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 1/0 -7/8 -6/11 -13/15 -1/2 -6/7 -5/11 -5/6 -2/5 -14/17 -1/3 -9/11 -1/2 -4/5 -1/3 -11/14 -8/27 -7/9 -1/4 -10/13 -1/3 -3/4 -2/7 -11/15 -1/4 -8/11 -5/21 -5/7 -3/14 -12/17 -1/5 -7/10 0/1 -9/13 -1/4 -2/3 -1/5 -11/17 -1/6 -9/14 -4/25 -16/25 -1/7 -7/11 -1/8 -5/8 -2/9 -18/29 -7/37 -13/21 -1/6 -8/13 -1/5 -3/5 -1/6 -10/17 -1/7 -17/29 -5/36 -7/12 0/1 -4/7 -1/9 -9/16 -2/7 -5/9 -1/6 -16/29 -3/23 -11/20 0/1 -6/11 -1/7 -7/13 -1/6 -8/15 -1/7 -1/2 0/1 -7/15 -1/6 -6/13 -1/7 -5/11 -1/6 -9/20 0/1 -4/9 -1/7 -7/16 -2/19 -10/23 -1/15 -13/30 0/1 -3/7 -1/4 -8/19 -3/19 -21/50 -2/13 -13/31 -5/34 -5/12 0/1 -2/5 -1/7 -7/18 -2/15 -12/31 -7/55 -29/75 -1/8 -17/44 -6/49 -5/13 -1/8 -8/21 -1/7 -19/50 -2/15 -11/29 -7/54 -3/8 -2/17 -7/19 -1/12 -11/30 0/1 -4/11 -1/5 -5/14 -4/27 -1/3 -1/8 -5/16 -6/53 -14/45 -1/9 -9/29 -7/64 -4/13 -1/9 -3/10 0/1 -5/17 -1/8 -7/24 0/1 -2/7 -3/25 -5/18 -2/17 -8/29 -13/113 -11/40 -4/35 -3/11 -5/44 -4/15 -1/9 -1/4 -2/19 -4/17 -1/9 -7/30 -2/19 -3/13 -1/10 -2/9 -1/9 -3/14 -8/77 -1/5 -1/10 -3/16 -10/103 -2/11 -1/11 -3/17 -1/10 -1/6 -2/21 -2/13 -1/11 -1/7 -5/54 -2/15 -1/11 -1/8 -6/67 0/1 -1/13 1/8 -6/89 2/15 -1/15 1/7 -5/76 1/6 -2/31 3/17 -1/16 2/11 -1/15 1/5 -1/16 3/14 -8/131 2/9 -1/17 3/13 -1/16 1/4 -2/33 4/15 -1/17 3/11 -5/86 2/7 -3/53 5/17 -1/18 3/10 0/1 4/13 -1/17 1/3 -1/18 6/17 -1/19 5/14 -4/77 9/25 -1/20 4/11 -1/21 3/8 -2/35 11/29 -7/128 8/21 -1/19 5/13 -1/18 2/5 -1/19 7/17 -1/20 12/29 -5/101 5/12 0/1 3/7 -1/22 7/16 -2/33 4/9 -1/19 13/29 -3/62 9/20 0/1 5/11 -1/20 6/13 -1/19 7/15 -1/20 1/2 0/1 8/15 -1/19 7/13 -1/20 6/11 -1/19 11/20 0/1 5/9 -1/20 9/16 -2/45 13/23 -1/28 17/30 0/1 4/7 -1/17 11/19 -3/58 29/50 -2/39 18/31 -5/99 7/12 0/1 3/5 -1/20 11/18 -2/41 19/31 -7/146 46/75 -1/21 27/44 -6/127 8/13 -1/21 13/21 -1/20 31/50 -2/41 18/29 -7/145 5/8 -2/43 12/19 -1/25 19/30 0/1 7/11 -1/18 9/14 -4/79 2/3 -1/21 11/16 -6/131 31/45 -1/22 20/29 -7/155 9/13 -1/22 7/10 0/1 12/17 -1/21 17/24 0/1 5/7 -3/64 13/18 -2/43 21/29 -13/282 29/40 -4/87 8/11 -5/109 11/15 -1/22 3/4 -2/45 13/17 -1/22 23/30 -2/45 10/13 -1/23 7/9 -1/22 11/14 -8/181 4/5 -1/23 13/16 -10/233 9/11 -1/24 14/17 -1/23 5/6 -2/47 11/13 -1/24 6/7 -5/119 13/15 -1/24 7/8 -6/145 1/1 -1/26 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,-26,1) Matrix(91,80,240,211) -> Matrix(7,4,-128,-73) Matrix(209,182,240,209) -> Matrix(23,12,-554,-289) Matrix(181,156,210,181) -> Matrix(21,10,-502,-239) Matrix(61,52,-210,-179) -> Matrix(5,2,-38,-15) Matrix(361,298,510,421) -> Matrix(5,2,-108,-43) Matrix(151,124,330,271) -> Matrix(5,2,-98,-39) Matrix(211,172,-330,-269) -> Matrix(5,2,-38,-15) Matrix(269,212,-420,-331) -> Matrix(13,4,-88,-27) Matrix(151,118,270,211) -> Matrix(7,2,-144,-41) Matrix(241,186,390,301) -> Matrix(7,2,-144,-41) Matrix(29,22,-120,-91) -> Matrix(1,0,-6,1) Matrix(89,66,120,89) -> Matrix(15,4,-334,-89) Matrix(241,176,330,241) -> Matrix(41,10,-898,-219) Matrix(89,64,-210,-151) -> Matrix(9,2,-50,-11) Matrix(31,22,-210,-149) -> Matrix(11,2,-116,-21) Matrix(91,64,300,211) -> Matrix(1,0,-12,1) Matrix(89,62,300,209) -> Matrix(1,0,-14,1) Matrix(59,40,-90,-61) -> Matrix(9,2,-50,-11) Matrix(359,232,-930,-601) -> Matrix(11,2,-94,-17) Matrix(89,56,-240,-151) -> Matrix(1,0,-4,1) Matrix(29,18,240,149) -> Matrix(21,4,-310,-59) Matrix(481,298,-870,-539) -> Matrix(11,2,-72,-13) Matrix(301,186,390,241) -> Matrix(11,2,-248,-45) Matrix(89,54,-150,-91) -> Matrix(11,2,-72,-13) Matrix(269,158,-870,-511) -> Matrix(13,2,-124,-19) Matrix(629,368,870,509) -> Matrix(17,2,-366,-43) Matrix(59,34,-210,-121) -> Matrix(15,2,-128,-17) Matrix(209,118,-480,-271) -> Matrix(1,0,-6,1) Matrix(211,118,270,151) -> Matrix(11,2,-248,-45) Matrix(1201,662,2070,1141) -> Matrix(17,2,-332,-39) Matrix(391,214,-930,-509) -> Matrix(11,2,-72,-13) Matrix(59,32,330,179) -> Matrix(13,2,-202,-31) Matrix(209,112,390,209) -> Matrix(13,2,-254,-39) Matrix(31,16,60,31) -> Matrix(1,0,-12,1) Matrix(29,14,60,29) -> Matrix(1,0,-14,1) Matrix(181,84,390,181) -> Matrix(13,2,-254,-39) Matrix(271,124,330,151) -> Matrix(13,2,-306,-47) Matrix(181,82,-660,-299) -> Matrix(29,4,-254,-35) Matrix(331,148,-870,-389) -> Matrix(13,2,-98,-15) Matrix(59,26,270,119) -> Matrix(15,2,-248,-33) Matrix(511,222,900,391) -> Matrix(1,0,-2,1) Matrix(509,220,900,389) -> Matrix(1,0,-24,1) Matrix(929,390,2070,869) -> Matrix(13,2,-280,-43) Matrix(569,238,930,389) -> Matrix(15,2,-308,-41) Matrix(59,24,-150,-61) -> Matrix(13,2,-98,-15) Matrix(541,210,930,361) -> Matrix(15,2,-308,-41) Matrix(2461,952,3570,1381) -> Matrix(111,14,-2450,-309) Matrix(2189,846,3180,1229) -> Matrix(97,12,-2126,-263) Matrix(89,34,390,149) -> Matrix(15,2,-248,-33) Matrix(1891,718,2610,991) -> Matrix(137,18,-2976,-391) Matrix(211,80,240,91) -> Matrix(31,4,-752,-97) Matrix(571,210,900,331) -> Matrix(1,0,-6,1) Matrix(569,208,900,329) -> Matrix(1,0,-20,1) Matrix(61,22,-330,-119) -> Matrix(11,2,-116,-21) Matrix(29,10,-90,-31) -> Matrix(15,2,-128,-17) Matrix(1951,608,3180,991) -> Matrix(107,12,-2256,-253) Matrix(2189,680,3570,1109) -> Matrix(127,14,-2658,-293) Matrix(211,64,300,91) -> Matrix(1,0,-12,1) Matrix(209,62,300,89) -> Matrix(1,0,-14,1) Matrix(89,26,510,149) -> Matrix(17,2,-264,-31) Matrix(361,100,870,241) -> Matrix(17,2,-366,-43) Matrix(1619,446,2610,719) -> Matrix(157,18,-3236,-371) Matrix(89,24,330,89) -> Matrix(89,10,-1522,-171) Matrix(31,8,120,31) -> Matrix(37,4,-620,-67) Matrix(691,162,900,211) -> Matrix(37,4,-842,-91) Matrix(689,160,900,209) -> Matrix(39,4,-868,-89) Matrix(149,34,390,89) -> Matrix(19,2,-352,-37) Matrix(119,26,270,59) -> Matrix(19,2,-352,-37) Matrix(29,6,-150,-31) -> Matrix(59,6,-600,-61) Matrix(179,32,330,59) -> Matrix(21,2,-410,-39) Matrix(151,26,180,31) -> Matrix(41,4,-974,-95) Matrix(149,24,180,29) -> Matrix(43,4,-1000,-93) Matrix(29,4,210,29) -> Matrix(109,10,-1646,-151) Matrix(31,4,240,31) -> Matrix(133,12,-1984,-179) Matrix(149,18,240,29) -> Matrix(45,4,-934,-83) Matrix(149,-22,210,-31) -> Matrix(31,2,-636,-41) Matrix(119,-22,330,-61) -> Matrix(31,2,-636,-41) Matrix(151,-32,420,-89) -> Matrix(65,4,-1284,-79) Matrix(91,-22,120,-29) -> Matrix(1,0,-6,1) Matrix(121,-34,210,-59) -> Matrix(35,2,-648,-37) Matrix(179,-52,210,-61) -> Matrix(37,2,-870,-47) Matrix(31,-10,90,-29) -> Matrix(35,2,-648,-37) Matrix(571,-202,930,-329) -> Matrix(37,2,-796,-43) Matrix(151,-56,240,-89) -> Matrix(1,0,-4,1) Matrix(389,-148,870,-331) -> Matrix(37,2,-722,-39) Matrix(61,-24,150,-59) -> Matrix(37,2,-722,-39) Matrix(601,-248,870,-359) -> Matrix(39,2,-878,-45) Matrix(151,-64,210,-89) -> Matrix(41,2,-882,-43) Matrix(271,-118,480,-209) -> Matrix(1,0,-6,1) Matrix(539,-244,930,-421) -> Matrix(37,2,-722,-39) Matrix(479,-262,660,-361) -> Matrix(81,4,-1762,-87) Matrix(539,-298,870,-481) -> Matrix(39,2,-800,-41) Matrix(91,-54,150,-89) -> Matrix(39,2,-800,-41) Matrix(269,-172,330,-211) -> Matrix(37,2,-870,-47) Matrix(61,-40,90,-59) -> Matrix(41,2,-882,-43) Matrix(121,-96,150,-119) -> Matrix(137,6,-3174,-139) 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): 48 Degree of the the map X: 48 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): 96 Minimal number of generators: 17 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: 16 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/6 1/5 1/4 2/7 1/3 2/5 1/2 3/5 2/3 11/15 23/30 4/5 5/6 13/15 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/13 1/6 -2/31 1/5 -1/16 2/9 -1/17 1/4 -2/33 2/7 -3/53 1/3 -1/18 3/8 -2/35 2/5 -1/19 3/7 -1/22 4/9 -1/19 1/2 0/1 6/11 -1/19 5/9 -1/20 9/16 -2/45 4/7 -1/17 3/5 -1/20 5/8 -2/43 2/3 -1/21 5/7 -3/64 8/11 -5/109 11/15 -1/22 3/4 -2/45 13/17 -1/22 23/30 -2/45 10/13 -1/23 7/9 -1/22 4/5 -1/23 5/6 -2/47 6/7 -5/119 13/15 -1/24 7/8 -6/145 1/1 -1/26 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(59,-9,105,-16) (0/1,1/6) -> (5/9,9/16) Hyperbolic Matrix(16,-3,75,-14) (1/6,1/5) -> (1/5,2/9) Parabolic Matrix(74,-17,135,-31) (2/9,1/4) -> (6/11,5/9) Hyperbolic Matrix(76,-21,105,-29) (1/4,2/7) -> (5/7,8/11) Hyperbolic Matrix(16,-5,45,-14) (2/7,1/3) -> (1/3,3/8) Parabolic Matrix(31,-12,75,-29) (3/8,2/5) -> (2/5,3/7) Parabolic Matrix(89,-39,105,-46) (3/7,4/9) -> (5/6,6/7) Hyperbolic Matrix(104,-47,135,-61) (4/9,1/2) -> (10/13,7/9) Hyperbolic Matrix(149,-81,195,-106) (1/2,6/11) -> (3/4,13/17) Hyperbolic Matrix(119,-67,135,-76) (9/16,4/7) -> (7/8,1/1) Hyperbolic Matrix(46,-27,75,-44) (4/7,3/5) -> (3/5,5/8) Parabolic Matrix(31,-20,45,-29) (5/8,2/3) -> (2/3,5/7) Parabolic Matrix(166,-121,225,-164) (8/11,11/15) -> (11/15,3/4) Parabolic Matrix(691,-529,900,-689) (13/17,23/30) -> (23/30,10/13) Parabolic Matrix(61,-48,75,-59) (7/9,4/5) -> (4/5,5/6) Parabolic Matrix(196,-169,225,-194) (6/7,13/15) -> (13/15,7/8) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-13,1) Matrix(59,-9,105,-16) -> Matrix(15,1,-331,-22) Matrix(16,-3,75,-14) -> Matrix(47,3,-768,-49) Matrix(74,-17,135,-31) -> Matrix(16,1,-337,-21) Matrix(76,-21,105,-29) -> Matrix(52,3,-1127,-65) Matrix(16,-5,45,-14) -> Matrix(17,1,-324,-19) Matrix(31,-12,75,-29) -> Matrix(18,1,-361,-20) Matrix(89,-39,105,-46) -> Matrix(17,1,-409,-24) Matrix(104,-47,135,-61) -> Matrix(18,1,-415,-23) Matrix(149,-81,195,-106) -> Matrix(21,1,-463,-22) Matrix(119,-67,135,-76) -> Matrix(23,1,-553,-24) Matrix(46,-27,75,-44) -> Matrix(19,1,-400,-21) Matrix(31,-20,45,-29) -> Matrix(20,1,-441,-22) Matrix(166,-121,225,-164) -> Matrix(153,7,-3388,-155) Matrix(691,-529,900,-689) -> Matrix(89,4,-2025,-91) Matrix(61,-48,75,-59) -> Matrix(68,3,-1587,-70) Matrix(196,-169,225,-194) -> Matrix(263,11,-6336,-265) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 1 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 1 Number of equivalence classes of cusps: 1 Genus: 0 Degree of H/liftables -> H/(image of liftables): 48 ----------------------------------------------------------------------- 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 -1/13 1 15 1/6 -2/31 1 5 1/5 -1/16 3 3 2/9 -1/17 1 5 1/4 -2/33 1 15 4/15 -1/17 7 1 2/7 -3/53 1 15 1/3 -1/18 1 5 3/8 -2/35 1 15 2/5 -1/19 1 3 3/7 -1/22 1 15 13/30 0/1 11 1 7/16 -2/33 1 15 4/9 -1/19 1 5 5/11 -1/20 1 15 7/15 -1/20 1 1 1/2 0/1 1 15 1/0 0/1 13 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(46,-7,105,-16) (0/1,1/6) -> (7/16,4/9) Glide Reflection Matrix(16,-3,75,-14) (1/6,1/5) -> (1/5,2/9) Parabolic Matrix(61,-14,135,-31) (2/9,1/4) -> (4/9,5/11) Glide Reflection Matrix(31,-8,120,-31) (1/4,4/15) -> (1/4,4/15) Reflection Matrix(29,-8,105,-29) (4/15,2/7) -> (4/15,2/7) Reflection Matrix(16,-5,45,-14) (2/7,1/3) -> (1/3,3/8) Parabolic Matrix(31,-12,75,-29) (3/8,2/5) -> (2/5,3/7) Parabolic 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(76,-35,165,-76) (5/11,7/15) -> (5/11,7/15) 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(-1,0,26,1) (0/1,1/0) -> (-1/13,0/1) Matrix(46,-7,105,-16) -> Matrix(15,1,-254,-17) Matrix(16,-3,75,-14) -> Matrix(47,3,-768,-49) -1/16 Matrix(61,-14,135,-31) -> Matrix(16,1,-287,-18) Matrix(31,-8,120,-31) -> Matrix(67,4,-1122,-67) (1/4,4/15) -> (-2/33,-1/17) Matrix(29,-8,105,-29) -> Matrix(52,3,-901,-52) (4/15,2/7) -> (-1/17,-3/53) Matrix(16,-5,45,-14) -> Matrix(17,1,-324,-19) -1/18 Matrix(31,-12,75,-29) -> Matrix(18,1,-361,-20) -1/19 Matrix(181,-78,420,-181) -> Matrix(-1,0,44,1) (3/7,13/30) -> (-1/22,0/1) Matrix(209,-91,480,-209) -> Matrix(-1,0,33,1) (13/30,7/16) -> (-2/33,0/1) Matrix(76,-35,165,-76) -> Matrix(19,1,-360,-19) (5/11,7/15) -> (-1/18,-1/20) Matrix(29,-14,60,-29) -> Matrix(-1,0,40,1) (7/15,1/2) -> (-1/20,0/1) Matrix(-1,1,0,1) -> Matrix(-1,0,39,1) (1/2,1/0) -> (-2/39,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.