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 -6/1 -4/1 -7/2 -13/4 -2/1 -7/4 -7/6 0/1 1/1 14/11 7/5 3/2 14/9 7/4 2/1 7/3 5/2 21/8 14/5 3/1 7/2 56/15 4/1 9/2 14/3 5/1 11/2 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 0/1 -6/1 -1/1 1/1 -5/1 -1/1 -14/3 0/1 -9/2 0/1 1/1 -13/3 1/1 -4/1 -1/1 1/1 -7/2 0/1 -10/3 1/3 1/1 -13/4 0/1 1/1 -3/1 1/1 -14/5 1/0 -11/4 -3/1 1/0 -19/7 -3/1 -8/3 -3/1 -1/1 -5/2 -1/1 0/1 -7/3 0/1 -9/4 0/1 1/1 -11/5 1/1 -13/6 0/1 1/1 -2/1 -1/1 1/1 -7/4 1/0 -12/7 -3/1 -1/1 -17/10 -1/1 1/0 -5/3 -1/1 -13/8 -1/1 0/1 -21/13 0/1 -8/5 -1/1 1/1 -27/17 -1/1 -19/12 -1/1 0/1 -11/7 -1/1 -14/9 0/1 -3/2 0/1 1/0 -7/5 1/0 -11/8 -3/1 1/0 -26/19 -5/1 -3/1 -41/30 -4/1 -3/1 -56/41 -3/1 -15/11 -3/1 -4/3 -3/1 -1/1 -13/10 -2/1 -1/1 -9/7 -1/1 -14/11 -1/1 -5/4 -1/1 0/1 -6/5 -3/1 -1/1 -7/6 -1/1 -8/7 -1/1 -1/3 -1/1 -1/1 0/1 0/1 1/1 1/1 5/4 0/1 1/1 14/11 1/1 9/7 1/1 4/3 1/1 3/1 15/11 3/1 11/8 3/1 1/0 7/5 1/0 3/2 0/1 1/0 14/9 0/1 11/7 1/1 19/12 0/1 1/1 8/5 -1/1 1/1 21/13 0/1 13/8 0/1 1/1 18/11 1/3 1/1 5/3 1/1 7/4 1/0 9/5 -1/1 11/6 -1/1 1/0 13/7 -1/1 2/1 -1/1 1/1 11/5 -1/1 9/4 -1/1 0/1 7/3 0/1 5/2 0/1 1/1 13/5 1/1 21/8 1/1 29/11 1/1 8/3 1/1 3/1 11/4 3/1 1/0 14/5 1/0 3/1 -1/1 7/2 0/1 11/3 1/1 26/7 1/3 1/1 41/11 1/1 56/15 1/1 15/4 0/1 1/1 4/1 -1/1 1/1 13/3 -1/1 22/5 -1/1 -1/3 9/2 -1/1 0/1 14/3 0/1 5/1 1/1 11/2 1/1 1/0 6/1 -1/1 1/1 7/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,14,0,1) (-7/1,1/0) -> (7/1,1/0) Parabolic Matrix(29,182,18,113) (-7/1,-6/1) -> (8/5,21/13) Hyperbolic Matrix(27,154,-10,-57) (-6/1,-5/1) -> (-19/7,-8/3) 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(111,490,-70,-309) (-9/2,-13/3) -> (-27/17,-19/12) Hyperbolic Matrix(27,112,20,83) (-13/3,-4/1) -> (4/3,15/11) Hyperbolic Matrix(27,98,-8,-29) (-4/1,-7/2) -> (-7/2,-10/3) Parabolic Matrix(111,364,68,223) (-10/3,-13/4) -> (13/8,18/11) Hyperbolic Matrix(57,182,-26,-83) (-13/4,-3/1) -> (-11/5,-13/6) 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(27,70,-22,-57) (-8/3,-5/2) -> (-5/4,-6/5) Hyperbolic Matrix(29,70,12,29) (-5/2,-7/3) -> (7/3,5/2) Hyperbolic Matrix(55,126,24,55) (-7/3,-9/4) -> (9/4,7/3) Hyperbolic 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(55,98,-32,-57) (-2/1,-7/4) -> (-7/4,-12/7) Parabolic Matrix(337,574,-246,-419) (-12/7,-17/10) -> (-11/8,-26/19) Hyperbolic Matrix(83,140,16,27) (-17/10,-5/3) -> (5/1,11/2) Hyperbolic Matrix(111,182,-86,-141) (-5/3,-13/8) -> (-13/10,-9/7) Hyperbolic Matrix(337,546,208,337) (-13/8,-21/13) -> (21/13,13/8) Hyperbolic Matrix(113,182,18,29) (-21/13,-8/5) -> (6/1,7/1) Hyperbolic Matrix(281,448,106,169) (-8/5,-27/17) -> (29/11,8/3) Hyperbolic Matrix(195,308,88,139) (-19/12,-11/7) -> (11/5,9/4) 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(29,42,20,29) (-3/2,-7/5) -> (7/5,3/2) Hyperbolic Matrix(111,154,80,111) (-7/5,-11/8) -> (11/8,7/5) Hyperbolic 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(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(83,98,-72,-85) (-6/5,-7/6) -> (-7/6,-8/7) Parabolic Matrix(113,126,26,29) (-8/7,-1/1) -> (13/3,22/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(141,-182,86,-111) (9/7,4/3) -> (18/11,5/3) Hyperbolic Matrix(225,-308,122,-167) (15/11,11/8) -> (11/6,13/7) Hyperbolic Matrix(309,-490,70,-111) (19/12,8/5) -> (22/5,9/2) Hyperbolic Matrix(57,-98,32,-55) (5/3,7/4) -> (7/4,9/5) Parabolic Matrix(223,-420,60,-113) (13/7,2/1) -> (26/7,41/11) Hyperbolic Matrix(141,-308,38,-83) (2/1,11/5) -> (11/3,26/7) Hyperbolic Matrix(337,-882,128,-335) (13/5,21/8) -> (21/8,29/11) Parabolic Matrix(57,-154,10,-27) (8/3,11/4) -> (11/2,6/1) Hyperbolic Matrix(29,-98,8,-27) (3/1,7/2) -> (7/2,11/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,14,0,1) -> Matrix(1,0,0,1) Matrix(29,182,18,113) -> Matrix(1,0,0,1) Matrix(27,154,-10,-57) -> Matrix(1,-2,0,1) Matrix(29,140,6,29) -> Matrix(1,0,2,1) Matrix(55,252,12,55) -> Matrix(1,0,-2,1) Matrix(111,490,-70,-309) -> Matrix(1,0,-2,1) Matrix(27,112,20,83) -> Matrix(1,2,0,1) Matrix(27,98,-8,-29) -> Matrix(1,0,2,1) Matrix(111,364,68,223) -> Matrix(1,0,0,1) Matrix(57,182,-26,-83) -> Matrix(1,0,0,1) Matrix(29,84,10,29) -> Matrix(1,-2,0,1) Matrix(111,308,40,111) -> Matrix(1,6,0,1) Matrix(113,308,62,169) -> Matrix(1,2,0,1) Matrix(27,70,-22,-57) -> Matrix(1,0,0,1) Matrix(29,70,12,29) -> Matrix(1,0,2,1) Matrix(55,126,24,55) -> Matrix(1,0,-2,1) Matrix(139,308,88,195) -> Matrix(1,0,0,1) Matrix(197,420,-144,-307) -> Matrix(1,-4,0,1) Matrix(55,98,-32,-57) -> Matrix(1,-2,0,1) Matrix(337,574,-246,-419) -> Matrix(1,-2,0,1) Matrix(83,140,16,27) -> Matrix(1,2,0,1) Matrix(111,182,-86,-141) -> Matrix(3,2,-2,-1) Matrix(337,546,208,337) -> Matrix(1,0,2,1) Matrix(113,182,18,29) -> Matrix(1,0,0,1) Matrix(281,448,106,169) -> Matrix(1,2,0,1) Matrix(195,308,88,139) -> Matrix(1,0,0,1) Matrix(197,308,126,197) -> Matrix(1,0,2,1) Matrix(55,84,36,55) -> Matrix(1,0,0,1) Matrix(29,42,20,29) -> Matrix(1,0,0,1) Matrix(111,154,80,111) -> Matrix(1,6,0,1) Matrix(2295,3136,614,839) -> Matrix(1,4,0,1) Matrix(2297,3136,616,841) -> Matrix(1,2,2,5) Matrix(83,112,20,27) -> Matrix(1,2,0,1) Matrix(85,112,22,29) -> Matrix(1,2,0,1) Matrix(197,252,154,197) -> Matrix(5,6,4,5) Matrix(111,140,88,111) -> Matrix(1,0,2,1) Matrix(83,98,-72,-85) -> Matrix(1,2,-2,-3) Matrix(113,126,26,29) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(1,0,2,1) Matrix(57,-70,22,-27) -> Matrix(1,0,0,1) Matrix(141,-182,86,-111) -> Matrix(1,-2,2,-3) Matrix(225,-308,122,-167) -> Matrix(1,-4,0,1) Matrix(309,-490,70,-111) -> Matrix(1,0,-2,1) Matrix(57,-98,32,-55) -> Matrix(1,-2,0,1) Matrix(223,-420,60,-113) -> Matrix(1,0,2,1) Matrix(141,-308,38,-83) -> Matrix(1,0,2,1) Matrix(337,-882,128,-335) -> Matrix(3,-2,2,-1) Matrix(57,-154,10,-27) -> Matrix(1,-2,0,1) Matrix(29,-98,8,-27) -> Matrix(1,0,2,1) 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): 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)(5,18,32,10,9,31,19)(8,25,37,16,15,41,26)(11,29,28,21,20,45,34)(14,38,27,24,17,43,39); (1,4,16,34,45,41,47,48,44,31,43,17,5,2)(3,10,32,35,20,40,39,46,25,8,7,24,33,11)(6,21,18,42,37,14,13,36,27,26,30,9,29,22)(12,23)(15,19)(28,38); (1,2,8,27,28,9,3)(4,14,40,20,6,5,15)(7,12,11,16,42,18,17)(19,44,29,33,24,36,41)(21,38,37,46,48,47,32)(23,39,31,30,26,45,35)) ----------------------------------------------------------------------- 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 0/1 1 1 1/1 1/1 1 14 5/4 (0/1,1/1) 0 14 14/11 1/1 3 1 9/7 1/1 1 14 4/3 0 7 15/11 3/1 1 14 11/8 (3/1,1/0) 0 14 7/5 1/0 3 2 3/2 (0/1,1/0) 0 14 14/9 0/1 1 1 11/7 1/1 1 14 19/12 (0/1,1/1) 0 14 8/5 0 7 21/13 0/1 1 2 13/8 (0/1,1/1) 0 14 18/11 0 7 5/3 1/1 1 14 7/4 1/0 2 2 9/5 -1/1 1 14 11/6 (-1/1,1/0) 0 14 13/7 -1/1 1 14 2/1 0 7 11/5 -1/1 1 14 9/4 (-1/1,0/1) 0 14 7/3 0/1 2 2 5/2 (0/1,1/1) 0 14 13/5 1/1 1 14 21/8 1/1 2 2 29/11 1/1 1 14 8/3 0 7 11/4 (3/1,1/0) 0 14 14/5 1/0 4 1 3/1 -1/1 1 14 7/2 0/1 2 2 11/3 1/1 1 14 26/7 0 7 41/11 1/1 1 14 56/15 1/1 1 1 15/4 (0/1,1/1) 0 14 4/1 0 7 13/3 -1/1 1 14 22/5 0 7 9/2 (-1/1,0/1) 0 14 14/3 0/1 2 1 5/1 1/1 1 14 11/2 (1/1,1/0) 0 14 6/1 0 7 7/1 0/1 1 2 1/0 (0/1,1/0) 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(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) 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(141,-182,86,-111) (9/7,4/3) -> (18/11,5/3) Hyperbolic Matrix(83,-112,20,-27) (4/3,15/11) -> (4/1,13/3) Glide Reflection Matrix(225,-308,122,-167) (15/11,11/8) -> (11/6,13/7) Hyperbolic Matrix(111,-154,80,-111) (11/8,7/5) -> (11/8,7/5) Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) Reflection 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(195,-308,88,-139) (11/7,19/12) -> (11/5,9/4) Glide Reflection Matrix(309,-490,70,-111) (19/12,8/5) -> (22/5,9/2) Hyperbolic Matrix(113,-182,18,-29) (8/5,21/13) -> (6/1,7/1) Glide Reflection Matrix(337,-546,208,-337) (21/13,13/8) -> (21/13,13/8) Reflection Matrix(197,-322,52,-85) (13/8,18/11) -> (15/4,4/1) Glide Reflection Matrix(57,-98,32,-55) (5/3,7/4) -> (7/4,9/5) Parabolic Matrix(85,-154,16,-29) (9/5,11/6) -> (5/1,11/2) Glide Reflection Matrix(223,-420,60,-113) (13/7,2/1) -> (26/7,41/11) Hyperbolic Matrix(141,-308,38,-83) (2/1,11/5) -> (11/3,26/7) Hyperbolic Matrix(55,-126,24,-55) (9/4,7/3) -> (9/4,7/3) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(337,-882,128,-335) (13/5,21/8) -> (21/8,29/11) Parabolic Matrix(281,-742,64,-169) (29/11,8/3) -> (13/3,22/5) Glide Reflection Matrix(57,-154,10,-27) (8/3,11/4) -> (11/2,6/1) Hyperbolic 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(29,-98,8,-27) (3/1,7/2) -> (7/2,11/3) Parabolic 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(-1,14,0,1) (7/1,1/0) -> (7/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Matrix(57,-70,22,-27) -> Matrix(1,0,0,1) Matrix(111,-140,88,-111) -> Matrix(1,0,2,-1) (5/4,14/11) -> (0/1,1/1) Matrix(197,-252,154,-197) -> Matrix(5,-6,4,-5) (14/11,9/7) -> (1/1,3/2) Matrix(141,-182,86,-111) -> Matrix(1,-2,2,-3) 1/1 Matrix(83,-112,20,-27) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(225,-308,122,-167) -> Matrix(1,-4,0,1) 1/0 Matrix(111,-154,80,-111) -> Matrix(-1,6,0,1) (11/8,7/5) -> (3/1,1/0) Matrix(29,-42,20,-29) -> Matrix(1,0,0,-1) (7/5,3/2) -> (0/1,1/0) Matrix(55,-84,36,-55) -> Matrix(1,0,0,-1) (3/2,14/9) -> (0/1,1/0) Matrix(197,-308,126,-197) -> Matrix(1,0,2,-1) (14/9,11/7) -> (0/1,1/1) Matrix(195,-308,88,-139) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(309,-490,70,-111) -> Matrix(1,0,-2,1) 0/1 Matrix(113,-182,18,-29) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(337,-546,208,-337) -> Matrix(1,0,2,-1) (21/13,13/8) -> (0/1,1/1) Matrix(197,-322,52,-85) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(57,-98,32,-55) -> Matrix(1,-2,0,1) 1/0 Matrix(85,-154,16,-29) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(223,-420,60,-113) -> Matrix(1,0,2,1) 0/1 Matrix(141,-308,38,-83) -> Matrix(1,0,2,1) 0/1 Matrix(55,-126,24,-55) -> Matrix(-1,0,2,1) (9/4,7/3) -> (-1/1,0/1) Matrix(29,-70,12,-29) -> Matrix(1,0,2,-1) (7/3,5/2) -> (0/1,1/1) Matrix(337,-882,128,-335) -> Matrix(3,-2,2,-1) 1/1 Matrix(281,-742,64,-169) -> Matrix(1,-2,-2,3) Matrix(57,-154,10,-27) -> Matrix(1,-2,0,1) 1/0 Matrix(111,-308,40,-111) -> Matrix(-1,6,0,1) (11/4,14/5) -> (3/1,1/0) Matrix(29,-84,10,-29) -> Matrix(1,2,0,-1) (14/5,3/1) -> (-1/1,1/0) Matrix(29,-98,8,-27) -> Matrix(1,0,2,1) 0/1 Matrix(1231,-4592,330,-1231) -> Matrix(3,-2,4,-3) (41/11,56/15) -> (1/2,1/1) Matrix(449,-1680,120,-449) -> Matrix(1,0,2,-1) (56/15,15/4) -> (0/1,1/1) Matrix(55,-252,12,-55) -> Matrix(-1,0,2,1) (9/2,14/3) -> (-1/1,0/1) Matrix(29,-140,6,-29) -> Matrix(1,0,2,-1) (14/3,5/1) -> (0/1,1/1) Matrix(-1,14,0,1) -> Matrix(1,0,0,-1) (7/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.