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): 96 Minimal number of generators: 17 Number of equivalence classes of cusps: 12 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -8/3 -2/1 -4/3 0/1 1/1 4/3 2/1 16/7 8/3 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/4 -7/2 -7/32 -3/1 -3/16 -8/3 -1/6 -5/2 -5/32 -7/3 -7/48 -2/1 -1/8 -9/5 -9/80 -16/9 -1/9 -7/4 -7/64 -5/3 -5/48 -8/5 -1/10 -3/2 -3/32 -4/3 -1/12 -5/4 -5/64 -1/1 -1/16 0/1 0/1 1/1 1/16 4/3 1/12 7/5 7/80 3/2 3/32 8/5 1/10 5/3 5/48 7/4 7/64 2/1 1/8 9/4 9/64 16/7 1/7 7/3 7/48 5/2 5/32 8/3 1/6 3/1 3/16 4/1 1/4 5/1 5/16 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,32,-2,-9) (-4/1,1/0) -> (-4/1,-7/2) Parabolic Matrix(19,64,8,27) (-7/2,-3/1) -> (7/3,5/2) Hyperbolic Matrix(23,64,14,39) (-3/1,-8/3) -> (8/5,5/3) Hyperbolic Matrix(25,64,16,41) (-8/3,-5/2) -> (3/2,8/5) Hyperbolic Matrix(13,32,2,5) (-5/2,-7/3) -> (5/1,1/0) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,62,111) (-9/5,-16/9) -> (16/7,7/3) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(37,64,26,45) (-7/4,-5/3) -> (7/5,3/2) Hyperbolic Matrix(39,64,14,23) (-5/3,-8/5) -> (8/3,3/1) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(23,32,-18,-25) (-3/2,-4/3) -> (-4/3,-5/4) Parabolic Matrix(27,32,16,19) (-5/4,-1/1) -> (5/3,7/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,32,-2,-9) -> Matrix(7,2,-32,-9) Matrix(19,64,8,27) -> Matrix(19,4,128,27) Matrix(23,64,14,39) -> Matrix(23,4,224,39) Matrix(25,64,16,41) -> Matrix(25,4,256,41) Matrix(13,32,2,5) -> Matrix(13,2,32,5) Matrix(15,32,-8,-17) -> Matrix(15,2,-128,-17) Matrix(143,256,62,111) -> Matrix(143,16,992,111) Matrix(145,256,64,113) -> Matrix(145,16,1024,113) Matrix(37,64,26,45) -> Matrix(37,4,416,45) Matrix(39,64,14,23) -> Matrix(39,4,224,23) Matrix(41,64,16,25) -> Matrix(41,4,256,25) Matrix(23,32,-18,-25) -> Matrix(23,2,-288,-25) Matrix(27,32,16,19) -> Matrix(27,2,256,19) Matrix(1,0,2,1) -> Matrix(1,0,32,1) Matrix(25,-32,18,-23) -> Matrix(25,-2,288,-23) Matrix(17,-32,8,-15) -> Matrix(17,-2,128,-15) Matrix(9,-32,2,-7) -> Matrix(9,-2,32,-7) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 96 Minimal number of generators: 17 Number of equivalence classes of cusps: 12 Genus: 3 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 16 Degree of the the map Y: 16 Permutation triple for Y: ((2,6,15,7)(3,11,12,4); (1,4,14,6,10,3,9,15,16,11,8,7,13,12,5,2); (1,2,8,11,10,6,5,12,16,15,14,4,13,7,9,3)) ----------------------------------------------------------------------- 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): 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: 6 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 4/3 2/1 8/3 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/16 4/3 1/12 3/2 3/32 2/1 1/8 5/2 5/32 8/3 1/6 3/1 3/16 4/1 1/4 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(13,-16,9,-11) (1/1,4/3) -> (4/3,3/2) Parabolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(25,-64,9,-23) (5/2,8/3) -> (8/3,3/1) Parabolic Matrix(5,-16,1,-3) (3/1,4/1) -> (4/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,16,1) Matrix(13,-16,9,-11) -> Matrix(13,-1,144,-11) Matrix(9,-16,4,-7) -> Matrix(9,-1,64,-7) Matrix(25,-64,9,-23) -> Matrix(25,-4,144,-23) Matrix(5,-16,1,-3) -> Matrix(5,-1,16,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM 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: 6 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 16 1 2/1 1/8 2 8 8/3 1/6 8 2 3/1 3/16 1 16 4/1 1/4 4 4 1/0 1/0 1 16 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(7,-16,3,-7) (2/1,8/3) -> (2/1,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(5,-16,1,-3) (3/1,4/1) -> (4/1,1/0) Parabolic 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(1,0,16,-1) (0/1,2/1) -> (0/1,1/8) Matrix(7,-16,3,-7) -> Matrix(7,-1,48,-7) (2/1,8/3) -> (1/8,1/6) Matrix(17,-48,6,-17) -> Matrix(17,-3,96,-17) (8/3,3/1) -> (1/6,3/16) Matrix(5,-16,1,-3) -> Matrix(5,-1,16,-3) 1/4 ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.