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 -4/5 -1/2 -2/5 -1/4 0/1 1/5 2/9 1/4 2/7 3/10 1/3 2/5 1/2 11/20 3/5 2/3 7/10 4/5 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -4/5 1/0 -7/9 1/0 -3/4 -3/2 1/0 -8/11 -3/2 -1/1 -5/7 -5/4 -7/10 -1/1 -2/3 -1/1 -1/2 -3/5 -1/2 -4/7 -1/2 0/1 -5/9 1/0 -1/2 -1/2 1/0 -2/5 -1/2 -3/8 -1/2 -3/8 -7/19 -3/8 -11/30 -1/3 -4/11 -1/2 -1/3 -1/3 -1/4 -3/10 0/1 -2/7 0/1 1/2 -3/11 1/0 -1/4 -1/2 1/0 -2/9 -1/1 -1/2 -1/5 -1/2 0/1 -1/2 0/1 1/5 -1/2 2/9 -1/2 -1/3 1/4 -1/2 -1/4 3/11 -1/4 2/7 -1/6 0/1 3/10 0/1 1/3 1/0 2/5 -1/2 3/7 -3/8 4/9 -1/2 -1/3 1/2 -1/2 -1/4 6/11 -1/2 -1/3 11/20 -1/3 5/9 -1/4 4/7 -1/2 0/1 3/5 -1/2 2/3 -1/2 -1/3 7/10 -1/3 5/7 -5/16 8/11 -1/3 -3/10 3/4 -3/10 -1/4 4/5 -1/4 5/6 -1/4 -1/6 1/1 -1/4 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(39,32,-50,-41) (-1/1,-4/5) -> (-4/5,-7/9) Parabolic Matrix(21,16,80,61) (-7/9,-3/4) -> (1/4,3/11) Hyperbolic Matrix(19,14,80,59) (-3/4,-8/11) -> (2/9,1/4) Hyperbolic Matrix(61,44,140,101) (-8/11,-5/7) -> (3/7,4/9) Hyperbolic Matrix(99,70,140,99) (-5/7,-7/10) -> (7/10,5/7) Hyperbolic Matrix(41,28,60,41) (-7/10,-2/3) -> (2/3,7/10) Hyperbolic Matrix(19,12,30,19) (-2/3,-3/5) -> (3/5,2/3) Hyperbolic Matrix(41,24,70,41) (-3/5,-4/7) -> (4/7,3/5) Hyperbolic Matrix(39,22,140,79) (-4/7,-5/9) -> (3/11,2/7) Hyperbolic Matrix(41,22,-110,-59) (-5/9,-1/2) -> (-3/8,-7/19) Hyperbolic Matrix(19,8,-50,-21) (-1/2,-2/5) -> (-2/5,-3/8) Parabolic Matrix(359,132,650,239) (-7/19,-11/30) -> (11/20,5/9) Hyperbolic Matrix(301,110,550,201) (-11/30,-4/11) -> (6/11,11/20) Hyperbolic Matrix(79,28,110,39) (-4/11,-1/3) -> (5/7,8/11) Hyperbolic Matrix(19,6,60,19) (-1/3,-3/10) -> (3/10,1/3) Hyperbolic Matrix(41,12,140,41) (-3/10,-2/7) -> (2/7,3/10) Hyperbolic Matrix(79,22,140,39) (-2/7,-3/11) -> (5/9,4/7) Hyperbolic Matrix(59,16,70,19) (-3/11,-1/4) -> (5/6,1/1) Hyperbolic Matrix(59,14,80,19) (-1/4,-2/9) -> (8/11,3/4) Hyperbolic Matrix(19,4,90,19) (-2/9,-1/5) -> (1/5,2/9) Hyperbolic Matrix(1,0,10,1) (-1/5,0/1) -> (0/1,1/5) Parabolic Matrix(21,-8,50,-19) (1/3,2/5) -> (2/5,3/7) Parabolic Matrix(21,-10,40,-19) (4/9,1/2) -> (1/2,6/11) Parabolic Matrix(41,-32,50,-39) (3/4,4/5) -> (4/5,5/6) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(39,32,-50,-41) -> Matrix(1,-2,0,1) Matrix(21,16,80,61) -> Matrix(1,2,-4,-7) Matrix(19,14,80,59) -> Matrix(1,2,-4,-7) Matrix(61,44,140,101) -> Matrix(1,2,-4,-7) Matrix(99,70,140,99) -> Matrix(9,10,-28,-31) Matrix(41,28,60,41) -> Matrix(3,2,-8,-5) Matrix(19,12,30,19) -> Matrix(3,2,-8,-5) Matrix(41,24,70,41) -> Matrix(1,0,0,1) Matrix(39,22,140,79) -> Matrix(1,0,-4,1) Matrix(41,22,-110,-59) -> Matrix(3,2,-8,-5) Matrix(19,8,-50,-21) -> Matrix(3,2,-8,-5) Matrix(359,132,650,239) -> Matrix(11,4,-36,-13) Matrix(301,110,550,201) -> Matrix(1,0,0,1) Matrix(79,28,110,39) -> Matrix(11,4,-36,-13) Matrix(19,6,60,19) -> Matrix(1,0,4,1) Matrix(41,12,140,41) -> Matrix(1,0,-8,1) Matrix(79,22,140,39) -> Matrix(1,0,-4,1) Matrix(59,16,70,19) -> Matrix(1,0,-4,1) Matrix(59,14,80,19) -> Matrix(1,2,-4,-7) Matrix(19,4,90,19) -> Matrix(3,2,-8,-5) Matrix(1,0,10,1) -> Matrix(1,0,0,1) Matrix(21,-8,50,-19) -> Matrix(3,2,-8,-5) Matrix(21,-10,40,-19) -> Matrix(1,0,0,1) Matrix(41,-32,50,-39) -> Matrix(7,2,-32,-9) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 4 Degree of the the map X: 8 Degree of the the map Y: 24 Permutation triple for Y: ((1,4,14,5,2)(3,10,20,8,7)(11,19,13,17,16)(15,21,23,24,18); (1,2,8,22,19,24,23,16,9,3)(4,12,21,20,10,18,6,5,17,13)(7,11)(14,15); (2,6,18,19,7)(3,11,23,12,4)(5,15,10,9,16)(8,21,14,13,22)) ----------------------------------------------------------------------- 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): 72 Minimal number of generators: 13 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: 12 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/10 0/1 1/5 1/4 1/3 2/5 1/2 3/5 2/3 4/5 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -4/5 1/0 -3/4 -3/2 1/0 -2/3 -1/1 -1/2 -3/5 -1/2 -1/2 -1/2 1/0 -2/5 -1/2 -1/3 -1/4 -3/10 0/1 -2/7 0/1 1/2 -1/4 -1/2 1/0 -1/5 -1/2 0/1 -1/2 0/1 1/5 -1/2 1/4 -1/2 -1/4 1/3 1/0 2/5 -1/2 1/2 -1/2 -1/4 3/5 -1/2 2/3 -1/2 -1/3 7/10 -1/3 5/7 -5/16 3/4 -3/10 -1/4 4/5 -1/4 1/1 -1/4 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(9,8,10,9) (-1/1,-4/5) -> (4/5,1/1) Hyperbolic Matrix(31,24,40,31) (-4/5,-3/4) -> (3/4,4/5) Hyperbolic Matrix(11,8,-40,-29) (-3/4,-2/3) -> (-2/7,-1/4) Hyperbolic Matrix(19,12,30,19) (-2/3,-3/5) -> (3/5,2/3) Hyperbolic Matrix(11,6,20,11) (-3/5,-1/2) -> (1/2,3/5) Hyperbolic Matrix(9,4,20,9) (-1/2,-2/5) -> (2/5,1/2) Hyperbolic Matrix(11,4,30,11) (-2/5,-1/3) -> (1/3,2/5) Hyperbolic Matrix(71,22,100,31) (-1/3,-3/10) -> (7/10,5/7) Hyperbolic Matrix(69,20,100,29) (-3/10,-2/7) -> (2/3,7/10) Hyperbolic Matrix(9,2,40,9) (-1/4,-1/5) -> (1/5,1/4) Hyperbolic Matrix(1,0,10,1) (-1/5,0/1) -> (0/1,1/5) Parabolic Matrix(29,-8,40,-11) (1/4,1/3) -> (5/7,3/4) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(9,8,10,9) -> Matrix(1,1,-4,-3) Matrix(31,24,40,31) -> Matrix(1,3,-4,-11) Matrix(11,8,-40,-29) -> Matrix(1,1,0,1) Matrix(19,12,30,19) -> Matrix(3,2,-8,-5) Matrix(11,6,20,11) -> Matrix(1,1,-4,-3) Matrix(9,4,20,9) -> Matrix(1,1,-4,-3) Matrix(11,4,30,11) -> Matrix(3,1,-4,-1) Matrix(71,22,100,31) -> Matrix(9,1,-28,-3) Matrix(69,20,100,29) -> Matrix(3,-1,-8,3) Matrix(9,2,40,9) -> Matrix(1,1,-4,-3) Matrix(1,0,10,1) -> Matrix(1,0,0,1) Matrix(29,-8,40,-11) -> Matrix(5,1,-16,-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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 4 ----------------------------------------------------------------------- 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 (-1/2,0/1) 0 10 1/5 -1/2 1 2 1/4 (-1/2,-1/4) 0 5 1/3 1/0 1 10 2/5 -1/2 2 2 1/2 (-1/2,-1/4) 0 5 3/5 -1/2 1 2 2/3 (-1/2,-1/3) 0 10 7/10 -1/3 6 1 5/7 -5/16 1 10 3/4 (-3/10,-1/4) 0 5 4/5 -1/4 2 2 1/1 -1/4 1 10 1/0 0/1 2 1 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(1,0,10,-1) (0/1,1/5) -> (0/1,1/5) Reflection Matrix(9,-2,40,-9) (1/5,1/4) -> (1/5,1/4) Reflection Matrix(29,-8,40,-11) (1/4,1/3) -> (5/7,3/4) Hyperbolic Matrix(11,-4,30,-11) (1/3,2/5) -> (1/3,2/5) Reflection Matrix(9,-4,20,-9) (2/5,1/2) -> (2/5,1/2) Reflection Matrix(11,-6,20,-11) (1/2,3/5) -> (1/2,3/5) Reflection Matrix(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(41,-28,60,-41) (2/3,7/10) -> (2/3,7/10) Reflection Matrix(99,-70,140,-99) (7/10,5/7) -> (7/10,5/7) Reflection Matrix(31,-24,40,-31) (3/4,4/5) -> (3/4,4/5) Reflection Matrix(9,-8,10,-9) (4/5,1/1) -> (4/5,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,4,1) (0/1,1/0) -> (-1/2,0/1) Matrix(1,0,10,-1) -> Matrix(-1,0,4,1) (0/1,1/5) -> (-1/2,0/1) Matrix(9,-2,40,-9) -> Matrix(3,1,-8,-3) (1/5,1/4) -> (-1/2,-1/4) Matrix(29,-8,40,-11) -> Matrix(5,1,-16,-3) -1/4 Matrix(11,-4,30,-11) -> Matrix(1,1,0,-1) (1/3,2/5) -> (-1/2,1/0) Matrix(9,-4,20,-9) -> Matrix(3,1,-8,-3) (2/5,1/2) -> (-1/2,-1/4) Matrix(11,-6,20,-11) -> Matrix(3,1,-8,-3) (1/2,3/5) -> (-1/2,-1/4) Matrix(19,-12,30,-19) -> Matrix(5,2,-12,-5) (3/5,2/3) -> (-1/2,-1/3) Matrix(41,-28,60,-41) -> Matrix(5,2,-12,-5) (2/3,7/10) -> (-1/2,-1/3) Matrix(99,-70,140,-99) -> Matrix(31,10,-96,-31) (7/10,5/7) -> (-1/3,-5/16) Matrix(31,-24,40,-31) -> Matrix(11,3,-40,-11) (3/4,4/5) -> (-3/10,-1/4) Matrix(9,-8,10,-9) -> Matrix(3,1,-8,-3) (4/5,1/1) -> (-1/2,-1/4) Matrix(-1,2,0,1) -> Matrix(-1,0,8,1) (1/1,1/0) -> (-1/4,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.