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): 576 Minimal number of generators: 97 Number of equivalence classes of cusps: 48 Genus: 25 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -9/2 -10/3 -3/1 -5/2 -12/5 -20/9 -9/5 -5/3 -3/2 -5/4 0/1 1/1 15/13 5/4 15/11 3/2 5/3 9/5 15/8 2/1 15/7 20/9 30/13 12/5 5/2 75/29 30/11 3/1 45/14 10/3 7/2 11/3 15/4 4/1 30/7 13/3 9/2 14/3 5/1 11/2 6/1 13/2 7/1 15/2 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 -2/3 -15/2 -1/2 -7/1 -1/2 0/1 -6/1 -2/3 0/1 -17/3 -1/1 -1/2 -11/2 -1/1 0/1 -5/1 -1/2 1/0 -14/3 -2/5 -9/2 -1/1 -1/2 0/1 -13/3 -1/1 -1/2 -4/1 0/1 -15/4 -1/2 1/0 -11/3 -1/1 0/1 -7/2 -1/1 0/1 -17/5 -1/1 1/0 -10/3 -1/1 -13/4 -2/3 -1/2 -3/1 0/1 -17/6 2/1 1/0 -14/5 -4/1 -25/9 -3/2 1/0 -11/4 -2/1 -1/1 -8/3 0/1 -29/11 1/1 1/0 -21/8 -1/1 0/1 1/0 -13/5 -1/1 1/0 -5/2 -1/2 1/0 -17/7 -1/1 1/0 -29/12 -1/1 -4/5 -12/5 -2/3 0/1 -7/3 -1/2 0/1 -16/7 2/1 -9/4 -1/1 0/1 1/0 -29/13 1/1 1/0 -20/9 -1/1 -11/5 -1/1 0/1 -13/6 -1/2 0/1 -15/7 0/1 -2/1 0/1 -15/8 1/0 -13/7 -1/1 1/0 -11/6 -1/1 0/1 -20/11 -1/1 -9/5 0/1 -16/9 -2/3 -23/13 -1/4 0/1 -30/17 0/1 -7/4 0/1 1/1 -19/11 0/1 1/1 -50/29 1/1 -31/18 2/3 1/1 -12/7 0/1 2/1 -5/3 1/0 -18/11 -2/1 0/1 -31/19 3/1 1/0 -75/46 1/0 -44/27 -8/1 -13/8 -2/1 1/0 -21/13 -2/1 -50/31 -1/1 -29/18 -1/1 0/1 -8/5 -2/1 -19/12 -4/3 -1/1 -30/19 -1/1 -11/7 -1/1 0/1 -14/9 -4/3 -3/2 -1/1 0/1 1/0 -16/11 -4/3 -45/31 -1/1 -29/20 -1/1 -6/7 -13/9 -1/1 -1/2 -10/7 -1/1 -17/12 -1/2 0/1 -24/17 -2/3 0/1 -7/5 -1/2 0/1 -18/13 -2/3 0/1 -29/21 -1/2 -3/7 -40/29 -1/3 -11/8 -1/3 0/1 -15/11 0/1 -4/3 0/1 -17/13 1/1 1/0 -30/23 1/0 -13/10 0/1 1/0 -9/7 0/1 -14/11 2/3 -5/4 1/0 -16/13 -4/3 -11/9 -1/1 0/1 -17/14 0/1 1/0 -6/5 -2/1 0/1 -13/11 -1/1 1/0 -7/6 -1/1 0/1 -15/13 0/1 -8/7 0/1 -1/1 -1/1 1/0 0/1 -1/1 1/1 -1/1 -1/2 8/7 0/1 15/13 0/1 7/6 -1/1 0/1 6/5 -2/3 0/1 17/14 -1/2 0/1 11/9 -1/1 0/1 5/4 -1/2 14/11 -2/7 9/7 0/1 13/10 -1/2 0/1 4/3 0/1 15/11 0/1 11/8 0/1 1/1 7/5 0/1 1/0 17/12 0/1 1/0 10/7 -1/1 13/9 -1/1 1/0 3/2 -1/1 -1/2 0/1 17/11 -1/1 1/0 14/9 -4/5 25/16 -1/2 11/7 -1/1 0/1 8/5 -2/3 29/18 -1/1 0/1 21/13 -2/3 13/8 -2/3 -1/2 5/3 -1/2 17/10 -1/2 0/1 29/17 -1/2 -5/11 12/7 -2/5 0/1 7/4 -1/3 0/1 16/9 -2/1 9/5 0/1 29/16 -2/1 -1/1 20/11 -1/1 11/6 -1/1 0/1 13/7 -1/1 -1/2 15/8 -1/2 2/1 0/1 15/7 0/1 13/6 0/1 1/0 11/5 -1/1 0/1 20/9 -1/1 9/4 -1/1 -1/2 0/1 16/7 -2/5 23/10 -1/5 0/1 30/13 0/1 7/3 0/1 1/0 19/8 0/1 1/1 50/21 1/1 31/13 3/1 1/0 12/5 -2/1 0/1 5/2 -1/2 1/0 18/7 -2/1 0/1 31/12 -4/3 -1/1 75/29 -1/1 44/17 -6/7 13/5 -1/1 -1/2 21/8 -1/1 -1/2 0/1 50/19 -1/1 29/11 -1/2 -1/3 8/3 0/1 19/7 -2/1 -1/1 30/11 -1/1 11/4 -1/1 -2/3 14/5 -4/7 3/1 0/1 16/5 4/1 45/14 1/0 29/9 -7/1 1/0 13/4 -2/1 1/0 10/3 -1/1 17/5 -1/1 -1/2 24/7 -2/3 0/1 7/2 -1/1 0/1 18/5 -2/1 0/1 29/8 -1/1 -2/3 40/11 -1/1 11/3 -1/1 0/1 15/4 -1/2 1/0 4/1 0/1 17/4 0/1 1/0 30/7 1/0 13/3 -1/1 1/0 9/2 -1/1 0/1 1/0 14/3 2/1 5/1 -1/2 1/0 16/3 2/1 11/2 -1/1 0/1 17/3 -1/1 1/0 6/1 -2/1 0/1 13/2 0/1 1/0 7/1 0/1 1/0 15/2 1/0 8/1 -2/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(29,240,18,149) (-8/1,1/0) -> (8/5,29/18) Hyperbolic Matrix(31,240,4,31) (-8/1,-15/2) -> (15/2,8/1) Hyperbolic Matrix(29,210,4,29) (-15/2,-7/1) -> (7/1,15/2) Hyperbolic Matrix(31,210,-22,-149) (-7/1,-6/1) -> (-24/17,-7/5) Hyperbolic Matrix(89,510,26,149) (-6/1,-17/3) -> (17/5,24/7) Hyperbolic Matrix(59,330,32,179) (-17/3,-11/2) -> (11/6,13/7) Hyperbolic Matrix(61,330,-22,-119) (-11/2,-5/1) -> (-25/9,-11/4) Hyperbolic Matrix(89,420,-32,-151) (-5/1,-14/3) -> (-14/5,-25/9) Hyperbolic Matrix(59,270,26,119) (-14/3,-9/2) -> (9/4,16/7) Hyperbolic Matrix(89,390,34,149) (-9/2,-13/3) -> (13/5,21/8) Hyperbolic Matrix(29,120,-22,-91) (-13/3,-4/1) -> (-4/3,-17/13) Hyperbolic Matrix(31,120,8,31) (-4/1,-15/4) -> (15/4,4/1) Hyperbolic Matrix(89,330,24,89) (-15/4,-11/3) -> (11/3,15/4) Hyperbolic Matrix(59,210,-34,-121) (-11/3,-7/2) -> (-7/4,-19/11) Hyperbolic Matrix(61,210,-52,-179) (-7/2,-17/5) -> (-13/11,-7/6) Hyperbolic Matrix(89,300,62,209) (-17/5,-10/3) -> (10/7,13/9) Hyperbolic Matrix(91,300,64,211) (-10/3,-13/4) -> (17/12,10/7) Hyperbolic Matrix(29,90,-10,-31) (-13/4,-3/1) -> (-3/1,-17/6) Parabolic Matrix(329,930,-202,-571) (-17/6,-14/5) -> (-44/27,-13/8) Hyperbolic Matrix(89,240,-56,-151) (-11/4,-8/3) -> (-8/5,-19/12) Hyperbolic Matrix(91,240,80,211) (-8/3,-29/11) -> (1/1,8/7) Hyperbolic Matrix(331,870,-148,-389) (-29/11,-21/8) -> (-9/4,-29/13) Hyperbolic Matrix(149,390,34,89) (-21/8,-13/5) -> (13/3,9/2) Hyperbolic Matrix(59,150,-24,-61) (-13/5,-5/2) -> (-5/2,-17/7) Parabolic Matrix(359,870,-248,-601) (-17/7,-29/12) -> (-29/20,-13/9) Hyperbolic Matrix(361,870,100,241) (-29/12,-12/5) -> (18/5,29/8) Hyperbolic Matrix(89,210,-64,-151) (-12/5,-7/3) -> (-7/5,-18/13) Hyperbolic Matrix(209,480,-118,-271) (-7/3,-16/7) -> (-16/9,-23/13) Hyperbolic Matrix(119,270,26,59) (-16/7,-9/4) -> (9/2,14/3) Hyperbolic Matrix(929,2070,390,869) (-29/13,-20/9) -> (50/21,31/13) Hyperbolic Matrix(421,930,-244,-539) (-20/9,-11/5) -> (-19/11,-50/29) Hyperbolic Matrix(151,330,124,271) (-11/5,-13/6) -> (17/14,11/9) Hyperbolic Matrix(181,390,84,181) (-13/6,-15/7) -> (15/7,13/6) Hyperbolic Matrix(29,60,14,29) (-15/7,-2/1) -> (2/1,15/7) Hyperbolic Matrix(31,60,16,31) (-2/1,-15/8) -> (15/8,2/1) Hyperbolic Matrix(209,390,112,209) (-15/8,-13/7) -> (13/7,15/8) Hyperbolic Matrix(179,330,32,59) (-13/7,-11/6) -> (11/2,17/3) Hyperbolic Matrix(361,660,-262,-479) (-11/6,-20/11) -> (-40/29,-11/8) Hyperbolic Matrix(481,870,-298,-539) (-20/11,-9/5) -> (-21/13,-50/31) Hyperbolic Matrix(151,270,118,211) (-9/5,-16/9) -> (14/11,9/7) Hyperbolic Matrix(509,900,220,389) (-23/13,-30/17) -> (30/13,7/3) Hyperbolic Matrix(511,900,222,391) (-30/17,-7/4) -> (23/10,30/13) Hyperbolic Matrix(1201,2070,662,1141) (-50/29,-31/18) -> (29/16,20/11) Hyperbolic Matrix(541,930,210,361) (-31/18,-12/7) -> (18/7,31/12) Hyperbolic Matrix(89,150,-54,-91) (-12/7,-5/3) -> (-5/3,-18/11) Parabolic Matrix(569,930,238,389) (-18/11,-31/19) -> (31/13,12/5) Hyperbolic Matrix(2189,3570,680,1109) (-31/19,-75/46) -> (45/14,29/9) Hyperbolic Matrix(1951,3180,608,991) (-75/46,-44/27) -> (16/5,45/14) Hyperbolic Matrix(241,390,186,301) (-13/8,-21/13) -> (9/7,13/10) Hyperbolic Matrix(1619,2610,446,719) (-50/31,-29/18) -> (29/8,40/11) Hyperbolic Matrix(149,240,18,29) (-29/18,-8/5) -> (8/1,1/0) Hyperbolic Matrix(569,900,208,329) (-19/12,-30/19) -> (30/11,11/4) Hyperbolic Matrix(571,900,210,331) (-30/19,-11/7) -> (19/7,30/11) Hyperbolic Matrix(211,330,-172,-269) (-11/7,-14/9) -> (-16/13,-11/9) Hyperbolic Matrix(59,90,-40,-61) (-14/9,-3/2) -> (-3/2,-16/11) Parabolic Matrix(2189,3180,846,1229) (-16/11,-45/31) -> (75/29,44/17) Hyperbolic Matrix(2461,3570,952,1381) (-45/31,-29/20) -> (31/12,75/29) Hyperbolic Matrix(209,300,62,89) (-13/9,-10/7) -> (10/3,17/5) Hyperbolic Matrix(211,300,64,91) (-10/7,-17/12) -> (13/4,10/3) Hyperbolic Matrix(361,510,298,421) (-17/12,-24/17) -> (6/5,17/14) Hyperbolic Matrix(629,870,368,509) (-18/13,-29/21) -> (29/17,12/7) Hyperbolic Matrix(1891,2610,718,991) (-29/21,-40/29) -> (50/19,29/11) Hyperbolic Matrix(241,330,176,241) (-11/8,-15/11) -> (15/11,11/8) Hyperbolic Matrix(89,120,66,89) (-15/11,-4/3) -> (4/3,15/11) Hyperbolic Matrix(689,900,160,209) (-17/13,-30/23) -> (30/7,13/3) Hyperbolic Matrix(691,900,162,211) (-30/23,-13/10) -> (17/4,30/7) Hyperbolic Matrix(301,390,186,241) (-13/10,-9/7) -> (21/13,13/8) Hyperbolic Matrix(211,270,118,151) (-9/7,-14/11) -> (16/9,9/5) Hyperbolic Matrix(119,150,-96,-121) (-14/11,-5/4) -> (-5/4,-16/13) Parabolic Matrix(271,330,124,151) (-11/9,-17/14) -> (13/6,11/5) Hyperbolic Matrix(149,180,24,29) (-17/14,-6/5) -> (6/1,13/2) Hyperbolic Matrix(151,180,26,31) (-6/5,-13/11) -> (17/3,6/1) Hyperbolic Matrix(181,210,156,181) (-7/6,-15/13) -> (15/13,7/6) Hyperbolic Matrix(209,240,182,209) (-15/13,-8/7) -> (8/7,15/13) Hyperbolic Matrix(211,240,80,91) (-8/7,-1/1) -> (29/11,8/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(179,-210,52,-61) (7/6,6/5) -> (24/7,7/2) Hyperbolic Matrix(269,-330,172,-211) (11/9,5/4) -> (25/16,11/7) Hyperbolic Matrix(331,-420,212,-269) (5/4,14/11) -> (14/9,25/16) Hyperbolic Matrix(91,-120,22,-29) (13/10,4/3) -> (4/1,17/4) Hyperbolic Matrix(151,-210,64,-89) (11/8,7/5) -> (7/3,19/8) Hyperbolic Matrix(149,-210,22,-31) (7/5,17/12) -> (13/2,7/1) Hyperbolic Matrix(61,-90,40,-59) (13/9,3/2) -> (3/2,17/11) Parabolic Matrix(601,-930,232,-359) (17/11,14/9) -> (44/17,13/5) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(539,-870,298,-481) (29/18,21/13) -> (9/5,29/16) Hyperbolic Matrix(91,-150,54,-89) (13/8,5/3) -> (5/3,17/10) Parabolic Matrix(511,-870,158,-269) (17/10,29/17) -> (29/9,13/4) Hyperbolic Matrix(121,-210,34,-59) (12/7,7/4) -> (7/2,18/5) Hyperbolic Matrix(271,-480,118,-209) (7/4,16/9) -> (16/7,23/10) Hyperbolic Matrix(509,-930,214,-391) (20/11,11/6) -> (19/8,50/21) Hyperbolic Matrix(299,-660,82,-181) (11/5,20/9) -> (40/11,11/3) Hyperbolic Matrix(389,-870,148,-331) (20/9,9/4) -> (21/8,50/19) Hyperbolic Matrix(61,-150,24,-59) (12/5,5/2) -> (5/2,18/7) Parabolic Matrix(119,-330,22,-61) (11/4,14/5) -> (16/3,11/2) Hyperbolic Matrix(31,-90,10,-29) (14/5,3/1) -> (3/1,16/5) Parabolic Matrix(31,-150,6,-29) (14/3,5/1) -> (5/1,16/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(29,240,18,149) -> Matrix(1,0,0,1) Matrix(31,240,4,31) -> Matrix(7,4,-2,-1) Matrix(29,210,4,29) -> Matrix(1,0,2,1) Matrix(31,210,-22,-149) -> Matrix(1,0,0,1) Matrix(89,510,26,149) -> Matrix(1,0,0,1) Matrix(59,330,32,179) -> Matrix(1,0,0,1) Matrix(61,330,-22,-119) -> Matrix(3,2,-2,-1) Matrix(89,420,-32,-151) -> Matrix(3,2,-2,-1) Matrix(59,270,26,119) -> Matrix(1,0,0,1) Matrix(89,390,34,149) -> Matrix(1,0,0,1) Matrix(29,120,-22,-91) -> Matrix(1,0,2,1) Matrix(31,120,8,31) -> Matrix(1,0,0,1) Matrix(89,330,24,89) -> Matrix(1,0,0,1) Matrix(59,210,-34,-121) -> Matrix(1,0,2,1) Matrix(61,210,-52,-179) -> Matrix(1,0,0,1) Matrix(89,300,62,209) -> Matrix(1,0,0,1) Matrix(91,300,64,211) -> Matrix(3,2,-2,-1) Matrix(29,90,-10,-31) -> Matrix(1,0,2,1) Matrix(329,930,-202,-571) -> Matrix(1,-4,0,1) Matrix(89,240,-56,-151) -> Matrix(3,2,-2,-1) Matrix(91,240,80,211) -> Matrix(1,0,-2,1) Matrix(331,870,-148,-389) -> Matrix(1,0,0,1) Matrix(149,390,34,89) -> Matrix(1,0,0,1) Matrix(59,150,-24,-61) -> Matrix(1,0,0,1) Matrix(359,870,-248,-601) -> Matrix(1,2,-2,-3) Matrix(361,870,100,241) -> Matrix(3,2,-2,-1) Matrix(89,210,-64,-151) -> Matrix(1,0,0,1) Matrix(209,480,-118,-271) -> Matrix(1,0,-2,1) Matrix(119,270,26,59) -> Matrix(1,0,0,1) Matrix(929,2070,390,869) -> Matrix(1,2,0,1) Matrix(421,930,-244,-539) -> Matrix(1,0,2,1) Matrix(151,330,124,271) -> Matrix(1,0,0,1) Matrix(181,390,84,181) -> Matrix(1,0,2,1) Matrix(29,60,14,29) -> Matrix(1,0,0,1) Matrix(31,60,16,31) -> Matrix(1,0,-2,1) Matrix(209,390,112,209) -> Matrix(1,2,-2,-3) Matrix(179,330,32,59) -> Matrix(1,0,0,1) Matrix(361,660,-262,-479) -> Matrix(1,0,-2,1) Matrix(481,870,-298,-539) -> Matrix(3,2,-2,-1) Matrix(151,270,118,211) -> Matrix(1,0,-2,1) Matrix(509,900,220,389) -> Matrix(1,0,4,1) Matrix(511,900,222,391) -> Matrix(1,0,-6,1) Matrix(1201,2070,662,1141) -> Matrix(1,0,-2,1) Matrix(541,930,210,361) -> Matrix(1,-2,0,1) Matrix(89,150,-54,-91) -> Matrix(1,-2,0,1) Matrix(569,930,238,389) -> Matrix(1,0,0,1) Matrix(2189,3570,680,1109) -> Matrix(1,-10,0,1) Matrix(1951,3180,608,991) -> Matrix(1,12,0,1) Matrix(241,390,186,301) -> Matrix(1,2,-2,-3) Matrix(1619,2610,446,719) -> Matrix(1,2,-2,-3) Matrix(149,240,18,29) -> Matrix(1,0,0,1) Matrix(569,900,208,329) -> Matrix(5,6,-6,-7) Matrix(571,900,210,331) -> Matrix(3,2,-2,-1) Matrix(211,330,-172,-269) -> Matrix(1,0,0,1) Matrix(59,90,-40,-61) -> Matrix(1,0,0,1) Matrix(2189,3180,846,1229) -> Matrix(9,10,-10,-11) Matrix(2461,3570,952,1381) -> Matrix(11,10,-10,-9) Matrix(209,300,62,89) -> Matrix(1,0,0,1) Matrix(211,300,64,91) -> Matrix(3,2,-2,-1) Matrix(361,510,298,421) -> Matrix(1,0,0,1) Matrix(629,870,368,509) -> Matrix(3,2,-8,-5) Matrix(1891,2610,718,991) -> Matrix(5,2,-8,-3) Matrix(241,330,176,241) -> Matrix(1,0,4,1) Matrix(89,120,66,89) -> Matrix(1,0,-2,1) Matrix(689,900,160,209) -> Matrix(1,-2,0,1) Matrix(691,900,162,211) -> Matrix(1,0,0,1) Matrix(301,390,186,241) -> Matrix(1,2,-2,-3) Matrix(211,270,118,151) -> Matrix(1,0,-2,1) Matrix(119,150,-96,-121) -> Matrix(1,-2,0,1) Matrix(271,330,124,151) -> Matrix(1,0,0,1) Matrix(149,180,24,29) -> Matrix(1,0,0,1) Matrix(151,180,26,31) -> Matrix(1,0,0,1) Matrix(181,210,156,181) -> Matrix(1,0,0,1) Matrix(209,240,182,209) -> Matrix(1,0,-4,1) Matrix(211,240,80,91) -> Matrix(1,0,-2,1) Matrix(1,0,2,1) -> Matrix(1,2,-2,-3) Matrix(179,-210,52,-61) -> Matrix(1,0,0,1) Matrix(269,-330,172,-211) -> Matrix(1,0,0,1) Matrix(331,-420,212,-269) -> Matrix(5,2,-8,-3) Matrix(91,-120,22,-29) -> Matrix(1,0,2,1) Matrix(151,-210,64,-89) -> Matrix(1,0,0,1) Matrix(149,-210,22,-31) -> Matrix(1,0,0,1) Matrix(61,-90,40,-59) -> Matrix(1,0,0,1) Matrix(601,-930,232,-359) -> Matrix(1,2,-2,-3) Matrix(151,-240,56,-89) -> Matrix(3,2,-2,-1) Matrix(539,-870,298,-481) -> Matrix(3,2,-2,-1) Matrix(91,-150,54,-89) -> Matrix(3,2,-8,-5) Matrix(511,-870,158,-269) -> Matrix(3,2,-2,-1) Matrix(121,-210,34,-59) -> Matrix(1,0,2,1) Matrix(271,-480,118,-209) -> Matrix(1,0,-2,1) Matrix(509,-930,214,-391) -> Matrix(1,0,2,1) Matrix(299,-660,82,-181) -> Matrix(1,0,0,1) Matrix(389,-870,148,-331) -> Matrix(1,0,0,1) Matrix(61,-150,24,-59) -> Matrix(1,0,0,1) Matrix(119,-330,22,-61) -> Matrix(3,2,-2,-1) Matrix(31,-90,10,-29) -> Matrix(1,0,2,1) Matrix(31,-150,6,-29) -> 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): 19 Degree of the the map X: 19 Degree of the the map Y: 96 ----------------------------------------------------------------------- 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 1 1 1/1 (-1/1,-1/2) 0 15 8/7 0/1 1 15 15/13 0/1 2 1 7/6 (-1/1,0/1) 0 15 6/5 0 5 17/14 (-1/2,0/1) 0 15 11/9 (-1/1,0/1) 0 15 5/4 -1/2 1 3 14/11 -2/7 1 15 9/7 0/1 1 5 13/10 (-1/2,0/1) 0 15 4/3 0/1 1 15 15/11 0/1 3 1 11/8 (0/1,1/1) 0 15 7/5 (0/1,1/0) 0 15 17/12 (0/1,1/0) 0 15 10/7 -1/1 1 3 13/9 (-1/1,1/0) 0 15 3/2 0 5 17/11 (-1/1,1/0) 0 15 14/9 -4/5 1 15 25/16 -1/2 1 3 11/7 (-1/1,0/1) 0 15 8/5 -2/3 1 15 29/18 (-1/1,0/1) 0 15 21/13 -2/3 1 5 13/8 (-2/3,-1/2) 0 15 5/3 -1/2 1 3 17/10 (-1/2,0/1) 0 15 29/17 (-1/2,-5/11) 0 15 12/7 0 5 7/4 (-1/3,0/1) 0 15 16/9 -2/1 1 15 9/5 0/1 1 5 29/16 (-2/1,-1/1) 0 15 20/11 -1/1 1 3 11/6 (-1/1,0/1) 0 15 13/7 (-1/1,-1/2) 0 15 15/8 -1/2 1 1 2/1 0/1 1 15 15/7 0/1 1 1 13/6 (0/1,1/0) 0 15 11/5 (-1/1,0/1) 0 15 20/9 -1/1 1 3 9/4 0 5 16/7 -2/5 1 15 23/10 (-1/5,0/1) 0 15 30/13 0/1 5 1 7/3 (0/1,1/0) 0 15 19/8 (0/1,1/1) 0 15 50/21 1/1 1 3 31/13 (3/1,1/0) 0 15 12/5 0 5 5/2 0 3 18/7 0 5 31/12 (-4/3,-1/1) 0 15 75/29 -1/1 10 1 44/17 -6/7 1 15 13/5 (-1/1,-1/2) 0 15 21/8 0 5 50/19 -1/1 1 3 29/11 (-1/2,-1/3) 0 15 8/3 0/1 1 15 19/7 (-2/1,-1/1) 0 15 30/11 -1/1 4 1 11/4 (-1/1,-2/3) 0 15 14/5 -4/7 1 15 3/1 0/1 1 5 16/5 4/1 1 15 45/14 1/0 11 1 29/9 (-7/1,1/0) 0 15 13/4 (-2/1,1/0) 0 15 10/3 -1/1 1 3 17/5 (-1/1,-1/2) 0 15 24/7 0 5 7/2 (-1/1,0/1) 0 15 18/5 0 5 29/8 (-1/1,-2/3) 0 15 40/11 -1/1 1 3 11/3 (-1/1,0/1) 0 15 15/4 (-1/1,0/1) 0 1 4/1 0/1 1 15 17/4 (0/1,1/0) 0 15 30/7 1/0 1 1 13/3 (-1/1,1/0) 0 15 9/2 0 5 14/3 2/1 1 15 5/1 0 3 16/3 2/1 1 15 11/2 (-1/1,0/1) 0 15 17/3 (-1/1,1/0) 0 15 6/1 0 5 13/2 (0/1,1/0) 0 15 7/1 (0/1,1/0) 0 15 15/2 1/0 2 1 8/1 -2/1 1 15 1/0 (-1/1,0/1) 0 15 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(211,-240,80,-91) (1/1,8/7) -> (29/11,8/3) Glide Reflection Matrix(209,-240,182,-209) (8/7,15/13) -> (8/7,15/13) Reflection Matrix(181,-210,156,-181) (15/13,7/6) -> (15/13,7/6) Reflection Matrix(179,-210,52,-61) (7/6,6/5) -> (24/7,7/2) Hyperbolic Matrix(149,-180,24,-29) (6/5,17/14) -> (6/1,13/2) Glide Reflection Matrix(271,-330,124,-151) (17/14,11/9) -> (13/6,11/5) Glide Reflection Matrix(269,-330,172,-211) (11/9,5/4) -> (25/16,11/7) Hyperbolic Matrix(331,-420,212,-269) (5/4,14/11) -> (14/9,25/16) Hyperbolic Matrix(211,-270,118,-151) (14/11,9/7) -> (16/9,9/5) Glide Reflection Matrix(301,-390,186,-241) (9/7,13/10) -> (21/13,13/8) Glide Reflection Matrix(91,-120,22,-29) (13/10,4/3) -> (4/1,17/4) Hyperbolic Matrix(89,-120,66,-89) (4/3,15/11) -> (4/3,15/11) Reflection Matrix(241,-330,176,-241) (15/11,11/8) -> (15/11,11/8) Reflection Matrix(151,-210,64,-89) (11/8,7/5) -> (7/3,19/8) Hyperbolic Matrix(149,-210,22,-31) (7/5,17/12) -> (13/2,7/1) Hyperbolic Matrix(211,-300,64,-91) (17/12,10/7) -> (13/4,10/3) Glide Reflection Matrix(209,-300,62,-89) (10/7,13/9) -> (10/3,17/5) Glide Reflection Matrix(61,-90,40,-59) (13/9,3/2) -> (3/2,17/11) Parabolic Matrix(601,-930,232,-359) (17/11,14/9) -> (44/17,13/5) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(149,-240,18,-29) (8/5,29/18) -> (8/1,1/0) Glide Reflection Matrix(539,-870,298,-481) (29/18,21/13) -> (9/5,29/16) Hyperbolic Matrix(91,-150,54,-89) (13/8,5/3) -> (5/3,17/10) Parabolic Matrix(511,-870,158,-269) (17/10,29/17) -> (29/9,13/4) Hyperbolic Matrix(421,-720,176,-301) (29/17,12/7) -> (31/13,12/5) Glide Reflection Matrix(121,-210,34,-59) (12/7,7/4) -> (7/2,18/5) Hyperbolic Matrix(271,-480,118,-209) (7/4,16/9) -> (16/7,23/10) Hyperbolic Matrix(959,-1740,264,-479) (29/16,20/11) -> (29/8,40/11) Glide Reflection Matrix(509,-930,214,-391) (20/11,11/6) -> (19/8,50/21) Hyperbolic Matrix(179,-330,32,-59) (11/6,13/7) -> (11/2,17/3) Glide Reflection Matrix(209,-390,112,-209) (13/7,15/8) -> (13/7,15/8) Reflection Matrix(31,-60,16,-31) (15/8,2/1) -> (15/8,2/1) Reflection Matrix(29,-60,14,-29) (2/1,15/7) -> (2/1,15/7) Reflection Matrix(181,-390,84,-181) (15/7,13/6) -> (15/7,13/6) Reflection Matrix(299,-660,82,-181) (11/5,20/9) -> (40/11,11/3) Hyperbolic Matrix(389,-870,148,-331) (20/9,9/4) -> (21/8,50/19) Hyperbolic Matrix(119,-270,26,-59) (9/4,16/7) -> (9/2,14/3) Glide Reflection Matrix(599,-1380,260,-599) (23/10,30/13) -> (23/10,30/13) Reflection Matrix(181,-420,78,-181) (30/13,7/3) -> (30/13,7/3) Reflection Matrix(1259,-3000,478,-1139) (50/21,31/13) -> (50/19,29/11) Glide Reflection Matrix(61,-150,24,-59) (12/5,5/2) -> (5/2,18/7) Parabolic Matrix(419,-1080,116,-299) (18/7,31/12) -> (18/5,29/8) Glide Reflection Matrix(1799,-4650,696,-1799) (31/12,75/29) -> (31/12,75/29) Reflection Matrix(2551,-6600,986,-2551) (75/29,44/17) -> (75/29,44/17) Reflection Matrix(149,-390,34,-89) (13/5,21/8) -> (13/3,9/2) Glide Reflection Matrix(419,-1140,154,-419) (19/7,30/11) -> (19/7,30/11) Reflection Matrix(241,-660,88,-241) (30/11,11/4) -> (30/11,11/4) Reflection Matrix(119,-330,22,-61) (11/4,14/5) -> (16/3,11/2) Hyperbolic Matrix(31,-90,10,-29) (14/5,3/1) -> (3/1,16/5) Parabolic Matrix(449,-1440,140,-449) (16/5,45/14) -> (16/5,45/14) Reflection Matrix(811,-2610,252,-811) (45/14,29/9) -> (45/14,29/9) Reflection Matrix(149,-510,26,-89) (17/5,24/7) -> (17/3,6/1) Glide Reflection Matrix(89,-330,24,-89) (11/3,15/4) -> (11/3,15/4) Reflection Matrix(31,-120,8,-31) (15/4,4/1) -> (15/4,4/1) Reflection Matrix(239,-1020,56,-239) (17/4,30/7) -> (17/4,30/7) Reflection Matrix(181,-780,42,-181) (30/7,13/3) -> (30/7,13/3) Reflection Matrix(31,-150,6,-29) (14/3,5/1) -> (5/1,16/3) Parabolic Matrix(29,-210,4,-29) (7/1,15/2) -> (7/1,15/2) Reflection Matrix(31,-240,4,-31) (15/2,8/1) -> (15/2,8/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(3,2,-4,-3) (0/1,1/1) -> (-1/1,-1/2) Matrix(211,-240,80,-91) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(209,-240,182,-209) -> Matrix(-1,0,6,1) (8/7,15/13) -> (-1/3,0/1) Matrix(181,-210,156,-181) -> Matrix(-1,0,2,1) (15/13,7/6) -> (-1/1,0/1) Matrix(179,-210,52,-61) -> Matrix(1,0,0,1) Matrix(149,-180,24,-29) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(271,-330,124,-151) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(269,-330,172,-211) -> Matrix(1,0,0,1) Matrix(331,-420,212,-269) -> Matrix(5,2,-8,-3) -1/2 Matrix(211,-270,118,-151) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(301,-390,186,-241) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(91,-120,22,-29) -> Matrix(1,0,2,1) 0/1 Matrix(89,-120,66,-89) -> Matrix(-1,0,4,1) (4/3,15/11) -> (-1/2,0/1) Matrix(241,-330,176,-241) -> Matrix(1,0,2,-1) (15/11,11/8) -> (0/1,1/1) Matrix(151,-210,64,-89) -> Matrix(1,0,0,1) Matrix(149,-210,22,-31) -> Matrix(1,0,0,1) Matrix(211,-300,64,-91) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(209,-300,62,-89) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(61,-90,40,-59) -> Matrix(1,0,0,1) Matrix(601,-930,232,-359) -> Matrix(1,2,-2,-3) -1/1 Matrix(151,-240,56,-89) -> Matrix(3,2,-2,-1) -1/1 Matrix(149,-240,18,-29) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(539,-870,298,-481) -> Matrix(3,2,-2,-1) -1/1 Matrix(91,-150,54,-89) -> Matrix(3,2,-8,-5) -1/2 Matrix(511,-870,158,-269) -> Matrix(3,2,-2,-1) -1/1 Matrix(421,-720,176,-301) -> Matrix(5,2,-2,-1) Matrix(121,-210,34,-59) -> Matrix(1,0,2,1) 0/1 Matrix(271,-480,118,-209) -> Matrix(1,0,-2,1) 0/1 Matrix(959,-1740,264,-479) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(509,-930,214,-391) -> Matrix(1,0,2,1) 0/1 Matrix(179,-330,32,-59) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(209,-390,112,-209) -> Matrix(3,2,-4,-3) (13/7,15/8) -> (-1/1,-1/2) Matrix(31,-60,16,-31) -> Matrix(-1,0,4,1) (15/8,2/1) -> (-1/2,0/1) Matrix(29,-60,14,-29) -> Matrix(-1,0,2,1) (2/1,15/7) -> (-1/1,0/1) Matrix(181,-390,84,-181) -> Matrix(1,0,0,-1) (15/7,13/6) -> (0/1,1/0) Matrix(299,-660,82,-181) -> Matrix(1,0,0,1) Matrix(389,-870,148,-331) -> Matrix(1,0,0,1) Matrix(119,-270,26,-59) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(599,-1380,260,-599) -> Matrix(-1,0,10,1) (23/10,30/13) -> (-1/5,0/1) Matrix(181,-420,78,-181) -> Matrix(1,0,0,-1) (30/13,7/3) -> (0/1,1/0) Matrix(1259,-3000,478,-1139) -> Matrix(1,-2,-2,3) Matrix(61,-150,24,-59) -> Matrix(1,0,0,1) Matrix(419,-1080,116,-299) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(1799,-4650,696,-1799) -> Matrix(7,8,-6,-7) (31/12,75/29) -> (-4/3,-1/1) Matrix(2551,-6600,986,-2551) -> Matrix(13,12,-14,-13) (75/29,44/17) -> (-1/1,-6/7) Matrix(149,-390,34,-89) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(419,-1140,154,-419) -> Matrix(3,4,-2,-3) (19/7,30/11) -> (-2/1,-1/1) Matrix(241,-660,88,-241) -> Matrix(5,4,-6,-5) (30/11,11/4) -> (-1/1,-2/3) Matrix(119,-330,22,-61) -> Matrix(3,2,-2,-1) -1/1 Matrix(31,-90,10,-29) -> Matrix(1,0,2,1) 0/1 Matrix(449,-1440,140,-449) -> Matrix(-1,8,0,1) (16/5,45/14) -> (4/1,1/0) Matrix(811,-2610,252,-811) -> Matrix(1,14,0,-1) (45/14,29/9) -> (-7/1,1/0) Matrix(149,-510,26,-89) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(89,-330,24,-89) -> Matrix(-1,0,2,1) (11/3,15/4) -> (-1/1,0/1) Matrix(31,-120,8,-31) -> Matrix(-1,0,2,1) (15/4,4/1) -> (-1/1,0/1) Matrix(239,-1020,56,-239) -> Matrix(1,0,0,-1) (17/4,30/7) -> (0/1,1/0) Matrix(181,-780,42,-181) -> Matrix(1,2,0,-1) (30/7,13/3) -> (-1/1,1/0) Matrix(31,-150,6,-29) -> Matrix(1,0,0,1) Matrix(29,-210,4,-29) -> Matrix(1,0,0,-1) (7/1,15/2) -> (0/1,1/0) Matrix(31,-240,4,-31) -> Matrix(1,4,0,-1) (15/2,8/1) -> (-2/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.