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): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 30 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -7/3 -2/1 -21/13 -7/5 0/1 1/1 7/6 14/11 7/5 3/2 14/9 7/4 9/5 2/1 7/3 5/2 8/3 14/5 3/1 7/2 11/3 56/15 4/1 13/3 14/3 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -3/1 -2/1 -14/3 -2/1 -9/2 -3/2 -4/1 -2/1 -1/1 1/0 -15/4 1/0 -11/3 -3/1 -2/1 -7/2 -2/1 -3/1 -2/1 -1/1 -14/5 -2/1 -11/4 -3/2 -19/7 -2/1 -3/2 -8/3 -2/1 -3/2 -1/1 -21/8 -1/1 -13/5 -1/1 1/0 -18/7 -2/1 -1/1 1/0 -5/2 -3/2 -7/3 -1/1 -9/4 1/0 -11/5 -1/1 0/1 -13/6 1/0 -2/1 -2/1 -1/1 1/0 -11/6 1/0 -9/5 -2/1 1/0 -7/4 -2/1 -5/3 -2/1 -1/1 -13/8 -3/2 -21/13 -3/2 -29/18 -3/2 -8/5 -3/2 -4/3 -1/1 -11/7 -4/3 -1/1 -14/9 -1/1 -3/2 1/0 -7/5 -2/1 -11/8 -3/2 -26/19 -2/1 -5/3 -3/2 -41/30 -3/2 -56/41 -3/2 -15/11 -3/2 -1/1 -4/3 -2/1 -3/2 -1/1 -13/10 -3/2 -22/17 -2/1 -3/2 -1/1 -9/7 -2/1 -3/2 -14/11 -3/2 -5/4 -3/2 -11/9 -4/3 -1/1 -6/5 -3/2 -4/3 -1/1 -7/6 -1/1 -1/1 -2/1 -1/1 0/1 -1/1 1/1 -1/1 0/1 7/6 -1/1 6/5 -1/1 -2/3 -1/2 5/4 -1/2 14/11 -1/2 9/7 -1/2 0/1 13/10 -1/2 4/3 -1/1 -1/2 0/1 7/5 0/1 10/7 0/1 1/1 1/0 13/9 1/1 1/0 3/2 1/0 14/9 -1/1 11/7 -1/1 -2/3 19/12 -1/2 8/5 -1/1 -2/3 -1/2 5/3 -1/1 0/1 7/4 0/1 9/5 0/1 1/0 11/6 1/0 13/7 1/1 1/0 2/1 -1/1 0/1 1/0 7/3 -1/1 12/5 -1/1 -2/3 -1/2 17/7 -1/1 0/1 5/2 -1/2 13/5 -1/1 1/0 21/8 -1/1 8/3 -1/1 -1/2 0/1 27/10 -1/2 19/7 -1/2 0/1 11/4 -1/2 14/5 0/1 3/1 -1/1 0/1 7/2 0/1 11/3 0/1 1/1 26/7 1/1 2/1 1/0 41/11 3/1 1/0 56/15 1/0 15/4 1/0 4/1 -1/1 0/1 1/0 13/3 -1/1 1/0 9/2 -1/2 14/3 0/1 5/1 0/1 1/1 6/1 1/1 2/1 1/0 7/1 1/0 8/1 -1/1 0/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(13,70,-8,-43) (-5/1,1/0) -> (-5/3,-13/8) Hyperbolic Matrix(29,140,6,29) (-5/1,-14/3) -> (14/3,5/1) Hyperbolic Matrix(55,252,12,55) (-14/3,-9/2) -> (9/2,14/3) Hyperbolic Matrix(41,182,-16,-71) (-9/2,-4/1) -> (-18/7,-5/2) Hyperbolic Matrix(29,112,22,85) (-4/1,-15/4) -> (13/10,4/3) Hyperbolic Matrix(83,308,-38,-141) (-15/4,-11/3) -> (-11/5,-13/6) Hyperbolic Matrix(43,154,12,43) (-11/3,-7/2) -> (7/2,11/3) Hyperbolic Matrix(13,42,4,13) (-7/2,-3/1) -> (3/1,7/2) Hyperbolic Matrix(29,84,10,29) (-3/1,-14/5) -> (14/5,3/1) Hyperbolic Matrix(111,308,40,111) (-14/5,-11/4) -> (11/4,14/5) Hyperbolic Matrix(113,308,62,169) (-11/4,-19/7) -> (9/5,11/6) Hyperbolic Matrix(181,490,-140,-379) (-19/7,-8/3) -> (-22/17,-9/7) Hyperbolic Matrix(69,182,58,153) (-8/3,-21/8) -> (7/6,6/5) Hyperbolic Matrix(209,546,80,209) (-21/8,-13/5) -> (13/5,21/8) Hyperbolic Matrix(141,364,98,253) (-13/5,-18/7) -> (10/7,13/9) Hyperbolic Matrix(41,98,-18,-43) (-5/2,-7/3) -> (-7/3,-9/4) Parabolic Matrix(139,308,88,195) (-9/4,-11/5) -> (11/7,19/12) Hyperbolic Matrix(197,420,-144,-307) (-13/6,-2/1) -> (-26/19,-41/30) Hyperbolic Matrix(167,308,-122,-225) (-2/1,-11/6) -> (-11/8,-26/19) Hyperbolic Matrix(169,308,62,113) (-11/6,-9/5) -> (19/7,11/4) Hyperbolic Matrix(71,126,40,71) (-9/5,-7/4) -> (7/4,9/5) Hyperbolic Matrix(41,70,24,41) (-7/4,-5/3) -> (5/3,7/4) Hyperbolic Matrix(545,882,-338,-547) (-13/8,-21/13) -> (-21/13,-29/18) Parabolic Matrix(279,448,104,167) (-29/18,-8/5) -> (8/3,27/10) Hyperbolic Matrix(97,154,-80,-127) (-8/5,-11/7) -> (-11/9,-6/5) Hyperbolic Matrix(197,308,126,197) (-11/7,-14/9) -> (14/9,11/7) Hyperbolic Matrix(55,84,36,55) (-14/9,-3/2) -> (3/2,14/9) Hyperbolic Matrix(69,98,-50,-71) (-3/2,-7/5) -> (-7/5,-11/8) Parabolic Matrix(2295,3136,614,839) (-41/30,-56/41) -> (56/15,15/4) Hyperbolic Matrix(2297,3136,616,841) (-56/41,-15/11) -> (41/11,56/15) Hyperbolic Matrix(83,112,20,27) (-15/11,-4/3) -> (4/1,13/3) Hyperbolic Matrix(85,112,22,29) (-4/3,-13/10) -> (15/4,4/1) Hyperbolic Matrix(97,126,10,13) (-13/10,-22/17) -> (8/1,1/0) Hyperbolic Matrix(197,252,154,197) (-9/7,-14/11) -> (14/11,9/7) Hyperbolic Matrix(111,140,88,111) (-14/11,-5/4) -> (5/4,14/11) Hyperbolic Matrix(113,140,46,57) (-5/4,-11/9) -> (17/7,5/2) Hyperbolic Matrix(153,182,58,69) (-6/5,-7/6) -> (21/8,8/3) Hyperbolic Matrix(13,14,12,13) (-7/6,-1/1) -> (1/1,7/6) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(127,-154,80,-97) (6/5,5/4) -> (19/12,8/5) Hyperbolic Matrix(379,-490,140,-181) (9/7,13/10) -> (27/10,19/7) Hyperbolic Matrix(71,-98,50,-69) (4/3,7/5) -> (7/5,10/7) Parabolic Matrix(125,-182,68,-99) (13/9,3/2) -> (11/6,13/7) Hyperbolic Matrix(43,-70,8,-13) (8/5,5/3) -> (5/1,6/1) Hyperbolic Matrix(223,-420,60,-113) (13/7,2/1) -> (26/7,41/11) Hyperbolic Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(237,-574,64,-155) (12/5,17/7) -> (11/3,26/7) Hyperbolic Matrix(71,-182,16,-41) (5/2,13/5) -> (13/3,9/2) Hyperbolic Matrix(15,-98,2,-13) (6/1,7/1) -> (7/1,8/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(13,70,-8,-43) -> Matrix(3,8,-2,-5) Matrix(29,140,6,29) -> Matrix(1,2,2,5) Matrix(55,252,12,55) -> Matrix(1,2,-4,-7) Matrix(41,182,-16,-71) -> Matrix(1,0,0,1) Matrix(29,112,22,85) -> Matrix(1,2,-2,-3) Matrix(83,308,-38,-141) -> Matrix(1,2,0,1) Matrix(43,154,12,43) -> Matrix(1,2,2,5) Matrix(13,42,4,13) -> Matrix(1,2,-2,-3) Matrix(29,84,10,29) -> Matrix(1,2,-2,-3) Matrix(111,308,40,111) -> Matrix(1,2,-4,-7) Matrix(113,308,62,169) -> Matrix(1,2,-2,-3) Matrix(181,490,-140,-379) -> Matrix(1,0,0,1) Matrix(69,182,58,153) -> Matrix(3,4,-4,-5) Matrix(209,546,80,209) -> Matrix(1,0,0,1) Matrix(141,364,98,253) -> Matrix(1,2,0,1) Matrix(41,98,-18,-43) -> Matrix(1,2,-2,-3) Matrix(139,308,88,195) -> Matrix(1,2,-2,-3) Matrix(197,420,-144,-307) -> Matrix(3,8,-2,-5) Matrix(167,308,-122,-225) -> Matrix(3,8,-2,-5) Matrix(169,308,62,113) -> Matrix(1,2,-2,-3) Matrix(71,126,40,71) -> Matrix(1,2,0,1) Matrix(41,70,24,41) -> Matrix(1,2,-2,-3) Matrix(545,882,-338,-547) -> Matrix(11,18,-8,-13) Matrix(279,448,104,167) -> Matrix(3,4,-4,-5) Matrix(97,154,-80,-127) -> Matrix(1,0,0,1) Matrix(197,308,126,197) -> Matrix(5,6,-6,-7) Matrix(55,84,36,55) -> Matrix(1,0,0,1) Matrix(69,98,-50,-71) -> Matrix(3,8,-2,-5) Matrix(2295,3136,614,839) -> Matrix(5,8,-2,-3) Matrix(2297,3136,616,841) -> Matrix(7,10,2,3) Matrix(83,112,20,27) -> Matrix(1,2,-2,-3) Matrix(85,112,22,29) -> Matrix(1,2,-2,-3) Matrix(97,126,10,13) -> Matrix(1,2,-2,-3) Matrix(197,252,154,197) -> Matrix(1,2,-4,-7) Matrix(111,140,88,111) -> Matrix(3,4,-4,-5) Matrix(113,140,46,57) -> Matrix(3,4,-4,-5) Matrix(153,182,58,69) -> Matrix(3,4,-4,-5) Matrix(13,14,12,13) -> Matrix(1,2,-2,-3) Matrix(1,0,2,1) -> Matrix(1,2,-2,-3) Matrix(127,-154,80,-97) -> Matrix(1,0,0,1) Matrix(379,-490,140,-181) -> Matrix(1,0,0,1) Matrix(71,-98,50,-69) -> Matrix(1,0,2,1) Matrix(125,-182,68,-99) -> Matrix(1,0,0,1) Matrix(43,-70,8,-13) -> Matrix(1,0,2,1) Matrix(223,-420,60,-113) -> Matrix(1,2,0,1) Matrix(43,-98,18,-41) -> Matrix(1,2,-2,-3) Matrix(237,-574,64,-155) -> Matrix(1,0,2,1) Matrix(71,-182,16,-41) -> Matrix(1,0,0,1) Matrix(15,-98,2,-13) -> Matrix(1,-2,0,1) 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): 9 Degree of the the map X: 9 Degree of the the map Y: 48 Permutation triple for Y: ((2,6,22,44,46,23,7)(3,12,35,47,36,13,4)(5,18,31,10,9,30,19)(8,26,40,16,15,39,27)(11,33,45,21,20,17,34)(14,28,43,25,24,42,37); (1,4,16,42,17,5,2)(3,10,8,7,25,32,11)(6,21,27,29,9,14,13)(18,41,40,33,22,36,43)(20,38,37,46,26,31,35)(24,39,47,48,44,30,34); (1,2,8,21,45,40,46,48,47,31,43,28,9,3)(4,14,38,20,6,5,19,44,33,32,25,36,39,15)(7,23,37,16,41,18,17,35,12,11,30,29,27,24)(10,26)(13,22)(34,42)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 2 2 1/1 (-1/1,0/1) 0 14 7/6 -1/1 3 2 6/5 0 14 5/4 -1/2 1 14 14/11 -1/2 2 2 9/7 (-1/2,0/1) 0 14 13/10 -1/2 1 14 4/3 0 14 7/5 0/1 2 2 10/7 0 14 13/9 (1/1,1/0) 0 14 3/2 1/0 1 14 14/9 -1/1 6 2 11/7 (-1/1,-2/3) 0 14 19/12 -1/2 1 14 8/5 0 14 5/3 (-1/1,0/1) 0 14 7/4 0/1 1 2 9/5 (0/1,1/0) 0 14 11/6 1/0 1 14 13/7 (1/1,1/0) 0 14 2/1 0 14 7/3 -1/1 2 2 12/5 0 14 17/7 (-1/1,0/1) 0 14 5/2 -1/2 1 14 13/5 (-1/1,1/0) 0 14 21/8 -1/1 3 2 8/3 0 14 27/10 -1/2 1 14 19/7 (-1/2,0/1) 0 14 11/4 -1/2 1 14 14/5 0/1 2 2 3/1 (-1/1,0/1) 0 14 7/2 0/1 2 2 11/3 (0/1,1/1) 0 14 26/7 0 14 41/11 (3/1,1/0) 0 14 56/15 1/0 6 2 15/4 1/0 1 14 4/1 0 14 13/3 (-1/1,1/0) 0 14 9/2 -1/2 1 14 14/3 0/1 6 2 5/1 (0/1,1/1) 0 14 6/1 0 14 7/1 1/0 2 2 8/1 0 14 1/0 1/0 1 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(13,-14,12,-13) (1/1,7/6) -> (1/1,7/6) Reflection Matrix(153,-182,58,-69) (7/6,6/5) -> (21/8,8/3) Glide Reflection Matrix(127,-154,80,-97) (6/5,5/4) -> (19/12,8/5) Hyperbolic Matrix(111,-140,88,-111) (5/4,14/11) -> (5/4,14/11) Reflection Matrix(197,-252,154,-197) (14/11,9/7) -> (14/11,9/7) Reflection Matrix(379,-490,140,-181) (9/7,13/10) -> (27/10,19/7) Hyperbolic Matrix(85,-112,22,-29) (13/10,4/3) -> (15/4,4/1) Glide Reflection Matrix(71,-98,50,-69) (4/3,7/5) -> (7/5,10/7) Parabolic Matrix(127,-182,30,-43) (10/7,13/9) -> (4/1,13/3) Glide Reflection Matrix(125,-182,68,-99) (13/9,3/2) -> (11/6,13/7) Hyperbolic Matrix(55,-84,36,-55) (3/2,14/9) -> (3/2,14/9) Reflection Matrix(197,-308,126,-197) (14/9,11/7) -> (14/9,11/7) Reflection Matrix(239,-378,98,-155) (11/7,19/12) -> (17/7,5/2) Glide Reflection Matrix(43,-70,8,-13) (8/5,5/3) -> (5/1,6/1) Hyperbolic Matrix(41,-70,24,-41) (5/3,7/4) -> (5/3,7/4) Reflection Matrix(71,-126,40,-71) (7/4,9/5) -> (7/4,9/5) Reflection Matrix(169,-308,62,-113) (9/5,11/6) -> (19/7,11/4) Glide Reflection Matrix(223,-420,60,-113) (13/7,2/1) -> (26/7,41/11) Hyperbolic Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(237,-574,64,-155) (12/5,17/7) -> (11/3,26/7) Hyperbolic Matrix(71,-182,16,-41) (5/2,13/5) -> (13/3,9/2) Hyperbolic Matrix(209,-546,80,-209) (13/5,21/8) -> (13/5,21/8) Reflection Matrix(83,-224,10,-27) (8/3,27/10) -> (8/1,1/0) Glide Reflection Matrix(111,-308,40,-111) (11/4,14/5) -> (11/4,14/5) Reflection Matrix(29,-84,10,-29) (14/5,3/1) -> (14/5,3/1) Reflection Matrix(13,-42,4,-13) (3/1,7/2) -> (3/1,7/2) Reflection Matrix(43,-154,12,-43) (7/2,11/3) -> (7/2,11/3) Reflection Matrix(1231,-4592,330,-1231) (41/11,56/15) -> (41/11,56/15) Reflection Matrix(449,-1680,120,-449) (56/15,15/4) -> (56/15,15/4) Reflection Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) Reflection Matrix(29,-140,6,-29) (14/3,5/1) -> (14/3,5/1) Reflection Matrix(15,-98,2,-13) (6/1,7/1) -> (7/1,8/1) Parabolic 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(-1,0,2,1) (0/1,1/1) -> (-1/1,0/1) Matrix(13,-14,12,-13) -> Matrix(-1,0,2,1) (1/1,7/6) -> (-1/1,0/1) Matrix(153,-182,58,-69) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(127,-154,80,-97) -> Matrix(1,0,0,1) Matrix(111,-140,88,-111) -> Matrix(3,2,-4,-3) (5/4,14/11) -> (-1/1,-1/2) Matrix(197,-252,154,-197) -> Matrix(-1,0,4,1) (14/11,9/7) -> (-1/2,0/1) Matrix(379,-490,140,-181) -> Matrix(1,0,0,1) Matrix(85,-112,22,-29) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(71,-98,50,-69) -> Matrix(1,0,2,1) 0/1 Matrix(127,-182,30,-43) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(125,-182,68,-99) -> Matrix(1,0,0,1) Matrix(55,-84,36,-55) -> Matrix(1,2,0,-1) (3/2,14/9) -> (-1/1,1/0) Matrix(197,-308,126,-197) -> Matrix(5,4,-6,-5) (14/9,11/7) -> (-1/1,-2/3) Matrix(239,-378,98,-155) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(43,-70,8,-13) -> Matrix(1,0,2,1) 0/1 Matrix(41,-70,24,-41) -> Matrix(-1,0,2,1) (5/3,7/4) -> (-1/1,0/1) Matrix(71,-126,40,-71) -> Matrix(1,0,0,-1) (7/4,9/5) -> (0/1,1/0) Matrix(169,-308,62,-113) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(223,-420,60,-113) -> Matrix(1,2,0,1) 1/0 Matrix(43,-98,18,-41) -> Matrix(1,2,-2,-3) -1/1 Matrix(237,-574,64,-155) -> Matrix(1,0,2,1) 0/1 Matrix(71,-182,16,-41) -> Matrix(1,0,0,1) Matrix(209,-546,80,-209) -> Matrix(1,2,0,-1) (13/5,21/8) -> (-1/1,1/0) Matrix(83,-224,10,-27) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(111,-308,40,-111) -> Matrix(-1,0,4,1) (11/4,14/5) -> (-1/2,0/1) Matrix(29,-84,10,-29) -> Matrix(-1,0,2,1) (14/5,3/1) -> (-1/1,0/1) Matrix(13,-42,4,-13) -> Matrix(-1,0,2,1) (3/1,7/2) -> (-1/1,0/1) Matrix(43,-154,12,-43) -> Matrix(1,0,2,-1) (7/2,11/3) -> (0/1,1/1) Matrix(1231,-4592,330,-1231) -> Matrix(-1,6,0,1) (41/11,56/15) -> (3/1,1/0) Matrix(449,-1680,120,-449) -> Matrix(1,0,0,-1) (56/15,15/4) -> (0/1,1/0) Matrix(55,-252,12,-55) -> Matrix(-1,0,4,1) (9/2,14/3) -> (-1/2,0/1) Matrix(29,-140,6,-29) -> Matrix(1,0,2,-1) (14/3,5/1) -> (0/1,1/1) Matrix(15,-98,2,-13) -> Matrix(1,-2,0,1) 1/0 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.