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 -4/9 -3/7 -3/8 -1/3 -2/7 -1/5 0/1 3/17 1/4 1/3 3/7 1/2 3/5 2/3 9/11 1/1 11/9 14/11 7/5 3/2 5/3 19/11 16/9 9/5 13/7 2/1 7/3 5/2 18/7 8/3 11/4 3/1 61/19 10/3 7/2 4/1 14/3 33/7 5/1 16/3 17/3 6/1 7/1 22/3 8/1 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 1/2 -5/11 1/2 5/8 -4/9 0/1 2/3 -11/25 1/2 5/8 -7/16 1/2 -10/23 2/3 1/1 -3/7 1/1 -8/19 3/4 1/1 -5/12 3/4 -2/5 1/2 1/1 -9/23 5/4 3/2 -7/18 3/2 -5/13 1/1 -8/21 0/1 1/1 -11/29 1/4 1/2 -3/8 1/1 -13/35 11/8 3/2 -10/27 5/3 2/1 -7/19 2/1 -4/11 0/1 1/0 -1/3 1/2 1/0 -6/19 -1/1 0/1 -5/16 -1/2 -4/13 0/1 1/2 -3/10 1/0 -2/7 0/1 -5/18 1/2 -3/11 1/1 -4/15 -1/1 1/0 -1/4 1/2 -2/9 0/1 1/2 -1/5 0/1 -3/16 1/0 -2/11 0/1 1/2 -3/17 3/2 1/0 -1/6 0/1 -1/7 1/2 1/0 -1/8 1/2 -1/9 0/1 0/1 0/1 1/1 1/7 0/1 1/6 1/0 3/17 0/1 2/11 0/1 1/3 1/5 1/2 1/0 3/14 1/4 2/9 1/2 1/1 3/13 1/1 1/4 0/1 5/19 1/3 4/15 1/2 2/3 3/11 1/2 3/4 8/29 7/8 1/1 5/18 5/4 2/7 0/1 1/0 1/3 0/1 5/14 1/2 4/11 0/1 2/3 3/8 1/2 5/13 1/2 1/0 2/5 0/1 1/0 5/12 1/0 3/7 0/1 7/16 1/6 18/41 1/4 4/15 11/25 1/3 4/9 0/1 1/3 1/2 1/0 7/13 0/1 6/11 0/1 1/7 5/9 1/4 1/2 4/7 1/3 1/2 11/19 1/2 7/12 1/2 17/29 1/2 5/8 10/17 2/3 1/1 3/5 1/1 2/3 0/1 5/7 1/3 13/18 3/8 8/11 2/5 1/2 11/15 3/8 1/2 3/4 1/2 7/9 0/1 4/5 0/1 1/0 9/11 0/1 14/17 0/1 1/8 5/6 1/4 1/1 1/2 1/0 9/8 7/12 8/7 2/3 15/13 2/3 7/6 3/4 6/5 3/4 1/1 11/9 1/1 5/4 1/0 19/15 3/2 1/0 14/11 0/1 2/1 37/29 3/2 1/0 23/18 1/0 9/7 0/1 4/3 0/1 1/2 19/14 1/2 15/11 1/2 5/8 11/8 1/2 29/21 1/2 18/13 1/2 4/7 25/18 11/18 7/5 2/3 24/17 2/3 1/1 17/12 1/2 27/19 1/2 5/8 10/7 2/3 5/7 3/2 1/1 14/9 2/1 3/1 25/16 1/0 61/39 1/0 36/23 -8/1 1/0 47/30 -7/2 11/7 -1/2 1/0 8/5 0/1 1/1 37/23 0/1 29/18 1/4 21/13 1/4 1/2 13/8 1/2 31/19 1/2 18/11 1/2 3/5 5/3 1/1 17/10 1/2 12/7 1/2 2/3 31/18 5/8 19/11 2/3 45/26 1/2 26/15 2/3 1/1 7/4 3/4 16/9 2/3 4/5 25/14 3/4 9/5 3/4 5/6 20/11 6/7 7/8 11/6 11/12 13/7 1/1 15/8 13/12 2/1 1/1 1/0 9/4 1/2 7/3 1/1 19/8 9/8 31/13 8/7 43/18 7/6 12/5 1/1 6/5 41/17 37/30 5/4 29/12 5/4 17/7 4/3 5/2 3/2 23/9 2/1 18/7 2/1 49/19 2/1 31/12 7/2 13/5 1/2 1/0 21/8 1/2 29/11 1/1 8/3 1/1 3/2 19/7 3/2 1/0 11/4 2/1 25/9 7/2 1/0 14/5 4/1 1/0 3/1 1/1 16/5 10/7 3/2 61/19 3/2 45/14 3/2 29/9 3/2 25/16 13/4 7/4 10/3 1/1 2/1 17/5 3/2 7/4 7/2 1/0 4/1 0/1 2/1 9/2 1/0 32/7 1/1 2/1 23/5 2/1 14/3 5/3 2/1 33/7 2/1 19/4 9/4 5/1 5/2 1/0 16/3 2/1 1/0 11/2 1/0 17/3 1/0 23/4 1/0 6/1 0/1 1/0 7/1 2/1 22/3 2/1 15/2 11/4 8/1 4/1 1/0 9/1 1/0 1/0 1/0 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(151,70,110,51) (-1/2,-5/11) -> (15/11,11/8) Hyperbolic Matrix(143,64,-324,-145) (-5/11,-4/9) -> (-4/9,-11/25) Parabolic Matrix(857,376,604,265) (-11/25,-7/16) -> (17/12,27/19) Hyperbolic Matrix(211,92,-672,-293) (-7/16,-10/23) -> (-6/19,-5/16) Hyperbolic Matrix(281,122,638,277) (-10/23,-3/7) -> (11/25,4/9) Hyperbolic Matrix(71,30,310,131) (-3/7,-8/19) -> (2/9,3/13) Hyperbolic Matrix(205,86,174,73) (-8/19,-5/12) -> (7/6,6/5) Hyperbolic Matrix(69,28,32,13) (-5/12,-2/5) -> (2/1,9/4) Hyperbolic Matrix(199,78,722,283) (-2/5,-9/23) -> (3/11,8/29) Hyperbolic Matrix(457,178,362,141) (-9/23,-7/18) -> (5/4,19/15) Hyperbolic Matrix(259,100,360,139) (-7/18,-5/13) -> (5/7,13/18) Hyperbolic Matrix(193,74,326,125) (-5/13,-8/21) -> (10/17,3/5) Hyperbolic Matrix(195,74,-614,-233) (-8/21,-11/29) -> (-1/3,-6/19) Hyperbolic Matrix(191,72,-512,-193) (-11/29,-3/8) -> (-3/8,-13/35) Parabolic Matrix(1527,566,634,235) (-13/35,-10/27) -> (12/5,41/17) Hyperbolic Matrix(887,328,192,71) (-10/27,-7/19) -> (23/5,14/3) Hyperbolic Matrix(191,70,30,11) (-7/19,-4/11) -> (6/1,7/1) Hyperbolic Matrix(61,22,158,57) (-4/11,-1/3) -> (5/13,2/5) Hyperbolic Matrix(289,90,350,109) (-5/16,-4/13) -> (14/17,5/6) Hyperbolic Matrix(59,18,-318,-97) (-4/13,-3/10) -> (-3/16,-2/11) 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(273,74,166,45) (-3/11,-4/15) -> (18/11,5/3) Hyperbolic Matrix(53,14,246,65) (-4/15,-1/4) -> (3/14,2/9) Hyperbolic Matrix(53,12,128,29) (-1/4,-2/9) -> (2/5,5/12) Hyperbolic Matrix(101,22,78,17) (-2/9,-1/5) -> (9/7,4/3) Hyperbolic Matrix(591,112,248,47) (-1/5,-3/16) -> (19/8,31/13) Hyperbolic Matrix(439,78,242,43) (-2/11,-3/17) -> (9/5,20/11) Hyperbolic Matrix(337,58,122,21) (-3/17,-1/6) -> (11/4,25/9) Hyperbolic Matrix(191,30,70,11) (-1/6,-1/7) -> (19/7,11/4) Hyperbolic Matrix(371,50,230,31) (-1/7,-1/8) -> (29/18,21/13) Hyperbolic Matrix(183,22,158,19) (-1/8,-1/9) -> (15/13,7/6) Hyperbolic Matrix(311,32,68,7) (-1/9,0/1) -> (32/7,23/5) Hyperbolic Matrix(175,-24,124,-17) (0/1,1/7) -> (7/5,24/17) Hyperbolic Matrix(173,-28,68,-11) (1/7,1/6) -> (5/2,23/9) Hyperbolic Matrix(715,-124,444,-77) (1/6,3/17) -> (37/23,29/18) Hyperbolic Matrix(543,-98,338,-61) (3/17,2/11) -> (8/5,37/23) Hyperbolic Matrix(237,-44,404,-75) (2/11,1/5) -> (17/29,10/17) Hyperbolic Matrix(165,-34,34,-7) (1/5,3/14) -> (19/4,5/1) Hyperbolic Matrix(33,-8,128,-31) (3/13,1/4) -> (1/4,5/19) Parabolic Matrix(507,-134,1154,-305) (5/19,4/15) -> (18/41,11/25) Hyperbolic Matrix(251,-68,48,-13) (4/15,3/11) -> (5/1,16/3) Hyperbolic Matrix(471,-130,250,-69) (8/29,5/18) -> (15/8,2/1) Hyperbolic Matrix(499,-140,360,-101) (5/18,2/7) -> (18/13,25/18) Hyperbolic Matrix(61,-18,78,-23) (2/7,1/3) -> (7/9,4/5) Hyperbolic Matrix(325,-114,134,-47) (1/3,5/14) -> (29/12,17/7) Hyperbolic Matrix(499,-180,280,-101) (5/14,4/11) -> (16/9,25/14) 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(2503,-1098,1598,-701) (7/16,18/41) -> (36/23,47/30) Hyperbolic Matrix(137,-62,42,-19) (4/9,1/2) -> (13/4,10/3) Hyperbolic Matrix(313,-166,66,-35) (1/2,7/13) -> (33/7,19/4) Hyperbolic Matrix(545,-296,116,-63) (7/13,6/11) -> (14/3,33/7) Hyperbolic Matrix(387,-212,272,-149) (6/11,5/9) -> (27/19,10/7) Hyperbolic Matrix(205,-116,76,-43) (5/9,4/7) -> (8/3,19/7) Hyperbolic Matrix(559,-322,342,-197) (4/7,11/19) -> (31/19,18/11) Hyperbolic Matrix(557,-324,404,-235) (11/19,7/12) -> (11/8,29/21) Hyperbolic Matrix(1111,-650,870,-509) (7/12,17/29) -> (37/29,23/18) Hyperbolic 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(485,-354,174,-127) (8/11,11/15) -> (25/9,14/5) Hyperbolic Matrix(345,-254,254,-187) (11/15,3/4) -> (19/14,15/11) Hyperbolic Matrix(113,-86,46,-35) (3/4,7/9) -> (17/7,5/2) Hyperbolic Matrix(199,-162,242,-197) (4/5,9/11) -> (9/11,14/17) Parabolic Matrix(109,-92,32,-27) (5/6,1/1) -> (17/5,7/2) Hyperbolic Matrix(135,-146,86,-93) (1/1,9/8) -> (47/30,11/7) Hyperbolic Matrix(281,-318,38,-43) (9/8,8/7) -> (22/3,15/2) Hyperbolic Matrix(335,-386,46,-53) (8/7,15/13) -> (7/1,22/3) Hyperbolic Matrix(217,-262,82,-99) (6/5,11/9) -> (29/11,8/3) Hyperbolic Matrix(305,-376,116,-143) (11/9,5/4) -> (21/8,29/11) Hyperbolic Matrix(617,-784,484,-615) (19/15,14/11) -> (14/11,37/29) Parabolic Matrix(859,-1100,360,-461) (23/18,9/7) -> (31/13,43/18) Hyperbolic Matrix(153,-206,26,-35) (4/3,19/14) -> (23/4,6/1) Hyperbolic Matrix(285,-394,34,-47) (29/21,18/13) -> (8/1,9/1) Hyperbolic Matrix(1037,-1442,402,-559) (25/18,7/5) -> (49/19,31/12) Hyperbolic Matrix(1077,-1522,622,-879) (24/17,17/12) -> (45/26,26/15) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(579,-902,242,-377) (14/9,25/16) -> (43/18,12/5) Hyperbolic Matrix(2731,-4270,850,-1329) (25/16,61/39) -> (61/19,45/14) Hyperbolic Matrix(2027,-3172,632,-989) (61/39,36/23) -> (16/5,61/19) Hyperbolic Matrix(155,-246,46,-73) (11/7,8/5) -> (10/3,17/5) Hyperbolic Matrix(397,-642,222,-359) (21/13,13/8) -> (25/14,9/5) Hyperbolic Matrix(91,-148,8,-13) (13/8,31/19) -> (9/1,1/0) Hyperbolic Matrix(237,-404,44,-75) (17/10,12/7) -> (16/3,11/2) Hyperbolic Matrix(837,-1444,484,-835) (31/18,19/11) -> (19/11,45/26) Parabolic Matrix(263,-458,58,-101) (26/15,7/4) -> (9/2,32/7) Hyperbolic Matrix(213,-388,28,-51) (20/11,11/6) -> (15/2,8/1) Hyperbolic Matrix(183,-338,98,-181) (11/6,13/7) -> (13/7,15/8) Parabolic Matrix(85,-196,36,-83) (9/4,7/3) -> (7/3,19/8) Parabolic Matrix(1113,-2686,346,-835) (41/17,29/12) -> (45/14,29/9) Hyperbolic Matrix(505,-1296,196,-503) (23/9,18/7) -> (18/7,49/19) Parabolic Matrix(413,-1068,128,-331) (31/12,13/5) -> (29/9,13/4) Hyperbolic Matrix(31,-90,10,-29) (14/5,3/1) -> (3/1,16/5) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic Matrix(103,-578,18,-101) (11/2,17/3) -> (17/3,23/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(151,70,110,51) -> Matrix(1,0,0,1) Matrix(143,64,-324,-145) -> Matrix(1,0,0,1) Matrix(857,376,604,265) -> Matrix(1,0,0,1) Matrix(211,92,-672,-293) -> Matrix(3,-2,-4,3) Matrix(281,122,638,277) -> Matrix(3,-2,8,-5) Matrix(71,30,310,131) -> Matrix(3,-2,2,-1) Matrix(205,86,174,73) -> Matrix(1,0,0,1) Matrix(69,28,32,13) -> Matrix(3,-2,2,-1) Matrix(199,78,722,283) -> Matrix(5,-6,6,-7) Matrix(457,178,362,141) -> Matrix(1,-2,2,-3) Matrix(259,100,360,139) -> Matrix(1,0,2,1) Matrix(193,74,326,125) -> Matrix(1,-2,2,-3) Matrix(195,74,-614,-233) -> Matrix(1,0,-2,1) Matrix(191,72,-512,-193) -> Matrix(5,-4,4,-3) Matrix(1527,566,634,235) -> Matrix(17,-28,14,-23) Matrix(887,328,192,71) -> Matrix(1,0,0,1) Matrix(191,70,30,11) -> Matrix(1,0,0,1) Matrix(61,22,158,57) -> Matrix(1,0,0,1) Matrix(289,90,350,109) -> Matrix(1,0,6,1) Matrix(59,18,-318,-97) -> Matrix(1,0,0,1) Matrix(55,16,-196,-57) -> Matrix(1,0,2,1) Matrix(277,76,164,45) -> Matrix(1,0,0,1) Matrix(273,74,166,45) -> Matrix(1,-2,2,-3) Matrix(53,14,246,65) -> Matrix(1,0,2,1) Matrix(53,12,128,29) -> Matrix(1,0,-2,1) Matrix(101,22,78,17) -> Matrix(1,0,0,1) Matrix(591,112,248,47) -> Matrix(9,-8,8,-7) Matrix(439,78,242,43) -> Matrix(5,-6,6,-7) Matrix(337,58,122,21) -> Matrix(1,2,0,1) Matrix(191,30,70,11) -> Matrix(3,-2,2,-1) Matrix(371,50,230,31) -> Matrix(1,0,2,1) Matrix(183,22,158,19) -> Matrix(1,-2,2,-3) Matrix(311,32,68,7) -> Matrix(3,-2,2,-1) Matrix(175,-24,124,-17) -> Matrix(1,-2,2,-3) Matrix(173,-28,68,-11) -> Matrix(3,-2,2,-1) Matrix(715,-124,444,-77) -> Matrix(1,0,4,1) Matrix(543,-98,338,-61) -> Matrix(1,0,-2,1) Matrix(237,-44,404,-75) -> Matrix(5,-2,8,-3) Matrix(165,-34,34,-7) -> Matrix(1,2,0,1) Matrix(33,-8,128,-31) -> Matrix(1,0,2,1) Matrix(507,-134,1154,-305) -> Matrix(5,-2,18,-7) Matrix(251,-68,48,-13) -> Matrix(7,-4,2,-1) Matrix(471,-130,250,-69) -> Matrix(9,-8,8,-7) Matrix(499,-140,360,-101) -> Matrix(1,-4,2,-7) Matrix(61,-18,78,-23) -> Matrix(1,0,0,1) Matrix(325,-114,134,-47) -> Matrix(13,-4,10,-3) Matrix(499,-180,280,-101) -> Matrix(1,-2,2,-3) Matrix(205,-76,116,-43) -> Matrix(1,-2,2,-3) Matrix(377,-144,144,-55) -> Matrix(1,0,0,1) Matrix(85,-36,196,-83) -> Matrix(1,0,6,1) Matrix(2503,-1098,1598,-701) -> Matrix(17,-4,-4,1) Matrix(137,-62,42,-19) -> Matrix(7,-2,4,-1) Matrix(313,-166,66,-35) -> Matrix(9,2,4,1) Matrix(545,-296,116,-63) -> Matrix(19,-2,10,-1) Matrix(387,-212,272,-149) -> Matrix(9,-2,14,-3) Matrix(205,-116,76,-43) -> Matrix(7,-2,4,-1) Matrix(559,-322,342,-197) -> Matrix(9,-4,16,-7) Matrix(557,-324,404,-235) -> Matrix(1,0,0,1) Matrix(1111,-650,870,-509) -> Matrix(7,-4,2,-1) Matrix(25,-16,36,-23) -> Matrix(1,0,2,1) Matrix(557,-404,324,-235) -> Matrix(9,-4,16,-7) Matrix(485,-354,174,-127) -> Matrix(3,-2,2,-1) Matrix(345,-254,254,-187) -> Matrix(9,-4,16,-7) Matrix(113,-86,46,-35) -> Matrix(5,-4,4,-3) Matrix(199,-162,242,-197) -> Matrix(1,0,8,1) Matrix(109,-92,32,-27) -> Matrix(7,-2,4,-1) Matrix(135,-146,86,-93) -> Matrix(1,0,-2,1) Matrix(281,-318,38,-43) -> Matrix(19,-12,8,-5) Matrix(335,-386,46,-53) -> Matrix(7,-4,2,-1) Matrix(217,-262,82,-99) -> Matrix(7,-6,6,-5) Matrix(305,-376,116,-143) -> Matrix(1,-2,2,-3) Matrix(617,-784,484,-615) -> Matrix(1,0,0,1) Matrix(859,-1100,360,-461) -> Matrix(7,8,6,7) Matrix(153,-206,26,-35) -> Matrix(1,0,-2,1) Matrix(285,-394,34,-47) -> Matrix(15,-8,2,-1) Matrix(1037,-1442,402,-559) -> Matrix(19,-12,8,-5) Matrix(1077,-1522,622,-879) -> Matrix(1,0,0,1) Matrix(49,-72,32,-47) -> Matrix(5,-4,4,-3) Matrix(579,-902,242,-377) -> Matrix(7,-20,6,-17) Matrix(2731,-4270,850,-1329) -> Matrix(3,-56,2,-37) Matrix(2027,-3172,632,-989) -> Matrix(3,34,2,23) Matrix(155,-246,46,-73) -> Matrix(3,-2,2,-1) Matrix(397,-642,222,-359) -> Matrix(11,-4,14,-5) Matrix(91,-148,8,-13) -> Matrix(3,-2,2,-1) Matrix(237,-404,44,-75) -> Matrix(7,-4,2,-1) Matrix(837,-1444,484,-835) -> Matrix(13,-8,18,-11) Matrix(263,-458,58,-101) -> Matrix(5,-4,4,-3) Matrix(213,-388,28,-51) -> Matrix(25,-22,8,-7) Matrix(183,-338,98,-181) -> Matrix(25,-24,24,-23) Matrix(85,-196,36,-83) -> Matrix(11,-10,10,-9) Matrix(1113,-2686,346,-835) -> Matrix(95,-118,62,-77) Matrix(505,-1296,196,-503) -> Matrix(5,-8,2,-3) Matrix(413,-1068,128,-331) -> Matrix(3,-14,2,-9) Matrix(31,-90,10,-29) -> Matrix(3,-2,2,-1) Matrix(17,-64,4,-15) -> Matrix(1,0,0,1) Matrix(103,-578,18,-101) -> Matrix(1,-10,0,1) 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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 29 Degree of the the map X: 29 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 1 1 0/1 (0/1,1/1) 0 20 1/7 0/1 1 5 1/6 1/0 1 20 3/17 0/1 3 1 2/11 (0/1,1/3) 0 20 1/5 0 10 2/9 (1/2,1/1) 0 20 1/4 0/1 1 4 4/15 (1/2,2/3) 0 20 3/11 0 10 5/18 5/4 1 20 2/7 (0/1,1/0) 0 20 1/3 0/1 1 5 5/14 1/2 1 20 4/11 0 4 3/8 1/2 1 20 2/5 (0/1,1/0) 0 20 3/7 0/1 3 2 4/9 (0/1,1/3) 0 20 1/2 1/0 1 20 7/13 0/1 7 1 6/11 (0/1,1/7) 0 20 5/9 0 10 4/7 (1/3,1/2) 0 20 11/19 1/2 5 2 7/12 1/2 1 20 3/5 1/1 1 5 2/3 0/1 2 4 5/7 1/3 1 5 8/11 (2/5,1/2) 0 20 11/15 0 10 3/4 1/2 1 20 7/9 0/1 1 5 4/5 (0/1,1/0) 0 20 9/11 0/1 4 1 5/6 1/4 1 20 1/1 0 10 6/5 (3/4,1/1) 0 20 11/9 1/1 4 1 5/4 1/0 1 20 4/3 (0/1,1/2) 0 20 11/8 1/2 1 20 29/21 1/2 5 2 18/13 (1/2,4/7) 0 20 25/18 11/18 1 20 7/5 2/3 1 5 24/17 (2/3,1/1) 0 20 17/12 1/2 1 20 10/7 (2/3,5/7) 0 20 3/2 1/1 2 4 14/9 (2/1,3/1) 0 20 25/16 1/0 1 20 61/39 1/0 15 1 36/23 (-8/1,1/0) 0 20 11/7 0 10 8/5 (0/1,1/1) 0 20 21/13 0 10 13/8 1/2 1 20 5/3 1/1 1 5 12/7 (1/2,2/3) 0 20 19/11 2/3 1 2 26/15 (2/3,1/1) 0 20 7/4 3/4 1 20 16/9 0 4 25/14 3/4 1 20 9/5 0 10 11/6 11/12 1 20 13/7 1/1 12 1 2/1 (1/1,1/0) 0 20 7/3 1/1 5 2 12/5 (1/1,6/5) 0 20 29/12 5/4 1 20 17/7 4/3 1 5 5/2 3/2 1 20 23/9 2/1 1 5 18/7 2/1 2 4 49/19 2/1 1 5 31/12 7/2 1 20 13/5 0 10 8/3 (1/1,3/2) 0 20 11/4 2/1 1 4 14/5 (4/1,1/0) 0 20 3/1 1/1 1 5 16/5 (10/7,3/2) 0 20 29/9 0 10 13/4 7/4 1 20 10/3 (1/1,2/1) 0 20 17/5 0 10 7/2 1/0 1 20 4/1 0 4 9/2 1/0 1 20 14/3 (5/3,2/1) 0 20 5/1 0 10 11/2 1/0 1 20 17/3 1/0 5 1 6/1 (0/1,1/0) 0 20 1/0 1/0 1 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(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(175,-24,124,-17) (0/1,1/7) -> (7/5,24/17) Hyperbolic Matrix(173,-28,68,-11) (1/7,1/6) -> (5/2,23/9) Hyperbolic Matrix(35,-6,204,-35) (1/6,3/17) -> (1/6,3/17) Reflection Matrix(67,-12,374,-67) (3/17,2/11) -> (3/17,2/11) Reflection Matrix(271,-50,168,-31) (2/11,1/5) -> (8/5,21/13) Glide Reflection Matrix(65,-14,116,-25) (1/5,2/9) -> (5/9,4/7) Glide Reflection Matrix(131,-30,48,-11) (2/9,1/4) -> (8/3,11/4) Glide Reflection Matrix(221,-58,80,-21) (1/4,4/15) -> (11/4,14/5) Glide Reflection Matrix(253,-68,346,-93) (4/15,3/11) -> (8/11,11/15) Glide Reflection Matrix(283,-78,156,-43) (3/11,5/18) -> (9/5,11/6) Glide Reflection Matrix(499,-140,360,-101) (5/18,2/7) -> (18/13,25/18) Hyperbolic Matrix(61,-18,78,-23) (2/7,1/3) -> (7/9,4/5) Hyperbolic Matrix(325,-114,134,-47) (1/3,5/14) -> (29/12,17/7) Hyperbolic Matrix(499,-180,280,-101) (5/14,4/11) -> (16/9,25/14) Hyperbolic Matrix(205,-76,116,-43) (4/11,3/8) -> (7/4,16/9) Hyperbolic Matrix(57,-22,44,-17) (3/8,2/5) -> (5/4,4/3) Glide Reflection Matrix(29,-12,70,-29) (2/5,3/7) -> (2/5,3/7) Reflection Matrix(55,-24,126,-55) (3/7,4/9) -> (3/7,4/9) Reflection Matrix(137,-62,42,-19) (4/9,1/2) -> (13/4,10/3) Hyperbolic Matrix(27,-14,52,-27) (1/2,7/13) -> (1/2,7/13) Reflection Matrix(155,-84,286,-155) (7/13,6/11) -> (7/13,6/11) Reflection Matrix(181,-100,38,-21) (6/11,5/9) -> (14/3,5/1) Glide Reflection Matrix(153,-88,266,-153) (4/7,11/19) -> (4/7,11/19) Reflection Matrix(557,-324,404,-235) (11/19,7/12) -> (11/8,29/21) Hyperbolic Matrix(125,-74,76,-45) (7/12,3/5) -> (13/8,5/3) Glide Reflection Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(139,-100,82,-59) (5/7,8/11) -> (5/3,12/7) Glide Reflection Matrix(185,-136,34,-25) (11/15,3/4) -> (5/1,11/2) Glide Reflection Matrix(113,-86,46,-35) (3/4,7/9) -> (17/7,5/2) Hyperbolic Matrix(89,-72,110,-89) (4/5,9/11) -> (4/5,9/11) 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(73,-86,28,-33) (1/1,6/5) -> (13/5,8/3) Glide Reflection Matrix(109,-132,90,-109) (6/5,11/9) -> (6/5,11/9) Reflection Matrix(89,-110,72,-89) (11/9,5/4) -> (11/9,5/4) Reflection Matrix(51,-70,8,-11) (4/3,11/8) -> (6/1,1/0) Glide Reflection Matrix(755,-1044,546,-755) (29/21,18/13) -> (29/21,18/13) Reflection Matrix(1037,-1442,402,-559) (25/18,7/5) -> (49/19,31/12) Hyperbolic Matrix(431,-610,248,-351) (24/17,17/12) -> (26/15,7/4) Glide Reflection Matrix(231,-328,50,-71) (17/12,10/7) -> (9/2,14/3) Glide Reflection Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(453,-706,188,-293) (14/9,25/16) -> (12/5,29/12) Glide Reflection Matrix(1951,-3050,1248,-1951) (25/16,61/39) -> (25/16,61/39) Reflection Matrix(2807,-4392,1794,-2807) (61/39,36/23) -> (61/39,36/23) Reflection Matrix(779,-1220,242,-379) (36/23,11/7) -> (16/5,29/9) Glide Reflection Matrix(155,-246,46,-73) (11/7,8/5) -> (10/3,17/5) Hyperbolic Matrix(397,-642,222,-359) (21/13,13/8) -> (25/14,9/5) Hyperbolic Matrix(265,-456,154,-265) (12/7,19/11) -> (12/7,19/11) Reflection Matrix(571,-988,330,-571) (19/11,26/15) -> (19/11,26/15) Reflection Matrix(155,-286,84,-155) (11/6,13/7) -> (11/6,13/7) Reflection Matrix(27,-52,14,-27) (13/7,2/1) -> (13/7,2/1) Reflection Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(505,-1296,196,-503) (23/9,18/7) -> (18/7,49/19) Parabolic Matrix(413,-1068,128,-331) (31/12,13/5) -> (29/9,13/4) Hyperbolic Matrix(31,-90,10,-29) (14/5,3/1) -> (3/1,16/5) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic Matrix(67,-374,12,-67) (11/2,17/3) -> (11/2,17/3) Reflection Matrix(35,-204,6,-35) (17/3,6/1) -> (17/3,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(-1,0,2,1) -> Matrix(1,0,2,-1) (-1/1,0/1) -> (0/1,1/1) Matrix(175,-24,124,-17) -> Matrix(1,-2,2,-3) 1/1 Matrix(173,-28,68,-11) -> Matrix(3,-2,2,-1) 1/1 Matrix(35,-6,204,-35) -> Matrix(1,0,0,-1) (1/6,3/17) -> (0/1,1/0) Matrix(67,-12,374,-67) -> Matrix(1,0,6,-1) (3/17,2/11) -> (0/1,1/3) Matrix(271,-50,168,-31) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(65,-14,116,-25) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(131,-30,48,-11) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(221,-58,80,-21) -> Matrix(5,-2,2,-1) Matrix(253,-68,346,-93) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(283,-78,156,-43) -> Matrix(7,-6,8,-7) *** -> (3/4,1/1) Matrix(499,-140,360,-101) -> Matrix(1,-4,2,-7) Matrix(61,-18,78,-23) -> Matrix(1,0,0,1) Matrix(325,-114,134,-47) -> Matrix(13,-4,10,-3) Matrix(499,-180,280,-101) -> Matrix(1,-2,2,-3) 1/1 Matrix(205,-76,116,-43) -> Matrix(1,-2,2,-3) 1/1 Matrix(57,-22,44,-17) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(29,-12,70,-29) -> Matrix(1,0,0,-1) (2/5,3/7) -> (0/1,1/0) Matrix(55,-24,126,-55) -> Matrix(1,0,6,-1) (3/7,4/9) -> (0/1,1/3) Matrix(137,-62,42,-19) -> Matrix(7,-2,4,-1) Matrix(27,-14,52,-27) -> Matrix(1,0,0,-1) (1/2,7/13) -> (0/1,1/0) Matrix(155,-84,286,-155) -> Matrix(1,0,14,-1) (7/13,6/11) -> (0/1,1/7) Matrix(181,-100,38,-21) -> Matrix(9,-2,4,-1) Matrix(153,-88,266,-153) -> Matrix(5,-2,12,-5) (4/7,11/19) -> (1/3,1/2) Matrix(557,-324,404,-235) -> Matrix(1,0,0,1) Matrix(125,-74,76,-45) -> Matrix(3,-2,4,-3) *** -> (1/2,1/1) Matrix(25,-16,36,-23) -> Matrix(1,0,2,1) 0/1 Matrix(139,-100,82,-59) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(185,-136,34,-25) -> Matrix(9,-4,2,-1) Matrix(113,-86,46,-35) -> Matrix(5,-4,4,-3) 1/1 Matrix(89,-72,110,-89) -> Matrix(1,0,0,-1) (4/5,9/11) -> (0/1,1/0) Matrix(109,-90,132,-109) -> Matrix(1,0,8,-1) (9/11,5/6) -> (0/1,1/4) Matrix(109,-92,32,-27) -> Matrix(7,-2,4,-1) Matrix(73,-86,28,-33) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(109,-132,90,-109) -> Matrix(7,-6,8,-7) (6/5,11/9) -> (3/4,1/1) Matrix(89,-110,72,-89) -> Matrix(-1,2,0,1) (11/9,5/4) -> (1/1,1/0) Matrix(51,-70,8,-11) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(755,-1044,546,-755) -> Matrix(15,-8,28,-15) (29/21,18/13) -> (1/2,4/7) Matrix(1037,-1442,402,-559) -> Matrix(19,-12,8,-5) Matrix(431,-610,248,-351) -> Matrix(5,-4,6,-5) *** -> (2/3,1/1) Matrix(231,-328,50,-71) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(49,-72,32,-47) -> Matrix(5,-4,4,-3) 1/1 Matrix(453,-706,188,-293) -> Matrix(5,-16,4,-13) Matrix(1951,-3050,1248,-1951) -> Matrix(-1,14,0,1) (25/16,61/39) -> (7/1,1/0) Matrix(2807,-4392,1794,-2807) -> Matrix(1,16,0,-1) (61/39,36/23) -> (-8/1,1/0) Matrix(779,-1220,242,-379) -> Matrix(3,14,2,9) Matrix(155,-246,46,-73) -> Matrix(3,-2,2,-1) 1/1 Matrix(397,-642,222,-359) -> Matrix(11,-4,14,-5) Matrix(265,-456,154,-265) -> Matrix(7,-4,12,-7) (12/7,19/11) -> (1/2,2/3) Matrix(571,-988,330,-571) -> Matrix(5,-4,6,-5) (19/11,26/15) -> (2/3,1/1) Matrix(155,-286,84,-155) -> Matrix(23,-22,24,-23) (11/6,13/7) -> (11/12,1/1) Matrix(27,-52,14,-27) -> Matrix(-1,2,0,1) (13/7,2/1) -> (1/1,1/0) Matrix(13,-28,6,-13) -> Matrix(-1,2,0,1) (2/1,7/3) -> (1/1,1/0) Matrix(71,-168,30,-71) -> Matrix(11,-12,10,-11) (7/3,12/5) -> (1/1,6/5) Matrix(505,-1296,196,-503) -> Matrix(5,-8,2,-3) 2/1 Matrix(413,-1068,128,-331) -> Matrix(3,-14,2,-9) Matrix(31,-90,10,-29) -> Matrix(3,-2,2,-1) 1/1 Matrix(17,-64,4,-15) -> Matrix(1,0,0,1) Matrix(67,-374,12,-67) -> Matrix(-1,10,0,1) (11/2,17/3) -> (5/1,1/0) Matrix(35,-204,6,-35) -> Matrix(1,0,0,-1) (17/3,6/1) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.