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 0/1 1/0 -15/2 0/1 -7/1 1/4 -6/1 0/1 -17/3 1/4 -11/2 1/4 1/3 -5/1 1/2 -14/3 1/2 1/1 -9/2 1/1 -13/3 1/0 -4/1 0/1 1/0 -15/4 0/1 -11/3 1/8 -7/2 1/5 1/4 -17/5 1/4 -10/3 1/3 -13/4 0/1 1/2 -3/1 1/2 -17/6 1/2 2/3 -14/5 1/2 1/1 -25/9 1/2 -11/4 3/4 1/1 -8/3 2/1 1/0 -29/11 1/0 -21/8 -1/1 -13/5 1/0 -5/2 0/1 -17/7 1/4 -29/12 0/1 1/2 -12/5 0/1 -7/3 1/4 -16/7 1/3 1/2 -9/4 1/3 -29/13 3/8 -20/9 2/5 -11/5 5/12 -13/6 6/13 1/2 -15/7 1/2 -2/1 1/2 1/1 -15/8 1/1 -13/7 1/0 -11/6 1/1 1/0 -20/11 0/1 -9/5 1/2 -16/9 1/2 1/1 -23/13 3/4 -30/17 1/1 -7/4 1/1 1/0 -19/11 1/4 -50/29 0/1 -31/18 0/1 1/2 -12/7 0/1 -5/3 1/2 -18/11 2/3 -31/19 3/4 -75/46 1/1 -44/27 1/2 1/1 -13/8 1/2 2/3 -21/13 1/2 -50/31 2/3 -29/18 1/2 2/3 -8/5 2/3 3/4 -19/12 7/8 1/1 -30/19 1/1 -11/7 1/0 -14/9 1/2 1/1 -3/2 1/1 -16/11 1/1 3/2 -45/31 3/2 -29/20 3/2 2/1 -13/9 1/0 -10/7 1/1 -17/12 4/3 3/2 -24/17 2/1 -7/5 1/0 -18/13 2/1 -29/21 1/0 -40/29 2/1 -11/8 3/1 1/0 -15/11 1/0 -4/3 0/1 1/0 -17/13 1/0 -30/23 0/1 -13/10 0/1 1/2 -9/7 1/2 -14/11 1/2 1/1 -5/4 1/1 -16/13 1/1 3/2 -11/9 5/4 -17/14 3/2 8/5 -6/5 2/1 -13/11 1/0 -7/6 5/1 1/0 -15/13 1/0 -8/7 -4/1 1/0 -1/1 1/0 0/1 0/1 1/1 1/4 8/7 4/17 1/4 15/13 1/4 7/6 1/4 5/19 6/5 2/7 17/14 8/27 3/10 11/9 5/16 5/4 1/3 14/11 1/3 1/2 9/7 1/2 13/10 0/1 1/2 4/3 0/1 1/4 15/11 1/4 11/8 1/4 3/11 7/5 1/4 17/12 3/10 4/13 10/7 1/3 13/9 1/4 3/2 1/3 17/11 3/8 14/9 1/3 1/2 25/16 1/3 11/7 1/4 8/5 3/8 2/5 29/18 2/5 1/2 21/13 1/2 13/8 2/5 1/2 5/3 1/2 17/10 0/1 1/2 29/17 1/0 12/7 0/1 7/4 1/4 1/3 16/9 1/3 1/2 9/5 1/2 29/16 0/1 1/2 20/11 0/1 11/6 1/4 1/3 13/7 1/4 15/8 1/3 2/1 1/3 1/2 15/7 1/2 13/6 1/2 6/11 11/5 5/8 20/9 2/3 9/4 1/1 16/7 1/2 1/1 23/10 3/4 1/1 30/13 1/1 7/3 1/0 19/8 -1/1 1/0 50/21 0/1 31/13 1/0 12/5 0/1 5/2 0/1 18/7 0/1 31/12 0/1 1/6 75/29 1/6 44/17 1/6 1/5 13/5 1/4 21/8 1/5 50/19 2/9 29/11 1/4 8/3 1/4 2/7 19/7 5/16 30/11 1/3 11/4 1/3 3/8 14/5 1/3 1/2 3/1 1/2 16/5 1/2 1/1 45/14 1/1 29/9 1/0 13/4 0/1 1/2 10/3 1/1 17/5 1/0 24/7 2/1 7/2 -1/1 1/0 18/5 0/1 29/8 -1/2 0/1 40/11 0/1 11/3 -1/4 15/4 0/1 4/1 0/1 1/4 17/4 0/1 1/2 30/7 0/1 13/3 1/4 9/2 1/3 14/3 1/3 1/2 5/1 1/2 16/3 1/2 1/1 11/2 1/1 1/0 17/3 1/0 6/1 0/1 13/2 1/2 2/3 7/1 1/0 15/2 0/1 8/1 0/1 1/4 1/0 0/1 1/2 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(3,-2,8,-5) Matrix(31,240,4,31) -> Matrix(1,0,4,1) Matrix(29,210,4,29) -> Matrix(1,0,-4,1) Matrix(31,210,-22,-149) -> Matrix(7,-2,4,-1) Matrix(89,510,26,149) -> Matrix(7,-2,4,-1) Matrix(59,330,32,179) -> Matrix(1,0,0,1) Matrix(61,330,-22,-119) -> Matrix(5,-2,8,-3) Matrix(89,420,-32,-151) -> Matrix(1,0,0,1) Matrix(59,270,26,119) -> Matrix(1,0,0,1) Matrix(89,390,34,149) -> Matrix(1,0,4,1) Matrix(29,120,-22,-91) -> Matrix(1,0,0,1) Matrix(31,120,8,31) -> Matrix(1,0,4,1) Matrix(89,330,24,89) -> Matrix(1,0,-12,1) Matrix(59,210,-34,-121) -> Matrix(1,0,-4,1) Matrix(61,210,-52,-179) -> Matrix(15,-4,4,-1) Matrix(89,300,62,209) -> Matrix(1,0,0,1) Matrix(91,300,64,211) -> Matrix(11,-4,36,-13) Matrix(29,90,-10,-31) -> Matrix(5,-2,8,-3) Matrix(329,930,-202,-571) -> Matrix(1,0,0,1) Matrix(89,240,-56,-151) -> Matrix(3,-4,4,-5) Matrix(91,240,80,211) -> Matrix(1,2,4,9) Matrix(331,870,-148,-389) -> Matrix(3,4,8,11) Matrix(149,390,34,89) -> Matrix(1,0,4,1) Matrix(59,150,-24,-61) -> Matrix(1,0,4,1) Matrix(359,870,-248,-601) -> Matrix(7,-2,4,-1) Matrix(361,870,100,241) -> Matrix(1,0,-4,1) Matrix(89,210,-64,-151) -> Matrix(7,-2,4,-1) Matrix(209,480,-118,-271) -> Matrix(5,-2,8,-3) Matrix(119,270,26,59) -> Matrix(1,0,0,1) Matrix(929,2070,390,869) -> Matrix(5,-2,8,-3) Matrix(421,930,-244,-539) -> Matrix(5,-2,8,-3) Matrix(151,330,124,271) -> Matrix(23,-10,76,-33) Matrix(181,390,84,181) -> Matrix(25,-12,48,-23) Matrix(29,60,14,29) -> Matrix(3,-2,8,-5) Matrix(31,60,16,31) -> Matrix(3,-2,8,-5) Matrix(209,390,112,209) -> Matrix(1,-2,4,-7) Matrix(179,330,32,59) -> Matrix(1,0,0,1) Matrix(361,660,-262,-479) -> Matrix(1,2,0,1) Matrix(481,870,-298,-539) -> Matrix(5,-2,8,-3) Matrix(151,270,118,211) -> Matrix(3,-2,8,-5) Matrix(509,900,220,389) -> Matrix(5,-4,4,-3) Matrix(511,900,222,391) -> Matrix(3,-4,4,-5) Matrix(1201,2070,662,1141) -> Matrix(1,0,0,1) Matrix(541,930,210,361) -> Matrix(1,0,4,1) Matrix(89,150,-54,-91) -> Matrix(5,-2,8,-3) Matrix(569,930,238,389) -> Matrix(3,-2,-4,3) Matrix(2189,3570,680,1109) -> Matrix(5,-4,4,-3) Matrix(1951,3180,608,991) -> Matrix(1,0,0,1) Matrix(241,390,186,301) -> Matrix(3,-2,8,-5) Matrix(1619,2610,446,719) -> Matrix(3,-2,-4,3) Matrix(149,240,18,29) -> Matrix(3,-2,8,-5) Matrix(569,900,208,329) -> Matrix(11,-10,32,-29) Matrix(571,900,210,331) -> Matrix(5,-6,16,-19) Matrix(211,330,-172,-269) -> Matrix(5,-4,4,-3) Matrix(59,90,-40,-61) -> Matrix(5,-4,4,-3) Matrix(2189,3180,846,1229) -> Matrix(3,-4,16,-21) Matrix(2461,3570,952,1381) -> Matrix(1,-2,8,-15) Matrix(209,300,62,89) -> Matrix(1,0,0,1) Matrix(211,300,64,91) -> Matrix(3,-4,4,-5) Matrix(361,510,298,421) -> Matrix(7,-12,24,-41) Matrix(629,870,368,509) -> Matrix(1,-2,0,1) Matrix(1891,2610,718,991) -> Matrix(1,0,4,1) Matrix(241,330,176,241) -> Matrix(1,-6,4,-23) Matrix(89,120,66,89) -> Matrix(1,0,4,1) Matrix(689,900,160,209) -> Matrix(1,0,4,1) Matrix(691,900,162,211) -> Matrix(1,0,0,1) Matrix(301,390,186,241) -> Matrix(3,-2,8,-5) Matrix(211,270,118,151) -> Matrix(3,-2,8,-5) Matrix(119,150,-96,-121) -> Matrix(5,-4,4,-3) Matrix(271,330,124,151) -> Matrix(7,-10,12,-17) Matrix(149,180,24,29) -> Matrix(1,-2,4,-7) Matrix(151,180,26,31) -> Matrix(1,-2,0,1) Matrix(181,210,156,181) -> Matrix(1,-10,4,-39) Matrix(209,240,182,209) -> Matrix(1,8,4,33) Matrix(211,240,80,91) -> Matrix(1,2,4,9) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(179,-210,52,-61) -> Matrix(15,-4,4,-1) Matrix(269,-330,172,-211) -> Matrix(13,-4,36,-11) Matrix(331,-420,212,-269) -> Matrix(1,0,0,1) Matrix(91,-120,22,-29) -> Matrix(1,0,0,1) Matrix(151,-210,64,-89) -> Matrix(7,-2,4,-1) Matrix(149,-210,22,-31) -> Matrix(7,-2,4,-1) Matrix(61,-90,40,-59) -> Matrix(13,-4,36,-11) Matrix(601,-930,232,-359) -> Matrix(5,-2,28,-11) Matrix(151,-240,56,-89) -> Matrix(11,-4,36,-13) Matrix(539,-870,298,-481) -> Matrix(5,-2,8,-3) Matrix(91,-150,54,-89) -> Matrix(5,-2,8,-3) Matrix(511,-870,158,-269) -> Matrix(1,0,0,1) Matrix(121,-210,34,-59) -> Matrix(1,0,-4,1) Matrix(271,-480,118,-209) -> Matrix(5,-2,8,-3) Matrix(509,-930,214,-391) -> Matrix(1,0,-4,1) Matrix(299,-660,82,-181) -> Matrix(3,-2,-4,3) Matrix(389,-870,148,-331) -> Matrix(5,-4,24,-19) Matrix(61,-150,24,-59) -> Matrix(1,0,4,1) Matrix(119,-330,22,-61) -> Matrix(5,-2,8,-3) Matrix(31,-90,10,-29) -> Matrix(5,-2,8,-3) Matrix(31,-150,6,-29) -> Matrix(5,-2,8,-3) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 16 Degree of the the map X: 32 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 -6/1 -5/1 -3/1 -5/2 -9/5 -5/3 -3/2 -5/4 0/1 1/1 15/13 5/4 15/11 3/2 5/3 9/5 2/1 15/7 5/2 75/29 3/1 11/3 15/4 4/1 30/7 13/3 5/1 6/1 7/1 15/2 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 0/1 -5/1 1/2 -9/2 1/1 -13/3 1/0 -4/1 0/1 1/0 -15/4 0/1 -11/3 1/8 -7/2 1/5 1/4 -3/1 1/2 -8/3 2/1 1/0 -21/8 -1/1 -13/5 1/0 -5/2 0/1 -7/3 1/4 -9/4 1/3 -11/5 5/12 -2/1 1/2 1/1 -9/5 1/2 -16/9 1/2 1/1 -7/4 1/1 1/0 -5/3 1/2 -8/5 2/3 3/4 -11/7 1/0 -14/9 1/2 1/1 -3/2 1/1 -7/5 1/0 -4/3 0/1 1/0 -9/7 1/2 -14/11 1/2 1/1 -5/4 1/1 -16/13 1/1 3/2 -11/9 5/4 -6/5 2/1 -7/6 5/1 1/0 -1/1 1/0 0/1 0/1 1/1 1/4 8/7 4/17 1/4 15/13 1/4 7/6 1/4 5/19 6/5 2/7 11/9 5/16 5/4 1/3 14/11 1/3 1/2 9/7 1/2 13/10 0/1 1/2 4/3 0/1 1/4 15/11 1/4 11/8 1/4 3/11 7/5 1/4 10/7 1/3 13/9 1/4 3/2 1/3 17/11 3/8 14/9 1/3 1/2 25/16 1/3 11/7 1/4 8/5 3/8 2/5 5/3 1/2 7/4 1/4 1/3 16/9 1/3 1/2 9/5 1/2 11/6 1/4 1/3 13/7 1/4 15/8 1/3 2/1 1/3 1/2 15/7 1/2 13/6 1/2 6/11 11/5 5/8 9/4 1/1 16/7 1/2 1/1 23/10 3/4 1/1 30/13 1/1 7/3 1/0 12/5 0/1 5/2 0/1 18/7 0/1 31/12 0/1 1/6 75/29 1/6 44/17 1/6 1/5 13/5 1/4 21/8 1/5 8/3 1/4 2/7 3/1 1/2 7/2 -1/1 1/0 11/3 -1/4 15/4 0/1 4/1 0/1 1/4 17/4 0/1 1/2 30/7 0/1 13/3 1/4 9/2 1/3 5/1 1/2 6/1 0/1 7/1 1/0 15/2 0/1 8/1 0/1 1/4 1/0 0/1 1/2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(16,105,7,46) (-6/1,1/0) -> (9/4,16/7) Hyperbolic Matrix(14,75,-3,-16) (-6/1,-5/1) -> (-5/1,-9/2) Parabolic Matrix(89,390,34,149) (-9/2,-13/3) -> (13/5,21/8) Hyperbolic Matrix(46,195,25,106) (-13/3,-4/1) -> (11/6,13/7) 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(46,165,-29,-104) (-11/3,-7/2) -> (-8/5,-11/7) Hyperbolic Matrix(14,45,-5,-16) (-7/2,-3/1) -> (-3/1,-8/3) Parabolic Matrix(74,195,-63,-166) (-8/3,-21/8) -> (-6/5,-7/6) Hyperbolic Matrix(149,390,34,89) (-21/8,-13/5) -> (13/3,9/2) Hyperbolic Matrix(76,195,53,136) (-13/5,-5/2) -> (10/7,13/9) Hyperbolic Matrix(44,105,31,74) (-5/2,-7/3) -> (7/5,10/7) Hyperbolic Matrix(46,105,7,16) (-7/3,-9/4) -> (6/1,7/1) Hyperbolic Matrix(74,165,-61,-136) (-9/4,-11/5) -> (-11/9,-6/5) Hyperbolic Matrix(76,165,35,76) (-11/5,-2/1) -> (13/6,11/5) Hyperbolic Matrix(74,135,57,104) (-2/1,-9/5) -> (9/7,13/10) Hyperbolic Matrix(151,270,118,211) (-9/5,-16/9) -> (14/11,9/7) Hyperbolic Matrix(76,135,9,16) (-16/9,-7/4) -> (8/1,1/0) Hyperbolic Matrix(44,75,-27,-46) (-7/4,-5/3) -> (-5/3,-8/5) Parabolic Matrix(211,330,-172,-269) (-11/7,-14/9) -> (-16/13,-11/9) Hyperbolic Matrix(224,345,87,134) (-14/9,-3/2) -> (18/7,31/12) Hyperbolic Matrix(74,105,31,44) (-3/2,-7/5) -> (7/3,12/5) Hyperbolic Matrix(76,105,55,76) (-7/5,-4/3) -> (11/8,7/5) Hyperbolic Matrix(104,135,57,74) (-4/3,-9/7) -> (9/5,11/6) 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(14,15,13,14) (-7/6,-1/1) -> (1/1,8/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(196,-225,169,-194) (8/7,15/13) -> (15/13,7/6) Parabolic Matrix(166,-195,63,-74) (7/6,6/5) -> (21/8,8/3) Hyperbolic Matrix(136,-165,61,-74) (6/5,11/9) -> (11/5,9/4) 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(166,-225,121,-164) (4/3,15/11) -> (15/11,11/8) Parabolic 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(104,-165,29,-46) (11/7,8/5) -> (7/2,11/3) Hyperbolic Matrix(46,-75,27,-44) (8/5,5/3) -> (5/3,7/4) Parabolic Matrix(271,-480,118,-209) (7/4,16/9) -> (16/7,23/10) Hyperbolic Matrix(314,-585,73,-136) (13/7,15/8) -> (30/7,13/3) Hyperbolic Matrix(166,-315,39,-74) (15/8,2/1) -> (17/4,30/7) Hyperbolic Matrix(106,-225,49,-104) (2/1,15/7) -> (15/7,13/6) Parabolic Matrix(254,-585,33,-76) (23/10,30/13) -> (15/2,8/1) Hyperbolic Matrix(136,-315,19,-44) (30/13,7/3) -> (7/1,15/2) Hyperbolic Matrix(61,-150,24,-59) (12/5,5/2) -> (5/2,18/7) Parabolic Matrix(2176,-5625,841,-2174) (31/12,75/29) -> (75/29,44/17) Parabolic Matrix(16,-45,5,-14) (8/3,3/1) -> (3/1,7/2) Parabolic Matrix(16,-75,3,-14) (9/2,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(16,105,7,46) -> Matrix(3,-1,4,-1) Matrix(14,75,-3,-16) -> Matrix(3,-1,4,-1) Matrix(89,390,34,149) -> Matrix(1,0,4,1) Matrix(46,195,25,106) -> Matrix(1,-1,4,-3) Matrix(31,120,8,31) -> Matrix(1,0,4,1) Matrix(89,330,24,89) -> Matrix(1,0,-12,1) Matrix(46,165,-29,-104) -> Matrix(7,-1,8,-1) Matrix(14,45,-5,-16) -> Matrix(3,-1,4,-1) Matrix(74,195,-63,-166) -> Matrix(1,3,0,1) Matrix(149,390,34,89) -> Matrix(1,0,4,1) Matrix(76,195,53,136) -> Matrix(1,-1,4,-3) Matrix(44,105,31,74) -> Matrix(5,-1,16,-3) Matrix(46,105,7,16) -> Matrix(3,-1,4,-1) Matrix(74,165,-61,-136) -> Matrix(13,-5,8,-3) Matrix(76,165,35,76) -> Matrix(11,-5,20,-9) Matrix(74,135,57,104) -> Matrix(1,-1,4,-3) Matrix(151,270,118,211) -> Matrix(3,-2,8,-5) Matrix(76,135,9,16) -> Matrix(1,-1,4,-3) Matrix(44,75,-27,-46) -> Matrix(3,-1,4,-1) Matrix(211,330,-172,-269) -> Matrix(5,-4,4,-3) Matrix(224,345,87,134) -> Matrix(1,-1,8,-7) Matrix(74,105,31,44) -> Matrix(1,-1,0,1) Matrix(76,105,55,76) -> Matrix(1,-3,4,-11) Matrix(104,135,57,74) -> Matrix(1,-1,4,-3) Matrix(211,270,118,151) -> Matrix(3,-2,8,-5) Matrix(119,150,-96,-121) -> Matrix(5,-4,4,-3) Matrix(14,15,13,14) -> Matrix(1,-1,4,-3) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(196,-225,169,-194) -> Matrix(37,-9,144,-35) Matrix(166,-195,63,-74) -> Matrix(11,-3,48,-13) Matrix(136,-165,61,-74) -> Matrix(17,-5,24,-7) Matrix(269,-330,172,-211) -> Matrix(13,-4,36,-11) Matrix(331,-420,212,-269) -> Matrix(1,0,0,1) Matrix(91,-120,22,-29) -> Matrix(1,0,0,1) Matrix(166,-225,121,-164) -> Matrix(13,-3,48,-11) Matrix(61,-90,40,-59) -> Matrix(13,-4,36,-11) Matrix(601,-930,232,-359) -> Matrix(5,-2,28,-11) Matrix(104,-165,29,-46) -> Matrix(3,-1,-8,3) Matrix(46,-75,27,-44) -> Matrix(3,-1,4,-1) Matrix(271,-480,118,-209) -> Matrix(5,-2,8,-3) Matrix(314,-585,73,-136) -> Matrix(3,-1,16,-5) Matrix(166,-315,39,-74) -> Matrix(3,-1,4,-1) Matrix(106,-225,49,-104) -> Matrix(15,-7,28,-13) Matrix(254,-585,33,-76) -> Matrix(1,-1,8,-7) Matrix(136,-315,19,-44) -> Matrix(1,-1,0,1) Matrix(61,-150,24,-59) -> Matrix(1,0,4,1) Matrix(2176,-5625,841,-2174) -> Matrix(7,-1,36,-5) Matrix(16,-45,5,-14) -> Matrix(3,-1,4,-1) Matrix(16,-75,3,-14) -> Matrix(3,-1,4,-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): 16 ----------------------------------------------------------------------- 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 2 1 1/1 1/4 1 15 15/13 1/4 9 1 7/6 (1/4,5/19) 0 15 6/5 2/7 2 5 11/9 5/16 1 15 5/4 1/3 2 3 14/11 (1/3,1/2) 0 15 9/7 1/2 1 5 4/3 (0/1,1/4) 0 15 15/11 1/4 3 1 7/5 1/4 1 15 3/2 1/3 2 5 14/9 (1/3,1/2) 0 15 25/16 1/3 2 3 11/7 1/4 1 15 8/5 (3/8,2/5) 0 15 5/3 1/2 1 3 7/4 (1/4,1/3) 0 15 16/9 (1/3,1/2) 0 15 9/5 1/2 1 5 11/6 (1/4,1/3) 0 15 2/1 (1/3,1/2) 0 15 15/7 1/2 7 1 11/5 5/8 1 15 9/4 1/1 2 5 7/3 1/0 1 15 12/5 0/1 2 5 5/2 0/1 2 3 18/7 0/1 2 5 31/12 (0/1,1/6) 0 15 75/29 1/6 1 1 13/5 1/4 1 15 21/8 1/5 2 5 8/3 (1/4,2/7) 0 15 3/1 1/2 1 5 7/2 (-1/1,1/0) 0 15 11/3 -1/4 1 15 15/4 0/1 8 1 4/1 (0/1,1/4) 0 15 17/4 (0/1,1/2) 0 15 30/7 0/1 2 1 13/3 1/4 1 15 9/2 1/3 2 5 5/1 1/2 1 3 6/1 0/1 2 5 7/1 1/0 1 15 15/2 0/1 4 1 8/1 (0/1,1/4) 0 15 1/0 (0/1,1/2) 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(14,-15,13,-14) (1/1,15/13) -> (1/1,15/13) Reflection Matrix(181,-210,156,-181) (15/13,7/6) -> (15/13,7/6) Reflection Matrix(166,-195,63,-74) (7/6,6/5) -> (21/8,8/3) Hyperbolic Matrix(136,-165,61,-74) (6/5,11/9) -> (11/5,9/4) 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(211,-270,118,-151) (14/11,9/7) -> (16/9,9/5) Glide Reflection Matrix(104,-135,57,-74) (9/7,4/3) -> (9/5,11/6) Glide Reflection Matrix(89,-120,66,-89) (4/3,15/11) -> (4/3,15/11) Reflection Matrix(76,-105,55,-76) (15/11,7/5) -> (15/11,7/5) Reflection Matrix(74,-105,31,-44) (7/5,3/2) -> (7/3,12/5) Glide Reflection Matrix(224,-345,87,-134) (3/2,14/9) -> (18/7,31/12) Glide Reflection Matrix(104,-165,29,-46) (11/7,8/5) -> (7/2,11/3) Hyperbolic Matrix(46,-75,27,-44) (8/5,5/3) -> (5/3,7/4) Parabolic Matrix(76,-135,9,-16) (7/4,16/9) -> (8/1,1/0) Glide Reflection Matrix(106,-195,25,-46) (11/6,2/1) -> (4/1,17/4) Glide Reflection Matrix(29,-60,14,-29) (2/1,15/7) -> (2/1,15/7) Reflection Matrix(76,-165,35,-76) (15/7,11/5) -> (15/7,11/5) Reflection Matrix(46,-105,7,-16) (9/4,7/3) -> (6/1,7/1) Glide Reflection Matrix(61,-150,24,-59) (12/5,5/2) -> (5/2,18/7) Parabolic Matrix(1799,-4650,696,-1799) (31/12,75/29) -> (31/12,75/29) Reflection Matrix(376,-975,145,-376) (75/29,13/5) -> (75/29,13/5) Reflection Matrix(149,-390,34,-89) (13/5,21/8) -> (13/3,9/2) Glide Reflection Matrix(16,-45,5,-14) (8/3,3/1) -> (3/1,7/2) Parabolic 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(16,-75,3,-14) (9/2,5/1) -> (5/1,6/1) 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(1,0,4,-1) (0/1,1/0) -> (0/1,1/2) Matrix(1,0,2,-1) -> Matrix(1,0,8,-1) (0/1,1/1) -> (0/1,1/4) Matrix(14,-15,13,-14) -> Matrix(3,-1,8,-3) (1/1,15/13) -> (1/4,1/2) Matrix(181,-210,156,-181) -> Matrix(39,-10,152,-39) (15/13,7/6) -> (1/4,5/19) Matrix(166,-195,63,-74) -> Matrix(11,-3,48,-13) 1/4 Matrix(136,-165,61,-74) -> Matrix(17,-5,24,-7) Matrix(269,-330,172,-211) -> Matrix(13,-4,36,-11) 1/3 Matrix(331,-420,212,-269) -> Matrix(1,0,0,1) Matrix(211,-270,118,-151) -> Matrix(5,-2,12,-5) *** -> (1/3,1/2) Matrix(104,-135,57,-74) -> Matrix(3,-1,8,-3) *** -> (1/4,1/2) Matrix(89,-120,66,-89) -> Matrix(1,0,8,-1) (4/3,15/11) -> (0/1,1/4) Matrix(76,-105,55,-76) -> Matrix(11,-3,40,-11) (15/11,7/5) -> (1/4,3/10) Matrix(74,-105,31,-44) -> Matrix(3,-1,-4,1) Matrix(224,-345,87,-134) -> Matrix(3,-1,20,-7) Matrix(104,-165,29,-46) -> Matrix(3,-1,-8,3) Matrix(46,-75,27,-44) -> Matrix(3,-1,4,-1) 1/2 Matrix(76,-135,9,-16) -> Matrix(3,-1,8,-3) *** -> (1/4,1/2) Matrix(106,-195,25,-46) -> Matrix(3,-1,8,-3) *** -> (1/4,1/2) Matrix(29,-60,14,-29) -> Matrix(5,-2,12,-5) (2/1,15/7) -> (1/3,1/2) Matrix(76,-165,35,-76) -> Matrix(9,-5,16,-9) (15/7,11/5) -> (1/2,5/8) Matrix(46,-105,7,-16) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(61,-150,24,-59) -> Matrix(1,0,4,1) 0/1 Matrix(1799,-4650,696,-1799) -> Matrix(1,0,12,-1) (31/12,75/29) -> (0/1,1/6) Matrix(376,-975,145,-376) -> Matrix(5,-1,24,-5) (75/29,13/5) -> (1/6,1/4) Matrix(149,-390,34,-89) -> Matrix(1,0,8,-1) *** -> (0/1,1/4) Matrix(16,-45,5,-14) -> Matrix(3,-1,4,-1) 1/2 Matrix(89,-330,24,-89) -> Matrix(-1,0,8,1) (11/3,15/4) -> (-1/4,0/1) Matrix(31,-120,8,-31) -> Matrix(1,0,8,-1) (15/4,4/1) -> (0/1,1/4) Matrix(239,-1020,56,-239) -> Matrix(1,0,4,-1) (17/4,30/7) -> (0/1,1/2) Matrix(181,-780,42,-181) -> Matrix(1,0,8,-1) (30/7,13/3) -> (0/1,1/4) Matrix(16,-75,3,-14) -> Matrix(3,-1,4,-1) 1/2 Matrix(29,-210,4,-29) -> Matrix(1,0,0,-1) (7/1,15/2) -> (0/1,1/0) Matrix(31,-240,4,-31) -> Matrix(1,0,8,-1) (15/2,8/1) -> (0/1,1/4) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.