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 -4/5 -2/3 -3/5 -5/9 -9/20 -13/30 -5/12 -2/5 -11/30 -1/3 -14/45 -3/10 -7/30 -2/9 -1/5 -1/6 -2/13 0/1 1/8 2/15 1/7 1/6 2/11 1/5 3/14 2/9 3/13 1/4 4/15 3/11 2/7 3/10 1/3 2/5 5/12 9/20 7/15 1/2 8/15 5/9 3/5 2/3 31/45 11/15 4/5 13/15 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/2 -7/8 1/1 -13/15 0/1 -6/7 0/1 1/3 -5/6 1/3 1/1 -14/17 1/4 1/3 -9/11 2/5 1/2 -4/5 1/2 -11/14 1/1 -7/9 0/1 1/2 1/1 -10/13 1/3 1/2 -3/4 1/1 -11/15 0/1 -8/11 0/1 1/2 -5/7 0/1 1/1 -12/17 1/1 1/0 -7/10 0/1 -9/13 1/3 1/2 -2/3 0/1 1/2 1/1 -11/17 1/3 1/2 -9/14 1/3 -16/25 1/2 -7/11 2/5 1/2 -5/8 1/1 -18/29 0/1 1/2 -13/21 0/1 1/3 1/2 -8/13 1/2 1/1 -3/5 1/2 -10/17 1/2 3/5 -17/29 1/2 2/3 -7/12 3/5 1/1 -4/7 2/3 1/1 -9/16 1/1 -5/9 1/2 2/3 1/1 -16/29 1/2 2/3 -11/20 2/3 -6/11 2/3 3/4 -7/13 5/6 1/1 -8/15 1/1 -1/2 1/1 -7/15 1/1 -6/13 1/1 5/4 -5/11 3/2 2/1 -9/20 2/1 -4/9 1/1 2/1 1/0 -7/16 1/1 -10/23 0/1 1/1 -13/30 1/1 -3/7 1/1 2/1 -8/19 2/1 1/0 -21/50 2/1 -13/31 2/1 1/0 -5/12 1/1 3/1 -2/5 1/0 -7/18 -1/1 1/1 -12/31 0/1 1/0 -29/75 0/1 -17/44 1/1 -5/13 1/1 1/0 -8/21 -1/1 0/1 1/0 -19/50 0/1 -11/29 0/1 1/0 -3/8 1/1 -7/19 4/1 1/0 -11/30 1/0 -4/11 -2/1 1/0 -5/14 -1/1 -1/3 0/1 1/1 1/0 -5/16 -1/1 -14/45 0/1 -9/29 0/1 1/0 -4/13 -1/1 1/0 -3/10 0/1 -5/17 1/2 1/1 -7/24 -1/1 1/1 -2/7 0/1 1/1 -5/18 -1/1 1/1 -8/29 0/1 1/0 -11/40 0/1 -3/11 0/1 1/0 -4/15 0/1 -1/4 1/1 -4/17 3/1 1/0 -7/30 1/0 -3/13 -1/1 1/0 -2/9 0/1 1/1 1/0 -3/14 1/1 -1/5 1/0 -3/16 -1/1 -2/11 -2/1 1/0 -3/17 -1/1 -1/2 -1/6 -1/1 1/1 -2/13 -1/1 -1/2 -1/7 -1/1 0/1 -2/15 0/1 -1/8 1/1 0/1 0/1 1/0 1/8 -1/1 2/15 0/1 1/7 0/1 1/1 1/6 -1/1 1/1 3/17 1/2 1/1 2/11 2/1 1/0 1/5 1/0 3/14 -1/1 2/9 -1/1 0/1 1/0 3/13 1/1 1/0 1/4 -1/1 4/15 0/1 3/11 0/1 1/0 2/7 -1/1 0/1 5/17 -1/1 -1/2 3/10 0/1 4/13 1/1 1/0 1/3 -1/1 0/1 1/0 6/17 1/1 1/0 5/14 1/1 9/25 1/0 4/11 2/1 1/0 3/8 -1/1 11/29 0/1 1/0 8/21 0/1 1/1 1/0 5/13 -1/1 1/0 2/5 1/0 7/17 -3/1 1/0 12/29 -2/1 1/0 5/12 -3/1 -1/1 3/7 -2/1 -1/1 7/16 -1/1 4/9 -2/1 -1/1 1/0 13/29 -2/1 1/0 9/20 -2/1 5/11 -2/1 -3/2 6/13 -5/4 -1/1 7/15 -1/1 1/2 -1/1 8/15 -1/1 7/13 -1/1 -5/6 6/11 -3/4 -2/3 11/20 -2/3 5/9 -1/1 -2/3 -1/2 9/16 -1/1 13/23 -1/1 0/1 17/30 -1/1 4/7 -1/1 -2/3 11/19 -2/3 -1/2 29/50 -2/3 18/31 -2/3 -1/2 7/12 -1/1 -3/5 3/5 -1/2 11/18 -1/1 -1/3 19/31 -1/2 0/1 46/75 0/1 27/44 -1/1 8/13 -1/1 -1/2 13/21 -1/2 -1/3 0/1 31/50 0/1 18/29 -1/2 0/1 5/8 -1/1 12/19 -4/7 -1/2 19/30 -1/2 7/11 -1/2 -2/5 9/14 -1/3 2/3 -1/1 -1/2 0/1 11/16 -1/3 31/45 0/1 20/29 -1/2 0/1 9/13 -1/2 -1/3 7/10 0/1 12/17 -1/1 1/0 17/24 -1/1 -1/3 5/7 -1/1 0/1 13/18 -1/1 -1/3 21/29 -1/2 0/1 29/40 0/1 8/11 -1/2 0/1 11/15 0/1 3/4 -1/1 13/17 -3/5 -1/2 23/30 -1/2 10/13 -1/2 -1/3 7/9 -1/1 -1/2 0/1 11/14 -1/1 4/5 -1/2 13/16 -1/3 9/11 -1/2 -2/5 14/17 -1/3 -1/4 5/6 -1/1 -1/3 11/13 -1/3 -1/4 6/7 -1/3 0/1 13/15 0/1 7/8 -1/1 1/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(91,80,240,211) (-1/1,-7/8) -> (3/8,11/29) Hyperbolic Matrix(209,182,240,209) (-7/8,-13/15) -> (13/15,7/8) Hyperbolic Matrix(181,156,210,181) (-13/15,-6/7) -> (6/7,13/15) Hyperbolic Matrix(61,52,-210,-179) (-6/7,-5/6) -> (-7/24,-2/7) Hyperbolic Matrix(361,298,510,421) (-5/6,-14/17) -> (12/17,17/24) Hyperbolic Matrix(151,124,330,271) (-14/17,-9/11) -> (5/11,6/13) Hyperbolic Matrix(211,172,-330,-269) (-9/11,-4/5) -> (-16/25,-7/11) Hyperbolic Matrix(269,212,-420,-331) (-4/5,-11/14) -> (-9/14,-16/25) Hyperbolic Matrix(151,118,270,211) (-11/14,-7/9) -> (5/9,9/16) Hyperbolic Matrix(241,186,390,301) (-7/9,-10/13) -> (8/13,13/21) Hyperbolic Matrix(29,22,-120,-91) (-10/13,-3/4) -> (-1/4,-4/17) Hyperbolic Matrix(89,66,120,89) (-3/4,-11/15) -> (11/15,3/4) Hyperbolic Matrix(241,176,330,241) (-11/15,-8/11) -> (8/11,11/15) Hyperbolic Matrix(89,64,-210,-151) (-8/11,-5/7) -> (-3/7,-8/19) Hyperbolic Matrix(31,22,-210,-149) (-5/7,-12/17) -> (-2/13,-1/7) Hyperbolic Matrix(91,64,300,211) (-12/17,-7/10) -> (3/10,4/13) Hyperbolic Matrix(89,62,300,209) (-7/10,-9/13) -> (5/17,3/10) Hyperbolic Matrix(59,40,-90,-61) (-9/13,-2/3) -> (-2/3,-11/17) Parabolic Matrix(359,232,-930,-601) (-11/17,-9/14) -> (-17/44,-5/13) Hyperbolic Matrix(89,56,-240,-151) (-7/11,-5/8) -> (-3/8,-7/19) Hyperbolic Matrix(29,18,240,149) (-5/8,-18/29) -> (0/1,1/8) Hyperbolic Matrix(481,298,-870,-539) (-18/29,-13/21) -> (-5/9,-16/29) Hyperbolic Matrix(301,186,390,241) (-13/21,-8/13) -> (10/13,7/9) Hyperbolic Matrix(89,54,-150,-91) (-8/13,-3/5) -> (-3/5,-10/17) Parabolic Matrix(269,158,-870,-511) (-10/17,-17/29) -> (-9/29,-4/13) Hyperbolic Matrix(629,368,870,509) (-17/29,-7/12) -> (13/18,21/29) Hyperbolic Matrix(59,34,-210,-121) (-7/12,-4/7) -> (-2/7,-5/18) Hyperbolic Matrix(209,118,-480,-271) (-4/7,-9/16) -> (-7/16,-10/23) Hyperbolic Matrix(211,118,270,151) (-9/16,-5/9) -> (7/9,11/14) Hyperbolic Matrix(1201,662,2070,1141) (-16/29,-11/20) -> (29/50,18/31) Hyperbolic Matrix(391,214,-930,-509) (-11/20,-6/11) -> (-8/19,-21/50) Hyperbolic Matrix(59,32,330,179) (-6/11,-7/13) -> (3/17,2/11) Hyperbolic Matrix(209,112,390,209) (-7/13,-8/15) -> (8/15,7/13) Hyperbolic Matrix(31,16,60,31) (-8/15,-1/2) -> (1/2,8/15) Hyperbolic Matrix(29,14,60,29) (-1/2,-7/15) -> (7/15,1/2) Hyperbolic Matrix(181,84,390,181) (-7/15,-6/13) -> (6/13,7/15) Hyperbolic Matrix(271,124,330,151) (-6/13,-5/11) -> (9/11,14/17) Hyperbolic Matrix(181,82,-660,-299) (-5/11,-9/20) -> (-11/40,-3/11) Hyperbolic Matrix(331,148,-870,-389) (-9/20,-4/9) -> (-8/21,-19/50) Hyperbolic Matrix(59,26,270,119) (-4/9,-7/16) -> (3/14,2/9) Hyperbolic Matrix(511,222,900,391) (-10/23,-13/30) -> (17/30,4/7) Hyperbolic Matrix(509,220,900,389) (-13/30,-3/7) -> (13/23,17/30) Hyperbolic Matrix(929,390,2070,869) (-21/50,-13/31) -> (13/29,9/20) Hyperbolic Matrix(569,238,930,389) (-13/31,-5/12) -> (11/18,19/31) Hyperbolic Matrix(59,24,-150,-61) (-5/12,-2/5) -> (-2/5,-7/18) Parabolic Matrix(541,210,930,361) (-7/18,-12/31) -> (18/31,7/12) Hyperbolic Matrix(2461,952,3570,1381) (-12/31,-29/75) -> (31/45,20/29) Hyperbolic Matrix(2189,846,3180,1229) (-29/75,-17/44) -> (11/16,31/45) Hyperbolic Matrix(89,34,390,149) (-5/13,-8/21) -> (2/9,3/13) Hyperbolic Matrix(1891,718,2610,991) (-19/50,-11/29) -> (21/29,29/40) Hyperbolic Matrix(211,80,240,91) (-11/29,-3/8) -> (7/8,1/1) Hyperbolic Matrix(571,210,900,331) (-7/19,-11/30) -> (19/30,7/11) Hyperbolic Matrix(569,208,900,329) (-11/30,-4/11) -> (12/19,19/30) Hyperbolic Matrix(61,22,-330,-119) (-4/11,-5/14) -> (-3/16,-2/11) Hyperbolic Matrix(29,10,-90,-31) (-5/14,-1/3) -> (-1/3,-5/16) Parabolic Matrix(1951,608,3180,991) (-5/16,-14/45) -> (46/75,27/44) Hyperbolic Matrix(2189,680,3570,1109) (-14/45,-9/29) -> (19/31,46/75) Hyperbolic Matrix(211,64,300,91) (-4/13,-3/10) -> (7/10,12/17) Hyperbolic Matrix(209,62,300,89) (-3/10,-5/17) -> (9/13,7/10) Hyperbolic Matrix(89,26,510,149) (-5/17,-7/24) -> (1/6,3/17) Hyperbolic Matrix(361,100,870,241) (-5/18,-8/29) -> (12/29,5/12) Hyperbolic Matrix(1619,446,2610,719) (-8/29,-11/40) -> (31/50,18/29) Hyperbolic Matrix(89,24,330,89) (-3/11,-4/15) -> (4/15,3/11) Hyperbolic Matrix(31,8,120,31) (-4/15,-1/4) -> (1/4,4/15) Hyperbolic Matrix(691,162,900,211) (-4/17,-7/30) -> (23/30,10/13) Hyperbolic Matrix(689,160,900,209) (-7/30,-3/13) -> (13/17,23/30) Hyperbolic Matrix(149,34,390,89) (-3/13,-2/9) -> (8/21,5/13) Hyperbolic Matrix(119,26,270,59) (-2/9,-3/14) -> (7/16,4/9) Hyperbolic Matrix(29,6,-150,-31) (-3/14,-1/5) -> (-1/5,-3/16) Parabolic Matrix(179,32,330,59) (-2/11,-3/17) -> (7/13,6/11) Hyperbolic Matrix(151,26,180,31) (-3/17,-1/6) -> (5/6,11/13) Hyperbolic Matrix(149,24,180,29) (-1/6,-2/13) -> (14/17,5/6) Hyperbolic Matrix(29,4,210,29) (-1/7,-2/15) -> (2/15,1/7) Hyperbolic Matrix(31,4,240,31) (-2/15,-1/8) -> (1/8,2/15) Hyperbolic Matrix(149,18,240,29) (-1/8,0/1) -> (18/29,5/8) Hyperbolic Matrix(149,-22,210,-31) (1/7,1/6) -> (17/24,5/7) Hyperbolic Matrix(119,-22,330,-61) (2/11,1/5) -> (9/25,4/11) Hyperbolic Matrix(151,-32,420,-89) (1/5,3/14) -> (5/14,9/25) Hyperbolic Matrix(91,-22,120,-29) (3/13,1/4) -> (3/4,13/17) Hyperbolic Matrix(121,-34,210,-59) (3/11,2/7) -> (4/7,11/19) Hyperbolic Matrix(179,-52,210,-61) (2/7,5/17) -> (11/13,6/7) Hyperbolic Matrix(31,-10,90,-29) (4/13,1/3) -> (1/3,6/17) Parabolic Matrix(571,-202,930,-329) (6/17,5/14) -> (27/44,8/13) Hyperbolic Matrix(151,-56,240,-89) (4/11,3/8) -> (5/8,12/19) Hyperbolic Matrix(389,-148,870,-331) (11/29,8/21) -> (4/9,13/29) Hyperbolic Matrix(61,-24,150,-59) (5/13,2/5) -> (2/5,7/17) Parabolic Matrix(601,-248,870,-359) (7/17,12/29) -> (20/29,9/13) Hyperbolic Matrix(151,-64,210,-89) (5/12,3/7) -> (5/7,13/18) Hyperbolic Matrix(271,-118,480,-209) (3/7,7/16) -> (9/16,13/23) Hyperbolic Matrix(539,-244,930,-421) (9/20,5/11) -> (11/19,29/50) Hyperbolic Matrix(479,-262,660,-361) (6/11,11/20) -> (29/40,8/11) Hyperbolic Matrix(539,-298,870,-481) (11/20,5/9) -> (13/21,31/50) Hyperbolic Matrix(91,-54,150,-89) (7/12,3/5) -> (3/5,11/18) Parabolic Matrix(269,-172,330,-211) (7/11,9/14) -> (13/16,9/11) Hyperbolic Matrix(61,-40,90,-59) (9/14,2/3) -> (2/3,11/16) Parabolic Matrix(121,-96,150,-119) (11/14,4/5) -> (4/5,13/16) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(91,80,240,211) -> Matrix(1,0,-2,1) Matrix(209,182,240,209) -> Matrix(1,0,-2,1) Matrix(181,156,210,181) -> Matrix(1,0,-6,1) Matrix(61,52,-210,-179) -> Matrix(1,0,-2,1) Matrix(361,298,510,421) -> Matrix(1,0,-4,1) Matrix(151,124,330,271) -> Matrix(11,-4,-8,3) Matrix(211,172,-330,-269) -> Matrix(1,0,0,1) Matrix(269,212,-420,-331) -> Matrix(3,-2,8,-5) Matrix(151,118,270,211) -> Matrix(3,-2,-4,3) Matrix(241,186,390,301) -> Matrix(1,0,-4,1) Matrix(29,22,-120,-91) -> Matrix(3,-2,2,-1) Matrix(89,66,120,89) -> Matrix(1,0,-2,1) Matrix(241,176,330,241) -> Matrix(1,0,-4,1) Matrix(89,64,-210,-151) -> Matrix(3,-2,2,-1) Matrix(31,22,-210,-149) -> Matrix(1,0,-2,1) Matrix(91,64,300,211) -> Matrix(1,0,0,1) Matrix(89,62,300,209) -> Matrix(1,0,-4,1) Matrix(59,40,-90,-61) -> Matrix(1,0,0,1) Matrix(359,232,-930,-601) -> Matrix(1,0,-2,1) Matrix(89,56,-240,-151) -> Matrix(3,-2,2,-1) Matrix(29,18,240,149) -> Matrix(1,0,-2,1) Matrix(481,298,-870,-539) -> Matrix(5,-2,8,-3) Matrix(301,186,390,241) -> Matrix(1,0,-4,1) Matrix(89,54,-150,-91) -> Matrix(5,-2,8,-3) Matrix(269,158,-870,-511) -> Matrix(3,-2,2,-1) Matrix(629,368,870,509) -> Matrix(3,-2,-4,3) Matrix(59,34,-210,-121) -> Matrix(3,-2,2,-1) Matrix(209,118,-480,-271) -> Matrix(3,-2,2,-1) Matrix(211,118,270,151) -> Matrix(3,-2,-4,3) Matrix(1201,662,2070,1141) -> Matrix(7,-4,-12,7) Matrix(391,214,-930,-509) -> Matrix(5,-4,4,-3) Matrix(59,32,330,179) -> Matrix(5,-4,4,-3) Matrix(209,112,390,209) -> Matrix(11,-10,-12,11) Matrix(31,16,60,31) -> Matrix(3,-2,-4,3) Matrix(29,14,60,29) -> Matrix(1,-2,0,1) Matrix(181,84,390,181) -> Matrix(9,-10,-8,9) Matrix(271,124,330,151) -> Matrix(3,-4,-8,11) Matrix(181,82,-660,-299) -> Matrix(1,-2,2,-3) Matrix(331,148,-870,-389) -> Matrix(1,-2,0,1) Matrix(59,26,270,119) -> Matrix(1,-2,0,1) Matrix(511,222,900,391) -> Matrix(3,-2,-4,3) Matrix(509,220,900,389) -> Matrix(1,-2,0,1) Matrix(929,390,2070,869) -> Matrix(1,-4,0,1) Matrix(569,238,930,389) -> Matrix(1,-2,-2,5) Matrix(59,24,-150,-61) -> Matrix(1,-2,0,1) Matrix(541,210,930,361) -> Matrix(1,2,-2,-3) Matrix(2461,952,3570,1381) -> Matrix(1,0,-2,1) Matrix(2189,846,3180,1229) -> Matrix(1,0,-4,1) Matrix(89,34,390,149) -> Matrix(1,0,0,1) Matrix(1891,718,2610,991) -> Matrix(1,0,-2,1) Matrix(211,80,240,91) -> Matrix(1,0,-2,1) Matrix(571,210,900,331) -> Matrix(1,-6,-2,13) Matrix(569,208,900,329) -> Matrix(1,6,-2,-11) Matrix(61,22,-330,-119) -> Matrix(1,0,0,1) Matrix(29,10,-90,-31) -> Matrix(1,0,0,1) Matrix(1951,608,3180,991) -> Matrix(1,0,0,1) Matrix(2189,680,3570,1109) -> Matrix(1,0,-2,1) Matrix(211,64,300,91) -> Matrix(1,0,0,1) Matrix(209,62,300,89) -> Matrix(1,0,-4,1) Matrix(89,26,510,149) -> Matrix(1,0,0,1) Matrix(361,100,870,241) -> Matrix(1,-2,0,1) Matrix(1619,446,2610,719) -> Matrix(1,0,-2,1) Matrix(89,24,330,89) -> Matrix(1,0,0,1) Matrix(31,8,120,31) -> Matrix(1,0,-2,1) Matrix(691,162,900,211) -> Matrix(1,-4,-2,9) Matrix(689,160,900,209) -> Matrix(1,4,-2,-7) Matrix(149,34,390,89) -> Matrix(1,0,0,1) Matrix(119,26,270,59) -> Matrix(1,-2,0,1) Matrix(29,6,-150,-31) -> Matrix(1,-2,0,1) Matrix(179,32,330,59) -> Matrix(3,4,-4,-5) Matrix(151,26,180,31) -> Matrix(1,0,-2,1) Matrix(149,24,180,29) -> Matrix(1,0,-2,1) Matrix(29,4,210,29) -> Matrix(1,0,2,1) Matrix(31,4,240,31) -> Matrix(1,0,-2,1) Matrix(149,18,240,29) -> Matrix(1,0,-2,1) Matrix(149,-22,210,-31) -> Matrix(1,0,-2,1) Matrix(119,-22,330,-61) -> Matrix(1,0,0,1) Matrix(151,-32,420,-89) -> Matrix(1,2,0,1) Matrix(91,-22,120,-29) -> Matrix(1,2,-2,-3) Matrix(121,-34,210,-59) -> Matrix(1,2,-2,-3) Matrix(179,-52,210,-61) -> Matrix(1,0,-2,1) Matrix(31,-10,90,-29) -> Matrix(1,0,0,1) Matrix(571,-202,930,-329) -> Matrix(1,0,-2,1) Matrix(151,-56,240,-89) -> Matrix(1,2,-2,-3) Matrix(389,-148,870,-331) -> Matrix(1,-2,0,1) Matrix(61,-24,150,-59) -> Matrix(1,-2,0,1) Matrix(601,-248,870,-359) -> Matrix(1,2,-2,-3) Matrix(151,-64,210,-89) -> Matrix(1,2,-2,-3) Matrix(271,-118,480,-209) -> Matrix(1,2,-2,-3) Matrix(539,-244,930,-421) -> Matrix(3,4,-4,-5) Matrix(479,-262,660,-361) -> Matrix(3,2,-2,-1) Matrix(539,-298,870,-481) -> Matrix(3,2,-8,-5) Matrix(91,-54,150,-89) -> Matrix(3,2,-8,-5) Matrix(269,-172,330,-211) -> Matrix(1,0,0,1) Matrix(61,-40,90,-59) -> Matrix(1,0,0,1) Matrix(121,-96,150,-119) -> Matrix(3,2,-8,-5) 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): 18 Degree of the the map X: 18 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. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/8 2/15 1/7 1/6 2/11 1/5 3/14 2/9 3/13 1/4 4/15 3/10 1/3 2/5 5/12 4/9 9/20 7/15 1/2 11/20 17/30 7/12 3/5 19/30 2/3 31/45 7/10 23/30 4/5 5/6 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/0 1/8 -1/1 2/15 0/1 1/7 0/1 1/1 1/6 -1/1 1/1 3/17 1/2 1/1 2/11 2/1 1/0 1/5 1/0 3/14 -1/1 2/9 -1/1 0/1 1/0 3/13 1/1 1/0 1/4 -1/1 4/15 0/1 3/11 0/1 1/0 2/7 -1/1 0/1 5/17 -1/1 -1/2 3/10 0/1 4/13 1/1 1/0 1/3 -1/1 0/1 1/0 6/17 1/1 1/0 5/14 1/1 9/25 1/0 4/11 2/1 1/0 3/8 -1/1 11/29 0/1 1/0 8/21 0/1 1/1 1/0 5/13 -1/1 1/0 2/5 1/0 7/17 -3/1 1/0 12/29 -2/1 1/0 5/12 -3/1 -1/1 3/7 -2/1 -1/1 7/16 -1/1 4/9 -2/1 -1/1 1/0 13/29 -2/1 1/0 9/20 -2/1 5/11 -2/1 -3/2 6/13 -5/4 -1/1 7/15 -1/1 1/2 -1/1 8/15 -1/1 7/13 -1/1 -5/6 6/11 -3/4 -2/3 11/20 -2/3 5/9 -1/1 -2/3 -1/2 9/16 -1/1 13/23 -1/1 0/1 17/30 -1/1 4/7 -1/1 -2/3 11/19 -2/3 -1/2 29/50 -2/3 18/31 -2/3 -1/2 7/12 -1/1 -3/5 3/5 -1/2 11/18 -1/1 -1/3 19/31 -1/2 0/1 46/75 0/1 27/44 -1/1 8/13 -1/1 -1/2 13/21 -1/2 -1/3 0/1 31/50 0/1 18/29 -1/2 0/1 5/8 -1/1 12/19 -4/7 -1/2 19/30 -1/2 7/11 -1/2 -2/5 9/14 -1/3 2/3 -1/1 -1/2 0/1 11/16 -1/3 31/45 0/1 20/29 -1/2 0/1 9/13 -1/2 -1/3 7/10 0/1 12/17 -1/1 1/0 17/24 -1/1 -1/3 5/7 -1/1 0/1 13/18 -1/1 -1/3 21/29 -1/2 0/1 29/40 0/1 8/11 -1/2 0/1 11/15 0/1 3/4 -1/1 13/17 -3/5 -1/2 23/30 -1/2 10/13 -1/2 -1/3 7/9 -1/1 -1/2 0/1 11/14 -1/1 4/5 -1/2 13/16 -1/3 9/11 -1/2 -2/5 14/17 -1/3 -1/4 5/6 -1/1 -1/3 11/13 -1/3 -1/4 6/7 -1/3 0/1 13/15 0/1 7/8 -1/1 1/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(91,-11,240,-29) (0/1,1/8) -> (3/8,11/29) Hyperbolic Matrix(209,-27,240,-31) (1/8,2/15) -> (13/15,7/8) Hyperbolic Matrix(181,-25,210,-29) (2/15,1/7) -> (6/7,13/15) Hyperbolic Matrix(149,-22,210,-31) (1/7,1/6) -> (17/24,5/7) Hyperbolic Matrix(361,-63,510,-89) (1/6,3/17) -> (12/17,17/24) Hyperbolic Matrix(151,-27,330,-59) (3/17,2/11) -> (5/11,6/13) Hyperbolic Matrix(119,-22,330,-61) (2/11,1/5) -> (9/25,4/11) Hyperbolic Matrix(151,-32,420,-89) (1/5,3/14) -> (5/14,9/25) Hyperbolic Matrix(151,-33,270,-59) (3/14,2/9) -> (5/9,9/16) Hyperbolic Matrix(241,-55,390,-89) (2/9,3/13) -> (8/13,13/21) Hyperbolic Matrix(91,-22,120,-29) (3/13,1/4) -> (3/4,13/17) Hyperbolic Matrix(89,-23,120,-31) (1/4,4/15) -> (11/15,3/4) Hyperbolic Matrix(241,-65,330,-89) (4/15,3/11) -> (8/11,11/15) Hyperbolic Matrix(121,-34,210,-59) (3/11,2/7) -> (4/7,11/19) Hyperbolic Matrix(179,-52,210,-61) (2/7,5/17) -> (11/13,6/7) Hyperbolic Matrix(91,-27,300,-89) (5/17,3/10) -> (3/10,4/13) Parabolic Matrix(31,-10,90,-29) (4/13,1/3) -> (1/3,6/17) Parabolic Matrix(571,-202,930,-329) (6/17,5/14) -> (27/44,8/13) Hyperbolic Matrix(151,-56,240,-89) (4/11,3/8) -> (5/8,12/19) Hyperbolic Matrix(389,-148,870,-331) (11/29,8/21) -> (4/9,13/29) Hyperbolic Matrix(301,-115,390,-149) (8/21,5/13) -> (10/13,7/9) Hyperbolic Matrix(61,-24,150,-59) (5/13,2/5) -> (2/5,7/17) Parabolic Matrix(601,-248,870,-359) (7/17,12/29) -> (20/29,9/13) Hyperbolic Matrix(629,-261,870,-361) (12/29,5/12) -> (13/18,21/29) Hyperbolic Matrix(151,-64,210,-89) (5/12,3/7) -> (5/7,13/18) Hyperbolic Matrix(271,-118,480,-209) (3/7,7/16) -> (9/16,13/23) Hyperbolic Matrix(211,-93,270,-119) (7/16,4/9) -> (7/9,11/14) Hyperbolic Matrix(1201,-539,2070,-929) (13/29,9/20) -> (29/50,18/31) Hyperbolic Matrix(539,-244,930,-421) (9/20,5/11) -> (11/19,29/50) Hyperbolic Matrix(209,-97,390,-181) (6/13,7/15) -> (8/15,7/13) Hyperbolic Matrix(31,-15,60,-29) (7/15,1/2) -> (1/2,8/15) Parabolic Matrix(271,-147,330,-179) (7/13,6/11) -> (9/11,14/17) Hyperbolic Matrix(479,-262,660,-361) (6/11,11/20) -> (29/40,8/11) Hyperbolic Matrix(539,-298,870,-481) (11/20,5/9) -> (13/21,31/50) Hyperbolic Matrix(511,-289,900,-509) (13/23,17/30) -> (17/30,4/7) Parabolic Matrix(569,-331,930,-541) (18/31,7/12) -> (11/18,19/31) Hyperbolic Matrix(91,-54,150,-89) (7/12,3/5) -> (3/5,11/18) Parabolic Matrix(2461,-1509,3570,-2189) (19/31,46/75) -> (31/45,20/29) Hyperbolic Matrix(2189,-1343,3180,-1951) (46/75,27/44) -> (11/16,31/45) Hyperbolic Matrix(1891,-1173,2610,-1619) (31/50,18/29) -> (21/29,29/40) Hyperbolic Matrix(211,-131,240,-149) (18/29,5/8) -> (7/8,1/1) Hyperbolic Matrix(571,-361,900,-569) (12/19,19/30) -> (19/30,7/11) Parabolic Matrix(269,-172,330,-211) (7/11,9/14) -> (13/16,9/11) Hyperbolic Matrix(61,-40,90,-59) (9/14,2/3) -> (2/3,11/16) Parabolic Matrix(211,-147,300,-209) (9/13,7/10) -> (7/10,12/17) Parabolic Matrix(691,-529,900,-689) (13/17,23/30) -> (23/30,10/13) Parabolic Matrix(121,-96,150,-119) (11/14,4/5) -> (4/5,13/16) Parabolic Matrix(151,-125,180,-149) (14/17,5/6) -> (5/6,11/13) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-2,1) Matrix(91,-11,240,-29) -> Matrix(1,0,0,1) Matrix(209,-27,240,-31) -> Matrix(1,0,0,1) Matrix(181,-25,210,-29) -> Matrix(1,0,-4,1) Matrix(149,-22,210,-31) -> Matrix(1,0,-2,1) Matrix(361,-63,510,-89) -> Matrix(1,0,-2,1) Matrix(151,-27,330,-59) -> Matrix(3,-4,-2,3) Matrix(119,-22,330,-61) -> Matrix(1,0,0,1) Matrix(151,-32,420,-89) -> Matrix(1,2,0,1) Matrix(151,-33,270,-59) -> Matrix(1,2,-2,-3) Matrix(241,-55,390,-89) -> Matrix(1,0,-2,1) Matrix(91,-22,120,-29) -> Matrix(1,2,-2,-3) Matrix(89,-23,120,-31) -> Matrix(1,0,0,1) Matrix(241,-65,330,-89) -> Matrix(1,0,-2,1) Matrix(121,-34,210,-59) -> Matrix(1,2,-2,-3) Matrix(179,-52,210,-61) -> Matrix(1,0,-2,1) Matrix(91,-27,300,-89) -> Matrix(1,0,2,1) Matrix(31,-10,90,-29) -> Matrix(1,0,0,1) Matrix(571,-202,930,-329) -> Matrix(1,0,-2,1) Matrix(151,-56,240,-89) -> Matrix(1,2,-2,-3) Matrix(389,-148,870,-331) -> Matrix(1,-2,0,1) Matrix(301,-115,390,-149) -> Matrix(1,0,-2,1) Matrix(61,-24,150,-59) -> Matrix(1,-2,0,1) Matrix(601,-248,870,-359) -> Matrix(1,2,-2,-3) Matrix(629,-261,870,-361) -> Matrix(1,2,-2,-3) Matrix(151,-64,210,-89) -> Matrix(1,2,-2,-3) Matrix(271,-118,480,-209) -> Matrix(1,2,-2,-3) Matrix(211,-93,270,-119) -> Matrix(1,2,-2,-3) Matrix(1201,-539,2070,-929) -> Matrix(1,4,-2,-7) Matrix(539,-244,930,-421) -> Matrix(3,4,-4,-5) Matrix(209,-97,390,-181) -> Matrix(9,10,-10,-11) Matrix(31,-15,60,-29) -> Matrix(1,2,-2,-3) Matrix(271,-147,330,-179) -> Matrix(5,4,-14,-11) Matrix(479,-262,660,-361) -> Matrix(3,2,-2,-1) Matrix(539,-298,870,-481) -> Matrix(3,2,-8,-5) Matrix(511,-289,900,-509) -> Matrix(1,2,-2,-3) Matrix(569,-331,930,-541) -> Matrix(3,2,-8,-5) Matrix(91,-54,150,-89) -> Matrix(3,2,-8,-5) Matrix(2461,-1509,3570,-2189) -> Matrix(1,0,0,1) Matrix(2189,-1343,3180,-1951) -> Matrix(1,0,-2,1) Matrix(1891,-1173,2610,-1619) -> Matrix(1,0,0,1) Matrix(211,-131,240,-149) -> Matrix(1,0,0,1) Matrix(571,-361,900,-569) -> Matrix(11,6,-24,-13) Matrix(269,-172,330,-211) -> Matrix(1,0,0,1) Matrix(61,-40,90,-59) -> Matrix(1,0,0,1) Matrix(211,-147,300,-209) -> Matrix(1,0,2,1) Matrix(691,-529,900,-689) -> Matrix(7,4,-16,-9) Matrix(121,-96,150,-119) -> Matrix(3,2,-8,-5) Matrix(151,-125,180,-149) -> 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 9 ----------------------------------------------------------------------- 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 (0/1,1/0) 0 30 1/8 -1/1 2 15 2/15 0/1 4 2 1/7 (0/1,1/1) 0 30 1/6 0 5 3/17 (1/2,1/1) 0 30 2/11 (2/1,1/0) 0 30 1/5 1/0 2 6 3/14 -1/1 2 15 2/9 0 10 3/13 (1/1,1/0) 0 30 7/30 1/0 4 1 1/4 -1/1 2 15 4/15 0/1 2 2 3/11 (0/1,1/0) 0 30 2/7 (-1/1,0/1) 0 30 7/24 0 5 5/17 (-1/1,-1/2) 0 30 3/10 0/1 2 3 4/13 (1/1,1/0) 0 30 1/3 0 10 6/17 (1/1,1/0) 0 30 5/14 1/1 2 15 9/25 1/0 2 6 4/11 (2/1,1/0) 0 30 11/30 1/0 6 1 3/8 -1/1 2 15 11/29 (0/1,1/0) 0 30 8/21 0 10 5/13 (-1/1,1/0) 0 30 2/5 1/0 2 6 7/17 (-3/1,1/0) 0 30 19/46 -3/1 2 15 31/75 -2/1 2 2 12/29 (-2/1,1/0) 0 30 5/12 0 5 13/31 (-2/1,1/0) 0 30 8/19 (-2/1,1/0) 0 30 3/7 (-2/1,-1/1) 0 30 13/30 -1/1 2 1 7/16 -1/1 2 15 4/9 0 10 13/29 (-2/1,1/0) 0 30 9/20 -2/1 2 3 14/31 (-2/1,-3/2) 0 30 5/11 (-2/1,-3/2) 0 30 6/13 (-5/4,-1/1) 0 30 7/15 -1/1 8 2 1/2 -1/1 2 15 1/0 0/1 2 1 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(91,-11,240,-29) (0/1,1/8) -> (3/8,11/29) Hyperbolic Matrix(31,-4,240,-31) (1/8,2/15) -> (1/8,2/15) Reflection Matrix(29,-4,210,-29) (2/15,1/7) -> (2/15,1/7) Reflection Matrix(61,-9,210,-31) (1/7,1/6) -> (2/7,7/24) Glide Reflection Matrix(149,-26,510,-89) (1/6,3/17) -> (7/24,5/17) Glide Reflection Matrix(151,-27,330,-59) (3/17,2/11) -> (5/11,6/13) Hyperbolic Matrix(119,-22,330,-61) (2/11,1/5) -> (9/25,4/11) Hyperbolic Matrix(151,-32,420,-89) (1/5,3/14) -> (5/14,9/25) Hyperbolic Matrix(119,-26,270,-59) (3/14,2/9) -> (7/16,4/9) Glide Reflection Matrix(149,-34,390,-89) (2/9,3/13) -> (8/21,5/13) Glide Reflection Matrix(181,-42,780,-181) (3/13,7/30) -> (3/13,7/30) Reflection Matrix(29,-7,120,-29) (7/30,1/4) -> (7/30,1/4) Reflection Matrix(31,-8,120,-31) (1/4,4/15) -> (1/4,4/15) Reflection Matrix(89,-24,330,-89) (4/15,3/11) -> (4/15,3/11) Reflection Matrix(89,-25,210,-59) (3/11,2/7) -> (8/19,3/7) Glide Reflection Matrix(91,-27,300,-89) (5/17,3/10) -> (3/10,4/13) Parabolic Matrix(31,-10,90,-29) (4/13,1/3) -> (1/3,6/17) Parabolic Matrix(421,-149,1020,-361) (6/17,5/14) -> (7/17,19/46) Glide Reflection Matrix(241,-88,660,-241) (4/11,11/30) -> (4/11,11/30) Reflection Matrix(89,-33,240,-89) (11/30,3/8) -> (11/30,3/8) Reflection Matrix(389,-148,870,-331) (11/29,8/21) -> (4/9,13/29) Hyperbolic Matrix(61,-24,150,-59) (5/13,2/5) -> (2/5,7/17) Parabolic Matrix(2851,-1178,6900,-2851) (19/46,31/75) -> (19/46,31/75) Reflection Matrix(1799,-744,4350,-1799) (31/75,12/29) -> (31/75,12/29) Reflection Matrix(301,-125,720,-299) (12/29,5/12) -> (5/12,13/31) Parabolic Matrix(421,-177,930,-391) (13/31,8/19) -> (14/31,5/11) Glide Reflection Matrix(181,-78,420,-181) (3/7,13/30) -> (3/7,13/30) Reflection Matrix(209,-91,480,-209) (13/30,7/16) -> (13/30,7/16) Reflection Matrix(541,-243,1200,-539) (13/29,9/20) -> (9/20,14/31) Parabolic Matrix(181,-84,390,-181) (6/13,7/15) -> (6/13,7/15) Reflection Matrix(29,-14,60,-29) (7/15,1/2) -> (7/15,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) 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(91,-11,240,-29) -> Matrix(1,0,0,1) Matrix(31,-4,240,-31) -> Matrix(-1,0,2,1) (1/8,2/15) -> (-1/1,0/1) Matrix(29,-4,210,-29) -> Matrix(1,0,2,-1) (2/15,1/7) -> (0/1,1/1) Matrix(61,-9,210,-31) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(149,-26,510,-89) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(151,-27,330,-59) -> Matrix(3,-4,-2,3) Matrix(119,-22,330,-61) -> Matrix(1,0,0,1) Matrix(151,-32,420,-89) -> Matrix(1,2,0,1) 1/0 Matrix(119,-26,270,-59) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(149,-34,390,-89) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(181,-42,780,-181) -> Matrix(-1,2,0,1) (3/13,7/30) -> (1/1,1/0) Matrix(29,-7,120,-29) -> Matrix(1,2,0,-1) (7/30,1/4) -> (-1/1,1/0) Matrix(31,-8,120,-31) -> Matrix(-1,0,2,1) (1/4,4/15) -> (-1/1,0/1) Matrix(89,-24,330,-89) -> Matrix(1,0,0,-1) (4/15,3/11) -> (0/1,1/0) Matrix(89,-25,210,-59) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(91,-27,300,-89) -> Matrix(1,0,2,1) 0/1 Matrix(31,-10,90,-29) -> Matrix(1,0,0,1) Matrix(421,-149,1020,-361) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(241,-88,660,-241) -> Matrix(-1,4,0,1) (4/11,11/30) -> (2/1,1/0) Matrix(89,-33,240,-89) -> Matrix(1,2,0,-1) (11/30,3/8) -> (-1/1,1/0) Matrix(389,-148,870,-331) -> Matrix(1,-2,0,1) 1/0 Matrix(61,-24,150,-59) -> Matrix(1,-2,0,1) 1/0 Matrix(2851,-1178,6900,-2851) -> Matrix(5,12,-2,-5) (19/46,31/75) -> (-3/1,-2/1) Matrix(1799,-744,4350,-1799) -> Matrix(1,4,0,-1) (31/75,12/29) -> (-2/1,1/0) Matrix(301,-125,720,-299) -> Matrix(1,0,0,1) Matrix(421,-177,930,-391) -> Matrix(3,4,-2,-3) *** -> (-2/1,-1/1) Matrix(181,-78,420,-181) -> Matrix(3,4,-2,-3) (3/7,13/30) -> (-2/1,-1/1) Matrix(209,-91,480,-209) -> Matrix(1,2,0,-1) (13/30,7/16) -> (-1/1,1/0) Matrix(541,-243,1200,-539) -> Matrix(3,8,-2,-5) -2/1 Matrix(181,-84,390,-181) -> Matrix(9,10,-8,-9) (6/13,7/15) -> (-5/4,-1/1) Matrix(29,-14,60,-29) -> Matrix(1,2,0,-1) (7/15,1/2) -> (-1/1,1/0) Matrix(-1,1,0,1) -> Matrix(-1,0,2,1) (1/2,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.