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: 6 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/11 -3/11 0/1 1/2 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -3/4 -4/11 -2/3 -3/11 -3/5 -5/22 -1/2 -2/11 -2/5 -5/33 -3/8 -8/55 -4/11 -1/7 -1/3 -3/22 -2/7 -7/55 -3/11 -1/8 -1/4 -4/33 0/1 -1/11 1/4 -4/55 1/3 -3/44 2/5 -5/77 1/2 -2/33 3/5 -5/88 5/8 -8/143 7/11 -1/18 2/3 -3/55 5/7 -7/132 8/11 -1/19 3/4 -4/77 1/1 -1/22 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(13,10,22,17) (-1/1,-3/4) -> (1/2,3/5) Hyperbolic Matrix(17,12,-44,-31) (-3/4,-2/3) -> (-2/5,-3/8) Hyperbolic Matrix(13,8,-44,-27) (-2/3,-3/5) -> (-1/3,-2/7) Hyperbolic Matrix(17,10,22,13) (-3/5,-1/2) -> (3/4,1/1) Hyperbolic Matrix(5,2,22,9) (-1/2,-2/5) -> (0/1,1/4) Hyperbolic Matrix(97,36,132,49) (-3/8,-4/11) -> (8/11,3/4) Hyperbolic Matrix(79,28,110,39) (-4/11,-1/3) -> (5/7,8/11) Hyperbolic Matrix(71,20,110,31) (-2/7,-3/11) -> (7/11,2/3) Hyperbolic Matrix(83,22,132,35) (-3/11,-1/4) -> (5/8,7/11) Hyperbolic Matrix(9,2,22,5) (-1/4,0/1) -> (2/5,1/2) Hyperbolic Matrix(27,-8,44,-13) (1/4,1/3) -> (3/5,5/8) Hyperbolic Matrix(31,-12,44,-17) (1/3,2/5) -> (2/3,5/7) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-22,1) Matrix(13,10,22,17) -> Matrix(5,2,-88,-35) Matrix(17,12,-44,-31) -> Matrix(13,4,-88,-27) Matrix(13,8,-44,-27) -> Matrix(17,4,-132,-31) Matrix(17,10,22,13) -> Matrix(9,2,-176,-39) Matrix(5,2,22,9) -> Matrix(13,2,-176,-27) Matrix(97,36,132,49) -> Matrix(83,12,-1584,-229) Matrix(79,28,110,39) -> Matrix(71,10,-1342,-189) Matrix(71,20,110,31) -> Matrix(79,10,-1430,-181) Matrix(83,22,132,35) -> Matrix(97,12,-1738,-215) Matrix(9,2,22,5) -> Matrix(17,2,-264,-31) Matrix(27,-8,44,-13) -> Matrix(57,4,-1012,-71) Matrix(31,-12,44,-17) -> Matrix(61,4,-1144,-75) 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: 6 Genus: 4 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: ((1,4,6,8,7,11,10,3,9,5,2); (1,2,8,10,4,9,12,6,5,7,3); (2,6,10,11,5,4,3,8,12,9,7)) ----------------------------------------------------------------------- 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): 12 Minimal number of generators: 3 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: 2 Genus: 1 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 -1/11 1/3 -3/44 1/2 -2/33 2/3 -3/55 1/1 -1/22 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(7,-2,11,-3) (0/1,1/3) -> (1/2,2/3) Hyperbolic Matrix(8,-3,11,-4) (1/3,1/2) -> (2/3,1/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-11,1) Matrix(7,-2,11,-3) -> Matrix(14,1,-253,-18) Matrix(8,-3,11,-4) -> Matrix(15,1,-286,-19) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 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: 2 Genus: 1 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 -1/11 1 11 1/3 -3/44 1 11 1/2 -2/33 1 11 1/0 0/1 11 1 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(4,-1,11,-3) (0/1,1/3) -> (1/3,1/2) Glide Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,22,1) (0/1,1/0) -> (-1/11,0/1) Matrix(4,-1,11,-3) -> Matrix(14,1,-209,-15) Matrix(-1,1,0,1) -> Matrix(-1,0,33,1) (1/2,1/0) -> (-2/33,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.