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: 16 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -2/1 -1/1 -1/2 0/1 1/3 1/2 2/3 1/1 4/3 3/2 2/1 3/1 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 1/1 2/1 1/0 -5/1 1/1 -4/1 2/1 -3/1 1/0 -8/3 1/1 2/1 1/0 -5/2 2/1 -2/1 1/1 2/1 1/0 -5/3 1/1 2/1 1/0 -8/5 2/1 -3/2 1/0 -4/3 1/1 2/1 1/0 -5/4 1/1 -6/5 1/1 2/1 1/0 -1/1 2/1 -4/5 3/1 -3/4 1/0 -2/3 2/1 3/1 1/0 -3/5 1/0 -4/7 2/1 -5/9 2/1 5/2 3/1 -6/11 2/1 5/2 3/1 -1/2 3/1 -2/5 3/1 4/1 1/0 -1/3 3/1 4/1 1/0 0/1 1/0 1/3 -1/1 0/1 1/0 2/5 -1/1 0/1 1/0 1/2 0/1 3/5 1/0 2/3 0/1 1/1 1/0 3/4 1/0 4/5 0/1 1/1 1/1 6/5 1/1 2/1 1/0 5/4 2/1 4/3 1/1 2/1 1/0 3/2 1/0 8/5 1/1 5/3 1/1 2/1 1/0 2/1 1/1 2/1 1/0 5/2 1/1 8/3 1/1 2/1 1/0 3/1 1/0 7/2 0/1 18/5 0/1 1/2 1/1 11/3 0/1 1/2 1/1 4/1 1/1 5/1 2/1 6/1 1/1 2/1 1/0 1/0 1/1 2/1 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(7,36,6,31) (-6/1,-5/1) -> (1/1,6/5) Hyperbolic Matrix(5,24,6,29) (-5/1,-4/1) -> (4/5,1/1) Hyperbolic Matrix(7,24,-12,-41) (-4/1,-3/1) -> (-3/5,-4/7) Hyperbolic Matrix(17,48,6,17) (-3/1,-8/3) -> (8/3,3/1) Hyperbolic Matrix(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) Hyperbolic Matrix(5,12,12,29) (-5/2,-2/1) -> (2/5,1/2) Hyperbolic Matrix(7,12,18,31) (-2/1,-5/3) -> (1/3,2/5) Hyperbolic Matrix(37,60,-66,-107) (-5/3,-8/5) -> (-4/7,-5/9) Hyperbolic Matrix(23,36,30,47) (-8/5,-3/2) -> (3/4,4/5) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(29,36,-54,-67) (-5/4,-6/5) -> (-6/11,-1/2) Hyperbolic Matrix(31,36,6,7) (-6/5,-1/1) -> (5/1,6/1) Hyperbolic Matrix(29,24,6,5) (-1/1,-4/5) -> (4/1,5/1) Hyperbolic Matrix(47,36,30,23) (-4/5,-3/4) -> (3/2,8/5) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(19,12,30,19) (-2/3,-3/5) -> (3/5,2/3) Hyperbolic Matrix(283,156,78,43) (-5/9,-6/11) -> (18/5,11/3) Hyperbolic Matrix(29,12,12,5) (-1/2,-2/5) -> (2/1,5/2) Hyperbolic Matrix(31,12,18,7) (-2/5,-1/3) -> (5/3,2/1) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(41,-24,12,-7) (1/2,3/5) -> (3/1,7/2) Hyperbolic Matrix(107,-132,30,-37) (6/5,5/4) -> (7/2,18/5) Hyperbolic Matrix(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,0,0,1) Matrix(7,36,6,31) -> Matrix(1,0,0,1) Matrix(5,24,6,29) -> Matrix(1,-2,2,-3) Matrix(7,24,-12,-41) -> Matrix(1,0,0,1) Matrix(17,48,6,17) -> Matrix(1,0,0,1) Matrix(23,60,18,47) -> Matrix(1,0,0,1) Matrix(5,12,12,29) -> Matrix(1,-2,0,1) Matrix(7,12,18,31) -> Matrix(1,-2,0,1) Matrix(37,60,-66,-107) -> Matrix(5,-8,2,-3) Matrix(23,36,30,47) -> Matrix(1,-2,0,1) Matrix(17,24,12,17) -> Matrix(1,0,0,1) Matrix(47,60,18,23) -> Matrix(1,0,0,1) Matrix(29,36,-54,-67) -> Matrix(5,-8,2,-3) Matrix(31,36,6,7) -> Matrix(1,0,0,1) Matrix(29,24,6,5) -> Matrix(3,-8,2,-5) Matrix(47,36,30,23) -> Matrix(1,-2,0,1) Matrix(17,12,24,17) -> Matrix(1,-2,0,1) Matrix(19,12,30,19) -> Matrix(1,-2,0,1) Matrix(283,156,78,43) -> Matrix(1,-2,0,1) Matrix(29,12,12,5) -> Matrix(1,-2,0,1) Matrix(31,12,18,7) -> Matrix(1,-2,0,1) Matrix(1,0,6,1) -> Matrix(1,-4,0,1) Matrix(41,-24,12,-7) -> Matrix(1,0,0,1) Matrix(107,-132,30,-37) -> Matrix(1,-2,2,-3) Matrix(67,-108,18,-29) -> Matrix(1,-2,2,-3) 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): 4 Degree of the the map X: 4 Degree of the the map Y: 24 Permutation triple for Y: ((1,7,2)(3,12,13)(4,8,5)(6,10,9)(11,17,19)(14,21,20)(15,18,16)(22,23,24); (1,5,19,23,20,6)(2,10,15,22,11,3)(4,16,9,14,13,17)(7,12,21,24,18,8); (1,3,14,23,15,4)(2,8,17,22,21,9)(5,18,10,20,12,11)(6,16,24,19,13,7)) ----------------------------------------------------------------------- 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): 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 2/3 1/1 4/3 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 1/1 2/1 1/0 -1/1 2/1 0/1 1/0 1/2 0/1 2/3 0/1 1/1 1/0 1/1 1/1 4/3 1/1 2/1 1/0 3/2 1/0 2/1 1/1 2/1 1/0 1/0 1/1 2/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(5,8,3,5) (-2/1,-1/1) -> (3/2,2/1) Hyperbolic Matrix(1,0,3,1) (-1/1,0/1) -> (0/1,1/2) Parabolic Matrix(7,-4,9,-5) (1/2,2/3) -> (2/3,1/1) Parabolic Matrix(13,-16,9,-11) (1/1,4/3) -> (4/3,3/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,-3,1,-2) Matrix(5,8,3,5) -> Matrix(1,-3,1,-2) Matrix(1,0,3,1) -> Matrix(1,-2,0,1) Matrix(7,-4,9,-5) -> Matrix(0,1,-1,1) Matrix(13,-16,9,-11) -> 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): 2 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 This is a reflection group. CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 1/0 2 3 2/3 (0/1,2/1).(1/2,1/0).(-1/1,1/1) 0 3 1/1 1/1 1 6 4/3 (1/1,3/1).(3/2,1/0).(0/1,2/1) 0 3 2/1 (0/1,2/1) 0 3 1/0 (1/1,3/1).(3/2,1/0).(0/1,2/1) 0 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(1,0,3,-1) (0/1,2/3) -> (0/1,2/3) Reflection Matrix(5,-4,6,-5) (2/3,1/1) -> (2/3,1/1) Reflection Matrix(7,-8,6,-7) (1/1,4/3) -> (1/1,4/3) Reflection Matrix(5,-8,3,-5) (4/3,2/1) -> (4/3,2/1) Reflection Matrix(-1,4,0,1) (2/1,1/0) -> (2/1,1/0) 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(1,0,3,-1) -> Matrix(-1,1,0,1) (0/1,2/3) -> (1/2,1/0) Matrix(5,-4,6,-5) -> Matrix(0,1,1,0) (2/3,1/1) -> (-1/1,1/1) Matrix(7,-8,6,-7) -> Matrix(2,-3,1,-2) (1/1,4/3) -> (1/1,3/1) Matrix(5,-8,3,-5) -> Matrix(1,0,1,-1) (4/3,2/1) -> (0/1,2/1) Matrix(-1,4,0,1) -> Matrix(1,0,1,-1) (2/1,1/0) -> (0/1,2/1)