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: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -4/1 -3/1 -5/2 -2/1 -3/2 -4/3 -1/1 -4/5 -2/3 -1/2 -2/5 0/1 1/3 2/5 1/2 2/3 4/5 5/6 1/1 6/5 4/3 3/2 5/3 2/1 12/5 5/2 3/1 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 1/0 -5/1 -3/1 -1/1 -4/1 -1/1 -11/3 0/1 -7/2 1/0 -3/1 -1/1 -8/3 -1/2 1/0 -5/2 -1/1 -1/2 0/1 -2/1 0/1 -7/4 0/1 1/1 1/0 -12/7 0/1 -5/3 0/1 -8/5 1/1 -3/2 0/1 1/1 1/0 -7/5 1/1 3/1 -18/13 1/0 -11/8 1/1 2/1 1/0 -4/3 1/0 -5/4 1/0 -11/9 0/1 -6/5 1/0 -1/1 -1/1 1/1 -6/7 1/0 -5/6 1/0 -4/5 -1/1 -3/4 -1/1 -1/2 0/1 -8/11 -1/1 -5/7 -1/1 -1/3 -2/3 0/1 -7/11 1/3 1/1 -12/19 0/1 -5/8 0/1 1/3 1/2 -3/5 1/1 -7/12 1/2 -4/7 1/1 -5/9 2/1 -6/11 1/0 -1/2 1/0 -4/9 -3/2 1/0 -3/7 -1/1 -5/12 1/0 -2/5 -1/1 -7/18 -3/4 -12/31 -2/3 -5/13 -1/1 -3/5 -3/8 -1/1 -1/2 0/1 -4/11 -1/2 -1/3 0/1 0/1 0/1 1/3 0/1 4/11 1/1 3/8 0/1 1/1 1/0 2/5 0/1 5/12 1/4 3/7 1/3 1/2 0/1 1/2 1/1 4/7 1/1 7/12 1/2 3/5 1/1 5/8 1/2 2/3 1/1 7/10 3/2 5/7 1/1 3/1 3/4 0/1 1/1 1/0 4/5 1/1 5/6 1/0 1/1 -1/1 1/1 7/6 1/0 6/5 1/0 5/4 0/1 1/1 1/0 4/3 1/2 1/0 11/8 1/0 7/5 -1/1 1/1 17/12 1/0 10/7 0/1 3/2 0/1 1/1 1/0 8/5 1/1 13/8 1/1 2/1 1/0 18/11 1/0 5/3 0/1 2/1 1/1 7/3 2/1 12/5 2/1 29/12 5/2 17/7 7/3 3/1 5/2 1/0 8/3 1/0 11/4 -1/1 0/1 1/0 3/1 1/1 7/2 2/1 3/1 1/0 18/5 1/0 11/3 2/1 4/1 1/0 9/2 0/1 1/1 1/0 5/1 1/1 3/1 11/2 1/0 6/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(25,132,-18,-95) (-6/1,-5/1) -> (-7/5,-18/13) Hyperbolic Matrix(13,60,-18,-83) (-5/1,-4/1) -> (-8/11,-5/7) Hyperbolic Matrix(13,48,36,133) (-4/1,-11/3) -> (1/3,4/11) Hyperbolic Matrix(37,132,-30,-107) (-11/3,-7/2) -> (-5/4,-11/9) Hyperbolic Matrix(11,36,18,59) (-7/2,-3/1) -> (3/5,5/8) Hyperbolic Matrix(13,36,-30,-83) (-3/1,-8/3) -> (-4/9,-3/7) Hyperbolic Matrix(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) Hyperbolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic Matrix(83,144,-132,-229) (-7/4,-12/7) -> (-12/19,-5/8) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(37,60,-66,-107) (-5/3,-8/5) -> (-4/7,-5/9) Hyperbolic Matrix(23,36,30,47) (-8/5,-3/2) -> (3/4,4/5) Hyperbolic Matrix(25,36,-66,-95) (-3/2,-7/5) -> (-5/13,-3/8) Hyperbolic Matrix(313,432,192,265) (-18/13,-11/8) -> (13/8,18/11) Hyperbolic Matrix(97,132,36,49) (-11/8,-4/3) -> (8/3,11/4) Hyperbolic Matrix(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(217,264,60,73) (-11/9,-6/5) -> (18/5,11/3) Hyperbolic Matrix(11,12,-12,-13) (-6/5,-1/1) -> (-1/1,-6/7) Parabolic Matrix(85,72,72,61) (-6/7,-5/6) -> (7/6,6/5) Hyperbolic Matrix(59,48,102,83) (-5/6,-4/5) -> (4/7,7/12) Hyperbolic Matrix(47,36,30,23) (-4/5,-3/4) -> (3/2,8/5) Hyperbolic Matrix(49,36,132,97) (-3/4,-8/11) -> (4/11,3/8) Hyperbolic Matrix(35,24,-54,-37) (-5/7,-2/3) -> (-2/3,-7/11) Parabolic Matrix(227,144,-588,-373) (-7/11,-12/19) -> (-12/31,-5/13) Hyperbolic Matrix(59,36,18,11) (-5/8,-3/5) -> (3/1,7/2) Hyperbolic Matrix(61,36,144,85) (-3/5,-7/12) -> (5/12,3/7) Hyperbolic Matrix(83,48,102,59) (-7/12,-4/7) -> (4/5,5/6) Hyperbolic Matrix(217,120,132,73) (-5/9,-6/11) -> (18/11,5/3) Hyperbolic Matrix(133,72,24,13) (-6/11,-1/2) -> (11/2,6/1) Hyperbolic Matrix(107,48,78,35) (-1/2,-4/9) -> (4/3,11/8) Hyperbolic Matrix(85,36,144,61) (-3/7,-5/12) -> (7/12,3/5) Hyperbolic Matrix(59,24,-150,-61) (-5/12,-2/5) -> (-2/5,-7/18) Parabolic Matrix(1115,432,462,179) (-7/18,-12/31) -> (12/5,29/12) Hyperbolic Matrix(131,48,30,11) (-3/8,-4/11) -> (4/1,9/2) Hyperbolic Matrix(133,48,36,13) (-4/11,-1/3) -> (11/3,4/1) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(95,-36,66,-25) (3/8,2/5) -> (10/7,3/2) Hyperbolic Matrix(205,-84,144,-59) (2/5,5/12) -> (17/12,10/7) Hyperbolic Matrix(83,-36,30,-13) (3/7,1/2) -> (11/4,3/1) Hyperbolic Matrix(107,-60,66,-37) (1/2,4/7) -> (8/5,13/8) Hyperbolic Matrix(37,-24,54,-35) (5/8,2/3) -> (2/3,7/10) Parabolic Matrix(205,-144,84,-59) (7/10,5/7) -> (17/7,5/2) Hyperbolic Matrix(83,-60,18,-13) (5/7,3/4) -> (9/2,5/1) Hyperbolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(107,-132,30,-37) (6/5,5/4) -> (7/2,18/5) Hyperbolic Matrix(95,-132,18,-25) (11/8,7/5) -> (5/1,11/2) Hyperbolic Matrix(349,-492,144,-203) (7/5,17/12) -> (29/12,17/7) Hyperbolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,2,0,1) Matrix(25,132,-18,-95) -> Matrix(1,4,0,1) Matrix(13,60,-18,-83) -> Matrix(1,2,-2,-3) Matrix(13,48,36,133) -> Matrix(1,0,2,1) Matrix(37,132,-30,-107) -> Matrix(1,0,0,1) Matrix(11,36,18,59) -> Matrix(1,0,2,1) Matrix(13,36,-30,-83) -> Matrix(3,2,-2,-1) Matrix(23,60,18,47) -> Matrix(1,0,2,1) Matrix(11,24,-6,-13) -> Matrix(1,0,2,1) Matrix(83,144,-132,-229) -> Matrix(1,0,2,1) Matrix(85,144,36,61) -> Matrix(3,-2,2,-1) Matrix(37,60,-66,-107) -> Matrix(3,-2,2,-1) Matrix(23,36,30,47) -> Matrix(1,0,0,1) Matrix(25,36,-66,-95) -> Matrix(1,0,-2,1) Matrix(313,432,192,265) -> Matrix(1,0,0,1) Matrix(97,132,36,49) -> Matrix(1,-2,0,1) Matrix(47,60,18,23) -> Matrix(1,2,0,1) Matrix(217,264,60,73) -> Matrix(1,2,0,1) Matrix(11,12,-12,-13) -> Matrix(1,0,0,1) Matrix(85,72,72,61) -> Matrix(1,2,0,1) Matrix(59,48,102,83) -> Matrix(1,0,2,1) Matrix(47,36,30,23) -> Matrix(1,0,2,1) Matrix(49,36,132,97) -> Matrix(1,0,2,1) Matrix(35,24,-54,-37) -> Matrix(1,0,4,1) Matrix(227,144,-588,-373) -> Matrix(3,-2,-4,3) Matrix(59,36,18,11) -> Matrix(3,-2,2,-1) Matrix(61,36,144,85) -> Matrix(1,0,2,1) Matrix(83,48,102,59) -> Matrix(3,-2,2,-1) Matrix(217,120,132,73) -> Matrix(1,-2,0,1) Matrix(133,72,24,13) -> Matrix(1,2,0,1) Matrix(107,48,78,35) -> Matrix(1,2,0,1) Matrix(85,36,144,61) -> Matrix(1,0,2,1) Matrix(59,24,-150,-61) -> Matrix(3,4,-4,-5) Matrix(1115,432,462,179) -> Matrix(23,16,10,7) Matrix(131,48,30,11) -> Matrix(1,0,2,1) Matrix(133,48,36,13) -> Matrix(5,2,2,1) Matrix(1,0,6,1) -> Matrix(1,0,2,1) Matrix(95,-36,66,-25) -> Matrix(1,0,0,1) Matrix(205,-84,144,-59) -> Matrix(1,0,-4,1) Matrix(83,-36,30,-13) -> Matrix(1,0,-2,1) Matrix(107,-60,66,-37) -> Matrix(3,-2,2,-1) Matrix(37,-24,54,-35) -> Matrix(5,-4,4,-3) Matrix(205,-144,84,-59) -> Matrix(5,-8,2,-3) Matrix(83,-60,18,-13) -> Matrix(1,0,0,1) Matrix(13,-12,12,-11) -> Matrix(1,0,0,1) Matrix(107,-132,30,-37) -> Matrix(1,2,0,1) Matrix(95,-132,18,-25) -> Matrix(1,2,0,1) Matrix(349,-492,144,-203) -> Matrix(5,2,2,1) Matrix(13,-24,6,-11) -> Matrix(3,-2,2,-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): 11 Degree of the the map X: 11 Degree of the the map Y: 48 Permutation triple for Y: ((1,7,2)(3,12,13)(4,18,5)(6,23,24)(8,29,30)(9,34,10)(11,17,38)(14,40,35)(15,28,16)(19,25,36)(20,31,21)(22,33,32)(26,46,27)(37,47,43)(39,44,48)(41,45,42); (1,5,21,47,22,6)(2,10,36,37,11,3)(4,16,9,33,26,17)(7,27,40,43,28,8)(12,32,45,19,18,39)(13,31,30,25,24,14)(15,42,20,46,44,23)(29,38,41,35,34,48); (1,3,14,41,15,4)(2,8,31,42,32,9)(5,19,10,35,27,20)(6,25,45,38,26,7)(11,29,28,23,22,12)(13,39,34,16,43,21)(17,37,40,24,44,18)(30,48,46,33,47,36)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- The image of the extended modular group liftables in PGL(2,Z) equals the image of the modular liftables. ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.