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): 72 Minimal number of generators: 13 Number of equivalence classes of cusps: 8 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 2/1 5/2 10/3 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -2/5 -3/1 -3/10 -5/2 -1/4 -2/1 -1/5 -5/3 -1/6 -8/5 -4/25 -3/2 -3/20 -10/7 -1/7 -7/5 -7/50 -4/3 -2/15 -5/4 -1/8 -1/1 -1/10 0/1 0/1 1/1 1/10 4/3 2/15 3/2 3/20 5/3 1/6 2/1 1/5 5/2 1/4 8/3 4/15 3/1 3/10 10/3 1/3 7/2 7/20 4/1 2/5 5/1 1/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,20,-2,-13) (-4/1,1/0) -> (-8/5,-3/2) Hyperbolic Matrix(11,40,-8,-29) (-4/1,-3/1) -> (-7/5,-4/3) Hyperbolic Matrix(7,20,-6,-17) (-3/1,-5/2) -> (-5/4,-1/1) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(11,20,6,11) (-2/1,-5/3) -> (5/3,2/1) Hyperbolic Matrix(37,60,8,13) (-5/3,-8/5) -> (4/1,5/1) Hyperbolic Matrix(69,100,20,29) (-3/2,-10/7) -> (10/3,7/2) Hyperbolic Matrix(71,100,22,31) (-10/7,-7/5) -> (3/1,10/3) Hyperbolic Matrix(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(17,-20,6,-7) (1/1,4/3) -> (8/3,3/1) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(13,-20,2,-3) (3/2,5/3) -> (5/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,20,-2,-13) -> Matrix(3,2,-20,-13) Matrix(11,40,-8,-29) -> Matrix(11,4,-80,-29) Matrix(7,20,-6,-17) -> Matrix(7,2,-60,-17) Matrix(9,20,4,9) -> Matrix(9,2,40,9) Matrix(11,20,6,11) -> Matrix(11,2,60,11) Matrix(37,60,8,13) -> Matrix(37,6,80,13) Matrix(69,100,20,29) -> Matrix(69,10,200,29) Matrix(71,100,22,31) -> Matrix(71,10,220,31) Matrix(47,60,18,23) -> Matrix(47,6,180,23) Matrix(1,0,2,1) -> Matrix(1,0,20,1) Matrix(17,-20,6,-7) -> Matrix(17,-2,60,-7) Matrix(29,-40,8,-11) -> Matrix(29,-4,80,-11) Matrix(13,-20,2,-3) -> Matrix(13,-2,20,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 72 Minimal number of generators: 13 Number of equivalence classes of cusps: 8 Genus: 3 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 12 Degree of the the map Y: 12 Permutation triple for Y: ((2,6,4,3,7)(5,11,10,9,8); (1,4,11,3,10,12,8,7,5,2)(6,9); (1,2,8,6,5,12,10,4,9,3)(7,11)) ----------------------------------------------------------------------- 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): 18 Minimal number of generators: 5 Number of equivalence classes of elliptic points of order 2: 2 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 4 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 5/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/10 3/2 3/20 5/3 1/6 2/1 1/5 5/2 1/4 3/1 3/10 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(7,-10,5,-7) (1/1,3/2) -> (1/1,3/2) Elliptic Matrix(19,-30,7,-11) (3/2,5/3) -> (5/2,3/1) Hyperbolic Matrix(11,-20,5,-9) (5/3,2/1) -> (2/1,5/2) Parabolic Matrix(3,-10,1,-3) (3/1,1/0) -> (3/1,1/0) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,10,1) Matrix(7,-10,5,-7) -> Matrix(7,-1,50,-7) Matrix(19,-30,7,-11) -> Matrix(19,-3,70,-11) Matrix(11,-20,5,-9) -> Matrix(11,-2,50,-9) Matrix(3,-10,1,-3) -> Matrix(3,-1,10,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 18 Minimal number of generators: 5 Number of equivalence classes of elliptic points of order 2: 2 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 4 Genus: 0 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 10 1 2/1 1/5 2 5 5/2 1/4 5 2 3/1 3/10 1 10 1/0 1/0 1 10 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(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(3,-10,1,-3) (3/1,1/0) -> (3/1,1/0) Elliptic 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(1,0,10,-1) (0/1,2/1) -> (0/1,1/5) Matrix(9,-20,4,-9) -> Matrix(9,-2,40,-9) (2/1,5/2) -> (1/5,1/4) Matrix(11,-30,4,-11) -> Matrix(11,-3,40,-11) (5/2,3/1) -> (1/4,3/10) Matrix(3,-10,1,-3) -> Matrix(3,-1,10,-3) (0/1,1/3).(1/4,1/2) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.