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): 120 Minimal number of generators: 21 Number of equivalence classes of cusps: 6 Genus: 8 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 2/1 19/8 19/7 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -5/19 -4/1 -4/19 -7/2 -7/38 -3/1 -3/19 -8/3 -8/57 -5/2 -5/38 -7/3 -7/57 -2/1 -2/19 -7/4 -7/76 -19/11 -1/11 -12/7 -12/133 -5/3 -5/57 -8/5 -8/95 -19/12 -1/12 -11/7 -11/133 -3/2 -3/38 -7/5 -7/95 -4/3 -4/57 -5/4 -5/76 -1/1 -1/19 0/1 0/1 1/1 1/19 5/4 5/76 4/3 4/57 7/5 7/95 3/2 3/38 8/5 8/95 5/3 5/57 7/4 7/76 2/1 2/19 7/3 7/57 19/8 1/8 12/5 12/95 5/2 5/38 8/3 8/57 19/7 1/7 11/4 11/76 3/1 3/19 7/2 7/38 4/1 4/19 5/1 5/19 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,38,2,11) (-5/1,1/0) -> (3/1,7/2) Hyperbolic Matrix(9,38,4,17) (-5/1,-4/1) -> (2/1,7/3) Hyperbolic Matrix(21,76,8,29) (-4/1,-7/2) -> (5/2,8/3) Hyperbolic Matrix(11,38,2,7) (-7/2,-3/1) -> (5/1,1/0) Hyperbolic Matrix(27,76,-16,-45) (-3/1,-8/3) -> (-12/7,-5/3) Hyperbolic Matrix(29,76,8,21) (-8/3,-5/2) -> (7/2,4/1) Hyperbolic Matrix(31,76,-20,-49) (-5/2,-7/3) -> (-11/7,-3/2) Hyperbolic Matrix(17,38,4,9) (-7/3,-2/1) -> (4/1,5/1) Hyperbolic Matrix(21,38,16,29) (-2/1,-7/4) -> (5/4,4/3) Hyperbolic Matrix(197,342,72,125) (-7/4,-19/11) -> (19/7,11/4) Hyperbolic Matrix(221,380,82,141) (-19/11,-12/7) -> (8/3,19/7) Hyperbolic Matrix(47,76,34,55) (-5/3,-8/5) -> (4/3,7/5) Hyperbolic Matrix(239,380,100,159) (-8/5,-19/12) -> (19/8,12/5) Hyperbolic Matrix(217,342,92,145) (-19/12,-11/7) -> (7/3,19/8) Hyperbolic Matrix(27,38,22,31) (-3/2,-7/5) -> (1/1,5/4) Hyperbolic Matrix(55,76,34,47) (-7/5,-4/3) -> (8/5,5/3) Hyperbolic Matrix(29,38,16,21) (-4/3,-5/4) -> (7/4,2/1) Hyperbolic Matrix(31,38,22,27) (-5/4,-1/1) -> (7/5,3/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(49,-76,20,-31) (3/2,8/5) -> (12/5,5/2) Hyperbolic Matrix(45,-76,16,-27) (5/3,7/4) -> (11/4,3/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,38,2,11) -> Matrix(7,2,38,11) Matrix(9,38,4,17) -> Matrix(9,2,76,17) Matrix(21,76,8,29) -> Matrix(21,4,152,29) Matrix(11,38,2,7) -> Matrix(11,2,38,7) Matrix(27,76,-16,-45) -> Matrix(27,4,-304,-45) Matrix(29,76,8,21) -> Matrix(29,4,152,21) Matrix(31,76,-20,-49) -> Matrix(31,4,-380,-49) Matrix(17,38,4,9) -> Matrix(17,2,76,9) Matrix(21,38,16,29) -> Matrix(21,2,304,29) Matrix(197,342,72,125) -> Matrix(197,18,1368,125) Matrix(221,380,82,141) -> Matrix(221,20,1558,141) Matrix(47,76,34,55) -> Matrix(47,4,646,55) Matrix(239,380,100,159) -> Matrix(239,20,1900,159) Matrix(217,342,92,145) -> Matrix(217,18,1748,145) Matrix(27,38,22,31) -> Matrix(27,2,418,31) Matrix(55,76,34,47) -> Matrix(55,4,646,47) Matrix(29,38,16,21) -> Matrix(29,2,304,21) Matrix(31,38,22,27) -> Matrix(31,2,418,27) Matrix(1,0,2,1) -> Matrix(1,0,38,1) Matrix(49,-76,20,-31) -> Matrix(49,-4,380,-31) Matrix(45,-76,16,-27) -> Matrix(45,-4,304,-27) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 120 Minimal number of generators: 21 Number of equivalence classes of cusps: 6 Genus: 8 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 20 Degree of the the map Y: 20 Permutation triple for Y: ((2,6,16,15,17,20,14,5,13,4,3,12,8,11,19,18,10,9,7); (1,4,16,14,13,11,3,10,12,20,17,9,8,7,18,6,15,5,2); (1,2,8,10,7,17,6,5,16,18,19,13,15,4,14,12,11,9,3)) ----------------------------------------------------------------------- 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): 20 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: 2 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 0/1 1/1 1/19 4/3 4/57 3/2 3/38 2/1 2/19 3/1 3/19 4/1 4/19 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(15,-19,4,-5) (1/1,4/3) -> (3/1,4/1) Hyperbolic Matrix(14,-19,3,-4) (4/3,3/2) -> (4/1,1/0) Hyperbolic Matrix(12,-19,7,-11) (3/2,7/4) -> (8/5,2/1) Elliptic Matrix(8,-19,3,-7) (2/1,8/3) -> (7/3,3/1) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,19,1) Matrix(15,-19,4,-5) -> Matrix(15,-1,76,-5) Matrix(14,-19,3,-4) -> Matrix(14,-1,57,-4) Matrix(12,-19,7,-11) -> Matrix(12,-1,133,-11) Matrix(8,-19,3,-7) -> Matrix(8,-1,57,-7) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 20 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: 2 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 0/1 19 1 2/1 2/19 1 19 3/1 3/19 1 19 4/1 4/19 1 19 1/0 1/0 1 19 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(8,-19,3,-7) (2/1,8/3) -> (7/3,3/1) Elliptic Matrix(5,-19,1,-4) (3/1,4/1) -> (4/1,1/0) Glide 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(1,0,19,-1) (0/1,2/1) -> (0/1,2/19) Matrix(8,-19,3,-7) -> Matrix(8,-1,57,-7) (1/8,1/6).(0/1,2/15).(1/9,1/7) Matrix(5,-19,1,-4) -> Matrix(5,-1,19,-4) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.