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 1/1 1/0 -4/1 -1/1 1/0 -7/2 -1/1 -3/1 -1/1 0/1 -8/3 -1/1 -2/3 -5/2 -1/1 -1/2 -7/3 -1/2 -2/1 -1/2 0/1 -7/4 -1/2 -5/3 -1/2 -1/3 -13/8 -1/2 -2/5 -21/13 -2/5 -8/5 -2/5 -1/3 -11/7 -4/11 -1/3 -14/9 -1/3 -3/2 -1/3 0/1 -7/5 -1/3 -4/3 -1/3 -1/4 -9/7 -3/11 -1/4 -14/11 -1/4 -5/4 -1/4 -1/5 -6/5 -1/5 0/1 -7/6 0/1 -1/1 -1/4 0/1 0/1 0/1 1/1 0/1 1/4 5/4 1/5 1/4 4/3 1/4 1/3 7/5 1/3 3/2 0/1 1/3 8/5 1/3 2/5 5/3 1/3 1/2 7/4 1/2 2/1 0/1 1/2 7/3 1/2 5/2 1/2 1/1 13/5 1/2 2/3 21/8 2/3 8/3 2/3 1/1 11/4 4/5 1/1 14/5 1/1 3/1 0/1 1/1 7/2 1/1 4/1 1/1 1/0 9/2 3/1 1/0 14/3 1/0 5/1 -1/1 1/0 6/1 -1/1 0/1 7/1 0/1 1/0 0/1 1/0 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(1,-2,-2,5) Matrix(13,56,-10,-43) -> Matrix(1,2,-4,-7) Matrix(15,56,4,15) -> Matrix(1,2,0,1) Matrix(13,42,4,13) -> Matrix(1,0,2,1) Matrix(41,112,-26,-71) -> Matrix(5,4,-14,-11) Matrix(27,70,-22,-57) -> Matrix(3,2,-14,-9) Matrix(29,70,12,29) -> Matrix(3,2,4,3) Matrix(13,28,6,13) -> Matrix(1,0,4,1) Matrix(15,28,8,15) -> Matrix(1,0,4,1) Matrix(41,70,24,41) -> Matrix(5,2,12,5) Matrix(69,112,8,13) -> Matrix(5,2,2,1) Matrix(113,182,18,29) -> Matrix(5,2,-8,-3) Matrix(125,196,44,69) -> Matrix(11,4,8,3) Matrix(127,196,46,71) -> Matrix(13,4,16,5) Matrix(29,42,20,29) -> Matrix(1,0,6,1) Matrix(41,56,30,41) -> Matrix(7,2,24,7) Matrix(153,196,32,41) -> Matrix(15,4,-4,-1) Matrix(155,196,34,43) -> Matrix(17,4,4,1) Matrix(153,182,58,69) -> Matrix(11,2,16,3) Matrix(99,112,38,43) -> Matrix(7,2,10,3) Matrix(1,0,2,1) -> Matrix(1,0,8,1) Matrix(57,-70,22,-27) -> Matrix(9,-2,14,-3) Matrix(43,-56,10,-13) -> Matrix(7,-2,4,-1) Matrix(71,-112,26,-41) -> Matrix(11,-4,14,-5) Matrix(43,-70,8,-13) -> Matrix(5,-2,-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): 6 Degree of the the map X: 6 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): 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: 4 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 7/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 0/1 1/4 3/2 0/1 1/3 2/1 0/1 1/2 7/3 1/2 5/2 1/2 1/1 3/1 0/1 1/1 7/2 1/1 4/1 1/1 1/0 1/0 0/1 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,-14,4,-5) (1/1,3/2) -> (5/2,3/1) Hyperbolic Matrix(9,-14,2,-3) (3/2,2/1) -> (4/1,1/0) Hyperbolic Matrix(19,-42,5,-11) (2/1,7/3) -> (7/2,4/1) Hyperbolic Matrix(23,-56,7,-17) (7/3,5/2) -> (3/1,7/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,4,1) Matrix(11,-14,4,-5) -> Matrix(4,-1,5,-1) Matrix(9,-14,2,-3) -> Matrix(3,-1,1,0) Matrix(19,-42,5,-11) -> Matrix(3,-1,1,0) Matrix(23,-56,7,-17) -> Matrix(1,-1,3,-2) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 2 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: 2 Number of equivalence classes of cusps: 1 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 0/1 0/1 4 1 2/1 (0/1,1/2) 0 7 3/1 (0/1,1/1) 0 14 7/2 1/1 2 2 4/1 (1/1,1/0) 0 7 1/0 (0/1,1/0) 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(5,-14,1,-3) (2/1,3/1) -> (4/1,1/0) Glide Reflection Matrix(13,-42,4,-13) (3/1,7/2) -> (3/1,7/2) Reflection Matrix(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) 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(1,0,4,-1) (0/1,2/1) -> (0/1,1/2) Matrix(5,-14,1,-3) -> Matrix(-1,1,1,0) Matrix(13,-42,4,-13) -> Matrix(1,0,2,-1) (3/1,7/2) -> (0/1,1/1) Matrix(15,-56,4,-15) -> Matrix(-1,2,0,1) (7/2,4/1) -> (1/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.