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): 84 Minimal number of generators: 15 Number of equivalence classes of cusps: 6 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 13/8 2/1 13/5 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -5/13 -4/1 -4/13 -3/1 -3/13 -8/3 -8/39 -13/5 -1/5 -5/2 -5/26 -2/1 -2/13 -5/3 -5/39 -13/8 -1/8 -8/5 -8/65 -3/2 -3/26 -4/3 -4/39 -5/4 -5/52 -1/1 -1/13 0/1 0/1 1/1 1/13 5/4 5/52 4/3 4/39 3/2 3/26 8/5 8/65 13/8 1/8 5/3 5/39 2/1 2/13 5/2 5/26 13/5 1/5 8/3 8/39 3/1 3/13 4/1 4/13 5/1 5/13 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,26,4,21) (-5/1,1/0) -> (1/1,5/4) Hyperbolic Matrix(11,52,4,19) (-5/1,-4/1) -> (8/3,3/1) Hyperbolic Matrix(7,26,4,15) (-4/1,-3/1) -> (5/3,2/1) Hyperbolic Matrix(19,52,4,11) (-3/1,-8/3) -> (4/1,5/1) Hyperbolic Matrix(79,208,30,79) (-8/3,-13/5) -> (13/5,8/3) Hyperbolic Matrix(51,130,20,51) (-13/5,-5/2) -> (5/2,13/5) Hyperbolic Matrix(11,26,8,19) (-5/2,-2/1) -> (4/3,3/2) Hyperbolic Matrix(15,26,4,7) (-2/1,-5/3) -> (3/1,4/1) Hyperbolic Matrix(79,130,48,79) (-5/3,-13/8) -> (13/8,5/3) Hyperbolic Matrix(129,208,80,129) (-13/8,-8/5) -> (8/5,13/8) Hyperbolic Matrix(33,52,26,41) (-8/5,-3/2) -> (5/4,4/3) Hyperbolic Matrix(19,26,8,11) (-3/2,-4/3) -> (2/1,5/2) Hyperbolic Matrix(41,52,26,33) (-4/3,-5/4) -> (3/2,8/5) Hyperbolic Matrix(21,26,4,5) (-5/4,-1/1) -> (5/1,1/0) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,26,4,21) -> Matrix(5,2,52,21) Matrix(11,52,4,19) -> Matrix(11,4,52,19) Matrix(7,26,4,15) -> Matrix(7,2,52,15) Matrix(19,52,4,11) -> Matrix(19,4,52,11) Matrix(79,208,30,79) -> Matrix(79,16,390,79) Matrix(51,130,20,51) -> Matrix(51,10,260,51) Matrix(11,26,8,19) -> Matrix(11,2,104,19) Matrix(15,26,4,7) -> Matrix(15,2,52,7) Matrix(79,130,48,79) -> Matrix(79,10,624,79) Matrix(129,208,80,129) -> Matrix(129,16,1040,129) Matrix(33,52,26,41) -> Matrix(33,4,338,41) Matrix(19,26,8,11) -> Matrix(19,2,104,11) Matrix(41,52,26,33) -> Matrix(41,4,338,33) Matrix(21,26,4,5) -> Matrix(21,2,52,5) Matrix(1,0,2,1) -> Matrix(1,0,26,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 84 Minimal number of generators: 15 Number of equivalence classes of cusps: 6 Genus: 5 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 14 Degree of the the map Y: 14 Permutation triple for Y: ((2,6,9,14,10,8,4,3,5,11,13,12,7); (1,4,12,10,3,6,13,11,8,7,9,5,2); (1,2,8,12,6,5,10,14,7,4,11,9,3)) ----------------------------------------------------------------------- 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): 14 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: 2 Number of equivalence classes of cusps: 2 Genus: 0 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 1/13 3/2 3/26 2/1 2/13 3/1 3/13 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(10,-13,7,-9) (1/1,7/5) -> (5/4,3/2) Elliptic Matrix(8,-13,5,-8) (3/2,2/1) -> (3/2,2/1) Elliptic Matrix(5,-13,2,-5) (2/1,3/1) -> (2/1,3/1) Elliptic Matrix(4,-13,1,-3) (3/1,5/1) -> (7/2,1/0) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,13,1) Matrix(10,-13,7,-9) -> Matrix(10,-1,91,-9) Matrix(8,-13,5,-8) -> Matrix(8,-1,65,-8) Matrix(5,-13,2,-5) -> Matrix(5,-1,26,-5) Matrix(4,-13,1,-3) -> Matrix(4,-1,13,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 14 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: 2 Number of equivalence classes of cusps: 2 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 13 1 2/1 2/13 1 13 3/1 3/13 1 13 1/0 1/0 1 13 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,-13,2,-5) (2/1,3/1) -> (2/1,3/1) Elliptic Matrix(4,-13,1,-3) (3/1,5/1) -> (7/2,1/0) 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(1,0,13,-1) (0/1,2/1) -> (0/1,2/13) Matrix(5,-13,2,-5) -> Matrix(5,-1,26,-5) (0/1,1/5).(1/6,1/4) Matrix(4,-13,1,-3) -> Matrix(4,-1,13,-3) (1/4,1/2).(0/1,2/7).(1/5,1/3) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.