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): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/6 -2/3 -1/2 -5/12 -3/8 -1/3 -11/36 -3/10 -7/24 -1/4 -1/6 0/1 1/6 1/5 1/4 3/10 1/3 3/8 2/5 1/2 2/3 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/10 -5/6 1/8 -4/5 1/8 -3/4 1/7 -5/7 1/6 -7/10 0/1 1/7 -2/3 1/6 -9/14 2/11 1/5 -7/11 3/14 -5/8 0/1 -8/13 3/20 -3/5 1/6 -1/2 0/1 1/5 -3/7 1/6 -5/12 1/5 -2/5 1/4 -7/18 1/4 -5/13 3/10 -3/8 0/1 -4/11 3/16 -1/3 1/4 -4/13 5/16 -11/36 1/3 -7/23 5/14 -3/10 0/1 1/3 -5/17 1/2 -7/24 0/1 -2/7 1/4 -1/4 1/3 -1/5 1/2 -1/6 1/2 -1/7 1/2 0/1 1/0 1/6 -1/2 1/5 -1/2 1/4 -1/3 2/7 -1/4 3/10 -1/3 0/1 1/3 -1/4 5/14 -2/9 -1/5 4/11 -3/16 3/8 0/1 5/13 -3/10 2/5 -1/4 1/2 -1/5 0/1 4/7 -1/4 7/12 -1/5 3/5 -1/6 11/18 -1/6 8/13 -3/20 5/8 0/1 7/11 -3/14 2/3 -1/6 9/13 -5/34 25/36 -1/7 16/23 -5/36 7/10 -1/7 0/1 12/17 -1/8 17/24 0/1 5/7 -1/6 3/4 -1/7 4/5 -1/8 5/6 -1/8 6/7 -1/8 1/1 -1/10 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(37,32,-96,-83) (-1/1,-5/6) -> (-7/18,-5/13) Hyperbolic Matrix(47,38,-120,-97) (-5/6,-4/5) -> (-2/5,-7/18) Hyperbolic Matrix(13,10,48,37) (-4/5,-3/4) -> (1/4,2/7) Hyperbolic Matrix(11,8,48,35) (-3/4,-5/7) -> (1/5,1/4) Hyperbolic Matrix(71,50,-240,-169) (-5/7,-7/10) -> (-3/10,-5/17) Hyperbolic Matrix(47,32,-72,-49) (-7/10,-2/3) -> (-2/3,-9/14) Parabolic Matrix(131,84,-432,-277) (-9/14,-7/11) -> (-7/23,-3/10) Hyperbolic Matrix(73,46,192,121) (-7/11,-5/8) -> (3/8,5/13) Hyperbolic Matrix(71,44,192,119) (-5/8,-8/13) -> (4/11,3/8) Hyperbolic Matrix(13,8,-96,-59) (-8/13,-3/5) -> (-1/7,0/1) Hyperbolic Matrix(11,6,-24,-13) (-3/5,-1/2) -> (-1/2,-3/7) Parabolic Matrix(85,36,144,61) (-3/7,-5/12) -> (7/12,3/5) Hyperbolic Matrix(83,34,144,59) (-5/12,-2/5) -> (4/7,7/12) Hyperbolic Matrix(121,46,192,73) (-5/13,-3/8) -> (5/8,7/11) Hyperbolic Matrix(119,44,192,71) (-3/8,-4/11) -> (8/13,5/8) Hyperbolic Matrix(23,8,-72,-25) (-4/11,-1/3) -> (-1/3,-4/13) Parabolic Matrix(901,276,1296,397) (-4/13,-11/36) -> (25/36,16/23) Hyperbolic Matrix(899,274,1296,395) (-11/36,-7/23) -> (9/13,25/36) Hyperbolic Matrix(409,120,576,169) (-5/17,-7/24) -> (17/24,5/7) Hyperbolic Matrix(407,118,576,167) (-7/24,-2/7) -> (12/17,17/24) Hyperbolic Matrix(37,10,48,13) (-2/7,-1/4) -> (3/4,4/5) Hyperbolic Matrix(35,8,48,11) (-1/4,-1/5) -> (5/7,3/4) Hyperbolic Matrix(11,2,-72,-13) (-1/5,-1/6) -> (-1/6,-1/7) Parabolic Matrix(59,-8,96,-13) (0/1,1/6) -> (11/18,8/13) Hyperbolic Matrix(73,-14,120,-23) (1/6,1/5) -> (3/5,11/18) Hyperbolic Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(25,-8,72,-23) (3/10,1/3) -> (1/3,5/14) Parabolic Matrix(301,-108,432,-155) (5/14,4/11) -> (16/23,7/10) Hyperbolic Matrix(83,-32,96,-37) (5/13,2/5) -> (6/7,1/1) Hyperbolic Matrix(13,-6,24,-11) (2/5,1/2) -> (1/2,4/7) Parabolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(61,-50,72,-59) (4/5,5/6) -> (5/6,6/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-20,1) Matrix(37,32,-96,-83) -> Matrix(17,-2,60,-7) Matrix(47,38,-120,-97) -> Matrix(15,-2,68,-9) Matrix(13,10,48,37) -> Matrix(15,-2,-52,7) Matrix(11,8,48,35) -> Matrix(13,-2,-32,5) Matrix(71,50,-240,-169) -> Matrix(1,0,-4,1) Matrix(47,32,-72,-49) -> Matrix(13,-2,72,-11) Matrix(131,84,-432,-277) -> Matrix(11,-2,28,-5) Matrix(73,46,192,121) -> Matrix(1,0,-8,1) Matrix(71,44,192,119) -> Matrix(1,0,-12,1) Matrix(13,8,-96,-59) -> Matrix(13,-2,20,-3) Matrix(11,6,-24,-13) -> Matrix(1,0,0,1) Matrix(85,36,144,61) -> Matrix(11,-2,-60,11) Matrix(83,34,144,59) -> Matrix(9,-2,-40,9) Matrix(121,46,192,73) -> Matrix(1,0,-8,1) Matrix(119,44,192,71) -> Matrix(1,0,-12,1) Matrix(23,8,-72,-25) -> Matrix(9,-2,32,-7) Matrix(901,276,1296,397) -> Matrix(31,-10,-220,71) Matrix(899,274,1296,395) -> Matrix(29,-10,-200,69) Matrix(409,120,576,169) -> Matrix(1,0,-8,1) Matrix(407,118,576,167) -> Matrix(1,0,-12,1) Matrix(37,10,48,13) -> Matrix(7,-2,-52,15) Matrix(35,8,48,11) -> Matrix(5,-2,-32,13) Matrix(11,2,-72,-13) -> Matrix(9,-4,16,-7) Matrix(59,-8,96,-13) -> Matrix(3,2,-20,-13) Matrix(73,-14,120,-23) -> Matrix(5,2,-28,-11) Matrix(169,-50,240,-71) -> Matrix(1,0,-4,1) Matrix(25,-8,72,-23) -> Matrix(7,2,-32,-9) Matrix(301,-108,432,-155) -> Matrix(9,2,-68,-15) Matrix(83,-32,96,-37) -> Matrix(7,2,-60,-17) Matrix(13,-6,24,-11) -> Matrix(1,0,0,1) Matrix(49,-32,72,-47) -> Matrix(11,2,-72,-13) Matrix(61,-50,72,-59) -> Matrix(31,4,-256,-33) 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): 8 Degree of the the map X: 16 Degree of the the map Y: 32 Permutation triple for Y: ((1,4,15,28,18,31,32,29,21,16,5,2)(3,10,27,11)(6,19,24,14,26,30,17,9,25,13,12,20)(7,22,23,8); (1,2,8,24,19,27,32,31,22,25,9,3)(4,13,29,14)(5,17,18,6)(7,20,12,11,28,15,23,30,26,10,16,21); (2,6,7)(3,12,4)(5,10,9)(8,15,14)(11,19,18)(13,22,21)(17,23,31)(26,29,27)) ----------------------------------------------------------------------- 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 0/1 1/6 1/5 1/4 1/3 1/2 7/12 2/3 3/4 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/6 -1/2 1/5 -1/2 1/4 -1/3 1/3 -1/4 3/8 0/1 2/5 -1/4 1/2 -1/5 0/1 4/7 -1/4 7/12 -1/5 3/5 -1/6 2/3 -1/6 5/7 -1/6 3/4 -1/7 4/5 -1/8 5/6 -1/8 1/1 -1/10 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(11,-1,12,-1) (0/1,1/6) -> (5/6,1/1) Hyperbolic Matrix(49,-9,60,-11) (1/6,1/5) -> (4/5,5/6) Hyperbolic Matrix(23,-5,60,-13) (1/5,1/4) -> (3/8,2/5) Hyperbolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(13,-6,24,-11) (2/5,1/2) -> (1/2,4/7) Parabolic Matrix(85,-49,144,-83) (4/7,7/12) -> (7/12,3/5) Parabolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(37,-27,48,-35) (5/7,3/4) -> (3/4,4/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-10,1) Matrix(11,-1,12,-1) -> Matrix(1,1,-10,-9) Matrix(49,-9,60,-11) -> Matrix(7,3,-54,-23) Matrix(23,-5,60,-13) -> Matrix(3,1,-10,-3) Matrix(13,-4,36,-11) -> Matrix(3,1,-16,-5) Matrix(13,-6,24,-11) -> Matrix(1,0,0,1) Matrix(85,-49,144,-83) -> Matrix(9,2,-50,-11) Matrix(25,-16,36,-23) -> Matrix(5,1,-36,-7) Matrix(37,-27,48,-35) -> Matrix(13,2,-98,-15) 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 1/0 1 12 1/6 -1/2 4 2 1/5 -1/2 1 12 1/4 -1/3 2 3 1/3 -1/4 1 4 3/8 0/1 2 3 2/5 -1/4 1 12 5/12 -1/5 2 1 1/2 (-1/5,0/1) 0 6 1/0 0/1 10 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(1,0,12,-1) (0/1,1/6) -> (0/1,1/6) Reflection Matrix(11,-2,60,-11) (1/6,1/5) -> (1/6,1/5) Reflection Matrix(23,-5,60,-13) (1/5,1/4) -> (3/8,2/5) Hyperbolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(49,-20,120,-49) (2/5,5/12) -> (2/5,5/12) Reflection Matrix(11,-5,24,-11) (5/12,1/2) -> (5/12,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,12,-1) -> Matrix(1,1,0,-1) (0/1,1/6) -> (-1/2,1/0) Matrix(11,-2,60,-11) -> Matrix(7,3,-16,-7) (1/6,1/5) -> (-1/2,-3/8) Matrix(23,-5,60,-13) -> Matrix(3,1,-10,-3) (-1/2,-1/4).(-1/3,0/1) Matrix(13,-4,36,-11) -> Matrix(3,1,-16,-5) -1/4 Matrix(49,-20,120,-49) -> Matrix(9,2,-40,-9) (2/5,5/12) -> (-1/4,-1/5) Matrix(11,-5,24,-11) -> Matrix(-1,0,10,1) (5/12,1/2) -> (-1/5,0/1) Matrix(-1,1,0,1) -> Matrix(-1,0,10,1) (1/2,1/0) -> (-1/5,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.