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: 8 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 2/1 7/3 7/2 4/1 14/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 0/1 -7/2 -1/3 0/1 -3/1 -1/2 -8/3 -2/7 -5/2 -2/7 -3/11 -7/3 -1/4 -2/1 -1/5 -7/4 -1/5 0/1 -5/3 -1/6 -3/2 -1/5 -2/11 -7/5 -1/6 -4/3 -2/13 -9/7 -5/34 -14/11 -1/7 -5/4 -1/7 -2/15 -1/1 -1/8 0/1 0/1 1/1 1/6 4/3 2/9 7/5 1/4 3/2 2/7 1/3 8/5 2/5 5/3 1/4 7/4 0/1 1/3 2/1 1/3 7/3 1/2 5/2 3/5 2/3 3/1 1/0 7/2 0/1 1/1 4/1 0/1 9/2 4/5 1/1 14/3 1/1 5/1 3/2 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,28,-2,-11) (-4/1,1/0) -> (-8/3,-5/2) Hyperbolic Matrix(15,56,4,15) (-4/1,-7/2) -> (7/2,4/1) Hyperbolic Matrix(17,56,10,33) (-7/2,-3/1) -> (5/3,7/4) Hyperbolic Matrix(31,84,-24,-65) (-3/1,-8/3) -> (-4/3,-9/7) Hyperbolic Matrix(23,56,16,39) (-5/2,-7/3) -> (7/5,3/2) Hyperbolic Matrix(13,28,6,13) (-7/3,-2/1) -> (2/1,7/3) Hyperbolic Matrix(15,28,8,15) (-2/1,-7/4) -> (7/4,2/1) Hyperbolic Matrix(33,56,10,17) (-7/4,-5/3) -> (3/1,7/2) Hyperbolic Matrix(17,28,-14,-23) (-5/3,-3/2) -> (-5/4,-1/1) Hyperbolic Matrix(39,56,16,23) (-3/2,-7/5) -> (7/3,5/2) Hyperbolic Matrix(41,56,30,41) (-7/5,-4/3) -> (4/3,7/5) Hyperbolic Matrix(153,196,32,41) (-9/7,-14/11) -> (14/3,5/1) Hyperbolic Matrix(155,196,34,43) (-14/11,-5/4) -> (9/2,14/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(23,-28,14,-17) (1/1,4/3) -> (8/5,5/3) Hyperbolic Matrix(53,-84,12,-19) (3/2,8/5) -> (4/1,9/2) Hyperbolic Matrix(11,-28,2,-5) (5/2,3/1) -> (5/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,28,-2,-11) -> Matrix(5,2,-18,-7) Matrix(15,56,4,15) -> Matrix(1,0,4,1) Matrix(17,56,10,33) -> Matrix(1,0,6,1) Matrix(31,84,-24,-65) -> Matrix(13,4,-88,-27) Matrix(23,56,16,39) -> Matrix(15,4,56,15) Matrix(13,28,6,13) -> Matrix(9,2,22,5) Matrix(15,28,8,15) -> Matrix(1,0,8,1) Matrix(33,56,10,17) -> Matrix(1,0,6,1) Matrix(17,28,-14,-23) -> Matrix(1,0,-2,1) Matrix(39,56,16,23) -> Matrix(23,4,40,7) Matrix(41,56,30,41) -> Matrix(25,4,106,17) Matrix(153,196,32,41) -> Matrix(55,8,48,7) Matrix(155,196,34,43) -> Matrix(43,6,50,7) Matrix(1,0,2,1) -> Matrix(1,0,14,1) Matrix(23,-28,14,-17) -> Matrix(1,0,-2,1) Matrix(53,-84,12,-19) -> Matrix(5,-2,8,-3) Matrix(11,-28,2,-5) -> Matrix(3,-2,2,-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): 8 Degree of the the map X: 8 Degree of the the map Y: 16 Permutation triple for Y: ((2,6,12,4,3,11,7)(5,15,8,10,9,13,14); (1,4,13,12,8,7,15,16,9,3,10,6,5,2)(11,14); (1,2,8,7,14,4,13,16,15,6,5,11,9,3)(10,12)) ----------------------------------------------------------------------- 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: 6 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 2/1 7/3 7/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -1/2 -2/1 -1/5 -7/4 -1/5 0/1 -5/3 -1/6 -3/2 -1/5 -2/11 -7/5 -1/6 -4/3 -2/13 -1/1 -1/8 0/1 0/1 1/1 1/6 3/2 2/7 1/3 2/1 1/3 7/3 1/2 5/2 3/5 2/3 3/1 1/0 7/2 0/1 1/1 4/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,14,-2,-9) (-3/1,1/0) -> (-5/3,-3/2) Hyperbolic Matrix(5,14,-4,-11) (-3/1,-2/1) -> (-4/3,-1/1) Hyperbolic Matrix(23,42,6,11) (-2/1,-7/4) -> (7/2,4/1) Hyperbolic Matrix(33,56,10,17) (-7/4,-5/3) -> (3/1,7/2) Hyperbolic Matrix(39,56,16,23) (-3/2,-7/5) -> (7/3,5/2) Hyperbolic Matrix(31,42,14,19) (-7/5,-4/3) -> (2/1,7/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(11,-14,4,-5) (1/1,3/2) -> (5/2,3/1) Hyperbolic Matrix(9,-14,2,-3) (3/2,2/1) -> (4/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,14,-2,-9) -> Matrix(3,1,-16,-5) Matrix(5,14,-4,-11) -> Matrix(3,1,-22,-7) Matrix(23,42,6,11) -> Matrix(5,1,4,1) Matrix(33,56,10,17) -> Matrix(1,0,6,1) Matrix(39,56,16,23) -> Matrix(23,4,40,7) Matrix(31,42,14,19) -> Matrix(19,3,44,7) Matrix(1,0,2,1) -> Matrix(1,0,14,1) Matrix(11,-14,4,-5) -> Matrix(5,-1,6,-1) Matrix(9,-14,2,-3) -> Matrix(3,-1,4,-1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 8 ----------------------------------------------------------------------- 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 7 1 1/1 1/6 1 14 3/2 (1/4,3/10).(2/7,1/3) 0 14 2/1 1/3 1 7 7/3 1/2 7 2 5/2 (1/2,5/8).(3/5,2/3) 0 14 3/1 1/0 1 14 7/2 (0/1,1/1).(1/2,1/0) 0 2 4/1 0/1 1 7 1/0 (-1/1,0/1).(-1/2,1/0) 0 14 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(11,-14,4,-5) (1/1,3/2) -> (5/2,3/1) Hyperbolic Matrix(9,-14,2,-3) (3/2,2/1) -> (4/1,1/0) Hyperbolic Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(13,-42,4,-13) (3/1,7/2) -> (3/1,7/2) Reflection Matrix(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(1,0,12,-1) (0/1,1/1) -> (0/1,1/6) Matrix(11,-14,4,-5) -> Matrix(5,-1,6,-1) Matrix(9,-14,2,-3) -> Matrix(3,-1,4,-1) 1/2 Matrix(13,-28,6,-13) -> Matrix(5,-2,12,-5) (2/1,7/3) -> (1/3,1/2) Matrix(29,-70,12,-29) -> Matrix(9,-5,16,-9) (7/3,5/2) -> (1/2,5/8) Matrix(13,-42,4,-13) -> Matrix(-1,1,0,1) (3/1,7/2) -> (1/2,1/0) Matrix(15,-56,4,-15) -> Matrix(1,0,2,-1) (7/2,4/1) -> (0/1,1/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.