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 6/73 1/12 -15/2 1/12 -7/1 1/12 -6/1 2/23 -17/3 3/34 -11/2 3/34 -5/1 1/11 -14/3 1/11 4/43 -9/2 1/11 -13/3 7/74 -4/1 2/21 1/10 -15/4 1/10 -11/3 1/10 -7/2 1/10 -17/5 1/10 -10/3 2/19 -13/4 3/28 -3/1 1/9 -17/6 1/10 -14/5 1/9 4/35 -25/9 1/9 -11/4 3/26 -8/3 2/17 1/8 -29/11 7/54 -21/8 1/7 -13/5 1/10 -5/2 1/8 -17/7 3/22 -29/12 5/34 -12/5 0/1 -7/3 1/8 -16/7 2/15 1/7 -9/4 1/7 -29/13 3/16 -20/9 0/1 -11/5 1/8 -13/6 1/8 -15/7 1/7 -2/1 0/1 1/7 -15/8 1/7 -13/7 1/6 -11/6 1/6 -20/11 0/1 -9/5 1/7 -16/9 1/7 2/13 -23/13 3/20 -30/17 2/13 -7/4 1/6 -19/11 1/6 -50/29 2/11 -31/18 5/26 -12/7 0/1 -5/3 1/6 -18/11 2/11 -31/19 7/36 -75/46 1/5 -44/27 1/5 6/29 -13/8 1/4 -21/13 1/7 -50/31 2/13 -29/18 7/44 -8/5 1/6 2/11 -19/12 5/28 -30/19 2/11 -11/7 3/16 -14/9 4/21 1/5 -3/2 1/5 -16/11 2/11 1/5 -45/31 1/5 -29/20 11/54 -13/9 3/14 -10/7 2/9 -17/12 1/4 -24/17 4/17 -7/5 1/4 -18/13 2/9 -29/21 13/56 -40/29 4/17 -11/8 1/4 -15/11 1/4 -4/3 1/4 2/7 -17/13 9/32 -30/23 2/7 -13/10 7/24 -9/7 1/3 -14/11 4/13 1/3 -5/4 1/3 -16/13 1/3 2/5 -11/9 3/8 -17/14 3/8 -6/5 2/5 -13/11 1/2 -7/6 1/2 -15/13 1/2 -8/7 1/2 6/11 -1/1 1/0 0/1 0/1 1/1 1/28 8/7 6/157 1/26 15/13 1/26 7/6 1/26 6/5 2/51 17/14 3/76 11/9 3/76 5/4 1/25 14/11 1/25 4/99 9/7 1/25 13/10 7/172 4/3 2/49 1/24 15/11 1/24 11/8 1/24 7/5 1/24 17/12 1/24 10/7 2/47 13/9 3/70 3/2 1/23 17/11 1/24 14/9 1/23 4/91 25/16 1/23 11/7 3/68 8/5 2/45 1/22 29/18 7/152 21/13 1/21 13/8 1/24 5/3 1/22 17/10 3/64 29/17 5/104 12/7 0/1 7/4 1/22 16/9 2/43 1/21 9/5 1/21 29/16 3/58 20/11 0/1 11/6 1/22 13/7 1/22 15/8 1/21 2/1 0/1 1/21 15/7 1/21 13/6 1/20 11/5 1/20 20/9 0/1 9/4 1/21 16/7 1/21 2/41 23/10 3/62 30/13 2/41 7/3 1/20 19/8 1/20 50/21 2/39 31/13 5/96 12/5 0/1 5/2 1/20 18/7 2/39 31/12 7/134 75/29 1/19 44/17 1/19 6/113 13/5 1/18 21/8 1/21 50/19 2/41 29/11 7/142 8/3 1/20 2/39 19/7 5/98 30/11 2/39 11/4 3/58 14/5 4/77 1/19 3/1 1/19 16/5 2/39 1/19 45/14 1/19 29/9 11/208 13/4 3/56 10/3 2/37 17/5 1/18 24/7 4/73 7/2 1/18 18/5 2/37 29/8 13/238 40/11 4/73 11/3 1/18 15/4 1/18 4/1 1/18 2/35 17/4 9/158 30/7 2/35 13/3 7/122 9/2 1/17 14/3 4/69 1/17 5/1 1/17 16/3 1/17 2/33 11/2 3/50 17/3 3/50 6/1 2/33 13/2 1/16 7/1 1/16 15/2 1/16 8/1 1/16 6/95 1/0 1/14 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(49,-4,1066,-87) Matrix(31,240,4,31) -> Matrix(145,-12,2308,-191) Matrix(29,210,4,29) -> Matrix(119,-10,1916,-161) Matrix(31,210,-22,-149) -> Matrix(71,-6,296,-25) Matrix(89,510,26,149) -> Matrix(113,-10,2068,-183) Matrix(59,330,32,179) -> Matrix(23,-2,472,-41) Matrix(61,330,-22,-119) -> Matrix(67,-6,592,-53) Matrix(89,420,-32,-151) -> Matrix(1,0,-2,1) Matrix(59,270,26,119) -> Matrix(21,-2,452,-43) Matrix(89,390,34,149) -> Matrix(21,-2,452,-43) Matrix(29,120,-22,-91) -> Matrix(41,-4,154,-15) Matrix(31,120,8,31) -> Matrix(41,-4,728,-71) Matrix(89,330,24,89) -> Matrix(99,-10,1792,-181) Matrix(59,210,-34,-121) -> Matrix(19,-2,124,-13) Matrix(61,210,-52,-179) -> Matrix(59,-6,128,-13) Matrix(89,300,62,209) -> Matrix(77,-8,1800,-187) Matrix(91,300,64,211) -> Matrix(75,-8,1772,-189) Matrix(29,90,-10,-31) -> Matrix(19,-2,162,-17) Matrix(329,930,-202,-571) -> Matrix(19,-2,86,-9) Matrix(89,240,-56,-151) -> Matrix(33,-4,190,-23) Matrix(91,240,80,211) -> Matrix(31,-4,814,-105) Matrix(331,870,-148,-389) -> Matrix(15,-2,98,-13) Matrix(149,390,34,89) -> Matrix(13,-2,228,-35) Matrix(59,150,-24,-61) -> Matrix(17,-2,128,-15) Matrix(359,870,-248,-601) -> Matrix(43,-6,208,-29) Matrix(361,870,100,241) -> Matrix(11,-2,204,-37) Matrix(89,210,-64,-151) -> Matrix(15,-2,68,-9) Matrix(209,480,-118,-271) -> Matrix(29,-4,196,-27) Matrix(119,270,26,59) -> Matrix(13,-2,228,-35) Matrix(929,2070,390,869) -> Matrix(9,-2,176,-39) Matrix(421,930,-244,-539) -> Matrix(17,-2,94,-11) Matrix(151,330,124,271) -> Matrix(13,-2,332,-51) Matrix(181,390,84,181) -> Matrix(15,-2,308,-41) Matrix(29,60,14,29) -> Matrix(1,0,14,1) Matrix(31,60,16,31) -> Matrix(1,0,14,1) Matrix(209,390,112,209) -> Matrix(13,-2,280,-43) Matrix(179,330,32,59) -> Matrix(15,-2,248,-33) Matrix(361,660,-262,-479) -> Matrix(23,-4,98,-17) Matrix(481,870,-298,-539) -> Matrix(15,-2,98,-13) Matrix(151,270,118,211) -> Matrix(15,-2,368,-49) Matrix(509,900,220,389) -> Matrix(53,-8,1080,-163) Matrix(511,900,222,391) -> Matrix(51,-8,1052,-165) Matrix(1201,2070,662,1141) -> Matrix(11,-2,204,-37) Matrix(541,930,210,361) -> Matrix(9,-2,176,-39) Matrix(89,150,-54,-91) -> Matrix(13,-2,72,-11) Matrix(569,930,238,389) -> Matrix(11,-2,204,-37) Matrix(2189,3570,680,1109) -> Matrix(91,-18,1724,-341) Matrix(1951,3180,608,991) -> Matrix(39,-8,746,-153) Matrix(241,390,186,301) -> Matrix(15,-2,368,-49) Matrix(1619,2610,446,719) -> Matrix(115,-18,2102,-329) Matrix(149,240,18,29) -> Matrix(25,-4,394,-63) Matrix(569,900,208,329) -> Matrix(89,-16,1730,-311) Matrix(571,900,210,331) -> Matrix(87,-16,1702,-313) Matrix(211,330,-172,-269) -> Matrix(31,-6,88,-17) Matrix(59,90,-40,-61) -> Matrix(11,-2,50,-9) Matrix(2189,3180,846,1229) -> Matrix(41,-8,774,-151) Matrix(2461,3570,952,1381) -> Matrix(89,-18,1696,-343) Matrix(209,300,62,89) -> Matrix(37,-8,680,-147) Matrix(211,300,64,91) -> Matrix(35,-8,652,-149) Matrix(361,510,298,421) -> Matrix(43,-10,1088,-253) Matrix(629,870,368,509) -> Matrix(9,-2,176,-39) Matrix(1891,2610,718,991) -> Matrix(77,-18,1570,-367) Matrix(241,330,176,241) -> Matrix(41,-10,980,-239) Matrix(89,120,66,89) -> Matrix(15,-4,364,-97) Matrix(689,900,160,209) -> Matrix(113,-32,1974,-559) Matrix(691,900,162,211) -> Matrix(111,-32,1946,-561) Matrix(301,390,186,241) -> Matrix(7,-2,144,-41) Matrix(211,270,118,151) -> Matrix(7,-2,144,-41) Matrix(119,150,-96,-121) -> Matrix(19,-6,54,-17) Matrix(271,330,124,151) -> Matrix(5,-2,108,-43) Matrix(149,180,24,29) -> Matrix(21,-8,344,-131) Matrix(151,180,26,31) -> Matrix(19,-8,316,-133) Matrix(181,210,156,181) -> Matrix(21,-10,544,-259) Matrix(209,240,182,209) -> Matrix(23,-12,600,-313) Matrix(211,240,80,91) -> Matrix(7,-4,142,-81) Matrix(1,0,2,1) -> Matrix(1,0,28,1) Matrix(179,-210,52,-61) -> Matrix(155,-6,2816,-109) Matrix(269,-330,172,-211) -> Matrix(151,-6,3448,-137) Matrix(331,-420,212,-269) -> Matrix(1,0,-2,1) Matrix(91,-120,22,-29) -> Matrix(97,-4,1722,-71) Matrix(151,-210,64,-89) -> Matrix(47,-2,964,-41) Matrix(149,-210,22,-31) -> Matrix(143,-6,2312,-97) Matrix(61,-90,40,-59) -> Matrix(47,-2,1058,-45) Matrix(601,-930,232,-359) -> Matrix(47,-2,870,-37) Matrix(151,-240,56,-89) -> Matrix(89,-4,1758,-79) Matrix(539,-870,298,-481) -> Matrix(43,-2,882,-41) Matrix(91,-150,54,-89) -> Matrix(45,-2,968,-43) Matrix(511,-870,158,-269) -> Matrix(127,-6,2392,-113) Matrix(121,-210,34,-59) -> Matrix(43,-2,796,-37) Matrix(271,-480,118,-209) -> Matrix(85,-4,1764,-83) Matrix(509,-930,214,-391) -> Matrix(45,-2,878,-39) Matrix(299,-660,82,-181) -> Matrix(79,-4,1442,-73) Matrix(389,-870,148,-331) -> Matrix(43,-2,882,-41) Matrix(61,-150,24,-59) -> Matrix(41,-2,800,-39) Matrix(119,-330,22,-61) -> Matrix(115,-6,1936,-101) Matrix(31,-90,10,-29) -> Matrix(39,-2,722,-37) Matrix(31,-150,6,-29) -> Matrix(103,-6,1734,-101) 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, lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z. ----------------------------------------------------------------------- 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 6/5 5/4 10/7 3/2 5/3 20/11 2/1 15/7 20/9 9/4 30/13 12/5 5/2 30/11 3/1 45/14 10/3 18/5 15/4 4/1 30/7 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 0/1 0/1 1/1 1/28 8/7 6/157 1/26 15/13 1/26 7/6 1/26 6/5 2/51 17/14 3/76 11/9 3/76 5/4 1/25 14/11 1/25 4/99 9/7 1/25 13/10 7/172 4/3 2/49 1/24 15/11 1/24 11/8 1/24 7/5 1/24 17/12 1/24 10/7 2/47 13/9 3/70 3/2 1/23 17/11 1/24 14/9 1/23 4/91 25/16 1/23 11/7 3/68 8/5 2/45 1/22 29/18 7/152 21/13 1/21 13/8 1/24 5/3 1/22 17/10 3/64 29/17 5/104 12/7 0/1 7/4 1/22 16/9 2/43 1/21 9/5 1/21 29/16 3/58 20/11 0/1 11/6 1/22 13/7 1/22 15/8 1/21 2/1 0/1 1/21 15/7 1/21 13/6 1/20 11/5 1/20 20/9 0/1 9/4 1/21 16/7 1/21 2/41 23/10 3/62 30/13 2/41 7/3 1/20 19/8 1/20 50/21 2/39 31/13 5/96 12/5 0/1 5/2 1/20 18/7 2/39 31/12 7/134 75/29 1/19 44/17 1/19 6/113 13/5 1/18 21/8 1/21 50/19 2/41 29/11 7/142 8/3 1/20 2/39 19/7 5/98 30/11 2/39 11/4 3/58 14/5 4/77 1/19 3/1 1/19 16/5 2/39 1/19 45/14 1/19 29/9 11/208 13/4 3/56 10/3 2/37 17/5 1/18 24/7 4/73 7/2 1/18 18/5 2/37 29/8 13/238 40/11 4/73 11/3 1/18 15/4 1/18 4/1 1/18 2/35 17/4 9/158 30/7 2/35 13/3 7/122 9/2 1/17 14/3 4/69 1/17 5/1 1/17 16/3 1/17 2/33 11/2 3/50 17/3 3/50 6/1 2/33 13/2 1/16 7/1 1/16 15/2 1/16 8/1 1/16 6/95 1/0 1/14 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(211,-240,131,-149) (1/1,8/7) -> (8/5,29/18) Hyperbolic Matrix(209,-240,27,-31) (8/7,15/13) -> (15/2,8/1) Hyperbolic Matrix(181,-210,25,-29) (15/13,7/6) -> (7/1,15/2) Hyperbolic Matrix(179,-210,52,-61) (7/6,6/5) -> (24/7,7/2) Hyperbolic Matrix(421,-510,123,-149) (6/5,17/14) -> (17/5,24/7) Hyperbolic Matrix(271,-330,147,-179) (17/14,11/9) -> (11/6,13/7) 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,-270,93,-119) (14/11,9/7) -> (9/4,16/7) Hyperbolic Matrix(301,-390,115,-149) (9/7,13/10) -> (13/5,21/8) Hyperbolic Matrix(91,-120,22,-29) (13/10,4/3) -> (4/1,17/4) Hyperbolic Matrix(89,-120,23,-31) (4/3,15/11) -> (15/4,4/1) Hyperbolic Matrix(241,-330,65,-89) (15/11,11/8) -> (11/3,15/4) 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(211,-300,147,-209) (17/12,10/7) -> (10/7,13/9) 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(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(241,-390,55,-89) (21/13,13/8) -> (13/3,9/2) 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(509,-870,141,-241) (29/17,12/7) -> (18/5,29/8) 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(151,-270,33,-59) (16/9,9/5) -> (9/2,14/3) Hyperbolic Matrix(1141,-2070,479,-869) (29/16,20/11) -> (50/21,31/13) Hyperbolic Matrix(509,-930,214,-391) (20/11,11/6) -> (19/8,50/21) Hyperbolic Matrix(209,-390,97,-181) (13/7,15/8) -> (15/7,13/6) Hyperbolic Matrix(31,-60,15,-29) (15/8,2/1) -> (2/1,15/7) Parabolic Matrix(151,-330,27,-59) (13/6,11/5) -> (11/2,17/3) 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(391,-900,169,-389) (23/10,30/13) -> (30/13,7/3) Parabolic Matrix(389,-930,151,-361) (31/13,12/5) -> (18/7,31/12) Hyperbolic Matrix(61,-150,24,-59) (12/5,5/2) -> (5/2,18/7) Parabolic Matrix(1381,-3570,429,-1109) (31/12,75/29) -> (45/14,29/9) Hyperbolic Matrix(1229,-3180,383,-991) (75/29,44/17) -> (16/5,45/14) Hyperbolic Matrix(991,-2610,273,-719) (50/19,29/11) -> (29/8,40/11) Hyperbolic Matrix(91,-240,11,-29) (29/11,8/3) -> (8/1,1/0) Hyperbolic Matrix(331,-900,121,-329) (19/7,30/11) -> (30/11,11/4) 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(91,-300,27,-89) (13/4,10/3) -> (10/3,17/5) Parabolic Matrix(211,-900,49,-209) (17/4,30/7) -> (30/7,13/3) Parabolic Matrix(31,-150,6,-29) (14/3,5/1) -> (5/1,16/3) Parabolic Matrix(31,-180,5,-29) (17/3,6/1) -> (6/1,13/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,14,1) Matrix(211,-240,131,-149) -> Matrix(105,-4,2284,-87) Matrix(209,-240,27,-31) -> Matrix(313,-12,4982,-191) Matrix(181,-210,25,-29) -> Matrix(259,-10,4170,-161) Matrix(179,-210,52,-61) -> Matrix(155,-6,2816,-109) Matrix(421,-510,123,-149) -> Matrix(253,-10,4630,-183) Matrix(271,-330,147,-179) -> Matrix(51,-2,1046,-41) Matrix(269,-330,172,-211) -> Matrix(151,-6,3448,-137) Matrix(331,-420,212,-269) -> Matrix(1,0,-2,1) Matrix(211,-270,93,-119) -> Matrix(49,-2,1054,-43) Matrix(301,-390,115,-149) -> Matrix(49,-2,1054,-43) Matrix(91,-120,22,-29) -> Matrix(97,-4,1722,-71) Matrix(89,-120,23,-31) -> Matrix(97,-4,1722,-71) Matrix(241,-330,65,-89) -> Matrix(239,-10,4326,-181) Matrix(151,-210,64,-89) -> Matrix(47,-2,964,-41) Matrix(149,-210,22,-31) -> Matrix(143,-6,2312,-97) Matrix(211,-300,147,-209) -> Matrix(189,-8,4418,-187) Matrix(61,-90,40,-59) -> Matrix(47,-2,1058,-45) Matrix(601,-930,232,-359) -> Matrix(47,-2,870,-37) Matrix(151,-240,56,-89) -> Matrix(89,-4,1758,-79) Matrix(539,-870,298,-481) -> Matrix(43,-2,882,-41) Matrix(241,-390,55,-89) -> Matrix(41,-2,718,-35) Matrix(91,-150,54,-89) -> Matrix(45,-2,968,-43) Matrix(511,-870,158,-269) -> Matrix(127,-6,2392,-113) Matrix(509,-870,141,-241) -> Matrix(39,-2,722,-37) Matrix(121,-210,34,-59) -> Matrix(43,-2,796,-37) Matrix(271,-480,118,-209) -> Matrix(85,-4,1764,-83) Matrix(151,-270,33,-59) -> Matrix(41,-2,718,-35) Matrix(1141,-2070,479,-869) -> Matrix(37,-2,722,-39) Matrix(509,-930,214,-391) -> Matrix(45,-2,878,-39) Matrix(209,-390,97,-181) -> Matrix(43,-2,882,-41) Matrix(31,-60,15,-29) -> Matrix(1,0,0,1) Matrix(151,-330,27,-59) -> Matrix(43,-2,710,-33) Matrix(299,-660,82,-181) -> Matrix(79,-4,1442,-73) Matrix(389,-870,148,-331) -> Matrix(43,-2,882,-41) Matrix(391,-900,169,-389) -> Matrix(165,-8,3362,-163) Matrix(389,-930,151,-361) -> Matrix(37,-2,722,-39) Matrix(61,-150,24,-59) -> Matrix(41,-2,800,-39) Matrix(1381,-3570,429,-1109) -> Matrix(343,-18,6498,-341) Matrix(1229,-3180,383,-991) -> Matrix(151,-8,2888,-153) Matrix(991,-2610,273,-719) -> Matrix(367,-18,6708,-329) Matrix(91,-240,11,-29) -> Matrix(81,-4,1276,-63) Matrix(331,-900,121,-329) -> Matrix(313,-16,6084,-311) Matrix(119,-330,22,-61) -> Matrix(115,-6,1936,-101) Matrix(31,-90,10,-29) -> Matrix(39,-2,722,-37) Matrix(91,-300,27,-89) -> Matrix(149,-8,2738,-147) Matrix(211,-900,49,-209) -> Matrix(561,-32,9800,-559) Matrix(31,-150,6,-29) -> Matrix(103,-6,1734,-101) Matrix(31,-180,5,-29) -> Matrix(133,-8,2178,-131) 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): 24 ----------------------------------------------------------------------- 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 14 1 2/1 (0/1,1/21) 0 15 15/7 1/21 1 1 13/6 1/20 1 15 11/5 1/20 1 15 20/9 0/1 4 3 9/4 1/21 1 5 16/7 (1/21,2/41) 0 15 30/13 2/41 2 1 7/3 1/20 1 15 12/5 0/1 2 5 5/2 1/20 1 3 18/7 2/39 2 5 31/12 7/134 1 15 13/5 1/18 1 15 21/8 1/21 1 5 50/19 2/41 4 3 29/11 7/142 1 15 8/3 (1/20,2/39) 0 15 30/11 2/39 4 1 11/4 3/58 1 15 14/5 (4/77,1/19) 0 15 3/1 1/19 1 5 16/5 (2/39,1/19) 0 15 45/14 1/19 13 1 29/9 11/208 1 15 13/4 3/56 1 15 10/3 2/37 2 3 17/5 1/18 1 15 7/2 1/18 1 15 18/5 2/37 2 5 29/8 13/238 1 15 40/11 4/73 4 3 11/3 1/18 1 15 15/4 1/18 7 1 4/1 (1/18,2/35) 0 15 30/7 2/35 8 1 13/3 7/122 1 15 9/2 1/17 1 5 14/3 (4/69,1/17) 0 15 5/1 1/17 3 3 16/3 (1/17,2/33) 0 15 11/2 3/50 1 15 17/3 3/50 1 15 6/1 2/33 2 5 13/2 1/16 1 15 7/1 1/16 1 15 15/2 1/16 11 1 8/1 (1/16,6/95) 0 15 1/0 1/14 1 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) 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(151,-330,27,-59) (13/6,11/5) -> (11/2,17/3) 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(119,-270,26,-59) (9/4,16/7) -> (9/2,14/3) Glide Reflection Matrix(209,-480,91,-209) (16/7,30/13) -> (16/7,30/13) Reflection Matrix(181,-420,78,-181) (30/13,7/3) -> (30/13,7/3) Reflection Matrix(89,-210,25,-59) (7/3,12/5) -> (7/2,18/5) 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(301,-780,93,-241) (31/12,13/5) -> (29/9,13/4) Glide Reflection Matrix(149,-390,34,-89) (13/5,21/8) -> (13/3,9/2) Glide Reflection Matrix(991,-2610,273,-719) (50/19,29/11) -> (29/8,40/11) Hyperbolic Matrix(91,-240,11,-29) (29/11,8/3) -> (8/1,1/0) Hyperbolic Matrix(89,-240,33,-89) (8/3,30/11) -> (8/3,30/11) 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(91,-300,27,-89) (13/4,10/3) -> (10/3,17/5) Parabolic Matrix(61,-210,9,-31) (17/5,7/2) -> (13/2,7/1) 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(29,-120,7,-29) (4/1,30/7) -> (4/1,30/7) 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(31,-180,5,-29) (17/3,6/1) -> (6/1,13/2) 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(1,0,28,-1) (0/1,1/0) -> (0/1,1/14) Matrix(1,0,1,-1) -> Matrix(1,0,42,-1) (0/1,2/1) -> (0/1,1/21) Matrix(29,-60,14,-29) -> Matrix(1,0,42,-1) (2/1,15/7) -> (0/1,1/21) Matrix(181,-390,84,-181) -> Matrix(41,-2,840,-41) (15/7,13/6) -> (1/21,1/20) Matrix(151,-330,27,-59) -> Matrix(43,-2,710,-33) Matrix(299,-660,82,-181) -> Matrix(79,-4,1442,-73) Matrix(389,-870,148,-331) -> Matrix(43,-2,882,-41) 1/21 Matrix(119,-270,26,-59) -> Matrix(43,-2,752,-35) Matrix(209,-480,91,-209) -> Matrix(83,-4,1722,-83) (16/7,30/13) -> (1/21,2/41) Matrix(181,-420,78,-181) -> Matrix(81,-4,1640,-81) (30/13,7/3) -> (2/41,1/20) Matrix(89,-210,25,-59) -> Matrix(41,-2,758,-37) Matrix(61,-150,24,-59) -> Matrix(41,-2,800,-39) 1/20 Matrix(419,-1080,116,-299) -> Matrix(155,-8,2848,-147) Matrix(301,-780,93,-241) -> Matrix(75,-4,1406,-75) *** -> (1/19,2/37) Matrix(149,-390,34,-89) -> Matrix(43,-2,752,-35) Matrix(991,-2610,273,-719) -> Matrix(367,-18,6708,-329) Matrix(91,-240,11,-29) -> Matrix(81,-4,1276,-63) Matrix(89,-240,33,-89) -> Matrix(79,-4,1560,-79) (8/3,30/11) -> (1/20,2/39) Matrix(241,-660,88,-241) -> Matrix(233,-12,4524,-233) (30/11,11/4) -> (2/39,3/58) Matrix(119,-330,22,-61) -> Matrix(115,-6,1936,-101) Matrix(31,-90,10,-29) -> Matrix(39,-2,722,-37) 1/19 Matrix(449,-1440,140,-449) -> Matrix(77,-4,1482,-77) (16/5,45/14) -> (2/39,1/19) Matrix(811,-2610,252,-811) -> Matrix(417,-22,7904,-417) (45/14,29/9) -> (1/19,11/208) Matrix(91,-300,27,-89) -> Matrix(149,-8,2738,-147) 2/37 Matrix(61,-210,9,-31) -> Matrix(109,-6,1762,-97) Matrix(89,-330,24,-89) -> Matrix(181,-10,3276,-181) (11/3,15/4) -> (5/91,1/18) Matrix(31,-120,8,-31) -> Matrix(71,-4,1260,-71) (15/4,4/1) -> (1/18,2/35) Matrix(29,-120,7,-29) -> Matrix(71,-4,1260,-71) (4/1,30/7) -> (1/18,2/35) Matrix(181,-780,42,-181) -> Matrix(489,-28,8540,-489) (30/7,13/3) -> (2/35,7/122) Matrix(31,-150,6,-29) -> Matrix(103,-6,1734,-101) 1/17 Matrix(31,-180,5,-29) -> Matrix(133,-8,2178,-131) 2/33 Matrix(29,-210,4,-29) -> Matrix(161,-10,2592,-161) (7/1,15/2) -> (5/81,1/16) Matrix(31,-240,4,-31) -> Matrix(191,-12,3040,-191) (15/2,8/1) -> (1/16,6/95) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.