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 -1/1 -4/9 -3/8 -1/3 -2/7 0/1 1/4 1/3 3/7 1/2 11/19 2/3 1/1 11/9 3/2 5/3 19/11 9/5 13/7 2/1 7/3 5/2 8/3 11/4 3/1 7/2 4/1 5/1 17/3 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 1/1 -1/2 -1/1 0/1 1/0 -5/11 -1/2 -1/3 0/1 -4/9 0/1 -3/7 0/1 1/1 1/0 -5/12 -1/1 -1/2 0/1 -2/5 1/0 -7/18 -1/1 -2/3 -1/2 -5/13 -1/2 -1/3 0/1 -3/8 0/1 -7/19 0/1 1/4 1/3 -4/11 1/2 -1/3 0/1 1/1 1/0 -3/10 0/1 1/1 1/0 -2/7 0/1 -5/18 0/1 1/3 1/2 -3/11 0/1 1/2 1/1 -4/15 1/2 -1/4 0/1 1/1 1/0 -2/9 1/0 -1/5 1/1 2/1 1/0 -1/6 1/0 -1/7 -1/1 0/1 1/0 0/1 1/0 1/6 -2/1 -1/1 1/0 1/5 -2/1 -1/1 1/0 1/4 1/0 3/11 -4/1 -3/1 1/0 5/18 -3/1 -5/2 -2/1 2/7 1/0 1/3 -2/1 -1/1 1/0 4/11 -2/1 3/8 -2/1 -3/2 -1/1 5/13 -2/1 -1/1 1/0 2/5 -3/2 3/7 -1/1 4/9 1/0 1/2 -2/1 -1/1 1/0 5/9 -1/1 0/1 1/0 4/7 1/0 11/19 -2/1 7/12 -2/1 -3/2 -1/1 3/5 -2/1 -3/2 -1/1 2/3 -1/1 5/7 -1/1 -2/3 -1/2 13/18 -3/5 -4/7 -1/2 8/11 -1/2 11/15 -1/2 -1/3 0/1 3/4 -1/1 -1/2 0/1 7/9 0/1 1/1 1/0 4/5 1/0 1/1 -1/1 0/1 1/0 6/5 1/0 11/9 -1/1 5/4 -1/1 0/1 1/0 4/3 1/0 3/2 -1/1 8/5 -3/4 29/18 -1/1 -3/4 -2/3 21/13 -3/4 -5/7 -2/3 13/8 -1/1 -2/3 -1/2 31/19 -2/3 18/11 -1/2 5/3 -1/1 -2/3 -1/2 17/10 -2/3 -3/5 -1/2 12/7 -1/2 19/11 -1/2 26/15 -1/2 7/4 -1/2 -1/3 0/1 16/9 0/1 9/5 -1/1 0/1 1/0 11/6 -2/1 -1/1 1/0 13/7 -1/1 2/1 -1/2 7/3 0/1 12/5 1/2 5/2 0/1 1/1 1/0 13/5 0/1 1/1 1/0 21/8 0/1 1/2 1/1 29/11 1/1 8/3 1/0 19/7 1/1 2/1 1/0 11/4 1/0 3/1 -1/1 0/1 1/0 10/3 1/0 7/2 -1/1 0/1 1/0 4/1 -1/1 9/2 -1/1 -2/3 -1/2 14/3 1/0 33/7 -1/1 19/4 -1/1 -3/4 -2/3 5/1 -1/1 -1/2 0/1 11/2 -1/1 -1/2 0/1 17/3 -1/1 -1/3 23/4 -1/1 -1/2 0/1 6/1 -1/2 7/1 -1/1 0/1 1/0 1/0 -1/1 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(185,86,114,53) (-1/2,-5/11) -> (21/13,13/8) Hyperbolic Matrix(257,116,144,65) (-5/11,-4/9) -> (16/9,9/5) Hyperbolic Matrix(37,16,104,45) (-4/9,-3/7) -> (1/3,4/11) Hyperbolic Matrix(105,44,136,57) (-3/7,-5/12) -> (3/4,7/9) Hyperbolic Matrix(169,70,70,29) (-5/12,-2/5) -> (12/5,5/2) Hyperbolic Matrix(297,116,64,25) (-2/5,-7/18) -> (9/2,14/3) Hyperbolic Matrix(259,100,360,139) (-7/18,-5/13) -> (5/7,13/18) Hyperbolic Matrix(95,36,-256,-97) (-5/13,-3/8) -> (-3/8,-7/19) Parabolic Matrix(191,70,30,11) (-7/19,-4/11) -> (6/1,7/1) Hyperbolic Matrix(61,22,158,57) (-4/11,-1/3) -> (5/13,2/5) Hyperbolic Matrix(145,44,56,17) (-1/3,-3/10) -> (5/2,13/5) Hyperbolic Matrix(55,16,-196,-57) (-3/10,-2/7) -> (-2/7,-5/18) Parabolic Matrix(277,76,164,45) (-5/18,-3/11) -> (5/3,17/10) Hyperbolic Matrix(273,74,166,45) (-3/11,-4/15) -> (18/11,5/3) Hyperbolic Matrix(83,22,298,79) (-4/15,-1/4) -> (5/18,2/7) Hyperbolic Matrix(25,6,54,13) (-1/4,-2/9) -> (4/9,1/2) Hyperbolic Matrix(77,16,24,5) (-2/9,-1/5) -> (3/1,10/3) Hyperbolic Matrix(73,14,26,5) (-1/5,-1/6) -> (11/4,3/1) Hyperbolic Matrix(191,30,70,11) (-1/6,-1/7) -> (19/7,11/4) Hyperbolic Matrix(67,8,92,11) (-1/7,0/1) -> (8/11,11/15) Hyperbolic Matrix(89,-12,52,-7) (0/1,1/6) -> (17/10,12/7) Hyperbolic Matrix(85,-16,16,-3) (1/6,1/5) -> (5/1,11/2) Hyperbolic Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(299,-82,62,-17) (3/11,5/18) -> (19/4,5/1) Hyperbolic Matrix(61,-18,78,-23) (2/7,1/3) -> (7/9,4/5) Hyperbolic Matrix(205,-76,116,-43) (4/11,3/8) -> (7/4,16/9) Hyperbolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(43,-18,98,-41) (2/5,3/7) -> (3/7,4/9) Parabolic Matrix(117,-64,64,-35) (1/2,5/9) -> (9/5,11/6) Hyperbolic Matrix(205,-116,76,-43) (5/9,4/7) -> (8/3,19/7) Hyperbolic Matrix(559,-322,342,-197) (4/7,11/19) -> (31/19,18/11) Hyperbolic Matrix(619,-360,380,-221) (11/19,7/12) -> (13/8,31/19) Hyperbolic Matrix(89,-52,12,-7) (7/12,3/5) -> (7/1,1/0) Hyperbolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(545,-394,314,-227) (13/18,8/11) -> (26/15,7/4) Hyperbolic Matrix(519,-382,322,-237) (11/15,3/4) -> (29/18,21/13) Hyperbolic Matrix(11,-10,10,-9) (4/5,1/1) -> (1/1,6/5) Parabolic Matrix(217,-262,82,-99) (6/5,11/9) -> (29/11,8/3) Hyperbolic Matrix(305,-376,116,-143) (11/9,5/4) -> (21/8,29/11) Hyperbolic Matrix(61,-78,18,-23) (5/4,4/3) -> (10/3,7/2) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(223,-358,38,-61) (8/5,29/18) -> (23/4,6/1) Hyperbolic Matrix(419,-722,242,-417) (12/7,19/11) -> (19/11,26/15) Parabolic Matrix(331,-610,70,-129) (11/6,13/7) -> (33/7,19/4) Hyperbolic Matrix(131,-248,28,-53) (13/7,2/1) -> (14/3,33/7) Hyperbolic Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic Matrix(103,-578,18,-101) (11/2,17/3) -> (17/3,23/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,0,1) Matrix(185,86,114,53) -> Matrix(1,2,-2,-3) Matrix(257,116,144,65) -> Matrix(1,0,2,1) Matrix(37,16,104,45) -> Matrix(1,-2,0,1) Matrix(105,44,136,57) -> Matrix(1,0,0,1) Matrix(169,70,70,29) -> Matrix(1,0,2,1) Matrix(297,116,64,25) -> Matrix(1,0,0,1) Matrix(259,100,360,139) -> Matrix(5,2,-8,-3) Matrix(95,36,-256,-97) -> Matrix(1,0,6,1) Matrix(191,70,30,11) -> Matrix(1,0,-4,1) Matrix(61,22,158,57) -> Matrix(1,-2,0,1) Matrix(145,44,56,17) -> Matrix(1,0,0,1) Matrix(55,16,-196,-57) -> Matrix(1,0,2,1) Matrix(277,76,164,45) -> Matrix(3,-2,-4,3) Matrix(273,74,166,45) -> Matrix(3,-2,-4,3) Matrix(83,22,298,79) -> Matrix(5,-2,-2,1) Matrix(25,6,54,13) -> Matrix(1,-2,0,1) Matrix(77,16,24,5) -> Matrix(1,-2,0,1) Matrix(73,14,26,5) -> Matrix(1,-2,0,1) Matrix(191,30,70,11) -> Matrix(1,2,0,1) Matrix(67,8,92,11) -> Matrix(1,0,-2,1) Matrix(89,-12,52,-7) -> Matrix(1,4,-2,-7) Matrix(85,-16,16,-3) -> Matrix(1,2,-2,-3) Matrix(17,-4,64,-15) -> Matrix(1,-2,0,1) Matrix(299,-82,62,-17) -> Matrix(1,4,-2,-7) Matrix(61,-18,78,-23) -> Matrix(1,2,0,1) Matrix(205,-76,116,-43) -> Matrix(1,2,-4,-7) Matrix(377,-144,144,-55) -> Matrix(1,2,0,1) Matrix(43,-18,98,-41) -> Matrix(1,2,-2,-3) Matrix(117,-64,64,-35) -> Matrix(1,0,0,1) Matrix(205,-116,76,-43) -> Matrix(1,2,0,1) Matrix(559,-322,342,-197) -> Matrix(1,4,-2,-7) Matrix(619,-360,380,-221) -> Matrix(3,4,-4,-5) Matrix(89,-52,12,-7) -> Matrix(1,2,-2,-3) Matrix(25,-16,36,-23) -> Matrix(3,4,-4,-5) Matrix(545,-394,314,-227) -> Matrix(7,4,-16,-9) Matrix(519,-382,322,-237) -> Matrix(1,2,-2,-3) Matrix(11,-10,10,-9) -> Matrix(1,0,0,1) Matrix(217,-262,82,-99) -> Matrix(1,2,0,1) Matrix(305,-376,116,-143) -> Matrix(1,0,2,1) Matrix(61,-78,18,-23) -> Matrix(1,0,0,1) Matrix(25,-36,16,-23) -> Matrix(3,4,-4,-5) Matrix(223,-358,38,-61) -> Matrix(3,2,-2,-1) Matrix(419,-722,242,-417) -> Matrix(11,6,-24,-13) Matrix(331,-610,70,-129) -> Matrix(3,4,-4,-5) Matrix(131,-248,28,-53) -> Matrix(3,2,-2,-1) Matrix(43,-98,18,-41) -> Matrix(1,0,4,1) Matrix(17,-64,4,-15) -> Matrix(1,2,-2,-3) Matrix(103,-578,18,-101) -> 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): 10 Degree of the the map X: 10 Degree of the the map Y: 48 Permutation triple for Y: ((1,6,22,39,38,44,43,23,7,2)(3,12,29,8,28,33,48,36,13,4)(5,17)(9,10)(11,34,18,45,40,46,21,16,42,35)(14,41,47,32,31,27,20,19,24,15)(25,26)(30,37); (1,4,15,16,5)(3,10,24,23,11)(6,20,30,29,21)(7,26,35,27,8)(9,32,22,46,33)(13,39,25,40,14)(17,44,28,31,18)(34,43,41,37,36); (1,2,8,30,41,40,45,31,9,3)(4,14)(5,18,36,48,46,25,7,24,19,6)(10,33,44,38,13,37,20,35,42,15)(11,26,39,32,47,43,17,16,29,12)(21,22)(23,34)(27,28)) ----------------------------------------------------------------------- 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.