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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 20 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -4/1 -3/1 -2/1 -1/1 -2/3 -1/2 0/1 1/3 1/2 2/3 1/1 3/2 5/3 2/1 3/1 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 0/1 1/1 1/0 -5/1 0/1 1/1 1/0 -4/1 1/1 -3/1 1/1 2/1 1/0 -5/2 2/1 -2/1 1/0 -7/4 0/1 -12/7 0/1 1/1 1/0 -5/3 0/1 1/1 1/0 -3/2 0/1 1/1 1/0 -1/1 1/1 2/1 1/0 -3/4 0/1 1/1 1/0 -5/7 0/1 1/1 1/0 -2/3 1/1 -7/11 1/1 3/2 2/1 -12/19 1/1 3/2 2/1 -5/8 2/1 -3/5 1/1 2/1 1/0 -1/2 1/1 -3/7 1/1 2/1 1/0 -2/5 1/1 -5/13 1/1 3/2 2/1 -3/8 1/1 3/2 2/1 -1/3 1/1 2/1 1/0 0/1 1/1 2/1 1/0 1/3 1/1 2/1 1/0 3/8 1/1 3/2 2/1 2/5 2/1 1/2 2/1 3/5 1/1 2/1 1/0 5/8 1/1 2/3 2/1 1/1 1/1 2/1 1/0 4/3 2/1 11/8 3/1 7/5 2/1 3/1 1/0 3/2 2/1 3/1 1/0 5/3 2/1 3/1 1/0 2/1 1/0 7/3 0/1 1/1 1/0 12/5 0/1 1/1 1/0 5/2 1/1 3/1 1/1 2/1 1/0 7/2 2/1 4/1 2/1 9/2 2/1 3/1 1/0 5/1 2/1 3/1 1/0 11/2 3/1 6/1 2/1 3/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(19,102,-30,-161) (-6/1,-5/1) -> (-7/11,-12/19) Hyperbolic Matrix(7,30,-18,-77) (-5/1,-4/1) -> (-2/5,-5/13) Hyperbolic Matrix(5,18,-12,-43) (-4/1,-3/1) -> (-3/7,-2/5) Hyperbolic Matrix(7,18,12,31) (-3/1,-5/2) -> (1/2,3/5) Hyperbolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic Matrix(83,144,-132,-229) (-7/4,-12/7) -> (-12/19,-5/8) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(5,6,-6,-7) (-3/2,-1/1) -> (-1/1,-3/4) Parabolic Matrix(41,30,-108,-79) (-3/4,-5/7) -> (-5/13,-3/8) Hyperbolic Matrix(35,24,-54,-37) (-5/7,-2/3) -> (-2/3,-7/11) Parabolic Matrix(59,36,18,11) (-5/8,-3/5) -> (3/1,7/2) Hyperbolic Matrix(31,18,12,7) (-3/5,-1/2) -> (5/2,3/1) Hyperbolic Matrix(41,18,66,29) (-1/2,-3/7) -> (3/5,5/8) Hyperbolic Matrix(17,6,48,17) (-3/8,-1/3) -> (1/3,3/8) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(77,-30,18,-7) (3/8,2/5) -> (4/1,9/2) Hyperbolic Matrix(43,-18,12,-5) (2/5,1/2) -> (7/2,4/1) Hyperbolic Matrix(65,-42,48,-31) (5/8,2/3) -> (4/3,11/8) Hyperbolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(95,-132,18,-25) (11/8,7/5) -> (5/1,11/2) Hyperbolic Matrix(55,-78,12,-17) (7/5,3/2) -> (9/2,5/1) Hyperbolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(67,-162,12,-29) (12/5,5/2) -> (11/2,6/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,2,0,1) Matrix(19,102,-30,-161) -> Matrix(3,-2,2,-1) Matrix(7,30,-18,-77) -> Matrix(3,-2,2,-1) Matrix(5,18,-12,-43) -> Matrix(1,0,0,1) Matrix(7,18,12,31) -> Matrix(1,0,0,1) Matrix(11,24,-6,-13) -> Matrix(1,-2,0,1) Matrix(83,144,-132,-229) -> Matrix(3,-2,2,-1) Matrix(85,144,36,61) -> Matrix(1,0,0,1) Matrix(19,30,12,19) -> Matrix(1,2,0,1) Matrix(5,6,-6,-7) -> Matrix(1,0,0,1) Matrix(41,30,-108,-79) -> Matrix(3,-2,2,-1) Matrix(35,24,-54,-37) -> Matrix(3,-2,2,-1) Matrix(59,36,18,11) -> Matrix(1,0,0,1) Matrix(31,18,12,7) -> Matrix(1,0,0,1) Matrix(41,18,66,29) -> Matrix(1,0,0,1) Matrix(17,6,48,17) -> Matrix(1,0,0,1) Matrix(1,0,6,1) -> Matrix(1,0,0,1) Matrix(77,-30,18,-7) -> Matrix(5,-8,2,-3) Matrix(43,-18,12,-5) -> Matrix(1,0,0,1) Matrix(65,-42,48,-31) -> Matrix(5,-8,2,-3) Matrix(7,-6,6,-5) -> Matrix(1,0,0,1) Matrix(95,-132,18,-25) -> Matrix(1,0,0,1) Matrix(55,-78,12,-17) -> Matrix(1,0,0,1) Matrix(13,-24,6,-11) -> Matrix(1,-2,0,1) Matrix(67,-162,12,-29) -> Matrix(1,2,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 3 Degree of the the map X: 3 Degree of the the map Y: 24 Permutation triple for Y: ((1,7,2)(3,11,12)(4,16,5)(6,17,18)(8,20,13)(9,21,10)(14,19,15)(22,24,23); (1,5,6)(2,10,3)(4,15,9)(7,19,8)(11,16,22)(12,18,13)(14,24,17)(20,21,23); (1,3,13,23,14,4)(2,8,18,24,16,9)(5,11,10,20,19,17)(6,12,22,21,15,7)) ----------------------------------------------------------------------- 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 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- 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): 48 Minimal number of generators: 9 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: 10 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/1 -2/1 0/1 1/1 3/2 2/1 3/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 1/1 2/1 1/0 -5/2 2/1 -2/1 1/0 -7/4 0/1 -12/7 0/1 1/1 1/0 -5/3 0/1 1/1 1/0 -3/2 0/1 1/1 1/0 -1/1 1/1 2/1 1/0 0/1 1/1 2/1 1/0 1/1 1/1 2/1 1/0 3/2 2/1 3/1 1/0 5/3 2/1 3/1 1/0 2/1 1/0 3/1 1/1 2/1 1/0 4/1 2/1 5/1 2/1 3/1 1/0 6/1 2/1 3/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,18,-2,-7) (-3/1,1/0) -> (-3/1,-5/2) Parabolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic Matrix(31,54,4,7) (-7/4,-12/7) -> (6/1,1/0) Hyperbolic Matrix(53,90,10,17) (-12/7,-5/3) -> (5/1,6/1) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(5,6,4,5) (-3/2,-1/1) -> (1/1,3/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(17,-30,4,-7) (5/3,2/1) -> (4/1,5/1) Hyperbolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,18,-2,-7) -> Matrix(2,-3,1,-1) Matrix(11,24,-6,-13) -> Matrix(1,-2,0,1) Matrix(31,54,4,7) -> Matrix(3,-1,1,0) Matrix(53,90,10,17) -> Matrix(2,-3,1,-1) Matrix(19,30,12,19) -> Matrix(1,2,0,1) Matrix(5,6,4,5) -> Matrix(2,-3,1,-1) Matrix(1,0,2,1) -> Matrix(2,-3,1,-1) Matrix(17,-30,4,-7) -> Matrix(2,-7,1,-3) Matrix(7,-18,2,-5) -> Matrix(2,-3,1,-1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 2 Minimal number of generators: 2 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: 1 Genus: 0 Degree of H/liftables -> H/(image of liftables): 3 Alas! An error occurred while computing the image of the pullback map: abort the drawing of the pullback map. ----------------------------------------------------------------------- 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 (1/1,3/1).(3/2,1/0).(0/1,2/1) 0 3 1/1 (1/1,3/1) 0 6 3/2 (2/1,4/1).(5/2,1/0).(1/1,3/1) 0 6 5/3 (5/2,1/0) 0 6 2/1 1/0 1 3 3/1 (1/1,3/1).(3/2,1/0).(0/1,2/1) 0 6 4/1 2/1 1 3 5/1 (2/1,4/1) 0 6 6/1 (2/1,4/1).(5/2,1/0).(1/1,3/1) 0 3 1/0 1/0 1 6 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(5,-6,4,-5) (1/1,3/2) -> (1/1,3/2) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(17,-30,4,-7) (5/3,2/1) -> (4/1,5/1) Hyperbolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,3,0,1) (0/1,1/0) -> (3/2,1/0) Matrix(1,0,2,-1) -> Matrix(2,-3,1,-2) (0/1,1/1) -> (1/1,3/1) Matrix(5,-6,4,-5) -> Matrix(2,-3,1,-2) (1/1,3/2) -> (1/1,3/1) Matrix(19,-30,12,-19) -> Matrix(-1,5,0,1) (3/2,5/3) -> (5/2,1/0) Matrix(17,-30,4,-7) -> Matrix(2,-7,1,-3) (2/1,4/1).(5/2,1/0).(1/1,3/1) Matrix(7,-18,2,-5) -> Matrix(2,-3,1,-1) (1/1,3/1).(3/2,1/0).(0/1,2/1) Matrix(11,-60,2,-11) -> Matrix(3,-8,1,-3) (5/1,6/1) -> (2/1,4/1) Matrix(-1,12,0,1) -> Matrix(-1,5,0,1) (6/1,1/0) -> (5/2,1/0)