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 -1/1 -6/13 -5/12 -4/11 -1/3 -3/10 -2/9 -1/8 0/1 1/6 3/11 1/3 2/5 1/2 5/9 3/4 1/1 4/3 3/2 17/11 5/3 9/5 2/1 11/5 29/13 7/3 5/2 71/27 8/3 3/1 43/13 10/3 17/5 7/2 11/3 4/1 13/3 9/2 23/5 5/1 47/9 11/2 6/1 7/1 8/1 25/3 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 1/13 -6/13 1/11 -5/11 4/43 2/21 -4/9 1/10 -7/16 5/49 -3/7 2/19 -8/19 1/10 -5/12 1/9 -12/29 5/44 -7/17 0/1 2/17 -2/5 1/8 -7/18 3/23 -5/13 0/1 -13/34 1/8 -8/21 1/8 -3/8 1/7 -7/19 2/15 -4/11 1/7 -9/25 0/1 -5/14 1/5 -1/3 0/1 2/13 -4/13 1/8 -3/10 1/6 -8/27 3/16 -5/17 2/11 -2/7 1/6 -5/18 1/5 -8/29 1/5 -3/11 0/1 -4/15 3/16 -1/4 1/5 -3/13 0/1 2/9 -2/9 1/4 -5/23 6/23 4/15 -3/14 1/3 -1/5 2/7 -2/11 5/16 -1/6 1/3 -1/7 2/5 4/9 -1/8 1/2 0/1 1/0 1/6 -1/2 1/5 -4/9 -2/5 2/9 -7/18 3/13 -4/11 1/4 -1/3 3/11 -1/3 5/18 -1/3 2/7 -5/16 5/17 -2/7 0/1 8/27 -1/3 3/10 -1/3 4/13 -3/10 1/3 -2/7 5/14 -1/5 4/11 -1/4 3/8 -1/3 2/5 -1/4 5/12 -5/21 8/19 -11/48 3/7 -2/9 0/1 7/16 -1/3 4/9 -1/4 5/11 -4/17 6/13 -7/30 1/2 -1/5 5/9 -1/5 9/16 -1/5 4/7 -3/16 15/26 -1/5 11/19 -4/21 -2/11 7/12 -5/27 17/29 -2/11 -12/67 10/17 -7/40 13/22 -1/6 3/5 0/1 8/13 -1/5 5/8 -1/5 12/19 -3/16 7/11 -4/21 -2/11 2/3 -1/6 9/13 -2/15 0/1 7/10 -1/3 19/27 -4/17 -2/9 12/17 -5/24 5/7 -2/11 3/4 -1/6 7/9 0/1 11/14 -3/19 15/19 -2/13 -4/27 4/5 -1/8 9/11 -2/11 5/6 -1/7 6/7 -1/6 1/1 -2/13 0/1 6/5 -1/6 11/9 -2/15 5/4 -1/5 4/3 -1/7 11/8 -3/23 18/13 -1/6 25/18 -1/7 7/5 -2/15 24/17 -1/8 41/29 -1/8 17/12 -5/41 27/19 -2/17 -4/35 10/7 -1/10 23/16 -1/1 13/9 -2/11 0/1 3/2 -1/7 17/11 -2/15 14/9 -5/38 11/7 -2/15 -4/31 19/12 -3/23 27/17 -6/47 8/5 -1/8 29/18 -1/9 21/13 -2/15 -4/31 13/8 -1/8 18/11 -7/58 41/25 -12/101 -2/17 23/14 -5/43 5/3 0/1 22/13 -1/7 17/10 -7/51 12/7 -5/38 19/11 -2/15 -4/31 7/4 -3/23 9/5 -1/8 11/6 -5/41 2/1 -1/8 15/7 -2/17 -8/69 13/6 -7/61 11/5 -4/35 20/9 -11/98 29/13 -1/9 9/4 -1/9 16/7 -1/10 39/17 0/1 23/10 -1/7 7/3 -2/17 0/1 5/2 -1/9 13/5 -6/55 -4/37 47/18 -25/231 81/31 -4/37 34/13 -19/176 21/8 -13/121 71/27 -3/28 50/19 -3/28 29/11 -8/75 8/3 -1/10 19/7 -2/19 0/1 30/11 -1/10 41/15 0/1 11/4 -1/9 14/5 -1/8 3/1 -2/19 13/4 -3/29 23/7 -2/19 -4/39 33/10 -3/29 43/13 -4/39 10/3 -1/10 27/8 -1/10 17/5 -2/19 0/1 41/12 -5/47 65/19 -2/19 24/7 -5/48 7/2 -5/49 11/3 -1/10 15/4 -11/111 4/1 -1/10 13/3 -4/41 35/8 -3/31 57/13 -3/31 22/5 -17/176 9/2 -7/73 23/5 -2/21 37/8 -29/305 14/3 -11/116 5/1 -2/21 -4/43 26/5 -11/118 47/9 -4/43 21/4 -9/97 16/3 -7/76 11/2 -1/11 6/1 -1/11 7/1 -4/45 8/1 -7/80 25/3 -2/23 17/2 -15/173 9/1 -2/23 -8/93 1/0 -1/13 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(265,124,156,73) (-1/2,-6/13) -> (22/13,17/10) Hyperbolic Matrix(153,70,518,237) (-6/13,-5/11) -> (5/17,8/27) Hyperbolic Matrix(103,46,150,67) (-5/11,-4/9) -> (2/3,9/13) Hyperbolic Matrix(199,88,52,23) (-4/9,-7/16) -> (15/4,4/1) Hyperbolic Matrix(51,22,146,63) (-7/16,-3/7) -> (1/3,5/14) Hyperbolic Matrix(47,20,148,63) (-3/7,-8/19) -> (4/13,1/3) Hyperbolic Matrix(239,100,-576,-241) (-8/19,-5/12) -> (-5/12,-12/29) Parabolic Matrix(953,394,670,277) (-12/29,-7/17) -> (27/19,10/7) Hyperbolic Matrix(239,98,378,155) (-7/17,-2/5) -> (12/19,7/11) Hyperbolic Matrix(91,36,48,19) (-2/5,-7/18) -> (11/6,2/1) Hyperbolic Matrix(227,88,276,107) (-7/18,-5/13) -> (9/11,5/6) Hyperbolic Matrix(271,104,456,175) (-5/13,-13/34) -> (13/22,3/5) Hyperbolic Matrix(907,346,270,103) (-13/34,-8/21) -> (10/3,27/8) Hyperbolic Matrix(179,68,408,155) (-8/21,-3/8) -> (7/16,4/9) Hyperbolic Matrix(43,16,180,67) (-3/8,-7/19) -> (3/13,1/4) Hyperbolic Matrix(175,64,-484,-177) (-7/19,-4/11) -> (-4/11,-9/25) Parabolic Matrix(479,172,220,79) (-9/25,-5/14) -> (13/6,11/5) Hyperbolic Matrix(175,62,302,107) (-5/14,-1/3) -> (11/19,7/12) Hyperbolic Matrix(283,88,164,51) (-1/3,-4/13) -> (12/7,19/11) Hyperbolic Matrix(119,36,-400,-121) (-4/13,-3/10) -> (-3/10,-8/27) Parabolic Matrix(237,70,518,153) (-8/27,-5/17) -> (5/11,6/13) Hyperbolic Matrix(159,46,38,11) (-5/17,-2/7) -> (4/1,13/3) Hyperbolic Matrix(351,98,154,43) (-2/7,-5/18) -> (9/4,16/7) Hyperbolic Matrix(231,64,776,215) (-5/18,-8/29) -> (8/27,3/10) Hyperbolic Matrix(387,106,230,63) (-8/29,-3/11) -> (5/3,22/13) Hyperbolic Matrix(231,62,190,51) (-3/11,-4/15) -> (6/5,11/9) Hyperbolic Matrix(151,40,268,71) (-4/15,-1/4) -> (9/16,4/7) Hyperbolic Matrix(291,68,184,43) (-1/4,-3/13) -> (11/7,19/12) Hyperbolic Matrix(71,16,-324,-73) (-3/13,-2/9) -> (-2/9,-5/23) Parabolic Matrix(427,92,608,131) (-5/23,-3/14) -> (7/10,19/27) Hyperbolic Matrix(107,22,34,7) (-3/14,-1/5) -> (3/1,13/4) Hyperbolic Matrix(103,20,36,7) (-1/5,-2/11) -> (14/5,3/1) Hyperbolic Matrix(67,12,240,43) (-2/11,-1/6) -> (5/18,2/7) Hyperbolic Matrix(99,16,68,11) (-1/6,-1/7) -> (13/9,3/2) Hyperbolic Matrix(325,44,96,13) (-1/7,-1/8) -> (27/8,17/5) Hyperbolic Matrix(93,10,158,17) (-1/8,0/1) -> (10/17,13/22) Hyperbolic Matrix(121,-18,74,-11) (0/1,1/6) -> (13/8,18/11) Hyperbolic Matrix(191,-34,118,-21) (1/6,1/5) -> (21/13,13/8) Hyperbolic Matrix(115,-24,24,-5) (1/5,2/9) -> (14/3,5/1) Hyperbolic Matrix(365,-82,138,-31) (2/9,3/13) -> (29/11,8/3) Hyperbolic Matrix(67,-18,242,-65) (1/4,3/11) -> (3/11,5/18) Parabolic Matrix(173,-50,218,-63) (2/7,5/17) -> (15/19,4/5) Hyperbolic Matrix(235,-72,408,-125) (3/10,4/13) -> (4/7,15/26) Hyperbolic Matrix(431,-156,268,-97) (5/14,4/11) -> (8/5,29/18) Hyperbolic Matrix(103,-38,122,-45) (4/11,3/8) -> (5/6,6/7) Hyperbolic Matrix(41,-16,100,-39) (3/8,2/5) -> (2/5,5/12) Parabolic Matrix(435,-182,98,-41) (5/12,8/19) -> (22/5,9/2) Hyperbolic Matrix(393,-166,670,-283) (8/19,3/7) -> (17/29,10/17) Hyperbolic Matrix(273,-118,118,-51) (3/7,7/16) -> (23/10,7/3) Hyperbolic Matrix(327,-148,232,-105) (4/9,5/11) -> (7/5,24/17) Hyperbolic Matrix(305,-142,58,-27) (6/13,1/2) -> (21/4,16/3) Hyperbolic Matrix(91,-50,162,-89) (1/2,5/9) -> (5/9,9/16) Parabolic Matrix(1225,-708,372,-215) (15/26,11/19) -> (23/7,33/10) Hyperbolic Matrix(1519,-890,582,-341) (7/12,17/29) -> (13/5,47/18) Hyperbolic Matrix(121,-74,18,-11) (3/5,8/13) -> (6/1,7/1) Hyperbolic Matrix(191,-118,34,-21) (8/13,5/8) -> (11/2,6/1) Hyperbolic Matrix(619,-390,446,-281) (5/8,12/19) -> (18/13,25/18) Hyperbolic Matrix(187,-120,120,-77) (7/11,2/3) -> (14/9,11/7) Hyperbolic Matrix(429,-298,298,-207) (9/13,7/10) -> (23/16,13/9) Hyperbolic Matrix(1183,-834,722,-509) (19/27,12/17) -> (18/11,41/25) Hyperbolic Matrix(327,-232,148,-105) (12/17,5/7) -> (11/5,20/9) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(549,-430,346,-271) (7/9,11/14) -> (19/12,27/17) Hyperbolic Matrix(1017,-802,298,-235) (11/14,15/19) -> (17/5,41/12) Hyperbolic Matrix(123,-100,16,-13) (4/5,9/11) -> (7/1,8/1) Hyperbolic Matrix(163,-144,60,-53) (6/7,1/1) -> (19/7,30/11) Hyperbolic Matrix(103,-122,38,-45) (1/1,6/5) -> (8/3,19/7) Hyperbolic Matrix(227,-278,138,-169) (11/9,5/4) -> (23/14,5/3) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(283,-390,82,-113) (11/8,18/13) -> (24/7,7/2) Hyperbolic Matrix(409,-570,94,-131) (25/18,7/5) -> (13/3,35/8) Hyperbolic Matrix(2657,-3754,1010,-1427) (24/17,41/29) -> (71/27,50/19) Hyperbolic Matrix(1461,-2068,556,-787) (41/29,17/12) -> (21/8,71/27) Hyperbolic Matrix(127,-180,12,-17) (17/12,27/19) -> (9/1,1/0) Hyperbolic Matrix(483,-692,104,-149) (10/7,23/16) -> (37/8,14/3) Hyperbolic Matrix(331,-508,144,-221) (3/2,17/11) -> (39/17,23/10) Hyperbolic Matrix(527,-818,230,-357) (17/11,14/9) -> (16/7,39/17) Hyperbolic Matrix(995,-1582,378,-601) (27/17,8/5) -> (50/19,29/11) Hyperbolic Matrix(439,-708,204,-329) (29/18,21/13) -> (15/7,13/6) Hyperbolic Matrix(551,-904,64,-105) (41/25,23/14) -> (17/2,9/1) Hyperbolic Matrix(487,-830,186,-317) (17/10,12/7) -> (34/13,21/8) Hyperbolic Matrix(235,-408,72,-125) (19/11,7/4) -> (13/4,23/7) Hyperbolic Matrix(91,-162,50,-89) (7/4,9/5) -> (9/5,11/6) Parabolic Matrix(187,-400,36,-77) (2/1,15/7) -> (5/1,26/5) Hyperbolic Matrix(799,-1778,182,-405) (20/9,29/13) -> (57/13,22/5) Hyperbolic Matrix(683,-1528,156,-349) (29/13,9/4) -> (35/8,57/13) Hyperbolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(977,-2552,116,-303) (47/18,81/31) -> (25/3,17/2) Hyperbolic Matrix(573,-1498,70,-183) (81/31,34/13) -> (8/1,25/3) Hyperbolic Matrix(623,-1700,188,-513) (30/11,41/15) -> (43/13,10/3) Hyperbolic Matrix(667,-1826,202,-553) (41/15,11/4) -> (33/10,43/13) Hyperbolic Matrix(119,-330,22,-61) (11/4,14/5) -> (16/3,11/2) Hyperbolic Matrix(963,-3292,184,-629) (41/12,65/19) -> (47/9,21/4) Hyperbolic Matrix(823,-2818,158,-541) (65/19,24/7) -> (26/5,47/9) Hyperbolic Matrix(67,-242,18,-65) (7/2,11/3) -> (11/3,15/4) Parabolic Matrix(231,-1058,50,-229) (9/2,23/5) -> (23/5,37/8) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,26,1) Matrix(265,124,156,73) -> Matrix(45,-4,-326,29) Matrix(153,70,518,237) -> Matrix(21,-2,-52,5) Matrix(103,46,150,67) -> Matrix(21,-2,-136,13) Matrix(199,88,52,23) -> Matrix(61,-6,-620,61) Matrix(51,22,146,63) -> Matrix(39,-4,-146,15) Matrix(47,20,148,63) -> Matrix(37,-4,-120,13) Matrix(239,100,-576,-241) -> Matrix(55,-6,486,-53) Matrix(953,394,670,277) -> Matrix(35,-4,-306,35) Matrix(239,98,378,155) -> Matrix(19,-2,-104,11) Matrix(91,36,48,19) -> Matrix(17,-2,-144,17) Matrix(227,88,276,107) -> Matrix(15,-2,-82,11) Matrix(271,104,456,175) -> Matrix(1,0,-14,1) Matrix(907,346,270,103) -> Matrix(31,-4,-302,39) Matrix(179,68,408,155) -> Matrix(15,-2,-52,7) Matrix(43,16,180,67) -> Matrix(13,-2,-32,5) Matrix(175,64,-484,-177) -> Matrix(15,-2,98,-13) Matrix(479,172,220,79) -> Matrix(27,-4,-236,35) Matrix(175,62,302,107) -> Matrix(15,-2,-82,11) Matrix(283,88,164,51) -> Matrix(11,-2,-82,15) Matrix(119,36,-400,-121) -> Matrix(13,-2,72,-11) Matrix(237,70,518,153) -> Matrix(13,-2,-58,9) Matrix(159,46,38,11) -> Matrix(13,-2,-136,21) Matrix(351,98,154,43) -> Matrix(11,-2,-104,19) Matrix(231,64,776,215) -> Matrix(21,-4,-68,13) Matrix(387,106,230,63) -> Matrix(1,0,-12,1) Matrix(231,62,190,51) -> Matrix(11,-2,-82,15) Matrix(151,40,268,71) -> Matrix(31,-6,-160,31) Matrix(291,68,184,43) -> Matrix(7,-2,-52,15) Matrix(71,16,-324,-73) -> Matrix(25,-6,96,-23) Matrix(427,92,608,131) -> Matrix(7,-2,-24,7) Matrix(107,22,34,7) -> Matrix(15,-4,-146,39) Matrix(103,20,36,7) -> Matrix(13,-4,-120,37) Matrix(67,12,240,43) -> Matrix(31,-10,-96,31) Matrix(99,16,68,11) -> Matrix(5,-2,-32,13) Matrix(325,44,96,13) -> Matrix(5,-2,-52,21) Matrix(93,10,158,17) -> Matrix(7,-4,-40,23) Matrix(121,-18,74,-11) -> Matrix(7,4,-58,-33) Matrix(191,-34,118,-21) -> Matrix(13,6,-102,-47) Matrix(115,-24,24,-5) -> Matrix(19,8,-202,-85) Matrix(365,-82,138,-31) -> Matrix(31,12,-292,-113) Matrix(67,-18,242,-65) -> Matrix(47,16,-144,-49) Matrix(173,-50,218,-63) -> Matrix(13,4,-88,-27) Matrix(235,-72,408,-125) -> Matrix(1,0,-2,1) Matrix(431,-156,268,-97) -> Matrix(1,0,-4,1) Matrix(103,-38,122,-45) -> Matrix(7,2,-46,-13) Matrix(41,-16,100,-39) -> Matrix(23,6,-96,-25) Matrix(435,-182,98,-41) -> Matrix(77,18,-800,-187) Matrix(393,-166,670,-283) -> Matrix(53,12,-296,-67) Matrix(273,-118,118,-51) -> Matrix(1,0,-4,1) Matrix(327,-148,232,-105) -> Matrix(25,6,-196,-47) Matrix(305,-142,58,-27) -> Matrix(61,14,-658,-151) Matrix(91,-50,162,-89) -> Matrix(39,8,-200,-41) Matrix(1225,-708,372,-215) -> Matrix(43,8,-414,-77) Matrix(1519,-890,582,-341) -> Matrix(167,30,-1542,-277) Matrix(121,-74,18,-11) -> Matrix(19,4,-214,-45) Matrix(191,-118,34,-21) -> Matrix(31,6,-336,-65) Matrix(619,-390,446,-281) -> Matrix(21,4,-142,-27) Matrix(187,-120,120,-77) -> Matrix(43,8,-328,-61) Matrix(429,-298,298,-207) -> Matrix(1,0,2,1) Matrix(1183,-834,722,-509) -> Matrix(37,8,-310,-67) Matrix(327,-232,148,-105) -> Matrix(31,6,-274,-53) Matrix(49,-36,64,-47) -> Matrix(11,2,-72,-13) Matrix(549,-430,346,-271) -> Matrix(37,6,-290,-47) Matrix(1017,-802,298,-235) -> Matrix(27,4,-250,-37) Matrix(123,-100,16,-13) -> Matrix(9,2,-104,-23) Matrix(163,-144,60,-53) -> Matrix(13,2,-124,-19) Matrix(103,-122,38,-45) -> Matrix(13,2,-124,-19) Matrix(227,-278,138,-169) -> Matrix(15,2,-128,-17) Matrix(49,-64,36,-47) -> Matrix(13,2,-98,-15) Matrix(283,-390,82,-113) -> Matrix(17,2,-162,-19) Matrix(409,-570,94,-131) -> Matrix(73,10,-752,-103) Matrix(2657,-3754,1010,-1427) -> Matrix(349,44,-3260,-411) Matrix(1461,-2068,556,-787) -> Matrix(227,28,-2116,-261) Matrix(127,-180,12,-17) -> Matrix(33,4,-388,-47) Matrix(483,-692,104,-149) -> Matrix(31,2,-326,-21) Matrix(331,-508,144,-221) -> Matrix(15,2,-98,-13) Matrix(527,-818,230,-357) -> Matrix(15,2,-188,-25) Matrix(995,-1582,378,-601) -> Matrix(205,26,-1916,-243) Matrix(439,-708,204,-329) -> Matrix(29,4,-254,-35) Matrix(551,-904,64,-105) -> Matrix(169,20,-1952,-231) Matrix(487,-830,186,-317) -> Matrix(133,18,-1234,-167) Matrix(235,-408,72,-125) -> Matrix(1,0,-2,1) Matrix(91,-162,50,-89) -> Matrix(63,8,-512,-65) Matrix(187,-400,36,-77) -> Matrix(155,18,-1662,-193) Matrix(799,-1778,182,-405) -> Matrix(447,50,-4622,-517) Matrix(683,-1528,156,-349) -> Matrix(201,22,-2074,-227) Matrix(41,-100,16,-39) -> Matrix(53,6,-486,-55) Matrix(977,-2552,116,-303) -> Matrix(1017,110,-11714,-1267) Matrix(573,-1498,70,-183) -> Matrix(611,66,-7008,-757) Matrix(623,-1700,188,-513) -> Matrix(41,4,-400,-39) Matrix(667,-1826,202,-553) -> Matrix(33,4,-322,-39) Matrix(119,-330,22,-61) -> Matrix(55,6,-596,-65) Matrix(963,-3292,184,-629) -> Matrix(359,38,-3864,-409) Matrix(823,-2818,158,-541) -> Matrix(401,42,-4306,-451) Matrix(67,-242,18,-65) -> Matrix(159,16,-1600,-161) Matrix(231,-1058,50,-229) -> Matrix(755,72,-7938,-757) 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: 30 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/6 1/3 2/5 3/4 1/1 4/3 7/5 17/11 5/3 9/5 2/1 11/5 29/13 7/3 5/2 81/31 41/15 3/1 65/19 7/2 11/3 4/1 9/2 23/5 5/1 11/2 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 1/0 1/6 -1/2 1/5 -4/9 -2/5 2/9 -7/18 1/4 -1/3 3/11 -1/3 2/7 -5/16 5/17 -2/7 0/1 3/10 -1/3 1/3 -2/7 4/11 -1/4 3/8 -1/3 2/5 -1/4 5/12 -5/21 8/19 -11/48 3/7 -2/9 0/1 4/9 -1/4 5/11 -4/17 1/2 -1/5 5/9 -1/5 4/7 -3/16 7/12 -5/27 17/29 -2/11 -12/67 10/17 -7/40 3/5 0/1 8/13 -1/5 5/8 -1/5 7/11 -4/21 -2/11 2/3 -1/6 7/10 -1/3 12/17 -5/24 5/7 -2/11 3/4 -1/6 7/9 0/1 11/14 -3/19 15/19 -2/13 -4/27 4/5 -1/8 5/6 -1/7 6/7 -1/6 1/1 -2/13 0/1 6/5 -1/6 5/4 -1/5 4/3 -1/7 11/8 -3/23 18/13 -1/6 7/5 -2/15 24/17 -1/8 41/29 -1/8 17/12 -5/41 10/7 -1/10 3/2 -1/7 17/11 -2/15 14/9 -5/38 11/7 -2/15 -4/31 8/5 -1/8 21/13 -2/15 -4/31 13/8 -1/8 18/11 -7/58 5/3 0/1 17/10 -7/51 12/7 -5/38 7/4 -3/23 9/5 -1/8 2/1 -1/8 11/5 -4/35 20/9 -11/98 29/13 -1/9 9/4 -1/9 7/3 -2/17 0/1 5/2 -1/9 13/5 -6/55 -4/37 47/18 -25/231 81/31 -4/37 34/13 -19/176 21/8 -13/121 8/3 -1/10 19/7 -2/19 0/1 30/11 -1/10 41/15 0/1 11/4 -1/9 3/1 -2/19 10/3 -1/10 17/5 -2/19 0/1 41/12 -5/47 65/19 -2/19 24/7 -5/48 7/2 -5/49 11/3 -1/10 4/1 -1/10 9/2 -7/73 23/5 -2/21 14/3 -11/116 5/1 -2/21 -4/43 11/2 -1/11 6/1 -1/11 7/1 -4/45 1/0 -1/13 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(121,-18,74,-11) (0/1,1/6) -> (13/8,18/11) Hyperbolic Matrix(191,-34,118,-21) (1/6,1/5) -> (21/13,13/8) Hyperbolic Matrix(115,-24,24,-5) (1/5,2/9) -> (14/3,5/1) Hyperbolic Matrix(46,-11,67,-16) (2/9,1/4) -> (2/3,7/10) Hyperbolic Matrix(88,-23,23,-6) (1/4,3/11) -> (11/3,4/1) Hyperbolic Matrix(154,-43,43,-12) (3/11,2/7) -> (7/2,11/3) Hyperbolic Matrix(173,-50,218,-63) (2/7,5/17) -> (15/19,4/5) Hyperbolic Matrix(346,-103,215,-64) (5/17,3/10) -> (8/5,21/13) Hyperbolic Matrix(22,-7,63,-20) (3/10,1/3) -> (1/3,4/11) Parabolic Matrix(103,-38,122,-45) (4/11,3/8) -> (5/6,6/7) Hyperbolic Matrix(41,-16,100,-39) (3/8,2/5) -> (2/5,5/12) Parabolic Matrix(394,-165,277,-116) (5/12,8/19) -> (17/12,10/7) Hyperbolic Matrix(393,-166,670,-283) (8/19,3/7) -> (17/29,10/17) Hyperbolic Matrix(98,-43,155,-68) (3/7,4/9) -> (5/8,7/11) Hyperbolic Matrix(327,-148,232,-105) (4/9,5/11) -> (7/5,24/17) Hyperbolic Matrix(212,-97,153,-70) (5/11,1/2) -> (18/13,7/5) Hyperbolic Matrix(36,-19,19,-10) (1/2,5/9) -> (9/5,2/1) Hyperbolic Matrix(126,-71,71,-40) (5/9,4/7) -> (7/4,9/5) Hyperbolic Matrix(88,-51,107,-62) (4/7,7/12) -> (4/5,5/6) Hyperbolic Matrix(1519,-890,582,-341) (7/12,17/29) -> (13/5,47/18) Hyperbolic Matrix(124,-73,17,-10) (10/17,3/5) -> (7/1,1/0) Hyperbolic Matrix(121,-74,18,-11) (3/5,8/13) -> (6/1,7/1) Hyperbolic Matrix(191,-118,34,-21) (8/13,5/8) -> (11/2,6/1) Hyperbolic Matrix(187,-120,120,-77) (7/11,2/3) -> (14/9,11/7) Hyperbolic Matrix(346,-243,131,-92) (7/10,12/17) -> (21/8,8/3) Hyperbolic Matrix(327,-232,148,-105) (12/17,5/7) -> (11/5,20/9) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(172,-135,79,-62) (7/9,11/14) -> (2/1,11/5) Hyperbolic Matrix(1017,-802,298,-235) (11/14,15/19) -> (17/5,41/12) Hyperbolic Matrix(163,-144,60,-53) (6/7,1/1) -> (19/7,30/11) Hyperbolic Matrix(103,-122,38,-45) (1/1,6/5) -> (8/3,19/7) Hyperbolic Matrix(88,-107,51,-62) (6/5,5/4) -> (12/7,7/4) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(283,-390,82,-113) (11/8,18/13) -> (24/7,7/2) Hyperbolic Matrix(754,-1065,337,-476) (24/17,41/29) -> (29/13,9/4) Hyperbolic Matrix(928,-1313,417,-590) (41/29,17/12) -> (20/9,29/13) Hyperbolic Matrix(46,-67,11,-16) (10/7,3/2) -> (4/1,9/2) Hyperbolic Matrix(188,-289,121,-186) (3/2,17/11) -> (17/11,14/9) Parabolic Matrix(98,-155,43,-68) (11/7,8/5) -> (9/4,7/3) Hyperbolic Matrix(106,-175,63,-104) (18/11,5/3) -> (5/3,17/10) Parabolic Matrix(487,-830,186,-317) (17/10,12/7) -> (34/13,21/8) Hyperbolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(2512,-6561,961,-2510) (47/18,81/31) -> (81/31,34/13) Parabolic Matrix(616,-1681,225,-614) (30/11,41/15) -> (41/15,11/4) Parabolic Matrix(22,-63,7,-20) (11/4,3/1) -> (3/1,10/3) Parabolic Matrix(70,-237,13,-44) (10/3,17/5) -> (5/1,11/2) Hyperbolic Matrix(1236,-4225,361,-1234) (41/12,65/19) -> (65/19,24/7) Parabolic Matrix(116,-529,25,-114) (9/2,23/5) -> (23/5,14/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,13,1) Matrix(121,-18,74,-11) -> Matrix(7,4,-58,-33) Matrix(191,-34,118,-21) -> Matrix(13,6,-102,-47) Matrix(115,-24,24,-5) -> Matrix(19,8,-202,-85) Matrix(46,-11,67,-16) -> Matrix(5,2,-33,-13) Matrix(88,-23,23,-6) -> Matrix(17,6,-173,-61) Matrix(154,-43,43,-12) -> Matrix(31,10,-307,-99) Matrix(173,-50,218,-63) -> Matrix(13,4,-88,-27) Matrix(346,-103,215,-64) -> Matrix(13,4,-101,-31) Matrix(22,-7,63,-20) -> Matrix(13,4,-49,-15) Matrix(103,-38,122,-45) -> Matrix(7,2,-46,-13) Matrix(41,-16,100,-39) -> Matrix(23,6,-96,-25) Matrix(394,-165,277,-116) -> Matrix(17,4,-149,-35) Matrix(393,-166,670,-283) -> Matrix(53,12,-296,-67) Matrix(98,-43,155,-68) -> Matrix(7,2,-39,-11) Matrix(327,-148,232,-105) -> Matrix(25,6,-196,-47) Matrix(212,-97,153,-70) -> Matrix(9,2,-59,-13) Matrix(36,-19,19,-10) -> Matrix(9,2,-77,-17) Matrix(126,-71,71,-40) -> Matrix(31,6,-243,-47) Matrix(88,-51,107,-62) -> Matrix(11,2,-61,-11) Matrix(1519,-890,582,-341) -> Matrix(167,30,-1542,-277) Matrix(124,-73,17,-10) -> Matrix(23,4,-259,-45) Matrix(121,-74,18,-11) -> Matrix(19,4,-214,-45) Matrix(191,-118,34,-21) -> Matrix(31,6,-336,-65) Matrix(187,-120,120,-77) -> Matrix(43,8,-328,-61) Matrix(346,-243,131,-92) -> Matrix(7,2,-67,-19) Matrix(327,-232,148,-105) -> Matrix(31,6,-274,-53) Matrix(49,-36,64,-47) -> Matrix(11,2,-72,-13) Matrix(172,-135,79,-62) -> Matrix(25,4,-219,-35) Matrix(1017,-802,298,-235) -> Matrix(27,4,-250,-37) Matrix(163,-144,60,-53) -> Matrix(13,2,-124,-19) Matrix(103,-122,38,-45) -> Matrix(13,2,-124,-19) Matrix(88,-107,51,-62) -> Matrix(15,2,-113,-15) Matrix(49,-64,36,-47) -> Matrix(13,2,-98,-15) Matrix(283,-390,82,-113) -> Matrix(17,2,-162,-19) Matrix(754,-1065,337,-476) -> Matrix(63,8,-575,-73) Matrix(928,-1313,417,-590) -> Matrix(129,16,-1153,-143) Matrix(46,-67,11,-16) -> Matrix(13,2,-137,-21) Matrix(188,-289,121,-186) -> Matrix(89,12,-675,-91) Matrix(98,-155,43,-68) -> Matrix(15,2,-143,-19) Matrix(106,-175,63,-104) -> Matrix(1,0,1,1) Matrix(487,-830,186,-317) -> Matrix(133,18,-1234,-167) Matrix(41,-100,16,-39) -> Matrix(53,6,-486,-55) Matrix(2512,-6561,961,-2510) -> Matrix(1627,176,-15059,-1629) Matrix(616,-1681,225,-614) -> Matrix(1,0,1,1) Matrix(22,-63,7,-20) -> Matrix(37,4,-361,-39) Matrix(70,-237,13,-44) -> Matrix(21,2,-221,-21) Matrix(1236,-4225,361,-1234) -> Matrix(189,20,-1805,-191) Matrix(116,-529,25,-114) -> Matrix(377,36,-3969,-379) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 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: 0 Number of equivalence classes of cusps: 2 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 -1/1 0/1 13 1 1/1 (-2/13,0/1) 0 7 6/5 -1/6 1 14 5/4 -1/5 1 14 4/3 -1/7 1 2 11/8 -3/23 1 14 18/13 -1/6 1 14 7/5 -2/15 1 7 17/12 -5/41 1 14 10/7 -1/10 1 14 3/2 -1/7 1 14 17/11 -2/15 3 1 11/7 (-2/15,-4/31) 0 7 8/5 -1/8 1 14 13/8 -1/8 5 2 5/3 0/1 1 7 17/10 -7/51 1 14 12/7 -5/38 1 14 7/4 -3/23 1 14 9/5 -1/8 4 1 2/1 -1/8 1 14 11/5 -4/35 1 7 20/9 -11/98 1 14 29/13 -1/9 12 1 9/4 -1/9 1 14 7/3 (-2/17,0/1) 0 7 5/2 -1/9 3 2 13/5 (-6/55,-4/37) 0 7 81/31 -4/37 11 1 34/13 -19/176 1 14 21/8 -13/121 1 14 8/3 -1/10 1 14 19/7 (-2/19,0/1) 0 7 41/15 0/1 1 1 11/4 -1/9 1 14 3/1 -2/19 1 7 10/3 -1/10 1 14 17/5 (-2/19,0/1) 0 7 65/19 -2/19 5 1 24/7 -5/48 1 14 7/2 -5/49 1 14 11/3 -1/10 8 1 4/1 -1/10 1 14 9/2 -7/73 1 14 23/5 -2/21 9 1 5/1 (-2/21,-4/43) 0 7 11/2 -1/11 1 14 6/1 -1/11 5 2 7/1 -4/45 1 7 1/0 -1/13 1 14 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(103,-122,38,-45) (1/1,6/5) -> (8/3,19/7) Hyperbolic Matrix(88,-107,51,-62) (6/5,5/4) -> (12/7,7/4) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(283,-390,82,-113) (11/8,18/13) -> (24/7,7/2) Hyperbolic Matrix(153,-212,70,-97) (18/13,7/5) -> (2/1,11/5) Glide Reflection Matrix(232,-327,105,-148) (7/5,17/12) -> (11/5,20/9) Glide Reflection Matrix(243,-346,92,-131) (17/12,10/7) -> (21/8,8/3) Glide Reflection Matrix(46,-67,11,-16) (10/7,3/2) -> (4/1,9/2) Hyperbolic Matrix(67,-102,44,-67) (3/2,17/11) -> (3/2,17/11) Reflection Matrix(120,-187,77,-120) (17/11,11/7) -> (17/11,11/7) Reflection Matrix(98,-155,43,-68) (11/7,8/5) -> (9/4,7/3) Hyperbolic Matrix(118,-191,21,-34) (8/5,13/8) -> (11/2,6/1) Glide Reflection Matrix(74,-121,11,-18) (13/8,5/3) -> (6/1,7/1) Glide Reflection Matrix(73,-124,10,-17) (5/3,17/10) -> (7/1,1/0) Glide Reflection Matrix(487,-830,186,-317) (17/10,12/7) -> (34/13,21/8) Hyperbolic Matrix(71,-126,40,-71) (7/4,9/5) -> (7/4,9/5) Reflection Matrix(19,-36,10,-19) (9/5,2/1) -> (9/5,2/1) Reflection Matrix(521,-1160,234,-521) (20/9,29/13) -> (20/9,29/13) Reflection Matrix(233,-522,104,-233) (29/13,9/4) -> (29/13,9/4) Reflection Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(404,-1053,155,-404) (13/5,81/31) -> (13/5,81/31) Reflection Matrix(2107,-5508,806,-2107) (81/31,34/13) -> (81/31,34/13) Reflection Matrix(286,-779,105,-286) (19/7,41/15) -> (19/7,41/15) Reflection Matrix(329,-902,120,-329) (41/15,11/4) -> (41/15,11/4) Reflection Matrix(22,-63,7,-20) (11/4,3/1) -> (3/1,10/3) Parabolic Matrix(70,-237,13,-44) (10/3,17/5) -> (5/1,11/2) Hyperbolic Matrix(324,-1105,95,-324) (17/5,65/19) -> (17/5,65/19) Reflection Matrix(911,-3120,266,-911) (65/19,24/7) -> (65/19,24/7) Reflection Matrix(43,-154,12,-43) (7/2,11/3) -> (7/2,11/3) Reflection Matrix(23,-88,6,-23) (11/3,4/1) -> (11/3,4/1) Reflection Matrix(91,-414,20,-91) (9/2,23/5) -> (9/2,23/5) Reflection Matrix(24,-115,5,-24) (23/5,5/1) -> (23/5,5/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,26,1) (-1/1,1/0) -> (-1/13,0/1) Matrix(0,1,1,0) -> Matrix(-1,0,13,1) (-1/1,1/1) -> (-2/13,0/1) Matrix(103,-122,38,-45) -> Matrix(13,2,-124,-19) Matrix(88,-107,51,-62) -> Matrix(15,2,-113,-15) (-1/7,-1/8).(-2/15,0/1) Matrix(49,-64,36,-47) -> Matrix(13,2,-98,-15) -1/7 Matrix(283,-390,82,-113) -> Matrix(17,2,-162,-19) -1/9 Matrix(153,-212,70,-97) -> Matrix(13,2,-110,-17) Matrix(232,-327,105,-148) -> Matrix(47,6,-415,-53) Matrix(243,-346,92,-131) -> Matrix(19,2,-180,-19) *** -> (-1/9,-1/10) Matrix(46,-67,11,-16) -> Matrix(13,2,-137,-21) Matrix(67,-102,44,-67) -> Matrix(29,4,-210,-29) (3/2,17/11) -> (-1/7,-2/15) Matrix(120,-187,77,-120) -> Matrix(61,8,-465,-61) (17/11,11/7) -> (-2/15,-4/31) Matrix(98,-155,43,-68) -> Matrix(15,2,-143,-19) Matrix(118,-191,21,-34) -> Matrix(47,6,-509,-65) Matrix(74,-121,11,-18) -> Matrix(33,4,-371,-45) Matrix(73,-124,10,-17) -> Matrix(29,4,-326,-45) Matrix(487,-830,186,-317) -> Matrix(133,18,-1234,-167) Matrix(71,-126,40,-71) -> Matrix(47,6,-368,-47) (7/4,9/5) -> (-3/23,-1/8) Matrix(19,-36,10,-19) -> Matrix(17,2,-144,-17) (9/5,2/1) -> (-1/8,-1/9) Matrix(521,-1160,234,-521) -> Matrix(197,22,-1764,-197) (20/9,29/13) -> (-11/98,-1/9) Matrix(233,-522,104,-233) -> Matrix(19,2,-180,-19) (29/13,9/4) -> (-1/9,-1/10) Matrix(41,-100,16,-39) -> Matrix(53,6,-486,-55) -1/9 Matrix(404,-1053,155,-404) -> Matrix(221,24,-2035,-221) (13/5,81/31) -> (-6/55,-4/37) Matrix(2107,-5508,806,-2107) -> Matrix(1407,152,-13024,-1407) (81/31,34/13) -> (-4/37,-19/176) Matrix(286,-779,105,-286) -> Matrix(-1,0,19,1) (19/7,41/15) -> (-2/19,0/1) Matrix(329,-902,120,-329) -> Matrix(-1,0,18,1) (41/15,11/4) -> (-1/9,0/1) Matrix(22,-63,7,-20) -> Matrix(37,4,-361,-39) -2/19 Matrix(70,-237,13,-44) -> Matrix(21,2,-221,-21) (-1/10,-1/11).(-2/21,0/1) Matrix(324,-1105,95,-324) -> Matrix(-1,0,19,1) (17/5,65/19) -> (-2/19,0/1) Matrix(911,-3120,266,-911) -> Matrix(191,20,-1824,-191) (65/19,24/7) -> (-2/19,-5/48) Matrix(43,-154,12,-43) -> Matrix(99,10,-980,-99) (7/2,11/3) -> (-5/49,-1/10) Matrix(23,-88,6,-23) -> Matrix(61,6,-620,-61) (11/3,4/1) -> (-1/10,-3/31) Matrix(91,-414,20,-91) -> Matrix(293,28,-3066,-293) (9/2,23/5) -> (-7/73,-2/21) Matrix(24,-115,5,-24) -> Matrix(85,8,-903,-85) (23/5,5/1) -> (-2/21,-4/43) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.