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): 384 Minimal number of generators: 65 Number of equivalence classes of cusps: 40 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -5/11 -3/7 -2/5 -1/3 -1/4 -1/5 0/1 1/5 1/3 1/2 13/23 3/5 5/7 4/5 1/1 13/11 5/4 7/5 3/2 5/3 2/1 25/11 7/3 17/7 5/2 13/5 8/3 3/1 7/2 25/7 11/3 19/5 4/1 9/2 5/1 6/1 7/1 8/1 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 0/1 -4/9 1/0 -11/25 0/1 -7/16 -1/1 -3/7 0/1 -8/19 1/0 -5/12 -1/1 0/1 1/0 -7/17 -1/1 -2/5 0/1 -5/13 1/1 -3/8 -1/1 0/1 1/0 -4/11 1/0 -9/25 -1/1 1/1 -5/14 -1/1 1/1 -1/3 0/1 -4/13 0/1 -7/23 1/3 1/1 -3/10 0/1 1/2 1/1 -2/7 1/0 -3/11 1/1 -1/4 -1/1 1/1 -3/13 1/1 -5/22 0/1 1/2 1/1 -2/9 1/0 -3/14 0/1 1/1 1/0 -1/5 0/1 2/1 -2/11 1/1 -3/17 2/1 -4/23 1/0 -1/6 1/1 2/1 1/0 -1/7 1/0 0/1 1/0 1/6 -1/1 0/1 1/0 1/5 1/0 1/4 -3/1 -2/1 1/0 3/11 -3/1 5/18 -3/1 -2/1 1/0 2/7 -2/1 1/3 -2/1 0/1 4/11 -2/1 7/19 -2/1 3/8 -2/1 -1/1 1/0 5/13 -2/1 7/18 -2/1 -7/4 -5/3 2/5 -3/2 3/7 -2/1 1/2 -1/1 5/9 0/1 9/16 -1/1 0/1 1/0 13/23 0/1 4/7 1/0 7/12 -3/1 -2/1 1/0 3/5 -1/1 8/13 0/1 13/21 1/1 5/8 -2/1 -1/1 1/0 2/3 1/0 5/7 -1/1 8/11 -1/2 3/4 -1/1 0/1 1/0 10/13 1/0 7/9 -3/1 -1/1 18/23 1/0 11/14 -2/1 -3/2 -1/1 15/19 -2/1 4/5 -1/1 9/11 -2/1 0/1 5/6 -2/1 -1/1 1/0 1/1 -1/1 7/6 -1/1 -1/2 0/1 13/11 -1/1 6/5 -1/2 11/9 0/1 5/4 -1/1 14/11 -1/2 9/7 -1/1 -1/3 4/3 -1/2 7/5 0/1 10/7 1/0 3/2 -1/1 0/1 1/0 11/7 1/0 19/12 -3/1 -2/1 1/0 8/5 1/0 37/23 -2/1 29/18 -2/1 -3/2 -1/1 21/13 -1/1 13/8 -3/1 -1/1 5/3 -1/1 12/7 -1/2 19/11 0/1 7/4 0/1 1/1 1/0 2/1 -1/1 9/4 -2/3 -3/5 -1/2 25/11 -1/2 16/7 -1/2 23/10 -1/3 7/3 0/1 12/5 1/0 17/7 -1/1 22/9 -3/4 5/2 -1/1 -2/3 -1/2 23/9 -1/1 -3/5 18/7 -1/2 13/5 -1/2 21/8 -1/2 -3/7 -2/5 8/3 -1/2 3/1 0/1 10/3 1/0 37/11 -1/1 27/8 -1/1 -1/2 0/1 17/5 0/1 7/2 -1/1 1/1 25/7 -1/1 1/1 18/5 1/0 47/13 1/0 29/8 -4/1 -3/1 1/0 11/3 -1/1 15/4 -1/1 0/1 1/0 19/5 -1/1 1/1 23/6 -1/1 0/1 1/0 4/1 1/0 9/2 -2/1 -1/1 1/0 5/1 -1/1 11/2 -1/1 -1/2 0/1 6/1 -1/2 7/1 -1/1 -1/3 8/1 0/1 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(87,40,224,103) (-1/2,-5/11) -> (5/13,7/18) Hyperbolic Matrix(303,136,176,79) (-5/11,-4/9) -> (12/7,19/11) Hyperbolic Matrix(127,56,-728,-321) (-4/9,-11/25) -> (-3/17,-4/23) Hyperbolic Matrix(687,302,298,131) (-11/25,-7/16) -> (23/10,7/3) Hyperbolic Matrix(211,92,172,75) (-7/16,-3/7) -> (11/9,5/4) Hyperbolic Matrix(251,106,206,87) (-3/7,-8/19) -> (6/5,11/9) Hyperbolic Matrix(457,192,288,121) (-8/19,-5/12) -> (19/12,8/5) Hyperbolic Matrix(121,50,-530,-219) (-5/12,-7/17) -> (-3/13,-5/22) Hyperbolic Matrix(201,82,326,133) (-7/17,-2/5) -> (8/13,13/21) Hyperbolic Matrix(119,46,194,75) (-2/5,-5/13) -> (3/5,8/13) Hyperbolic Matrix(115,44,196,75) (-5/13,-3/8) -> (7/12,3/5) Hyperbolic Matrix(113,42,78,29) (-3/8,-4/11) -> (10/7,3/2) Hyperbolic Matrix(527,190,674,243) (-4/11,-9/25) -> (7/9,18/23) Hyperbolic Matrix(525,188,148,53) (-9/25,-5/14) -> (7/2,25/7) Hyperbolic Matrix(259,92,76,27) (-5/14,-1/3) -> (17/5,7/2) Hyperbolic Matrix(103,32,280,87) (-1/3,-4/13) -> (4/11,7/19) Hyperbolic Matrix(275,84,36,11) (-4/13,-7/23) -> (7/1,8/1) Hyperbolic Matrix(345,104,136,41) (-7/23,-3/10) -> (5/2,23/9) Hyperbolic Matrix(101,30,138,41) (-3/10,-2/7) -> (8/11,3/4) Hyperbolic Matrix(167,46,98,27) (-2/7,-3/11) -> (5/3,12/7) Hyperbolic Matrix(163,44,100,27) (-3/11,-1/4) -> (13/8,5/3) Hyperbolic Matrix(253,60,156,37) (-1/4,-3/13) -> (21/13,13/8) Hyperbolic Matrix(529,120,216,49) (-5/22,-2/9) -> (22/9,5/2) Hyperbolic Matrix(183,40,32,7) (-2/9,-3/14) -> (11/2,6/1) Hyperbolic Matrix(151,32,184,39) (-3/14,-1/5) -> (9/11,5/6) Hyperbolic Matrix(119,22,146,27) (-1/5,-2/11) -> (4/5,9/11) Hyperbolic Matrix(233,42,294,53) (-2/11,-3/17) -> (15/19,4/5) Hyperbolic Matrix(553,96,144,25) (-4/23,-1/6) -> (23/6,4/1) Hyperbolic Matrix(225,34,86,13) (-1/6,-1/7) -> (13/5,21/8) Hyperbolic Matrix(139,18,54,7) (-1/7,0/1) -> (18/7,13/5) Hyperbolic Matrix(63,-10,82,-13) (0/1,1/6) -> (3/4,10/13) Hyperbolic Matrix(161,-30,102,-19) (1/6,1/5) -> (11/7,19/12) Hyperbolic Matrix(59,-14,38,-9) (1/5,1/4) -> (3/2,11/7) Hyperbolic Matrix(209,-56,56,-15) (1/4,3/11) -> (11/3,15/4) Hyperbolic Matrix(697,-192,432,-119) (3/11,5/18) -> (29/18,21/13) Hyperbolic Matrix(151,-42,18,-5) (5/18,2/7) -> (8/1,1/0) Hyperbolic Matrix(19,-6,54,-17) (2/7,1/3) -> (1/3,4/11) Parabolic Matrix(649,-240,192,-71) (7/19,3/8) -> (27/8,17/5) Hyperbolic Matrix(243,-92,140,-53) (3/8,5/13) -> (19/11,7/4) Hyperbolic Matrix(379,-148,484,-189) (7/18,2/5) -> (18/23,11/14) Hyperbolic Matrix(119,-50,50,-21) (2/5,3/7) -> (7/3,12/5) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(339,-190,430,-241) (5/9,9/16) -> (11/14,15/19) Hyperbolic Matrix(1259,-710,782,-441) (9/16,13/23) -> (37/23,29/18) Hyperbolic Matrix(443,-252,276,-157) (13/23,4/7) -> (8/5,37/23) Hyperbolic Matrix(243,-140,92,-53) (4/7,7/12) -> (21/8,8/3) Hyperbolic Matrix(697,-432,192,-119) (13/21,5/8) -> (29/8,11/3) Hyperbolic Matrix(59,-38,14,-9) (5/8,2/3) -> (4/1,9/2) Hyperbolic Matrix(71,-50,98,-69) (2/3,5/7) -> (5/7,8/11) Parabolic Matrix(487,-376,136,-105) (10/13,7/9) -> (25/7,18/5) Hyperbolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(519,-610,154,-181) (7/6,13/11) -> (37/11,27/8) Hyperbolic Matrix(295,-352,88,-105) (13/11,6/5) -> (10/3,37/11) Hyperbolic Matrix(317,-402,138,-175) (5/4,14/11) -> (16/7,23/10) Hyperbolic Matrix(379,-484,148,-189) (14/11,9/7) -> (23/9,18/7) Hyperbolic Matrix(63,-82,10,-13) (9/7,4/3) -> (6/1,7/1) Hyperbolic Matrix(71,-98,50,-69) (4/3,7/5) -> (7/5,10/7) Parabolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(507,-1148,140,-317) (9/4,25/11) -> (47/13,29/8) Hyperbolic Matrix(527,-1202,146,-333) (25/11,16/7) -> (18/5,47/13) Hyperbolic Matrix(239,-578,98,-237) (12/5,17/7) -> (17/7,22/9) Parabolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(191,-722,50,-189) (15/4,19/5) -> (19/5,23/6) Parabolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,0,1) Matrix(87,40,224,103) -> Matrix(7,2,-4,-1) Matrix(303,136,176,79) -> Matrix(1,0,-2,1) Matrix(127,56,-728,-321) -> Matrix(1,2,0,1) Matrix(687,302,298,131) -> Matrix(1,0,-2,1) Matrix(211,92,172,75) -> Matrix(1,0,0,1) Matrix(251,106,206,87) -> Matrix(1,0,-2,1) Matrix(457,192,288,121) -> Matrix(1,-2,0,1) Matrix(121,50,-530,-219) -> Matrix(1,0,2,1) Matrix(201,82,326,133) -> Matrix(1,0,2,1) Matrix(119,46,194,75) -> Matrix(1,0,-2,1) Matrix(115,44,196,75) -> Matrix(1,-2,0,1) Matrix(113,42,78,29) -> Matrix(1,0,0,1) Matrix(527,190,674,243) -> Matrix(1,-2,0,1) Matrix(525,188,148,53) -> Matrix(1,0,0,1) Matrix(259,92,76,27) -> Matrix(1,0,0,1) Matrix(103,32,280,87) -> Matrix(1,-2,0,1) Matrix(275,84,36,11) -> Matrix(1,0,-4,1) Matrix(345,104,136,41) -> Matrix(3,-2,-4,3) Matrix(101,30,138,41) -> Matrix(1,0,-2,1) Matrix(167,46,98,27) -> Matrix(1,0,-2,1) Matrix(163,44,100,27) -> Matrix(1,-2,0,1) Matrix(253,60,156,37) -> Matrix(1,-2,0,1) Matrix(529,120,216,49) -> Matrix(3,-2,-4,3) Matrix(183,40,32,7) -> Matrix(1,0,-2,1) Matrix(151,32,184,39) -> Matrix(1,-2,0,1) Matrix(119,22,146,27) -> Matrix(1,-2,0,1) Matrix(233,42,294,53) -> Matrix(3,-4,-2,3) Matrix(553,96,144,25) -> Matrix(1,-2,0,1) Matrix(225,34,86,13) -> Matrix(1,-4,-2,9) Matrix(139,18,54,7) -> Matrix(1,2,-2,-3) Matrix(63,-10,82,-13) -> Matrix(1,0,0,1) Matrix(161,-30,102,-19) -> Matrix(1,-2,0,1) Matrix(59,-14,38,-9) -> Matrix(1,2,0,1) Matrix(209,-56,56,-15) -> Matrix(1,2,0,1) Matrix(697,-192,432,-119) -> Matrix(3,8,-2,-5) Matrix(151,-42,18,-5) -> Matrix(1,2,0,1) Matrix(19,-6,54,-17) -> Matrix(1,0,0,1) Matrix(649,-240,192,-71) -> Matrix(1,2,-2,-3) Matrix(243,-92,140,-53) -> Matrix(1,2,0,1) Matrix(379,-148,484,-189) -> Matrix(5,8,-2,-3) Matrix(119,-50,50,-21) -> Matrix(1,2,-2,-3) Matrix(17,-8,32,-15) -> Matrix(1,2,-2,-3) Matrix(339,-190,430,-241) -> Matrix(3,2,-2,-1) Matrix(1259,-710,782,-441) -> Matrix(3,2,-2,-1) Matrix(443,-252,276,-157) -> Matrix(1,-2,0,1) Matrix(243,-140,92,-53) -> Matrix(1,0,-2,1) Matrix(697,-432,192,-119) -> Matrix(1,-2,0,1) Matrix(59,-38,14,-9) -> Matrix(1,0,0,1) Matrix(71,-50,98,-69) -> Matrix(1,2,-2,-3) Matrix(487,-376,136,-105) -> Matrix(1,2,0,1) Matrix(13,-12,12,-11) -> Matrix(1,2,-2,-3) Matrix(519,-610,154,-181) -> Matrix(1,0,0,1) Matrix(295,-352,88,-105) -> Matrix(3,2,-2,-1) Matrix(317,-402,138,-175) -> Matrix(3,2,-8,-5) Matrix(379,-484,148,-189) -> Matrix(5,2,-8,-3) Matrix(63,-82,10,-13) -> Matrix(1,0,0,1) Matrix(71,-98,50,-69) -> Matrix(1,0,2,1) Matrix(17,-32,8,-15) -> Matrix(1,2,-2,-3) Matrix(507,-1148,140,-317) -> Matrix(11,6,-2,-1) Matrix(527,-1202,146,-333) -> Matrix(5,2,2,1) Matrix(239,-578,98,-237) -> Matrix(3,4,-4,-5) Matrix(19,-54,6,-17) -> Matrix(1,0,2,1) Matrix(191,-722,50,-189) -> Matrix(1,0,0,1) Matrix(21,-100,4,-19) -> Matrix(1,2,-2,-3) 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): 13 Degree of the the map X: 13 Degree of the the map Y: 64 Permutation triple for Y: ((1,6,24,50,36,62,41,45,39,25,7,2)(3,12,13,4)(5,18,28,27,42,32,60,61,35,10,9,19)(8,30,51,31)(11,37,63,47,46,44,14,43,64,52,20,38)(15,29,59,53,22,21,40,57,26,34,33,16)(17,48,56,49)(23,54,58,55); (1,4,16,47,17,5)(3,10,11)(6,22,23)(7,28,54,37,29,8)(9,34)(12,21,52,56,60,41)(13,27,14)(19,30,20)(24,44)(25,48,26)(31,46,32)(33,58,45)(35,55,43,57,51,36)(38,39)(42,53)(49,59,50); (1,2,8,32,53,49,52,64,55,33,9,3)(4,14,24,23,35,61,56,25,38,30,29,15)(5,20,21,6)(7,26,43,27)(10,36,59,37)(11,39,58,28,18,17,50,44,31,57,40,12)(13,41,62,51,19,34,48,47,63,54,22,42)(16,45,60,46)) ----------------------------------------------------------------------- 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.