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 -9/1 -7/1 -5/1 -4/1 -7/2 -3/1 -43/18 -9/4 -2/1 -1/1 -9/14 -7/12 -1/2 -4/9 -3/8 -2/7 -1/4 -1/6 0/1 1/4 3/7 1/2 2/3 3/4 11/14 9/11 1/1 3/2 7/4 11/6 13/7 2/1 13/6 7/3 5/2 3/1 19/6 31/9 7/2 4/1 9/2 5/1 11/2 17/3 13/2 7/1 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -9/1 0/1 -8/1 -1/2 -7/1 1/1 -6/1 1/0 -5/1 -2/1 -9/2 -3/2 -1/1 -4/1 -1/2 -7/2 1/0 -10/3 -5/2 -13/4 1/0 -16/5 -3/2 -3/1 -1/1 -5/2 -1/1 1/0 -12/5 -5/6 -43/18 -3/4 -74/31 -7/10 -31/13 -2/3 -19/8 -3/4 -7/3 -1/1 -23/10 -2/3 -1/2 -16/7 -1/2 -9/4 -1/2 -2/1 -1/2 -1/1 0/1 -2/3 1/2 -9/14 1/2 -16/25 1/2 -7/11 1/1 -12/19 -1/2 -5/8 1/4 -18/29 1/2 -13/21 1/3 -8/13 1/2 -3/5 0/1 -10/17 1/2 -7/12 1/2 -18/31 1/2 -11/19 1/1 -4/7 1/2 -9/16 1/4 -5/9 0/1 -1/2 0/1 1/2 -5/11 0/1 -4/9 1/2 -11/25 1/1 -7/16 1/0 -10/23 -1/2 -3/7 0/1 -5/12 1/0 -12/29 -1/2 -7/17 0/1 -9/22 0/1 1/6 -11/27 1/5 -2/5 1/2 -7/18 1/4 1/3 -5/13 1/3 -3/8 1/2 -7/19 1/1 -18/49 1/2 -11/30 1/2 1/1 -4/11 1/2 -5/14 0/1 1/0 -1/3 0/1 -3/10 1/3 1/2 -5/17 2/5 -2/7 1/2 -7/25 3/5 -12/43 1/2 -5/18 1/2 2/3 -3/11 1/1 -10/37 1/2 -7/26 1/2 1/1 -18/67 1/2 -11/41 1/1 -4/15 1/2 -5/19 2/3 -1/4 1/0 -4/17 -1/6 -3/13 0/1 -5/22 0/1 1/12 -2/9 1/6 -3/14 1/5 1/4 -1/5 1/3 -1/6 1/2 -1/7 1/1 0/1 1/2 1/8 1/0 1/7 0/1 1/6 1/2 1/1 1/5 0/1 1/4 1/2 3/11 2/3 8/29 9/14 5/18 2/3 3/4 2/7 1/2 3/10 1/2 2/3 1/3 1/1 5/14 1/2 1/1 9/25 1/1 4/11 1/2 3/8 3/4 5/13 2/3 7/18 3/4 2/5 5/6 5/12 11/12 3/7 1/1 7/16 13/12 4/9 7/6 1/2 1/1 1/0 3/5 1/1 11/18 1/1 1/0 8/13 3/2 21/34 3/2 13/21 2/1 5/8 1/0 12/19 3/2 7/11 2/1 2/3 1/0 9/13 0/1 25/36 1/0 16/23 -1/2 7/10 0/1 1/0 12/17 1/2 5/7 0/1 13/18 0/1 1/0 8/11 1/2 11/15 1/1 3/4 1/0 7/9 0/1 11/14 1/2 15/19 2/3 4/5 1/2 9/11 1/1 14/17 3/2 5/6 1/1 1/0 6/7 1/2 1/1 1/1 3/2 1/0 5/3 -1/1 17/10 -1/2 0/1 12/7 -1/2 31/18 -1/4 0/1 19/11 0/1 26/15 1/2 7/4 1/0 16/9 -1/2 9/5 0/1 11/6 -1/4 0/1 13/7 0/1 2/1 1/2 13/6 1/2 1/1 11/5 1/1 20/9 1/2 9/4 3/4 7/3 1/1 19/8 9/8 31/13 1/1 43/18 1/1 7/6 12/5 7/6 5/2 1/1 3/2 13/5 2/1 21/8 1/0 50/19 1/2 29/11 1/1 8/3 3/2 11/4 3/2 14/5 3/2 17/6 7/4 2/1 3/1 2/1 19/6 5/2 16/5 5/2 13/4 11/4 10/3 7/2 17/5 4/1 24/7 7/2 31/9 4/1 7/2 4/1 1/0 18/5 9/2 29/8 21/4 11/3 6/1 4/1 1/0 13/3 -5/1 9/2 -3/1 1/0 14/3 -5/2 5/1 -1/1 11/2 -1/2 0/1 17/3 0/1 6/1 1/2 13/2 1/0 7/1 0/1 8/1 1/2 9/1 1/1 10/1 3/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(25,234,36,337) (-9/1,1/0) -> (9/13,25/36) Hyperbolic Matrix(105,914,-44,-383) (-9/1,-8/1) -> (-74/31,-31/13) Hyperbolic Matrix(13,102,-32,-251) (-8/1,-7/1) -> (-11/27,-2/5) Hyperbolic Matrix(17,106,4,25) (-7/1,-6/1) -> (4/1,13/3) Hyperbolic Matrix(15,86,4,23) (-6/1,-5/1) -> (11/3,4/1) Hyperbolic Matrix(3,14,16,75) (-5/1,-9/2) -> (1/6,1/5) Hyperbolic Matrix(29,128,12,53) (-9/2,-4/1) -> (12/5,5/2) Hyperbolic Matrix(27,98,-8,-29) (-4/1,-7/2) -> (-7/2,-10/3) Parabolic Matrix(61,200,-140,-459) (-10/3,-13/4) -> (-7/16,-10/23) Hyperbolic Matrix(139,450,80,259) (-13/4,-16/5) -> (26/15,7/4) Hyperbolic Matrix(47,148,-168,-529) (-16/5,-3/1) -> (-7/25,-12/43) Hyperbolic Matrix(7,18,12,31) (-3/1,-5/2) -> (1/2,3/5) Hyperbolic Matrix(13,32,28,69) (-5/2,-12/5) -> (4/9,1/2) Hyperbolic Matrix(1547,3698,-648,-1549) (-12/5,-43/18) -> (-43/18,-74/31) Parabolic Matrix(21,50,160,381) (-31/13,-19/8) -> (1/8,1/7) Hyperbolic Matrix(109,258,-248,-587) (-19/8,-7/3) -> (-11/25,-7/16) Hyperbolic Matrix(137,316,-336,-775) (-7/3,-23/10) -> (-9/22,-11/27) Hyperbolic Matrix(155,356,-556,-1277) (-23/10,-16/7) -> (-12/43,-5/18) Hyperbolic Matrix(133,302,48,109) (-16/7,-9/4) -> (11/4,14/5) Hyperbolic Matrix(43,94,16,35) (-9/4,-2/1) -> (8/3,11/4) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(251,162,-392,-253) (-2/3,-9/14) -> (-9/14,-16/25) Parabolic Matrix(495,316,224,143) (-16/25,-7/11) -> (11/5,20/9) Hyperbolic Matrix(249,158,52,33) (-7/11,-12/19) -> (14/3,5/1) Hyperbolic Matrix(51,32,-212,-133) (-12/19,-5/8) -> (-1/4,-4/17) Hyperbolic Matrix(505,314,156,97) (-5/8,-18/29) -> (16/5,13/4) Hyperbolic Matrix(581,360,-1580,-979) (-18/29,-13/21) -> (-7/19,-18/49) Hyperbolic Matrix(133,82,-472,-291) (-13/21,-8/13) -> (-2/7,-7/25) Hyperbolic Matrix(75,46,-256,-157) (-8/13,-3/5) -> (-5/17,-2/7) Hyperbolic Matrix(339,200,100,59) (-3/5,-10/17) -> (10/3,17/5) Hyperbolic Matrix(335,196,-576,-337) (-10/17,-7/12) -> (-7/12,-18/31) Parabolic Matrix(283,164,-1044,-605) (-18/31,-11/19) -> (-3/11,-10/37) Hyperbolic Matrix(121,70,140,81) (-11/19,-4/7) -> (6/7,1/1) Hyperbolic Matrix(227,128,-548,-309) (-4/7,-9/16) -> (-5/12,-12/29) Hyperbolic Matrix(253,142,408,229) (-9/16,-5/9) -> (13/21,5/8) Hyperbolic Matrix(19,10,-40,-21) (-5/9,-1/2) -> (-1/2,-5/11) Parabolic Matrix(257,116,144,65) (-5/11,-4/9) -> (16/9,9/5) Hyperbolic Matrix(181,80,500,221) (-4/9,-11/25) -> (9/25,4/11) Hyperbolic Matrix(125,54,456,197) (-10/23,-3/7) -> (3/11,8/29) Hyperbolic Matrix(105,44,136,57) (-3/7,-5/12) -> (3/4,7/9) Hyperbolic Matrix(595,246,104,43) (-12/29,-7/17) -> (17/3,6/1) Hyperbolic Matrix(561,230,100,41) (-7/17,-9/22) -> (11/2,17/3) Hyperbolic Matrix(297,116,64,25) (-2/5,-7/18) -> (9/2,14/3) Hyperbolic Matrix(197,76,324,125) (-7/18,-5/13) -> (3/5,11/18) Hyperbolic Matrix(95,36,-256,-97) (-5/13,-3/8) -> (-3/8,-7/19) Parabolic Matrix(883,324,-3284,-1205) (-18/49,-11/30) -> (-7/26,-18/67) Hyperbolic Matrix(203,74,96,35) (-11/30,-4/11) -> (2/1,13/6) Hyperbolic Matrix(329,118,92,33) (-4/11,-5/14) -> (7/2,18/5) Hyperbolic Matrix(109,38,152,53) (-5/14,-1/3) -> (5/7,13/18) Hyperbolic Matrix(145,44,56,17) (-1/3,-3/10) -> (5/2,13/5) Hyperbolic Matrix(27,8,172,51) (-3/10,-5/17) -> (1/7,1/6) Hyperbolic Matrix(277,76,164,45) (-5/18,-3/11) -> (5/3,17/10) Hyperbolic Matrix(1903,514,796,215) (-10/37,-7/26) -> (43/18,12/5) Hyperbolic Matrix(1013,272,108,29) (-18/67,-11/41) -> (9/1,10/1) Hyperbolic Matrix(463,124,56,15) (-11/41,-4/15) -> (8/1,9/1) Hyperbolic Matrix(301,80,380,101) (-4/15,-5/19) -> (15/19,4/5) Hyperbolic Matrix(595,156,164,43) (-5/19,-1/4) -> (29/8,11/3) Hyperbolic Matrix(247,58,132,31) (-4/17,-3/13) -> (13/7,2/1) Hyperbolic Matrix(429,98,232,53) (-3/13,-5/22) -> (11/6,13/7) Hyperbolic Matrix(221,50,800,181) (-5/22,-2/9) -> (8/29,5/18) Hyperbolic Matrix(129,28,152,33) (-2/9,-3/14) -> (5/6,6/7) Hyperbolic Matrix(227,48,52,11) (-3/14,-1/5) -> (13/3,9/2) Hyperbolic Matrix(11,2,-72,-13) (-1/5,-1/6) -> (-1/6,-1/7) Parabolic Matrix(67,8,92,11) (-1/7,0/1) -> (8/11,11/15) Hyperbolic Matrix(421,-50,160,-19) (0/1,1/8) -> (21/8,50/19) Hyperbolic Matrix(17,-4,64,-15) (1/5,1/4) -> (1/4,3/11) Parabolic Matrix(337,-94,484,-135) (5/18,2/7) -> (16/23,7/10) Hyperbolic Matrix(239,-70,140,-41) (2/7,3/10) -> (17/10,12/7) Hyperbolic Matrix(83,-26,16,-5) (3/10,1/3) -> (5/1,11/2) Hyperbolic Matrix(193,-68,88,-31) (1/3,5/14) -> (13/6,11/5) Hyperbolic Matrix(1509,-542,632,-227) (5/14,9/25) -> (31/13,43/18) Hyperbolic Matrix(205,-76,116,-43) (4/11,3/8) -> (7/4,16/9) Hyperbolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(295,-114,44,-17) (5/13,7/18) -> (13/2,7/1) Hyperbolic Matrix(173,-68,28,-11) (7/18,2/5) -> (6/1,13/2) Hyperbolic Matrix(185,-76,56,-23) (2/5,5/12) -> (13/4,10/3) Hyperbolic Matrix(85,-36,196,-83) (5/12,3/7) -> (3/7,7/16) Parabolic Matrix(549,-242,152,-67) (7/16,4/9) -> (18/5,29/8) Hyperbolic Matrix(317,-194,384,-235) (11/18,8/13) -> (14/17,5/6) Hyperbolic Matrix(791,-488,248,-153) (8/13,21/34) -> (19/6,16/5) Hyperbolic Matrix(501,-310,160,-99) (21/34,13/21) -> (3/1,19/6) Hyperbolic Matrix(331,-208,148,-93) (5/8,12/19) -> (20/9,9/4) Hyperbolic Matrix(587,-372,172,-109) (12/19,7/11) -> (17/5,24/7) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(383,-266,36,-25) (25/36,16/23) -> (10/1,1/0) Hyperbolic Matrix(429,-302,152,-107) (7/10,12/17) -> (14/5,17/6) Hyperbolic Matrix(175,-124,24,-17) (12/17,5/7) -> (7/1,8/1) Hyperbolic Matrix(557,-404,324,-235) (13/18,8/11) -> (12/7,31/18) Hyperbolic Matrix(409,-302,172,-127) (11/15,3/4) -> (19/8,31/13) Hyperbolic Matrix(309,-242,392,-307) (7/9,11/14) -> (11/14,15/19) Parabolic Matrix(233,-188,88,-71) (4/5,9/11) -> (29/11,8/3) Hyperbolic Matrix(1043,-856,396,-325) (9/11,14/17) -> (50/19,29/11) Hyperbolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(635,-1094,184,-317) (31/18,19/11) -> (31/9,7/2) Hyperbolic Matrix(729,-1262,212,-367) (19/11,26/15) -> (24/7,31/9) Hyperbolic Matrix(103,-186,36,-65) (9/5,11/6) -> (17/6,3/1) Hyperbolic Matrix(85,-196,36,-83) (9/4,7/3) -> (7/3,19/8) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(25,234,36,337) -> Matrix(1,0,0,1) Matrix(105,914,-44,-383) -> Matrix(3,-2,-4,3) Matrix(13,102,-32,-251) -> Matrix(1,0,4,1) Matrix(17,106,4,25) -> Matrix(1,-6,0,1) Matrix(15,86,4,23) -> Matrix(1,8,0,1) Matrix(3,14,16,75) -> Matrix(1,2,0,1) Matrix(29,128,12,53) -> Matrix(5,6,4,5) Matrix(27,98,-8,-29) -> Matrix(1,-2,0,1) Matrix(61,200,-140,-459) -> Matrix(1,2,0,1) Matrix(139,450,80,259) -> Matrix(1,2,0,1) Matrix(47,148,-168,-529) -> Matrix(5,8,8,13) Matrix(7,18,12,31) -> Matrix(1,2,0,1) Matrix(13,32,28,69) -> Matrix(1,2,0,1) Matrix(1547,3698,-648,-1549) -> Matrix(23,18,-32,-25) Matrix(21,50,160,381) -> Matrix(3,2,4,3) Matrix(109,258,-248,-587) -> Matrix(3,2,4,3) Matrix(137,316,-336,-775) -> Matrix(3,2,16,11) Matrix(155,356,-556,-1277) -> Matrix(7,4,12,7) Matrix(133,302,48,109) -> Matrix(13,8,8,5) Matrix(43,94,16,35) -> Matrix(1,2,0,1) Matrix(3,4,-4,-5) -> Matrix(1,0,4,1) Matrix(251,162,-392,-253) -> Matrix(5,-2,8,-3) Matrix(495,316,224,143) -> Matrix(1,0,0,1) Matrix(249,158,52,33) -> Matrix(1,-2,0,1) Matrix(51,32,-212,-133) -> Matrix(1,0,-4,1) Matrix(505,314,156,97) -> Matrix(21,-8,8,-3) Matrix(581,360,-1580,-979) -> Matrix(5,-2,8,-3) Matrix(133,82,-472,-291) -> Matrix(9,-4,16,-7) Matrix(75,46,-256,-157) -> Matrix(3,-2,8,-5) Matrix(339,200,100,59) -> Matrix(15,-4,4,-1) Matrix(335,196,-576,-337) -> Matrix(5,-2,8,-3) Matrix(283,164,-1044,-605) -> Matrix(1,0,0,1) Matrix(121,70,140,81) -> Matrix(1,0,0,1) Matrix(227,128,-548,-309) -> Matrix(1,0,-4,1) Matrix(253,142,408,229) -> Matrix(7,-2,4,-1) Matrix(19,10,-40,-21) -> Matrix(1,0,0,1) Matrix(257,116,144,65) -> Matrix(1,0,-4,1) Matrix(181,80,500,221) -> Matrix(1,0,0,1) Matrix(125,54,456,197) -> Matrix(5,-2,8,-3) Matrix(105,44,136,57) -> Matrix(1,0,0,1) Matrix(595,246,104,43) -> Matrix(1,0,4,1) Matrix(561,230,100,41) -> Matrix(1,0,-8,1) Matrix(297,116,64,25) -> Matrix(9,-2,-4,1) Matrix(197,76,324,125) -> Matrix(7,-2,4,-1) Matrix(95,36,-256,-97) -> Matrix(5,-2,8,-3) Matrix(883,324,-3284,-1205) -> Matrix(1,0,0,1) Matrix(203,74,96,35) -> Matrix(1,0,0,1) Matrix(329,118,92,33) -> Matrix(1,4,0,1) Matrix(109,38,152,53) -> Matrix(1,0,0,1) Matrix(145,44,56,17) -> Matrix(7,-2,4,-1) Matrix(27,8,172,51) -> Matrix(5,-2,8,-3) Matrix(277,76,164,45) -> Matrix(3,-2,-4,3) Matrix(1903,514,796,215) -> Matrix(9,-8,8,-7) Matrix(1013,272,108,29) -> Matrix(5,-4,4,-3) Matrix(463,124,56,15) -> Matrix(1,0,0,1) Matrix(301,80,380,101) -> Matrix(1,0,0,1) Matrix(595,156,164,43) -> Matrix(21,-16,4,-3) Matrix(247,58,132,31) -> Matrix(1,0,8,1) Matrix(429,98,232,53) -> Matrix(1,0,-16,1) Matrix(221,50,800,181) -> Matrix(21,-2,32,-3) Matrix(129,28,152,33) -> Matrix(1,0,-4,1) Matrix(227,48,52,11) -> Matrix(17,-4,-4,1) Matrix(11,2,-72,-13) -> Matrix(5,-2,8,-3) Matrix(67,8,92,11) -> Matrix(1,0,0,1) Matrix(421,-50,160,-19) -> Matrix(1,0,0,1) Matrix(17,-4,64,-15) -> Matrix(5,-2,8,-3) Matrix(337,-94,484,-135) -> Matrix(3,-2,-4,3) Matrix(239,-70,140,-41) -> Matrix(3,-2,-4,3) Matrix(83,-26,16,-5) -> Matrix(3,-2,-4,3) Matrix(193,-68,88,-31) -> Matrix(1,0,0,1) Matrix(1509,-542,632,-227) -> Matrix(9,-8,8,-7) Matrix(205,-76,116,-43) -> Matrix(3,-2,-4,3) Matrix(377,-144,144,-55) -> Matrix(5,-4,4,-3) Matrix(295,-114,44,-17) -> Matrix(3,-2,-4,3) Matrix(173,-68,28,-11) -> Matrix(5,-4,4,-3) Matrix(185,-76,56,-23) -> Matrix(25,-22,8,-7) Matrix(85,-36,196,-83) -> Matrix(25,-24,24,-23) Matrix(549,-242,152,-67) -> Matrix(39,-44,8,-9) Matrix(317,-194,384,-235) -> Matrix(1,0,0,1) Matrix(791,-488,248,-153) -> Matrix(11,-14,4,-5) Matrix(501,-310,160,-99) -> Matrix(9,-16,4,-7) Matrix(331,-208,148,-93) -> Matrix(3,-4,4,-5) Matrix(587,-372,172,-109) -> Matrix(1,2,0,1) Matrix(49,-32,72,-47) -> Matrix(1,-2,0,1) Matrix(383,-266,36,-25) -> Matrix(1,2,0,1) Matrix(429,-302,152,-107) -> Matrix(7,-2,4,-1) Matrix(175,-124,24,-17) -> Matrix(1,0,0,1) Matrix(557,-404,324,-235) -> Matrix(1,0,-4,1) Matrix(409,-302,172,-127) -> Matrix(9,-8,8,-7) Matrix(309,-242,392,-307) -> Matrix(5,-2,8,-3) Matrix(233,-188,88,-71) -> Matrix(5,-4,4,-3) Matrix(1043,-856,396,-325) -> Matrix(3,-4,4,-5) Matrix(13,-18,8,-11) -> Matrix(1,-2,0,1) Matrix(635,-1094,184,-317) -> Matrix(17,4,4,1) Matrix(729,-1262,212,-367) -> Matrix(15,-4,4,-1) Matrix(103,-186,36,-65) -> Matrix(1,2,0,1) Matrix(85,-196,36,-83) -> Matrix(13,-12,12,-11) 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 -4/1 -7/2 -3/1 -9/4 -2/1 -1/1 -9/14 -7/12 -1/2 -4/9 -3/8 -2/7 -1/4 -1/6 0/1 1/4 3/7 1/2 2/3 3/4 9/11 1/1 3/2 7/4 2/1 7/3 5/2 3/1 7/2 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -2/1 -9/2 -3/2 -1/1 -4/1 -1/2 -7/2 1/0 -10/3 -5/2 -3/1 -1/1 -5/2 -1/1 1/0 -12/5 -5/6 -7/3 -1/1 -9/4 -1/2 -2/1 -1/2 -1/1 0/1 -2/3 1/2 -9/14 1/2 -7/11 1/1 -12/19 -1/2 -5/8 1/4 -3/5 0/1 -7/12 1/2 -11/19 1/1 -4/7 1/2 -9/16 1/4 -5/9 0/1 -1/2 0/1 1/2 -5/11 0/1 -4/9 1/2 -3/7 0/1 -5/12 1/0 -2/5 1/2 -3/8 1/2 -4/11 1/2 -5/14 0/1 1/0 -1/3 0/1 -3/10 1/3 1/2 -2/7 1/2 -5/18 1/2 2/3 -3/11 1/1 -1/4 1/0 -2/9 1/6 -1/5 1/3 -1/6 1/2 -1/7 1/1 0/1 1/2 1/4 1/2 2/7 1/2 3/10 1/2 2/3 1/3 1/1 4/11 1/2 3/8 3/4 5/13 2/3 2/5 5/6 5/12 11/12 3/7 1/1 7/16 13/12 4/9 7/6 1/2 1/1 1/0 3/5 1/1 2/3 1/0 5/7 0/1 13/18 0/1 1/0 8/11 1/2 11/15 1/1 3/4 1/0 7/9 0/1 11/14 1/2 4/5 1/2 9/11 1/1 5/6 1/1 1/0 1/1 1/1 3/2 1/0 5/3 -1/1 17/10 -1/2 0/1 12/7 -1/2 31/18 -1/4 0/1 19/11 0/1 7/4 1/0 16/9 -1/2 9/5 0/1 2/1 1/2 9/4 3/4 7/3 1/1 19/8 9/8 12/5 7/6 5/2 1/1 3/2 13/5 2/1 21/8 1/0 29/11 1/1 8/3 3/2 11/4 3/2 3/1 2/1 10/3 7/2 17/5 4/1 7/2 4/1 1/0 4/1 1/0 9/2 -3/1 1/0 14/3 -5/2 5/1 -1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,10,0,1) (-5/1,1/0) -> (5/1,1/0) Parabolic Matrix(5,24,16,77) (-5/1,-9/2) -> (3/10,1/3) Hyperbolic Matrix(29,128,12,53) (-9/2,-4/1) -> (12/5,5/2) Hyperbolic Matrix(27,98,-8,-29) (-4/1,-7/2) -> (-7/2,-10/3) Parabolic Matrix(17,56,44,145) (-10/3,-3/1) -> (5/13,2/5) Hyperbolic Matrix(7,18,12,31) (-3/1,-5/2) -> (1/2,3/5) Hyperbolic Matrix(13,32,28,69) (-5/2,-12/5) -> (4/9,1/2) Hyperbolic Matrix(51,122,28,67) (-12/5,-7/3) -> (9/5,2/1) Hyperbolic Matrix(45,104,16,37) (-7/3,-9/4) -> (11/4,3/1) Hyperbolic Matrix(43,94,16,35) (-9/4,-2/1) -> (8/3,11/4) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(89,58,112,73) (-2/3,-9/14) -> (11/14,4/5) Hyperbolic Matrix(219,140,280,179) (-9/14,-7/11) -> (7/9,11/14) Hyperbolic Matrix(249,158,52,33) (-7/11,-12/19) -> (14/3,5/1) Hyperbolic Matrix(165,104,376,237) (-12/19,-5/8) -> (7/16,4/9) Hyperbolic Matrix(29,18,-108,-67) (-5/8,-3/5) -> (-3/11,-1/4) Hyperbolic Matrix(167,98,-288,-169) (-3/5,-7/12) -> (-7/12,-11/19) Parabolic Matrix(309,178,92,53) (-11/19,-4/7) -> (10/3,17/5) Hyperbolic Matrix(325,184,136,77) (-4/7,-9/16) -> (19/8,12/5) Hyperbolic Matrix(203,114,276,155) (-9/16,-5/9) -> (11/15,3/4) Hyperbolic Matrix(19,10,-40,-21) (-5/9,-1/2) -> (-1/2,-5/11) Parabolic Matrix(257,116,144,65) (-5/11,-4/9) -> (16/9,9/5) Hyperbolic Matrix(37,16,104,45) (-4/9,-3/7) -> (1/3,4/11) Hyperbolic Matrix(105,44,136,57) (-3/7,-5/12) -> (3/4,7/9) Hyperbolic Matrix(69,28,32,13) (-5/12,-2/5) -> (2/1,9/4) Hyperbolic Matrix(47,18,-128,-49) (-2/5,-3/8) -> (-3/8,-4/11) Parabolic Matrix(295,106,64,23) (-4/11,-5/14) -> (9/2,14/3) Hyperbolic Matrix(109,38,152,53) (-5/14,-1/3) -> (5/7,13/18) Hyperbolic Matrix(145,44,56,17) (-1/3,-3/10) -> (5/2,13/5) Hyperbolic Matrix(55,16,-196,-57) (-3/10,-2/7) -> (-2/7,-5/18) Parabolic Matrix(277,76,164,45) (-5/18,-3/11) -> (5/3,17/10) Hyperbolic Matrix(53,12,128,29) (-1/4,-2/9) -> (2/5,5/12) Hyperbolic Matrix(77,16,24,5) (-2/9,-1/5) -> (3/1,10/3) Hyperbolic Matrix(11,2,-72,-13) (-1/5,-1/6) -> (-1/6,-1/7) Parabolic Matrix(67,8,92,11) (-1/7,0/1) -> (8/11,11/15) Hyperbolic Matrix(9,-2,32,-7) (0/1,1/4) -> (1/4,2/7) Parabolic Matrix(239,-70,140,-41) (2/7,3/10) -> (17/10,12/7) Hyperbolic Matrix(205,-76,116,-43) (4/11,3/8) -> (7/4,16/9) Hyperbolic Matrix(377,-144,144,-55) (3/8,5/13) -> (13/5,21/8) Hyperbolic Matrix(85,-36,196,-83) (5/12,3/7) -> (3/7,7/16) Parabolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(557,-404,324,-235) (13/18,8/11) -> (12/7,31/18) Hyperbolic Matrix(233,-188,88,-71) (4/5,9/11) -> (29/11,8/3) Hyperbolic Matrix(455,-374,264,-217) (9/11,5/6) -> (31/18,19/11) Hyperbolic Matrix(109,-92,32,-27) (5/6,1/1) -> (17/5,7/2) Hyperbolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(347,-602,132,-229) (19/11,7/4) -> (21/8,29/11) Hyperbolic Matrix(85,-196,36,-83) (9/4,7/3) -> (7/3,19/8) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,10,0,1) -> Matrix(1,1,0,1) Matrix(5,24,16,77) -> Matrix(3,5,4,7) Matrix(29,128,12,53) -> Matrix(5,6,4,5) Matrix(27,98,-8,-29) -> Matrix(1,-2,0,1) Matrix(17,56,44,145) -> Matrix(3,5,4,7) Matrix(7,18,12,31) -> Matrix(1,2,0,1) Matrix(13,32,28,69) -> Matrix(1,2,0,1) Matrix(51,122,28,67) -> Matrix(1,1,-4,-3) Matrix(45,104,16,37) -> Matrix(7,5,4,3) Matrix(43,94,16,35) -> Matrix(1,2,0,1) Matrix(3,4,-4,-5) -> Matrix(1,0,4,1) Matrix(89,58,112,73) -> Matrix(3,-1,4,-1) Matrix(219,140,280,179) -> Matrix(1,-1,4,-3) Matrix(249,158,52,33) -> Matrix(1,-2,0,1) Matrix(165,104,376,237) -> Matrix(9,1,8,1) Matrix(29,18,-108,-67) -> Matrix(3,-1,4,-1) Matrix(167,98,-288,-169) -> Matrix(3,-1,4,-1) Matrix(309,178,92,53) -> Matrix(1,3,0,1) Matrix(325,184,136,77) -> Matrix(5,1,4,1) Matrix(203,114,276,155) -> Matrix(3,-1,4,-1) Matrix(19,10,-40,-21) -> Matrix(1,0,0,1) Matrix(257,116,144,65) -> Matrix(1,0,-4,1) Matrix(37,16,104,45) -> Matrix(3,-1,4,-1) Matrix(105,44,136,57) -> Matrix(1,0,0,1) Matrix(69,28,32,13) -> Matrix(3,-1,4,-1) Matrix(47,18,-128,-49) -> Matrix(3,-1,4,-1) Matrix(295,106,64,23) -> Matrix(1,-3,0,1) Matrix(109,38,152,53) -> Matrix(1,0,0,1) Matrix(145,44,56,17) -> Matrix(7,-2,4,-1) Matrix(55,16,-196,-57) -> Matrix(7,-3,12,-5) Matrix(277,76,164,45) -> Matrix(3,-2,-4,3) Matrix(53,12,128,29) -> Matrix(11,-1,12,-1) Matrix(77,16,24,5) -> Matrix(11,-3,4,-1) Matrix(11,2,-72,-13) -> Matrix(5,-2,8,-3) Matrix(67,8,92,11) -> Matrix(1,0,0,1) Matrix(9,-2,32,-7) -> Matrix(3,-1,4,-1) Matrix(239,-70,140,-41) -> Matrix(3,-2,-4,3) Matrix(205,-76,116,-43) -> Matrix(3,-2,-4,3) Matrix(377,-144,144,-55) -> Matrix(5,-4,4,-3) Matrix(85,-36,196,-83) -> Matrix(25,-24,24,-23) Matrix(25,-16,36,-23) -> Matrix(1,-1,0,1) Matrix(557,-404,324,-235) -> Matrix(1,0,-4,1) Matrix(233,-188,88,-71) -> Matrix(5,-4,4,-3) Matrix(455,-374,264,-217) -> Matrix(1,-1,-4,5) Matrix(109,-92,32,-27) -> Matrix(1,3,0,1) Matrix(13,-18,8,-11) -> Matrix(1,-2,0,1) Matrix(347,-602,132,-229) -> Matrix(1,1,0,1) Matrix(85,-196,36,-83) -> Matrix(13,-12,12,-11) Matrix(17,-64,4,-15) -> Matrix(1,-7,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 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 -1/1 0/1 2 2 -1/2 (0/1,1/2) 0 20 -4/9 1/2 1 4 -3/7 0/1 2 10 -5/12 1/0 2 20 -2/5 1/2 1 20 -3/8 1/2 2 4 -4/11 1/2 1 20 -1/3 0/1 2 10 -3/10 (1/3,1/2) 0 20 -2/7 1/2 3 4 -1/4 1/0 2 20 -2/9 1/6 1 20 -1/5 1/3 2 10 -1/6 1/2 4 4 0/1 1/2 1 20 1/4 1/2 2 4 2/7 1/2 1 20 1/3 1/1 2 10 4/11 1/2 1 4 3/8 3/4 2 20 2/5 5/6 1 20 5/12 11/12 2 20 3/7 1/1 12 2 1/2 (1/1,1/0) 0 20 2/3 1/0 1 4 3/4 1/0 2 20 7/9 0/1 2 10 11/14 1/2 4 4 4/5 1/2 1 20 9/11 1/1 4 2 5/6 (1/1,1/0) 0 20 1/1 1/1 2 10 3/2 1/0 4 4 5/3 -1/1 2 10 12/7 -1/2 1 20 19/11 0/1 4 2 7/4 1/0 2 20 2/1 1/2 1 20 9/4 3/4 2 20 7/3 1/1 6 2 5/2 (1/1,3/2) 0 20 13/5 2/1 2 10 8/3 3/2 1 20 3/1 2/1 2 10 10/3 7/2 1 20 17/5 4/1 2 10 7/2 (4/1,1/0) 0 20 4/1 1/0 7 4 1/0 1/0 2 20 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(3,2,-4,-3) (-1/1,-1/2) -> (-1/1,-1/2) Reflection Matrix(17,8,-36,-17) (-1/2,-4/9) -> (-1/2,-4/9) Reflection Matrix(37,16,104,45) (-4/9,-3/7) -> (1/3,4/11) Hyperbolic Matrix(105,44,136,57) (-3/7,-5/12) -> (3/4,7/9) Hyperbolic Matrix(69,28,32,13) (-5/12,-2/5) -> (2/1,9/4) Hyperbolic Matrix(47,18,-128,-49) (-2/5,-3/8) -> (-3/8,-4/11) Parabolic Matrix(217,78,64,23) (-4/11,-1/3) -> (10/3,17/5) Glide Reflection Matrix(145,44,56,17) (-1/3,-3/10) -> (5/2,13/5) Hyperbolic Matrix(41,12,-140,-41) (-3/10,-2/7) -> (-3/10,-2/7) Reflection Matrix(15,4,-56,-15) (-2/7,-1/4) -> (-2/7,-1/4) Reflection Matrix(53,12,128,29) (-1/4,-2/9) -> (2/5,5/12) Hyperbolic Matrix(77,16,24,5) (-2/9,-1/5) -> (3/1,10/3) Hyperbolic Matrix(97,18,124,23) (-1/5,-1/6) -> (7/9,11/14) Glide Reflection Matrix(35,4,44,5) (-1/6,0/1) -> (11/14,4/5) Glide Reflection Matrix(9,-2,32,-7) (0/1,1/4) -> (1/4,2/7) Parabolic Matrix(45,-14,16,-5) (2/7,1/3) -> (8/3,3/1) Glide Reflection Matrix(65,-24,176,-65) (4/11,3/8) -> (4/11,3/8) Reflection Matrix(51,-20,28,-11) (3/8,2/5) -> (7/4,2/1) Glide Reflection Matrix(71,-30,168,-71) (5/12,3/7) -> (5/12,3/7) Reflection Matrix(13,-6,28,-13) (3/7,1/2) -> (3/7,1/2) Reflection Matrix(7,-4,12,-7) (1/2,2/3) -> (1/2,2/3) Reflection Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(227,-184,132,-107) (4/5,9/11) -> (12/7,19/11) Glide Reflection Matrix(109,-90,132,-109) (9/11,5/6) -> (9/11,5/6) Reflection Matrix(109,-92,32,-27) (5/6,1/1) -> (17/5,7/2) Hyperbolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(115,-196,44,-75) (5/3,12/7) -> (13/5,8/3) Glide Reflection Matrix(153,-266,88,-153) (19/11,7/4) -> (19/11,7/4) Reflection Matrix(55,-126,24,-55) (9/4,7/3) -> (9/4,7/3) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) Reflection Matrix(-1,8,0,1) (4/1,1/0) -> (4/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(3,2,-4,-3) -> Matrix(1,0,4,-1) (-1/1,-1/2) -> (0/1,1/2) Matrix(17,8,-36,-17) -> Matrix(1,0,4,-1) (-1/2,-4/9) -> (0/1,1/2) Matrix(37,16,104,45) -> Matrix(3,-1,4,-1) 1/2 Matrix(105,44,136,57) -> Matrix(1,0,0,1) Matrix(69,28,32,13) -> Matrix(3,-1,4,-1) 1/2 Matrix(47,18,-128,-49) -> Matrix(3,-1,4,-1) 1/2 Matrix(217,78,64,23) -> Matrix(-1,4,0,1) *** -> (2/1,1/0) Matrix(145,44,56,17) -> Matrix(7,-2,4,-1) Matrix(41,12,-140,-41) -> Matrix(5,-2,12,-5) (-3/10,-2/7) -> (1/3,1/2) Matrix(15,4,-56,-15) -> Matrix(-1,1,0,1) (-2/7,-1/4) -> (1/2,1/0) Matrix(53,12,128,29) -> Matrix(11,-1,12,-1) Matrix(77,16,24,5) -> Matrix(11,-3,4,-1) Matrix(97,18,124,23) -> Matrix(3,-1,8,-3) *** -> (1/4,1/2) Matrix(35,4,44,5) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(9,-2,32,-7) -> Matrix(3,-1,4,-1) 1/2 Matrix(45,-14,16,-5) -> Matrix(7,-5,4,-3) Matrix(65,-24,176,-65) -> Matrix(5,-3,8,-5) (4/11,3/8) -> (1/2,3/4) Matrix(51,-20,28,-11) -> Matrix(1,-1,-4,3) Matrix(71,-30,168,-71) -> Matrix(23,-22,24,-23) (5/12,3/7) -> (11/12,1/1) Matrix(13,-6,28,-13) -> Matrix(-1,2,0,1) (3/7,1/2) -> (1/1,1/0) Matrix(7,-4,12,-7) -> Matrix(-1,2,0,1) (1/2,2/3) -> (1/1,1/0) Matrix(17,-12,24,-17) -> Matrix(-1,1,0,1) (2/3,3/4) -> (1/2,1/0) Matrix(227,-184,132,-107) -> Matrix(1,-1,-4,3) Matrix(109,-90,132,-109) -> Matrix(-1,2,0,1) (9/11,5/6) -> (1/1,1/0) Matrix(109,-92,32,-27) -> Matrix(1,3,0,1) 1/0 Matrix(13,-18,8,-11) -> Matrix(1,-2,0,1) 1/0 Matrix(115,-196,44,-75) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(153,-266,88,-153) -> Matrix(1,0,0,-1) (19/11,7/4) -> (0/1,1/0) Matrix(55,-126,24,-55) -> Matrix(7,-6,8,-7) (9/4,7/3) -> (3/4,1/1) Matrix(29,-70,12,-29) -> Matrix(5,-6,4,-5) (7/3,5/2) -> (1/1,3/2) Matrix(15,-56,4,-15) -> Matrix(-1,8,0,1) (7/2,4/1) -> (4/1,1/0) Matrix(-1,8,0,1) -> Matrix(-1,1,0,1) (4/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.