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): 720 Minimal number of generators: 121 Number of equivalence classes of cusps: 50 Genus: 36 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -9/20 -7/18 -1/3 -5/16 -3/14 0/1 2/9 5/17 1/3 3/8 1/2 4/7 9/13 23/32 5/6 1/1 6/5 37/29 13/9 3/2 5/3 7/4 2/1 15/7 11/5 23/10 7/3 5/2 8/3 3/1 17/5 7/2 69/19 15/4 87/23 4/1 9/2 19/4 5/1 11/2 6/1 19/3 13/2 7/1 15/2 8/1 9/1 10/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/2 -7/15 1/1 -6/13 1/3 -5/11 1/3 -9/20 1/2 -13/29 1/1 -4/9 1/1 -11/25 1/3 -7/16 0/1 1/0 -10/23 1/5 -13/30 0/1 1/3 -3/7 1/3 -5/12 1/2 1/1 -2/5 1/3 -7/18 1/2 -12/31 3/5 -17/44 3/5 2/3 -5/13 1/1 -3/8 1/2 1/1 -10/27 1/1 -17/46 0/1 1/2 -7/19 1/1 -11/30 0/1 1/2 -4/11 1/3 -9/25 1/1 -14/39 1/1 -5/14 0/1 1/2 -1/3 1/1 -5/16 0/1 -9/29 1/3 -4/13 1/1 -3/10 1/2 1/1 -2/7 1/1 -5/18 -2/1 1/0 -8/29 -1/1 -11/40 -1/2 0/1 -14/51 1/1 -3/11 -1/1 -1/4 0/1 1/2 -6/25 1/1 -5/21 1/1 -4/17 1/3 -3/13 1/1 -2/9 1/1 -3/14 0/1 -4/19 1/5 -5/24 0/1 1/4 -1/5 1/3 -1/6 1/2 1/1 -1/7 1/1 0/1 1/1 1/8 3/2 2/1 1/7 3/1 1/6 -1/1 1/0 2/11 -1/1 1/5 1/1 4/19 1/1 3/14 0/1 1/1 2/9 1/1 5/22 1/1 2/1 3/13 1/1 1/4 1/1 1/0 3/11 1/1 5/18 1/1 1/0 7/25 1/1 2/7 3/1 5/17 1/0 3/10 -4/1 1/0 4/13 -1/1 5/16 -1/1 1/0 1/3 -1/1 3/8 0/1 5/13 1/3 2/5 1/1 7/17 1/1 5/12 2/3 1/1 3/7 1/1 4/9 -1/1 9/20 0/1 1/0 14/31 1/1 5/11 1/1 1/2 0/1 1/0 5/9 -1/3 4/7 0/1 11/19 1/9 18/31 1/7 7/12 0/1 1/6 3/5 1/3 11/18 0/1 1/4 19/31 1/3 8/13 1/3 5/8 2/5 1/2 7/11 1/1 9/14 1/2 1/1 2/3 1/1 9/13 1/1 7/10 1/1 4/3 26/37 1/1 19/27 1/1 12/17 1/1 5/7 1/1 23/32 1/1 18/25 1/1 13/18 3/2 2/1 8/11 3/1 3/4 1/1 1/0 10/13 -1/1 7/9 -1/1 11/14 0/1 1/2 15/19 1/1 4/5 1/1 5/6 0/1 6/7 1/3 13/15 1/1 7/8 0/1 1/2 1/1 1/1 7/6 2/1 1/0 6/5 1/0 17/14 -2/1 1/0 11/9 -1/1 5/4 0/1 1/0 24/19 1/1 43/34 1/1 19/15 1/1 14/11 1/1 37/29 1/0 60/47 -1/1 23/18 0/1 1/0 9/7 1/1 13/10 1/1 1/0 17/13 1/1 4/3 -1/1 15/11 1/1 41/30 0/1 1/1 26/19 1/1 11/8 0/1 1/0 29/21 1/3 18/13 1/1 25/18 0/1 1/2 7/5 1/1 17/12 0/1 1/0 44/31 -1/1 27/19 1/1 37/26 0/1 1/0 10/7 1/1 13/9 1/1 16/11 1/1 3/2 1/1 1/0 11/7 3/1 19/12 6/1 1/0 27/17 1/0 8/5 -1/1 21/13 1/1 55/34 2/1 1/0 34/21 1/0 47/29 1/1 13/8 1/1 1/0 44/27 1/1 31/19 1/1 18/11 3/1 41/25 1/0 23/14 -3/1 1/0 5/3 1/1 7/4 1/0 9/5 -3/1 11/6 -2/1 -1/1 13/7 -1/1 2/1 -1/1 15/7 0/1 13/6 0/1 1/6 11/5 1/3 20/9 1/3 29/13 1/3 9/4 0/1 1/2 16/7 1/1 23/10 1/1 7/3 1/1 19/8 2/3 1/1 31/13 1/1 12/5 1/1 5/2 1/1 1/0 13/5 3/1 8/3 1/0 19/7 -3/1 49/18 -3/1 1/0 79/29 -3/1 30/11 -3/1 41/15 -3/1 11/4 -2/1 1/0 25/9 -1/1 14/5 -1/1 31/11 -1/1 17/6 -1/1 1/0 3/1 -1/1 19/6 0/1 1/0 35/11 1/1 51/16 1/1 1/0 16/5 -1/1 29/9 -1/1 13/4 -1/1 0/1 23/7 -1/3 33/10 -1/2 0/1 10/3 -1/3 17/5 0/1 24/7 1/7 7/2 0/1 1/2 32/9 1/1 25/7 1/1 43/12 1/1 18/5 1/1 29/8 3/1 1/0 69/19 1/0 40/11 -3/1 11/3 -1/1 15/4 0/1 1/2 34/9 1/3 87/23 1/2 140/37 3/5 53/14 1/2 1/1 19/5 1/1 23/6 0/1 1/2 4/1 1/1 9/2 1/0 14/3 -1/1 19/4 -1/1 1/0 5/1 -1/1 21/4 -1/4 0/1 37/7 0/1 16/3 1/3 11/2 0/1 1/0 17/3 -1/1 6/1 -1/1 19/3 0/1 32/5 1/5 13/2 0/1 1/2 7/1 1/1 15/2 -1/1 0/1 8/1 1/1 9/1 1/1 10/1 1/0 1/0 0/1 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(541,254,164,77) (-1/2,-7/15) -> (23/7,33/10) Hyperbolic Matrix(249,116,-908,-423) (-7/15,-6/13) -> (-14/51,-3/11) Hyperbolic Matrix(363,166,164,75) (-6/13,-5/11) -> (11/5,20/9) Hyperbolic Matrix(359,162,-800,-361) (-5/11,-9/20) -> (-9/20,-13/29) Parabolic Matrix(1555,696,954,427) (-13/29,-4/9) -> (44/27,31/19) Hyperbolic Matrix(549,242,946,417) (-4/9,-11/25) -> (11/19,18/31) Hyperbolic Matrix(41,18,312,137) (-11/25,-7/16) -> (1/8,1/7) Hyperbolic Matrix(545,238,158,69) (-7/16,-10/23) -> (24/7,7/2) Hyperbolic Matrix(585,254,76,33) (-10/23,-13/30) -> (15/2,8/1) Hyperbolic Matrix(737,318,540,233) (-13/30,-3/7) -> (15/11,41/30) Hyperbolic Matrix(191,80,74,31) (-3/7,-5/12) -> (5/2,13/5) Hyperbolic Matrix(111,46,152,63) (-5/12,-2/5) -> (8/11,3/4) Hyperbolic Matrix(251,98,-648,-253) (-2/5,-7/18) -> (-7/18,-12/31) Parabolic Matrix(1361,526,1938,749) (-12/31,-17/44) -> (7/10,26/37) Hyperbolic Matrix(503,194,70,27) (-17/44,-5/13) -> (7/1,15/2) Hyperbolic Matrix(37,14,140,53) (-5/13,-3/8) -> (1/4,3/11) Hyperbolic Matrix(209,78,142,53) (-3/8,-10/27) -> (16/11,3/2) Hyperbolic Matrix(805,298,208,77) (-10/27,-17/46) -> (23/6,4/1) Hyperbolic Matrix(2095,774,766,283) (-17/46,-7/19) -> (41/15,11/4) Hyperbolic Matrix(659,242,836,307) (-7/19,-11/30) -> (11/14,15/19) Hyperbolic Matrix(175,64,-834,-305) (-11/30,-4/11) -> (-4/19,-5/24) Hyperbolic Matrix(791,286,484,175) (-4/11,-9/25) -> (31/19,18/11) Hyperbolic Matrix(345,124,-1444,-519) (-9/25,-14/39) -> (-6/25,-5/21) Hyperbolic Matrix(721,258,204,73) (-14/39,-5/14) -> (7/2,32/9) Hyperbolic Matrix(169,60,138,49) (-5/14,-1/3) -> (11/9,5/4) Hyperbolic Matrix(159,50,-512,-161) (-1/3,-5/16) -> (-5/16,-9/29) Parabolic Matrix(1627,504,1146,355) (-9/29,-4/13) -> (44/31,27/19) Hyperbolic Matrix(33,10,188,57) (-4/13,-3/10) -> (1/6,2/11) Hyperbolic Matrix(155,46,64,19) (-3/10,-2/7) -> (12/5,5/2) Hyperbolic Matrix(93,26,304,85) (-2/7,-5/18) -> (3/10,4/13) Hyperbolic Matrix(701,194,972,269) (-5/18,-8/29) -> (18/25,13/18) Hyperbolic Matrix(821,226,1820,501) (-8/29,-11/40) -> (9/20,14/31) Hyperbolic Matrix(1943,534,302,83) (-11/40,-14/51) -> (32/5,13/2) Hyperbolic Matrix(31,8,58,15) (-3/11,-1/4) -> (1/2,5/9) Hyperbolic Matrix(815,198,638,155) (-1/4,-6/25) -> (60/47,23/18) Hyperbolic Matrix(347,82,402,95) (-5/21,-4/17) -> (6/7,13/15) Hyperbolic Matrix(145,34,516,121) (-4/17,-3/13) -> (7/25,2/7) Hyperbolic Matrix(201,46,284,65) (-3/13,-2/9) -> (12/17,5/7) Hyperbolic Matrix(83,18,-392,-85) (-2/9,-3/14) -> (-3/14,-4/19) Parabolic Matrix(167,34,54,11) (-5/24,-1/5) -> (3/1,19/6) Hyperbolic Matrix(81,14,52,9) (-1/5,-1/6) -> (3/2,11/7) Hyperbolic Matrix(25,4,106,17) (-1/6,-1/7) -> (3/13,1/4) Hyperbolic Matrix(77,10,100,13) (-1/7,0/1) -> (10/13,7/9) Hyperbolic Matrix(197,-24,156,-19) (0/1,1/8) -> (5/4,24/19) Hyperbolic Matrix(191,-28,116,-17) (1/7,1/6) -> (23/14,5/3) Hyperbolic Matrix(409,-76,296,-55) (2/11,1/5) -> (29/21,18/13) Hyperbolic Matrix(165,-34,364,-75) (1/5,4/19) -> (14/31,5/11) Hyperbolic Matrix(1143,-242,836,-177) (4/19,3/14) -> (41/30,26/19) Hyperbolic Matrix(73,-16,324,-71) (3/14,2/9) -> (2/9,5/22) Parabolic Matrix(807,-184,250,-57) (5/22,3/13) -> (29/9,13/4) Hyperbolic Matrix(449,-124,344,-95) (3/11,5/18) -> (13/10,17/13) Hyperbolic Matrix(1171,-326,722,-201) (5/18,7/25) -> (47/29,13/8) Hyperbolic Matrix(325,-94,204,-59) (2/7,5/17) -> (27/17,8/5) Hyperbolic Matrix(593,-176,374,-111) (5/17,3/10) -> (19/12,27/17) Hyperbolic Matrix(873,-272,536,-167) (4/13,5/16) -> (13/8,44/27) Hyperbolic Matrix(451,-144,166,-53) (5/16,1/3) -> (19/7,49/18) Hyperbolic Matrix(49,-18,128,-47) (1/3,3/8) -> (3/8,5/13) Parabolic Matrix(127,-50,94,-37) (5/13,2/5) -> (4/3,15/11) Hyperbolic Matrix(187,-76,32,-13) (2/5,7/17) -> (17/3,6/1) Hyperbolic Matrix(343,-142,186,-77) (7/17,5/12) -> (11/6,13/7) Hyperbolic Matrix(217,-92,92,-39) (5/12,3/7) -> (7/3,19/8) Hyperbolic Matrix(245,-108,152,-67) (3/7,4/9) -> (8/5,21/13) Hyperbolic Matrix(1033,-464,728,-327) (4/9,9/20) -> (17/12,44/31) Hyperbolic Matrix(271,-124,212,-97) (5/11,1/2) -> (23/18,9/7) Hyperbolic Matrix(113,-64,196,-111) (5/9,4/7) -> (4/7,11/19) Parabolic Matrix(1267,-736,890,-517) (18/31,7/12) -> (37/26,10/7) Hyperbolic Matrix(207,-122,56,-33) (7/12,3/5) -> (11/3,15/4) Hyperbolic Matrix(191,-116,28,-17) (3/5,11/18) -> (13/2,7/1) Hyperbolic Matrix(451,-276,518,-317) (11/18,19/31) -> (13/15,7/8) Hyperbolic Matrix(873,-536,272,-167) (19/31,8/13) -> (16/5,29/9) Hyperbolic Matrix(245,-152,108,-67) (8/13,5/8) -> (9/4,16/7) Hyperbolic Matrix(297,-188,188,-119) (5/8,7/11) -> (11/7,19/12) Hyperbolic Matrix(229,-146,80,-51) (7/11,9/14) -> (17/6,3/1) Hyperbolic Matrix(253,-164,54,-35) (9/14,2/3) -> (14/3,19/4) Hyperbolic Matrix(249,-170,104,-71) (2/3,9/13) -> (31/13,12/5) Hyperbolic Matrix(557,-388,234,-163) (9/13,7/10) -> (19/8,31/13) Hyperbolic Matrix(3733,-2624,1370,-963) (26/37,19/27) -> (79/29,30/11) Hyperbolic Matrix(1033,-728,464,-327) (19/27,12/17) -> (20/9,29/13) Hyperbolic Matrix(909,-652,718,-515) (5/7,23/32) -> (43/34,19/15) Hyperbolic Matrix(1843,-1326,1458,-1049) (23/32,18/25) -> (24/19,43/34) Hyperbolic Matrix(409,-296,76,-55) (13/18,8/11) -> (16/3,11/2) Hyperbolic Matrix(449,-344,124,-95) (3/4,10/13) -> (18/5,29/8) Hyperbolic Matrix(271,-212,124,-97) (7/9,11/14) -> (13/6,11/5) Hyperbolic Matrix(197,-156,24,-19) (15/19,4/5) -> (8/1,9/1) Hyperbolic Matrix(61,-50,72,-59) (4/5,5/6) -> (5/6,6/7) Parabolic Matrix(175,-156,46,-41) (7/8,1/1) -> (19/5,23/6) Hyperbolic Matrix(113,-130,20,-23) (1/1,7/6) -> (11/2,17/3) Hyperbolic Matrix(121,-144,100,-119) (7/6,6/5) -> (6/5,17/14) Parabolic Matrix(487,-592,218,-265) (17/14,11/9) -> (29/13,9/4) Hyperbolic Matrix(901,-1144,330,-419) (19/15,14/11) -> (30/11,41/15) Hyperbolic Matrix(2147,-2738,1682,-2145) (14/11,37/29) -> (37/29,60/47) Parabolic Matrix(183,-236,38,-49) (9/7,13/10) -> (19/4,5/1) Hyperbolic Matrix(553,-724,152,-199) (17/13,4/3) -> (40/11,11/3) Hyperbolic Matrix(557,-764,148,-203) (26/19,11/8) -> (15/4,34/9) Hyperbolic Matrix(481,-664,92,-127) (11/8,29/21) -> (5/1,21/4) Hyperbolic Matrix(609,-844,184,-255) (18/13,25/18) -> (33/10,10/3) Hyperbolic Matrix(653,-910,404,-563) (25/18,7/5) -> (21/13,55/34) Hyperbolic Matrix(355,-502,128,-181) (7/5,17/12) -> (11/4,25/9) Hyperbolic Matrix(1271,-1808,400,-569) (27/19,37/26) -> (19/6,35/11) Hyperbolic Matrix(235,-338,162,-233) (10/7,13/9) -> (13/9,16/11) Parabolic Matrix(361,-584,34,-55) (55/34,34/21) -> (10/1,1/0) Hyperbolic Matrix(479,-776,50,-81) (34/21,47/29) -> (9/1,10/1) Hyperbolic Matrix(1759,-2882,484,-793) (18/11,41/25) -> (69/19,40/11) Hyperbolic Matrix(1691,-2776,466,-765) (41/25,23/14) -> (29/8,69/19) Hyperbolic Matrix(57,-98,32,-55) (5/3,7/4) -> (7/4,9/5) Parabolic Matrix(203,-370,62,-113) (9/5,11/6) -> (13/4,23/7) Hyperbolic Matrix(129,-244,46,-87) (13/7,2/1) -> (14/5,31/11) Hyperbolic Matrix(149,-314,28,-59) (2/1,15/7) -> (37/7,16/3) Hyperbolic Matrix(369,-796,70,-151) (15/7,13/6) -> (21/4,37/7) Hyperbolic Matrix(481,-1102,134,-307) (16/7,23/10) -> (43/12,18/5) Hyperbolic Matrix(379,-876,106,-245) (23/10,7/3) -> (25/7,43/12) Hyperbolic Matrix(97,-256,36,-95) (13/5,8/3) -> (8/3,19/7) Parabolic Matrix(2109,-5744,662,-1803) (49/18,79/29) -> (35/11,51/16) Hyperbolic Matrix(413,-1150,116,-323) (25/9,14/5) -> (32/9,25/7) Hyperbolic Matrix(697,-1966,184,-519) (31/11,17/6) -> (53/14,19/5) Hyperbolic Matrix(2505,-7988,662,-2111) (51/16,16/5) -> (140/37,53/14) Hyperbolic Matrix(171,-578,50,-169) (10/3,17/5) -> (17/5,24/7) Parabolic Matrix(4003,-15138,1058,-4001) (34/9,87/23) -> (87/23,140/37) Parabolic Matrix(37,-162,8,-35) (4/1,9/2) -> (9/2,14/3) Parabolic Matrix(115,-722,18,-113) (6/1,19/3) -> (19/3,32/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(541,254,164,77) -> Matrix(1,0,-4,1) Matrix(249,116,-908,-423) -> Matrix(1,0,-2,1) Matrix(363,166,164,75) -> Matrix(1,0,0,1) Matrix(359,162,-800,-361) -> Matrix(5,-2,8,-3) Matrix(1555,696,954,427) -> Matrix(3,-2,2,-1) Matrix(549,242,946,417) -> Matrix(1,0,6,1) Matrix(41,18,312,137) -> Matrix(3,-2,2,-1) Matrix(545,238,158,69) -> Matrix(1,0,2,1) Matrix(585,254,76,33) -> Matrix(1,0,-4,1) Matrix(737,318,540,233) -> Matrix(1,0,-2,1) Matrix(191,80,74,31) -> Matrix(3,-2,2,-1) Matrix(111,46,152,63) -> Matrix(3,-2,2,-1) Matrix(251,98,-648,-253) -> Matrix(9,-4,16,-7) Matrix(1361,526,1938,749) -> Matrix(17,-10,12,-7) Matrix(503,194,70,27) -> Matrix(3,-2,2,-1) Matrix(37,14,140,53) -> Matrix(3,-2,2,-1) Matrix(209,78,142,53) -> Matrix(3,-2,2,-1) Matrix(805,298,208,77) -> Matrix(1,0,0,1) Matrix(2095,774,766,283) -> Matrix(5,-2,-2,1) Matrix(659,242,836,307) -> Matrix(1,0,0,1) Matrix(175,64,-834,-305) -> Matrix(1,0,2,1) Matrix(791,286,484,175) -> Matrix(3,-2,2,-1) Matrix(345,124,-1444,-519) -> Matrix(1,0,0,1) Matrix(721,258,204,73) -> Matrix(1,0,0,1) Matrix(169,60,138,49) -> Matrix(1,0,-2,1) Matrix(159,50,-512,-161) -> Matrix(1,0,2,1) Matrix(1627,504,1146,355) -> Matrix(1,0,-2,1) Matrix(33,10,188,57) -> Matrix(1,0,-2,1) Matrix(155,46,64,19) -> Matrix(3,-2,2,-1) Matrix(93,26,304,85) -> Matrix(1,-2,0,1) Matrix(701,194,972,269) -> Matrix(3,4,2,3) Matrix(821,226,1820,501) -> Matrix(1,0,2,1) Matrix(1943,534,302,83) -> Matrix(1,0,4,1) Matrix(31,8,58,15) -> Matrix(1,0,-2,1) Matrix(815,198,638,155) -> Matrix(1,0,-2,1) Matrix(347,82,402,95) -> Matrix(1,0,0,1) Matrix(145,34,516,121) -> Matrix(3,-2,2,-1) Matrix(201,46,284,65) -> Matrix(3,-2,2,-1) Matrix(83,18,-392,-85) -> Matrix(1,0,4,1) Matrix(167,34,54,11) -> Matrix(1,0,-4,1) Matrix(81,14,52,9) -> Matrix(3,-2,2,-1) Matrix(25,4,106,17) -> Matrix(3,-2,2,-1) Matrix(77,10,100,13) -> Matrix(1,0,-2,1) Matrix(197,-24,156,-19) -> Matrix(1,-2,2,-3) Matrix(191,-28,116,-17) -> Matrix(1,-2,0,1) Matrix(409,-76,296,-55) -> Matrix(1,0,2,1) Matrix(165,-34,364,-75) -> Matrix(1,0,0,1) Matrix(1143,-242,836,-177) -> Matrix(1,0,0,1) Matrix(73,-16,324,-71) -> Matrix(3,-2,2,-1) Matrix(807,-184,250,-57) -> Matrix(1,-2,0,1) Matrix(449,-124,344,-95) -> Matrix(1,0,0,1) Matrix(1171,-326,722,-201) -> Matrix(1,0,0,1) Matrix(325,-94,204,-59) -> Matrix(1,-4,0,1) Matrix(593,-176,374,-111) -> Matrix(1,10,0,1) Matrix(873,-272,536,-167) -> Matrix(1,2,0,1) Matrix(451,-144,166,-53) -> Matrix(1,-2,0,1) Matrix(49,-18,128,-47) -> Matrix(1,0,4,1) Matrix(127,-50,94,-37) -> Matrix(1,0,-2,1) Matrix(187,-76,32,-13) -> Matrix(1,0,-2,1) Matrix(343,-142,186,-77) -> Matrix(1,0,-2,1) Matrix(217,-92,92,-39) -> Matrix(1,0,0,1) Matrix(245,-108,152,-67) -> Matrix(1,0,0,1) Matrix(1033,-464,728,-327) -> Matrix(1,0,0,1) Matrix(271,-124,212,-97) -> Matrix(1,0,0,1) Matrix(113,-64,196,-111) -> Matrix(1,0,12,1) Matrix(1267,-736,890,-517) -> Matrix(1,0,-6,1) Matrix(207,-122,56,-33) -> Matrix(1,0,-4,1) Matrix(191,-116,28,-17) -> Matrix(1,0,-2,1) Matrix(451,-276,518,-317) -> Matrix(1,0,-2,1) Matrix(873,-536,272,-167) -> Matrix(1,0,-4,1) Matrix(245,-152,108,-67) -> Matrix(5,-2,8,-3) Matrix(297,-188,188,-119) -> Matrix(7,-4,2,-1) Matrix(229,-146,80,-51) -> Matrix(1,0,-2,1) Matrix(253,-164,54,-35) -> Matrix(1,0,-2,1) Matrix(249,-170,104,-71) -> Matrix(5,-4,4,-3) Matrix(557,-388,234,-163) -> Matrix(5,-6,6,-7) Matrix(3733,-2624,1370,-963) -> Matrix(7,-10,-2,3) Matrix(1033,-728,464,-327) -> Matrix(1,-2,4,-7) Matrix(909,-652,718,-515) -> Matrix(1,0,0,1) Matrix(1843,-1326,1458,-1049) -> Matrix(5,-6,6,-7) Matrix(409,-296,76,-55) -> Matrix(1,-2,2,-3) Matrix(449,-344,124,-95) -> Matrix(1,2,0,1) Matrix(271,-212,124,-97) -> Matrix(1,0,4,1) Matrix(197,-156,24,-19) -> Matrix(1,0,0,1) Matrix(61,-50,72,-59) -> Matrix(1,0,2,1) Matrix(175,-156,46,-41) -> Matrix(1,0,0,1) Matrix(113,-130,20,-23) -> Matrix(1,-2,0,1) Matrix(121,-144,100,-119) -> Matrix(1,-4,0,1) Matrix(487,-592,218,-265) -> Matrix(1,2,2,5) Matrix(901,-1144,330,-419) -> Matrix(1,-4,0,1) Matrix(2147,-2738,1682,-2145) -> Matrix(1,-2,0,1) Matrix(183,-236,38,-49) -> Matrix(1,-2,0,1) Matrix(553,-724,152,-199) -> Matrix(1,-2,0,1) Matrix(557,-764,148,-203) -> Matrix(1,0,2,1) Matrix(481,-664,92,-127) -> Matrix(1,0,-4,1) Matrix(609,-844,184,-255) -> Matrix(1,0,-4,1) Matrix(653,-910,404,-563) -> Matrix(3,-2,2,-1) Matrix(355,-502,128,-181) -> Matrix(1,-2,0,1) Matrix(1271,-1808,400,-569) -> Matrix(1,0,0,1) Matrix(235,-338,162,-233) -> Matrix(3,-2,2,-1) Matrix(361,-584,34,-55) -> Matrix(1,-2,0,1) Matrix(479,-776,50,-81) -> Matrix(1,0,0,1) Matrix(1759,-2882,484,-793) -> Matrix(1,-6,0,1) Matrix(1691,-2776,466,-765) -> Matrix(1,6,0,1) Matrix(57,-98,32,-55) -> Matrix(1,-4,0,1) Matrix(203,-370,62,-113) -> Matrix(1,2,-2,-3) Matrix(129,-244,46,-87) -> Matrix(1,0,0,1) Matrix(149,-314,28,-59) -> Matrix(1,0,4,1) Matrix(369,-796,70,-151) -> Matrix(1,0,-10,1) Matrix(481,-1102,134,-307) -> Matrix(5,-4,4,-3) Matrix(379,-876,106,-245) -> Matrix(1,-2,2,-3) Matrix(97,-256,36,-95) -> Matrix(1,-6,0,1) Matrix(2109,-5744,662,-1803) -> Matrix(1,4,0,1) Matrix(413,-1150,116,-323) -> Matrix(1,0,2,1) Matrix(697,-1966,184,-519) -> Matrix(1,0,2,1) Matrix(2505,-7988,662,-2111) -> Matrix(1,-2,2,-3) Matrix(171,-578,50,-169) -> Matrix(1,0,10,1) Matrix(4003,-15138,1058,-4001) -> Matrix(9,-4,16,-7) Matrix(37,-162,8,-35) -> Matrix(1,-2,0,1) Matrix(115,-722,18,-113) -> Matrix(1,0,6,1) 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): 28 Degree of the the map X: 28 Degree of the the map Y: 120 ----------------------------------------------------------------------- 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 1/1 1 22 1/7 3/1 1 11 1/6 (-1/1,1/0) 0 22 1/5 1/1 1 11 4/19 1/1 1 22 3/14 (0/1,1/1) 0 22 2/9 1/1 1 2 1/4 (1/1,1/0) 0 22 2/7 3/1 1 22 5/17 1/0 7 1 3/10 (-4/1,1/0) 0 22 1/3 -1/1 1 11 3/8 0/1 4 2 5/13 1/3 1 11 2/5 1/1 1 22 7/17 1/1 1 11 5/12 (2/3,1/1) 0 22 3/7 1/1 1 11 4/9 -1/1 1 22 9/20 (0/1,1/0) 0 22 5/11 1/1 1 11 1/2 (0/1,1/0) 0 22 4/7 0/1 6 2 7/12 (0/1,1/6) 0 22 3/5 1/3 1 11 11/18 (0/1,1/4) 0 22 19/31 1/3 1 11 8/13 1/3 1 22 5/8 (2/5,1/2) 0 22 7/11 1/1 1 11 9/14 (1/2,1/1) 0 22 2/3 1/1 1 22 9/13 1/1 5 1 7/10 (1/1,4/3) 0 22 5/7 1/1 1 11 3/4 (1/1,1/0) 0 22 7/9 -1/1 1 11 11/14 (0/1,1/2) 0 22 4/5 1/1 1 22 5/6 0/1 2 2 6/7 1/3 1 22 7/8 (0/1,1/2) 0 22 1/1 1/1 1 11 7/6 (2/1,1/0) 0 22 6/5 1/0 2 2 5/4 (0/1,1/0) 0 22 19/15 1/1 1 11 14/11 1/1 1 22 37/29 1/0 1 1 23/18 (0/1,1/0) 0 22 9/7 1/1 1 11 13/10 (1/1,1/0) 0 22 4/3 -1/1 1 22 15/11 1/1 1 11 26/19 1/1 1 22 11/8 (0/1,1/0) 0 22 18/13 1/1 1 22 25/18 (0/1,1/2) 0 22 7/5 1/1 1 11 17/12 (0/1,1/0) 0 22 27/19 1/1 1 11 10/7 1/1 1 22 13/9 1/1 1 1 3/2 (1/1,1/0) 0 22 8/5 -1/1 1 22 21/13 1/1 1 11 55/34 (2/1,1/0) 0 22 34/21 1/0 1 2 13/8 (1/1,1/0) 0 22 18/11 3/1 1 22 41/25 1/0 6 1 23/14 (-3/1,1/0) 0 22 5/3 1/1 1 11 7/4 1/0 4 2 9/5 -3/1 1 11 11/6 (-2/1,-1/1) 0 22 13/7 -1/1 1 11 2/1 -1/1 1 22 15/7 0/1 7 1 13/6 (0/1,1/6) 0 22 11/5 1/3 1 11 9/4 (0/1,1/2) 0 22 16/7 1/1 1 22 23/10 1/1 6 2 7/3 1/1 1 11 5/2 (1/1,1/0) 0 22 8/3 1/0 3 2 11/4 (-2/1,1/0) 0 22 25/9 -1/1 1 11 14/5 -1/1 1 22 31/11 -1/1 1 11 17/6 (-1/1,1/0) 0 22 3/1 -1/1 1 11 16/5 -1/1 1 22 13/4 (-1/1,0/1) 0 22 23/7 -1/3 1 11 10/3 -1/3 1 22 17/5 0/1 5 1 7/2 (0/1,1/2) 0 22 25/7 1/1 1 11 43/12 1/1 6 2 18/5 1/1 1 22 11/3 -1/1 1 11 15/4 (0/1,1/2) 0 22 34/9 1/3 1 22 87/23 1/2 2 1 53/14 (1/2,1/1) 0 22 19/5 1/1 1 11 4/1 1/1 1 22 9/2 1/0 2 2 14/3 -1/1 1 22 19/4 (-1/1,1/0) 0 22 5/1 -1/1 1 11 11/2 (0/1,1/0) 0 22 17/3 -1/1 1 11 6/1 -1/1 1 22 19/3 0/1 3 1 13/2 (0/1,1/2) 0 22 7/1 1/1 1 11 8/1 1/1 1 22 1/0 (0/1,1/0) 0 22 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(137,-18,38,-5) (0/1,1/7) -> (18/5,11/3) Glide Reflection Matrix(191,-28,116,-17) (1/7,1/6) -> (23/14,5/3) Hyperbolic Matrix(57,-10,74,-13) (1/6,1/5) -> (3/4,7/9) Glide Reflection Matrix(401,-84,654,-137) (1/5,4/19) -> (19/31,8/13) Glide Reflection Matrix(471,-100,146,-31) (4/19,3/14) -> (16/5,13/4) Glide Reflection Matrix(55,-12,252,-55) (3/14,2/9) -> (3/14,2/9) Reflection Matrix(17,-4,72,-17) (2/9,1/4) -> (2/9,1/4) Reflection Matrix(53,-14,34,-9) (1/4,2/7) -> (3/2,8/5) Glide Reflection Matrix(69,-20,238,-69) (2/7,5/17) -> (2/7,5/17) Reflection Matrix(101,-30,340,-101) (5/17,3/10) -> (5/17,3/10) Reflection Matrix(85,-26,134,-41) (3/10,1/3) -> (5/8,7/11) Glide Reflection Matrix(49,-18,128,-47) (1/3,3/8) -> (3/8,5/13) Parabolic Matrix(127,-50,94,-37) (5/13,2/5) -> (4/3,15/11) Hyperbolic Matrix(187,-76,32,-13) (2/5,7/17) -> (17/3,6/1) Hyperbolic Matrix(343,-142,186,-77) (7/17,5/12) -> (11/6,13/7) Hyperbolic Matrix(109,-46,154,-65) (5/12,3/7) -> (7/10,5/7) Glide Reflection Matrix(245,-108,152,-67) (3/7,4/9) -> (8/5,21/13) Hyperbolic Matrix(183,-82,212,-95) (4/9,9/20) -> (6/7,7/8) Glide Reflection Matrix(501,-226,818,-369) (9/20,5/11) -> (11/18,19/31) Glide Reflection Matrix(271,-124,212,-97) (5/11,1/2) -> (23/18,9/7) Hyperbolic Matrix(15,-8,28,-15) (1/2,4/7) -> (1/2,4/7) Reflection Matrix(97,-56,168,-97) (4/7,7/12) -> (4/7,7/12) Reflection Matrix(207,-122,56,-33) (7/12,3/5) -> (11/3,15/4) Hyperbolic Matrix(191,-116,28,-17) (3/5,11/18) -> (13/2,7/1) Hyperbolic Matrix(245,-152,108,-67) (8/13,5/8) -> (9/4,16/7) Hyperbolic Matrix(229,-146,80,-51) (7/11,9/14) -> (17/6,3/1) Hyperbolic Matrix(253,-164,54,-35) (9/14,2/3) -> (14/3,19/4) Hyperbolic Matrix(53,-36,78,-53) (2/3,9/13) -> (2/3,9/13) Reflection Matrix(181,-126,260,-181) (9/13,7/10) -> (9/13,7/10) Reflection Matrix(63,-46,26,-19) (5/7,3/4) -> (7/3,5/2) Glide Reflection Matrix(271,-212,124,-97) (7/9,11/14) -> (13/6,11/5) Hyperbolic Matrix(307,-242,222,-175) (11/14,4/5) -> (11/8,18/13) Glide Reflection Matrix(61,-50,72,-59) (4/5,5/6) -> (5/6,6/7) Parabolic Matrix(233,-206,164,-145) (7/8,1/1) -> (17/12,27/19) Glide Reflection Matrix(113,-130,20,-23) (1/1,7/6) -> (11/2,17/3) Hyperbolic Matrix(71,-84,60,-71) (7/6,6/5) -> (7/6,6/5) Reflection Matrix(49,-60,40,-49) (6/5,5/4) -> (6/5,5/4) Reflection Matrix(205,-258,58,-73) (5/4,19/15) -> (7/2,25/7) Glide Reflection Matrix(485,-616,174,-221) (19/15,14/11) -> (25/9,14/5) Glide Reflection Matrix(813,-1036,638,-813) (14/11,37/29) -> (14/11,37/29) Reflection Matrix(1333,-1702,1044,-1333) (37/29,23/18) -> (37/29,23/18) Reflection Matrix(183,-236,38,-49) (9/7,13/10) -> (19/4,5/1) Hyperbolic Matrix(219,-286,134,-175) (13/10,4/3) -> (13/8,18/11) Glide Reflection Matrix(233,-318,74,-101) (15/11,26/19) -> (3/1,16/5) Glide Reflection Matrix(557,-764,148,-203) (26/19,11/8) -> (15/4,34/9) Hyperbolic Matrix(157,-218,18,-25) (18/13,25/18) -> (8/1,1/0) Glide Reflection Matrix(653,-910,404,-563) (25/18,7/5) -> (21/13,55/34) Hyperbolic Matrix(355,-502,128,-181) (7/5,17/12) -> (11/4,25/9) Hyperbolic Matrix(209,-298,54,-77) (27/19,10/7) -> (19/5,4/1) Glide Reflection Matrix(181,-260,126,-181) (10/7,13/9) -> (10/7,13/9) Reflection Matrix(53,-78,36,-53) (13/9,3/2) -> (13/9,3/2) Reflection Matrix(2311,-3740,1428,-2311) (55/34,34/21) -> (55/34,34/21) Reflection Matrix(545,-884,336,-545) (34/21,13/8) -> (34/21,13/8) Reflection Matrix(901,-1476,550,-901) (18/11,41/25) -> (18/11,41/25) Reflection Matrix(1149,-1886,700,-1149) (41/25,23/14) -> (41/25,23/14) Reflection Matrix(57,-98,32,-55) (5/3,7/4) -> (7/4,9/5) Parabolic Matrix(203,-370,62,-113) (9/5,11/6) -> (13/4,23/7) Hyperbolic Matrix(129,-244,46,-87) (13/7,2/1) -> (14/5,31/11) Hyperbolic 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(75,-166,14,-31) (11/5,9/4) -> (5/1,11/2) Glide Reflection Matrix(481,-1102,134,-307) (16/7,23/10) -> (43/12,18/5) Hyperbolic Matrix(379,-876,106,-245) (23/10,7/3) -> (25/7,43/12) Hyperbolic Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(65,-176,24,-65) (8/3,11/4) -> (8/3,11/4) Reflection Matrix(697,-1966,184,-519) (31/11,17/6) -> (53/14,19/5) Hyperbolic Matrix(77,-254,10,-33) (23/7,10/3) -> (7/1,8/1) Glide Reflection Matrix(101,-340,30,-101) (10/3,17/5) -> (10/3,17/5) Reflection Matrix(69,-238,20,-69) (17/5,7/2) -> (17/5,7/2) Reflection Matrix(1565,-5916,414,-1565) (34/9,87/23) -> (34/9,87/23) Reflection Matrix(2437,-9222,644,-2437) (87/23,53/14) -> (87/23,53/14) Reflection Matrix(37,-162,8,-35) (4/1,9/2) -> (9/2,14/3) Parabolic Matrix(37,-228,6,-37) (6/1,19/3) -> (6/1,19/3) Reflection Matrix(77,-494,12,-77) (19/3,13/2) -> (19/3,13/2) 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(137,-18,38,-5) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(191,-28,116,-17) -> Matrix(1,-2,0,1) 1/0 Matrix(57,-10,74,-13) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(401,-84,654,-137) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(471,-100,146,-31) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(55,-12,252,-55) -> Matrix(1,0,2,-1) (3/14,2/9) -> (0/1,1/1) Matrix(17,-4,72,-17) -> Matrix(-1,2,0,1) (2/9,1/4) -> (1/1,1/0) Matrix(53,-14,34,-9) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(69,-20,238,-69) -> Matrix(-1,6,0,1) (2/7,5/17) -> (3/1,1/0) Matrix(101,-30,340,-101) -> Matrix(1,8,0,-1) (5/17,3/10) -> (-4/1,1/0) Matrix(85,-26,134,-41) -> Matrix(1,2,2,3) Matrix(49,-18,128,-47) -> Matrix(1,0,4,1) 0/1 Matrix(127,-50,94,-37) -> Matrix(1,0,-2,1) 0/1 Matrix(187,-76,32,-13) -> Matrix(1,0,-2,1) 0/1 Matrix(343,-142,186,-77) -> Matrix(1,0,-2,1) 0/1 Matrix(109,-46,154,-65) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(245,-108,152,-67) -> Matrix(1,0,0,1) Matrix(183,-82,212,-95) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(501,-226,818,-369) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(271,-124,212,-97) -> Matrix(1,0,0,1) Matrix(15,-8,28,-15) -> Matrix(1,0,0,-1) (1/2,4/7) -> (0/1,1/0) Matrix(97,-56,168,-97) -> Matrix(1,0,12,-1) (4/7,7/12) -> (0/1,1/6) Matrix(207,-122,56,-33) -> Matrix(1,0,-4,1) 0/1 Matrix(191,-116,28,-17) -> Matrix(1,0,-2,1) 0/1 Matrix(245,-152,108,-67) -> Matrix(5,-2,8,-3) 1/2 Matrix(229,-146,80,-51) -> Matrix(1,0,-2,1) 0/1 Matrix(253,-164,54,-35) -> Matrix(1,0,-2,1) 0/1 Matrix(53,-36,78,-53) -> Matrix(3,-2,4,-3) (2/3,9/13) -> (1/2,1/1) Matrix(181,-126,260,-181) -> Matrix(7,-8,6,-7) (9/13,7/10) -> (1/1,4/3) Matrix(63,-46,26,-19) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(271,-212,124,-97) -> Matrix(1,0,4,1) 0/1 Matrix(307,-242,222,-175) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(61,-50,72,-59) -> Matrix(1,0,2,1) 0/1 Matrix(233,-206,164,-145) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(113,-130,20,-23) -> Matrix(1,-2,0,1) 1/0 Matrix(71,-84,60,-71) -> Matrix(-1,4,0,1) (7/6,6/5) -> (2/1,1/0) Matrix(49,-60,40,-49) -> Matrix(1,0,0,-1) (6/5,5/4) -> (0/1,1/0) Matrix(205,-258,58,-73) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(485,-616,174,-221) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(813,-1036,638,-813) -> Matrix(-1,2,0,1) (14/11,37/29) -> (1/1,1/0) Matrix(1333,-1702,1044,-1333) -> Matrix(1,0,0,-1) (37/29,23/18) -> (0/1,1/0) Matrix(183,-236,38,-49) -> Matrix(1,-2,0,1) 1/0 Matrix(219,-286,134,-175) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(233,-318,74,-101) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(557,-764,148,-203) -> Matrix(1,0,2,1) 0/1 Matrix(157,-218,18,-25) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(653,-910,404,-563) -> Matrix(3,-2,2,-1) 1/1 Matrix(355,-502,128,-181) -> Matrix(1,-2,0,1) 1/0 Matrix(209,-298,54,-77) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(181,-260,126,-181) -> Matrix(1,0,2,-1) (10/7,13/9) -> (0/1,1/1) Matrix(53,-78,36,-53) -> Matrix(-1,2,0,1) (13/9,3/2) -> (1/1,1/0) Matrix(2311,-3740,1428,-2311) -> Matrix(-1,4,0,1) (55/34,34/21) -> (2/1,1/0) Matrix(545,-884,336,-545) -> Matrix(-1,2,0,1) (34/21,13/8) -> (1/1,1/0) Matrix(901,-1476,550,-901) -> Matrix(-1,6,0,1) (18/11,41/25) -> (3/1,1/0) Matrix(1149,-1886,700,-1149) -> Matrix(1,6,0,-1) (41/25,23/14) -> (-3/1,1/0) Matrix(57,-98,32,-55) -> Matrix(1,-4,0,1) 1/0 Matrix(203,-370,62,-113) -> Matrix(1,2,-2,-3) -1/1 Matrix(129,-244,46,-87) -> Matrix(1,0,0,1) Matrix(29,-60,14,-29) -> Matrix(-1,0,2,1) (2/1,15/7) -> (-1/1,0/1) Matrix(181,-390,84,-181) -> Matrix(1,0,12,-1) (15/7,13/6) -> (0/1,1/6) Matrix(75,-166,14,-31) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(481,-1102,134,-307) -> Matrix(5,-4,4,-3) 1/1 Matrix(379,-876,106,-245) -> Matrix(1,-2,2,-3) 1/1 Matrix(31,-80,12,-31) -> Matrix(-1,2,0,1) (5/2,8/3) -> (1/1,1/0) Matrix(65,-176,24,-65) -> Matrix(1,4,0,-1) (8/3,11/4) -> (-2/1,1/0) Matrix(697,-1966,184,-519) -> Matrix(1,0,2,1) 0/1 Matrix(77,-254,10,-33) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(101,-340,30,-101) -> Matrix(-1,0,6,1) (10/3,17/5) -> (-1/3,0/1) Matrix(69,-238,20,-69) -> Matrix(1,0,4,-1) (17/5,7/2) -> (0/1,1/2) Matrix(1565,-5916,414,-1565) -> Matrix(5,-2,12,-5) (34/9,87/23) -> (1/3,1/2) Matrix(2437,-9222,644,-2437) -> Matrix(3,-2,4,-3) (87/23,53/14) -> (1/2,1/1) Matrix(37,-162,8,-35) -> Matrix(1,-2,0,1) 1/0 Matrix(37,-228,6,-37) -> Matrix(-1,0,2,1) (6/1,19/3) -> (-1/1,0/1) Matrix(77,-494,12,-77) -> Matrix(1,0,4,-1) (19/3,13/2) -> (0/1,1/2) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.