/2) -> (-5/4,-6/5) Hyperbolic Matrix(29,70,12,29) (-5/2,-7/3) -> (7/3,5/2) Hyperbolic Matrix(13,28,6,13) (-7/3,-2/1) -> (2/1,7/3) Hyperbolic Matrix(15,28,8,15) (-2/1,-7/4) -> (7/4,2/1) Hyperbolic Matrix(41,70,24,41) (-7/4,-5/3) -> (5/3,7/4) Hyperbolic Matrix(69,112,8,13) (-13/8,-21/13) -> (7/1,1/0) Hyperbolic Matrix(113,182,18,29) (-21/13,-8/5) -> (6/1,7/1) Hyperbolic Matrix(125,196,44,69) (-11/7,-14/9) -> (14/5,3/1) Hyperbolic Matrix(127,196,46,71) (-14/9,-3/2) -> (11/4,14/5) Hyperbolic Matrix(29,42,20,29) (-3/2,-7/5) -> (7/5,3/2) Hyperbolic Matrix(41,56,30,41) (-7/5,-4/3) -> (4/3,7/5) Hyperbolic Matrix(153,196,32,41) (-9/7,-14/11) -> (14/3,5/1) Hyperbolic Matrix(155,196,34,43) (-14/11,-5/4) -> (9/2,14/3) Hyperbolic Matrix(153,182,58,69) (-6/5,-7/6) -> (21/8,8/3) Hyperbolic Matrix(99,112,38,43) (-7/6,-1/1) -> (13/5,21/8) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(43,-56,10,-13) (5/4,4/3) -> (4/1,9/2) Hyperbolic Matrix(71,-112,26,-41) (3/2,8/5) -> (8/3,11/4) Hyperbolic Matrix(43,-70,8,-13) (8/5,5/3) -> (5/1,6/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(13,70,-8,-43) -> Matrix(7,4,-2,-1) Matrix(13,56,-10,-43) -> Matrix(3,2,-2,-1) Matrix(15,56,4,15) -> Matrix(1,0,2,1) Matrix(13,42,4,13) -> Matrix(1,0,-2,1) Matrix(41,112,-26,-71) -> Matrix(1,-2,0,1) Matrix(27,70,-22,-57) -> Matrix(1,-2,0,1) Matrix(29,70,12,29) -> Matrix(1,0,-2,1) Matrix(13,28,6,13) -> Matrix(1,0,-2,1) Matrix(15,28,8,15) -> Matrix(1,0,-2,1) Matrix(41,70,24,41) -> Matrix(1,8,-2,-15) Matrix(69,112,8,13) -> Matrix(1,4,-2,-7) Matrix(113,182,18,29) -> Matrix(3,8,-2,-5) Matrix(125,196,44,69) -> Matrix(1,2,-2,-3) Matrix(127,196,46,71) -> Matrix(1,2,-2,-3) Matrix(29,42,20,29) -> Matrix(1,4,-2,-7) Matrix(41,56,30,41) -> Matrix(1,4,-2,-7) Matrix(153,196,32,41) -> Matrix(1,2,0,1) Matrix(155,196,34,43) -> Matrix(1,2,0,1) Matrix(153,182,58,69) -> Matrix(1,2,0,1) Matrix(99,112,38,43) -> Matrix(1,2,-4,-7) Matrix(1,0,2,1) -> Matrix(3,4,-4,-5) Matrix(57,-70,22,-27) -> Matrix(3,2,-8,-5) Matrix(43,-56,10,-13) -> Matrix(3,2,-2,-1) Matrix(71,-112,26,-41) -> Matrix(3,2,-8,-5) Matrix(43,-70,8,-13) -> Matrix(7,4,-2,-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): 4 Degree of the the map X: 4 Degree of the the map Y: 24 Permutation triple for Y: ((2,6,13,4,3,12,7)(5,18,8,10,9,16,15)(11,14,21,20,19,17,24); (1,4,16,24,17,5,2)(3,10,8,7,19,23,11)(6,20,18,22,9,14,13); (1,2,8,20,21,9,3)(4,14,23,19,6,5,15)(7,12,11,16,22,18,17)) ----------------------------------------------------------------------- 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): 72 Minimal number of generators: 13 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: 12 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 7/5 3/2 7/4 2/1 7/3 5/2 21/8 3/1 7/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 0/1 1/0 -5/2 0/1 1/0 -7/3 -1/1 1/1 -2/1 0/1 1/0 -3/2 -2/1 1/0 -7/5 -3/1 -1/1 -4/3 -2/1 1/0 -1/1 -2/1 -1/1 0/1 -1/1 1/1 -1/1 -2/3 5/4 -2/3 -1/2 4/3 -2/3 -1/2 7/5 -1/1 -3/5 3/2 -2/3 -1/2 5/3 -4/7 -1/2 7/4 -1/2 2/1 -1/2 0/1 7/3 -1/1 -1/3 5/2 -1/2 0/1 13/5 -1/3 0/1 21/8 0/1 8/3 -1/1 0/1 3/1 -1/2 0/1 7/2 0/1 4/1 0/1 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,7,0,1) (-3/1,1/0) -> (4/1,1/0) Parabolic Matrix(13,35,10,27) (-3/1,-5/2) -> (5/4,4/3) Hyperbolic Matrix(29,70,12,29) (-5/2,-7/3) -> (7/3,5/2) Hyperbolic Matrix(13,28,6,13) (-7/3,-2/1) -> (2/1,7/3) Hyperbolic Matrix(13,21,8,13) (-2/1,-3/2) -> (3/2,5/3) Hyperbolic Matrix(29,42,20,29) (-3/2,-7/5) -> (7/5,3/2) Hyperbolic Matrix(41,56,30,41) (-7/5,-4/3) -> (4/3,7/5) Hyperbolic Matrix(27,35,10,13) (-4/3,-1/1) -> (8/3,3/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(29,-49,16,-27) (5/3,7/4) -> (7/4,2/1) Parabolic Matrix(169,-441,64,-167) (13/5,21/8) -> (21/8,8/3) Parabolic Matrix(15,-49,4,-13) (3/1,7/2) -> (7/2,4/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,7,0,1) -> Matrix(1,0,0,1) Matrix(13,35,10,27) -> Matrix(1,2,-2,-3) Matrix(29,70,12,29) -> Matrix(1,0,-2,1) Matrix(13,28,6,13) -> Matrix(1,0,-2,1) Matrix(13,21,8,13) -> Matrix(1,4,-2,-7) Matrix(29,42,20,29) -> Matrix(1,4,-2,-7) Matrix(41,56,30,41) -> Matrix(1,4,-2,-7) Matrix(27,35,10,13) -> Matrix(1,2,-2,-3) Matrix(1,0,2,1) -> Matrix(3,4,-4,-5) Matrix(57,-70,22,-27) -> Matrix(3,2,-8,-5) Matrix(29,-49,16,-27) -> Matrix(7,4,-16,-9) Matrix(169,-441,64,-167) -> Matrix(1,0,2,1) Matrix(15,-49,4,-13) -> Matrix(1,0,2,1) 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): 2 ----------------------------------------------------------------------- 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 -1/1 4 2 1/1 (-1/1,-2/3) 0 14 4/3 (-2/3,-1/2) 0 14 7/5 (-2/3,-1/2) 0 2 3/2 (-2/3,-1/2) 0 14 7/4 -1/2 8 2 2/1 (-1/2,0/1) 0 14 7/3 (-1/2,0/1) 0 2 5/2 (-1/2,0/1) 0 14 21/8 0/1 4 2 8/3 (-1/1,0/1) 0 14 3/1 (-1/2,0/1) 0 14 7/2 0/1 4 2 1/0 (-1/1,0/1) 0 14 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(27,-35,10,-13) (1/1,4/3) -> (8/3,3/1) Glide Reflection Matrix(41,-56,30,-41) (4/3,7/5) -> (4/3,7/5) Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) Reflection Matrix(13,-21,8,-13) (3/2,7/4) -> (3/2,7/4) Reflection Matrix(15,-28,8,-15) (7/4,2/1) -> (7/4,2/1) Reflection Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(41,-105,16,-41) (5/2,21/8) -> (5/2,21/8) Reflection Matrix(127,-336,48,-127) (21/8,8/3) -> (21/8,8/3) Reflection Matrix(13,-42,4,-13) (3/1,7/2) -> (3/1,7/2) Reflection Matrix(-1,7,0,1) (7/2,1/0) -> (7/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(5,4,-6,-5) (0/1,1/1) -> (-1/1,-2/3) Matrix(27,-35,10,-13) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(41,-56,30,-41) -> Matrix(7,4,-12,-7) (4/3,7/5) -> (-2/3,-1/2) Matrix(29,-42,20,-29) -> Matrix(7,4,-12,-7) (7/5,3/2) -> (-2/3,-1/2) Matrix(13,-21,8,-13) -> Matrix(7,4,-12,-7) (3/2,7/4) -> (-2/3,-1/2) Matrix(15,-28,8,-15) -> Matrix(-1,0,4,1) (7/4,2/1) -> (-1/2,0/1) Matrix(13,-28,6,-13) -> Matrix(-1,0,4,1) (2/1,7/3) -> (-1/2,0/1) Matrix(29,-70,12,-29) -> Matrix(-1,0,4,1) (7/3,5/2) -> (-1/2,0/1) Matrix(41,-105,16,-41) -> Matrix(-1,0,4,1) (5/2,21/8) -> (-1/2,0/1) Matrix(127,-336,48,-127) -> Matrix(-1,0,2,1) (21/8,8/3) -> (-1/1,0/1) Matrix(13,-42,4,-13) -> Matrix(-1,0,4,1) (3/1,7/2) -> (-1/2,0/1) Matrix(-1,7,0,1) -> Matrix(-1,0,2,1) (7/2,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.