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 -2/1 -5/3 0/1 1/1 5/4 10/7 3/2 5/3 2/1 20/9 7/3 5/2 3/1 10/3 11/3 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -3/2 1/0 -9/2 1/0 -4/1 -2/1 -1/1 1/0 -11/3 -2/1 1/0 -7/2 1/0 -10/3 -2/1 -3/1 -2/1 -1/1 -5/2 -3/2 1/0 -7/3 -2/1 -1/1 -9/4 1/0 -2/1 -2/1 -3/2 -1/1 -5/3 -3/2 1/0 -8/5 -2/1 -3/2 -1/1 -19/12 1/0 -30/19 -2/1 -11/7 -2/1 -3/2 -3/2 -3/2 -10/7 -1/1 -7/5 -2/1 -1/1 -11/8 -3/2 -4/3 -2/1 -3/2 -1/1 -9/7 -2/1 -3/2 -5/4 -3/2 -1/1 -3/2 -1/1 0/1 -1/1 1/1 -1/1 -1/2 5/4 -1/2 9/7 -1/2 0/1 4/3 -1/1 -1/2 0/1 11/8 -1/2 7/5 -1/1 0/1 10/7 -1/1 3/2 -1/2 5/3 -1/2 1/0 7/4 -1/2 9/5 -1/2 0/1 2/1 -1/1 -1/2 0/1 11/5 -1/2 0/1 20/9 0/1 9/4 1/0 7/3 -1/1 0/1 5/2 -1/2 1/0 3/1 -1/1 0/1 10/3 0/1 7/2 1/0 11/3 0/1 1/0 4/1 -1/1 0/1 1/0 5/1 -1/2 1/0 6/1 -1/1 0/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(9,50,-2,-11) (-5/1,1/0) -> (-5/1,-9/2) Parabolic Matrix(19,80,14,59) (-9/2,-4/1) -> (4/3,11/8) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(39,140,22,79) (-11/3,-7/2) -> (7/4,9/5) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(19,60,6,19) (-10/3,-3/1) -> (3/1,10/3) Hyperbolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(29,70,12,29) (-5/2,-7/3) -> (7/3,5/2) Hyperbolic Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(51,110,-32,-69) (-9/4,-2/1) -> (-8/5,-19/12) Hyperbolic Matrix(29,50,-18,-31) (-2/1,-5/3) -> (-5/3,-8/5) Parabolic Matrix(411,650,184,291) (-19/12,-30/19) -> (20/9,9/4) Hyperbolic Matrix(349,550,158,249) (-30/19,-11/7) -> (11/5,20/9) Hyperbolic Matrix(71,110,20,31) (-11/7,-3/2) -> (7/2,11/3) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(99,140,70,99) (-10/7,-7/5) -> (7/5,10/7) Hyperbolic Matrix(101,140,44,61) (-7/5,-11/8) -> (9/4,7/3) Hyperbolic Matrix(51,70,8,11) (-11/8,-4/3) -> (6/1,1/0) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(71,90,56,71) (-9/7,-5/4) -> (5/4,9/7) Hyperbolic Matrix(9,10,8,9) (-5/4,-1/1) -> (1/1,5/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(9,50,-2,-11) -> Matrix(1,0,0,1) Matrix(19,80,14,59) -> Matrix(1,2,-2,-3) Matrix(21,80,16,61) -> Matrix(1,2,-2,-3) Matrix(39,140,22,79) -> Matrix(1,2,-2,-3) Matrix(41,140,12,41) -> Matrix(1,2,0,1) Matrix(19,60,6,19) -> Matrix(1,2,-2,-3) Matrix(11,30,4,11) -> Matrix(1,2,-2,-3) Matrix(29,70,12,29) -> Matrix(1,2,-2,-3) Matrix(61,140,44,101) -> Matrix(1,2,-2,-3) Matrix(51,110,-32,-69) -> Matrix(1,0,0,1) Matrix(29,50,-18,-31) -> Matrix(1,0,0,1) Matrix(411,650,184,291) -> Matrix(1,2,0,1) Matrix(349,550,158,249) -> Matrix(1,2,-4,-7) Matrix(71,110,20,31) -> Matrix(1,2,-2,-3) Matrix(41,60,28,41) -> Matrix(3,4,-4,-5) Matrix(99,140,70,99) -> Matrix(1,2,-2,-3) Matrix(101,140,44,61) -> Matrix(1,2,-2,-3) Matrix(51,70,8,11) -> Matrix(1,2,-2,-3) Matrix(61,80,16,21) -> Matrix(1,2,-2,-3) Matrix(71,90,56,71) -> Matrix(1,2,-4,-7) Matrix(9,10,8,9) -> Matrix(3,4,-4,-5) Matrix(1,0,2,1) -> Matrix(3,4,-4,-5) Matrix(31,-50,18,-29) -> Matrix(1,0,0,1) Matrix(21,-40,10,-19) -> Matrix(1,0,0,1) Matrix(11,-50,2,-9) -> Matrix(1,0,0,1) 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: ((2,6,18,19,7)(3,11,23,12,4)(5,15,10,9,16)(8,21,14,13,22); (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)) ----------------------------------------------------------------------- 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 0/1 1/1 5/4 3/2 5/3 2/1 5/2 3/1 10/3 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -3/2 1/0 -4/1 -2/1 -1/1 1/0 -3/1 -2/1 -1/1 -5/2 -3/2 1/0 -2/1 -2/1 -3/2 -1/1 -5/3 -3/2 1/0 -3/2 -3/2 -10/7 -1/1 -7/5 -2/1 -1/1 -4/3 -2/1 -3/2 -1/1 -5/4 -3/2 -1/1 -3/2 -1/1 0/1 -1/1 1/1 -1/1 -1/2 5/4 -1/2 4/3 -1/1 -1/2 0/1 3/2 -1/2 5/3 -1/2 1/0 2/1 -1/1 -1/2 0/1 5/2 -1/2 1/0 3/1 -1/1 0/1 10/3 0/1 7/2 1/0 4/1 -1/1 0/1 1/0 5/1 -1/2 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,10,0,1) (-5/1,1/0) -> (5/1,1/0) Parabolic Matrix(9,40,2,9) (-5/1,-4/1) -> (4/1,5/1) Hyperbolic Matrix(11,40,-8,-29) (-4/1,-3/1) -> (-7/5,-4/3) Hyperbolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(11,20,6,11) (-2/1,-5/3) -> (5/3,2/1) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(69,100,20,29) (-3/2,-10/7) -> (10/3,7/2) Hyperbolic Matrix(71,100,22,31) (-10/7,-7/5) -> (3/1,10/3) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(9,10,8,9) (-5/4,-1/1) -> (1/1,5/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,10,0,1) -> Matrix(1,1,0,1) Matrix(9,40,2,9) -> Matrix(1,1,0,1) Matrix(11,40,-8,-29) -> Matrix(3,5,-2,-3) Matrix(11,30,4,11) -> Matrix(1,2,-2,-3) Matrix(9,20,4,9) -> Matrix(1,1,0,1) Matrix(11,20,6,11) -> Matrix(1,1,0,1) Matrix(19,30,12,19) -> Matrix(1,1,0,1) Matrix(69,100,20,29) -> Matrix(1,1,2,3) Matrix(71,100,22,31) -> Matrix(1,1,0,1) Matrix(31,40,24,31) -> Matrix(1,1,0,1) Matrix(9,10,8,9) -> Matrix(3,4,-4,-5) Matrix(1,0,2,1) -> Matrix(3,4,-4,-5) Matrix(29,-40,8,-11) -> Matrix(1,1,-2,-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): 3 ----------------------------------------------------------------------- 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 4 2 1/1 (-1/1,-1/2) 0 10 5/4 -1/2 1 2 4/3 (-1/2,1/0) 0 10 3/2 -1/2 1 10 5/3 (-1/2,1/0) 0 2 2/1 (-1/2,1/0) 0 10 5/2 (-1/1,0/1).(-1/2,1/0) 0 2 3/1 (-1/1,0/1) 0 10 10/3 0/1 2 2 7/2 1/0 1 10 4/1 (-1/2,1/0) 0 10 5/1 (-1/2,1/0) 0 2 1/0 1/0 1 10 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(9,-10,8,-9) (1/1,5/4) -> (1/1,5/4) Reflection Matrix(31,-40,24,-31) (5/4,4/3) -> (5/4,4/3) Reflection Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) Reflection Matrix(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) Reflection Matrix(41,-140,12,-41) (10/3,7/2) -> (10/3,7/2) Reflection Matrix(9,-40,2,-9) (4/1,5/1) -> (4/1,5/1) Reflection Matrix(-1,10,0,1) (5/1,1/0) -> (5/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,2,0,-1) (0/1,1/0) -> (-1/1,1/0) Matrix(1,0,2,-1) -> Matrix(3,2,-4,-3) (0/1,1/1) -> (-1/1,-1/2) Matrix(9,-10,8,-9) -> Matrix(3,2,-4,-3) (1/1,5/4) -> (-1/1,-1/2) Matrix(31,-40,24,-31) -> Matrix(1,1,0,-1) (5/4,4/3) -> (-1/2,1/0) Matrix(29,-40,8,-11) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(19,-30,12,-19) -> Matrix(1,1,0,-1) (3/2,5/3) -> (-1/2,1/0) Matrix(11,-20,6,-11) -> Matrix(1,1,0,-1) (5/3,2/1) -> (-1/2,1/0) Matrix(9,-20,4,-9) -> Matrix(1,1,0,-1) (2/1,5/2) -> (-1/2,1/0) Matrix(11,-30,4,-11) -> Matrix(-1,0,2,1) (5/2,3/1) -> (-1/1,0/1) Matrix(19,-60,6,-19) -> Matrix(-1,0,2,1) (3/1,10/3) -> (-1/1,0/1) Matrix(41,-140,12,-41) -> Matrix(1,0,0,-1) (10/3,7/2) -> (0/1,1/0) Matrix(9,-40,2,-9) -> Matrix(1,1,0,-1) (4/1,5/1) -> (-1/2,1/0) Matrix(-1,10,0,1) -> Matrix(1,1,0,-1) (5/1,1/0) -> (-1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.