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): 864 Minimal number of generators: 145 Number of equivalence classes of cusps: 64 Genus: 41 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -5/11 -7/16 -2/5 -5/14 -1/3 -1/4 -2/11 -1/7 0/1 1/5 2/7 1/3 4/11 5/13 1/2 19/35 7/11 5/7 4/5 11/13 1/1 13/11 5/4 7/5 3/2 11/7 28/17 5/3 17/10 23/13 2/1 37/17 25/11 7/3 19/8 5/2 13/5 8/3 11/4 3/1 16/5 55/17 13/4 10/3 7/2 11/3 4/1 29/7 13/3 22/5 9/2 14/3 5/1 11/2 17/3 6/1 31/5 19/3 13/2 7/1 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 1/1 -6/13 1/0 -5/11 1/0 -9/20 -1/1 -4/9 1/2 -11/25 0/1 2/3 -7/16 1/1 -17/39 6/5 4/3 -27/62 1/1 -10/23 3/2 -3/7 2/1 -11/26 1/1 3/1 -19/45 2/1 8/3 -8/19 1/0 -5/12 -1/1 -7/17 -2/1 0/1 -9/22 1/1 -2/5 1/2 1/0 -9/23 0/1 2/1 -7/18 1/1 -19/49 1/0 -12/31 1/0 -5/13 0/1 -13/34 1/1 3/1 -8/21 1/0 -3/8 -1/1 -7/19 -2/1 0/1 -4/11 -1/2 -9/25 0/1 -5/14 0/1 -6/17 1/2 -1/3 0/1 2/1 -5/16 -1/1 -4/13 0/1 -7/23 0/1 -3/10 1/3 -5/17 0/1 2/3 -2/7 1/2 -5/18 1/1 -8/29 3/2 1/0 -3/11 0/1 -7/26 1/1 -4/15 1/0 -9/34 1/1 -5/19 0/1 2/1 -1/4 -1/1 1/1 -4/17 1/0 -7/30 1/3 -3/13 0/1 2/3 -2/9 1/2 -3/14 1/1 -7/33 8/5 2/1 -4/19 3/2 1/0 -1/5 2/1 -3/16 3/1 -2/11 1/0 -5/28 -3/1 -8/45 1/0 -3/17 -2/1 0/1 -1/6 -1/1 -2/13 1/2 -3/20 1/1 -1/7 1/1 -1/8 1/1 0/1 1/0 1/7 -4/1 2/13 -5/2 1/0 3/19 -8/3 -2/1 1/6 -1/1 2/11 -3/2 1/5 -1/1 2/9 -1/2 5/22 0/1 3/13 0/1 7/30 -1/1 11/47 -2/3 0/1 4/17 -1/2 1/0 5/21 -2/5 0/1 1/4 1/1 3/11 0/1 2/1 2/7 1/0 5/17 -6/1 -4/1 8/27 1/0 3/10 -3/1 4/13 1/0 5/16 -3/1 -1/1 1/3 -2/1 6/17 1/0 5/14 -1/1 4/11 -3/2 1/0 11/30 -3/1 7/19 -2/1 -8/5 10/27 -3/2 3/8 -1/1 5/13 -1/1 7/18 -1/1 2/5 -1/2 5/12 -1/1 13/31 -4/3 8/19 -1/1 11/26 -1/1 14/33 -3/4 3/7 -2/3 0/1 4/9 1/0 9/20 -1/1 5/11 0/1 1/2 -1/1 1/1 7/13 0/1 13/24 1/1 19/35 1/0 6/11 1/0 5/9 -2/1 0/1 4/7 1/0 11/19 -1/1 7/12 -1/1 10/17 -1/1 3/5 0/1 14/23 1/0 11/18 -3/1 19/31 -2/1 -4/3 8/13 -3/2 1/0 21/34 -1/1 13/21 -2/1 0/1 18/29 -3/2 23/37 -2/1 5/8 -1/1 7/11 -1/1 9/14 -1/1 2/3 -1/2 7/10 -1/1 12/17 -3/4 5/7 -2/3 0/1 13/18 -1/1 21/29 -2/3 0/1 8/11 -1/2 11/15 -2/5 0/1 3/4 -1/3 13/17 -1/3 23/30 -1/3 10/13 -1/4 17/22 -1/3 -1/5 7/9 0/1 4/5 0/1 9/11 0/1 14/17 1/4 5/6 1/1 16/19 1/0 27/32 -1/1 11/13 0/1 6/7 1/0 7/8 -1/1 1/1 8/9 1/0 1/1 -2/1 0/1 8/7 -1/2 1/0 7/6 -1/1 20/17 -1/2 13/11 0/1 6/5 -1/2 5/4 0/1 14/11 1/4 23/18 1/1 32/25 1/4 1/2 9/7 0/1 22/17 1/4 1/2 13/10 1/3 17/13 1/2 4/3 1/2 15/11 0/1 2/3 26/19 1/1 11/8 1/1 18/13 1/0 7/5 0/1 2/1 10/7 1/0 53/37 1/0 43/30 -1/1 33/23 0/1 23/16 -1/1 1/1 13/9 0/1 2/1 3/2 1/1 11/7 1/0 19/12 -7/1 8/5 1/0 37/23 -2/1 29/18 -3/1 21/13 -2/1 0/1 13/8 -3/1 -1/1 44/27 1/0 31/19 -4/1 -2/1 49/30 -7/3 67/41 -2/1 18/11 -3/2 23/14 -1/1 28/17 -1/1 5/3 0/1 22/13 1/0 17/10 1/0 46/27 1/0 29/17 -6/1 -4/1 12/7 1/0 7/4 -1/1 23/13 -2/1 0/1 39/22 -1/1 16/9 1/0 9/5 -2/1 0/1 20/11 1/0 11/6 -1/1 2/1 -1/2 1/0 13/6 -1/1 37/17 -1/1 24/11 -1/2 11/5 0/1 9/4 -1/1 25/11 0/1 41/18 1/5 16/7 1/2 7/3 0/1 2/1 33/14 3/1 59/25 4/1 26/11 1/0 19/8 1/0 50/21 1/0 31/13 -4/1 12/5 1/0 17/7 -2/1 0/1 5/2 1/1 13/5 1/0 21/8 -9/1 8/3 1/0 35/13 -4/1 62/23 -7/2 27/10 -3/1 19/7 -8/3 -2/1 49/18 -11/5 79/29 -2/1 30/11 -3/2 41/15 -2/1 52/19 1/0 11/4 -3/1 -1/1 25/9 -2/1 0/1 39/14 -1/1 53/19 -2/1 0/1 14/5 1/0 31/11 1/0 17/6 -1/1 3/1 -2/1 16/5 -3/2 1/0 29/9 -2/1 -4/3 42/13 -3/2 55/17 -2/1 -4/3 13/4 -1/1 23/7 -1/1 10/3 -3/2 7/2 -1/1 18/5 -3/4 47/13 -2/3 76/21 -1/2 29/8 -1/1 69/19 -2/3 40/11 -3/4 -1/2 11/3 -2/3 0/1 26/7 -1/2 1/0 41/11 0/1 15/4 -1/1 4/1 -1/2 29/7 0/1 25/6 1/5 21/5 0/1 2/3 17/4 -1/1 1/1 30/7 1/0 73/17 -1/1 43/10 -1/1 13/3 0/1 22/5 0/1 9/2 1/1 14/3 1/0 5/1 1/0 16/3 1/0 11/2 -3/1 17/3 -2/1 6/1 1/0 31/5 -2/1 25/4 -9/5 19/3 -2/1 -8/5 32/5 -3/2 13/2 -5/3 -1/1 7/1 -4/3 8/1 -1/1 9/1 -4/5 -2/3 1/0 -1/1 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(335,156,204,95) (-1/2,-6/13) -> (18/11,23/14) Hyperbolic Matrix(265,122,202,93) (-6/13,-5/11) -> (17/13,4/3) Hyperbolic Matrix(457,206,-1178,-531) (-5/11,-9/20) -> (-7/18,-19/49) Hyperbolic Matrix(325,146,394,177) (-9/20,-4/9) -> (14/17,5/6) Hyperbolic Matrix(581,256,320,141) (-4/9,-11/25) -> (9/5,20/11) Hyperbolic Matrix(447,196,-1024,-449) (-11/25,-7/16) -> (-7/16,-17/39) Parabolic Matrix(955,416,4084,1779) (-17/39,-27/62) -> (7/30,11/47) Hyperbolic Matrix(4465,1944,1656,721) (-27/62,-10/23) -> (62/23,27/10) Hyperbolic Matrix(65,28,188,81) (-10/23,-3/7) -> (1/3,6/17) Hyperbolic Matrix(61,26,190,81) (-3/7,-11/26) -> (5/16,1/3) Hyperbolic Matrix(1311,554,310,131) (-11/26,-19/45) -> (21/5,17/4) Hyperbolic Matrix(687,290,1618,683) (-19/45,-8/19) -> (14/33,3/7) Hyperbolic Matrix(495,208,188,79) (-8/19,-5/12) -> (21/8,8/3) Hyperbolic Matrix(183,76,248,103) (-5/12,-7/17) -> (11/15,3/4) Hyperbolic Matrix(185,76,-796,-327) (-7/17,-9/22) -> (-7/30,-3/13) Hyperbolic Matrix(549,224,424,173) (-9/22,-2/5) -> (22/17,13/10) Hyperbolic Matrix(653,256,176,69) (-2/5,-9/23) -> (11/3,26/7) Hyperbolic Matrix(297,116,-1124,-439) (-9/23,-7/18) -> (-9/34,-5/19) Hyperbolic Matrix(1647,638,586,227) (-19/49,-12/31) -> (14/5,31/11) Hyperbolic Matrix(1117,432,468,181) (-12/31,-5/13) -> (31/13,12/5) Hyperbolic Matrix(407,156,60,23) (-5/13,-13/34) -> (13/2,7/1) Hyperbolic Matrix(351,134,406,155) (-13/34,-8/21) -> (6/7,7/8) Hyperbolic Matrix(463,176,292,111) (-8/21,-3/8) -> (19/12,8/5) Hyperbolic Matrix(287,106,398,147) (-3/8,-7/19) -> (5/7,13/18) Hyperbolic Matrix(283,104,400,147) (-7/19,-4/11) -> (12/17,5/7) Hyperbolic Matrix(337,122,58,21) (-4/11,-9/25) -> (17/3,6/1) Hyperbolic Matrix(167,60,732,263) (-9/25,-5/14) -> (5/22,3/13) Hyperbolic Matrix(113,40,500,177) (-5/14,-6/17) -> (2/9,5/22) Hyperbolic Matrix(275,96,444,155) (-6/17,-1/3) -> (13/21,18/29) Hyperbolic Matrix(423,134,262,83) (-1/3,-5/16) -> (29/18,21/13) Hyperbolic Matrix(469,146,106,33) (-5/16,-4/13) -> (22/5,9/2) Hyperbolic Matrix(675,206,154,47) (-4/13,-7/23) -> (13/3,22/5) Hyperbolic Matrix(257,78,570,173) (-7/23,-3/10) -> (9/20,5/11) Hyperbolic Matrix(255,76,104,31) (-3/10,-5/17) -> (17/7,5/2) Hyperbolic Matrix(355,104,256,75) (-5/17,-2/7) -> (18/13,7/5) Hyperbolic Matrix(99,28,152,43) (-2/7,-5/18) -> (9/14,2/3) Hyperbolic Matrix(347,96,300,83) (-5/18,-8/29) -> (8/7,7/6) Hyperbolic Matrix(51,14,346,95) (-8/29,-3/11) -> (1/7,2/13) Hyperbolic Matrix(393,106,938,253) (-3/11,-7/26) -> (5/12,13/31) Hyperbolic Matrix(97,26,-638,-171) (-7/26,-4/15) -> (-2/13,-3/20) Hyperbolic Matrix(1129,300,636,169) (-4/15,-9/34) -> (39/22,16/9) Hyperbolic Matrix(489,128,340,89) (-5/19,-1/4) -> (23/16,13/9) Hyperbolic Matrix(329,78,426,101) (-1/4,-4/17) -> (10/13,17/22) Hyperbolic Matrix(1177,276,516,121) (-4/17,-7/30) -> (41/18,16/7) Hyperbolic Matrix(187,42,138,31) (-3/13,-2/9) -> (4/3,15/11) Hyperbolic Matrix(91,20,232,51) (-2/9,-3/14) -> (7/18,2/5) Hyperbolic Matrix(1187,252,504,107) (-3/14,-7/33) -> (7/3,33/14) Hyperbolic Matrix(227,48,960,203) (-7/33,-4/19) -> (4/17,5/21) Hyperbolic Matrix(137,28,44,9) (-4/19,-1/5) -> (3/1,16/5) Hyperbolic Matrix(133,26,46,9) (-1/5,-3/16) -> (17/6,3/1) Hyperbolic Matrix(87,16,-484,-89) (-3/16,-2/11) -> (-2/11,-5/28) Parabolic Matrix(1663,296,1972,351) (-5/28,-8/45) -> (16/19,27/32) Hyperbolic Matrix(1399,248,220,39) (-8/45,-3/17) -> (19/3,32/5) Hyperbolic Matrix(347,60,480,83) (-3/17,-1/6) -> (13/18,21/29) Hyperbolic Matrix(383,60,300,47) (-1/6,-2/13) -> (14/11,23/18) Hyperbolic Matrix(421,62,550,81) (-3/20,-1/7) -> (13/17,23/30) Hyperbolic Matrix(125,16,164,21) (-1/7,-1/8) -> (3/4,13/17) Hyperbolic Matrix(123,14,202,23) (-1/8,0/1) -> (14/23,11/18) Hyperbolic Matrix(159,-22,94,-13) (0/1,1/7) -> (5/3,22/13) Hyperbolic Matrix(219,-34,934,-145) (2/13,3/19) -> (11/47,4/17) Hyperbolic Matrix(589,-94,94,-15) (3/19,1/6) -> (25/4,19/3) Hyperbolic Matrix(123,-22,274,-49) (1/6,2/11) -> (4/9,9/20) Hyperbolic Matrix(303,-56,92,-17) (2/11,1/5) -> (23/7,10/3) Hyperbolic Matrix(119,-26,206,-45) (1/5,2/9) -> (4/7,11/19) Hyperbolic Matrix(1549,-360,1080,-251) (3/13,7/30) -> (43/30,33/23) Hyperbolic Matrix(609,-146,146,-35) (5/21,1/4) -> (25/6,21/5) Hyperbolic Matrix(85,-22,58,-15) (1/4,3/11) -> (13/9,3/2) Hyperbolic Matrix(57,-16,196,-55) (3/11,2/7) -> (2/7,5/17) Parabolic Matrix(1585,-468,972,-287) (5/17,8/27) -> (44/27,31/19) Hyperbolic Matrix(389,-116,332,-99) (8/27,3/10) -> (7/6,20/17) Hyperbolic Matrix(303,-92,56,-17) (3/10,4/13) -> (16/3,11/2) Hyperbolic Matrix(219,-68,248,-77) (4/13,5/16) -> (7/8,8/9) Hyperbolic Matrix(531,-188,692,-245) (6/17,5/14) -> (23/30,10/13) Hyperbolic Matrix(343,-124,556,-201) (5/14,4/11) -> (8/13,21/34) Hyperbolic Matrix(1213,-444,948,-347) (4/11,11/30) -> (23/18,32/25) Hyperbolic Matrix(1501,-552,552,-203) (11/30,7/19) -> (19/7,49/18) Hyperbolic Matrix(1657,-612,972,-359) (7/19,10/27) -> (46/27,29/17) Hyperbolic Matrix(263,-98,314,-117) (10/27,3/8) -> (5/6,16/19) Hyperbolic Matrix(131,-50,338,-129) (3/8,5/13) -> (5/13,7/18) Parabolic Matrix(179,-74,254,-105) (2/5,5/12) -> (7/10,12/17) Hyperbolic Matrix(963,-404,584,-245) (13/31,8/19) -> (28/17,5/3) Hyperbolic Matrix(1165,-492,708,-299) (8/19,11/26) -> (23/14,28/17) Hyperbolic Matrix(2933,-1242,810,-343) (11/26,14/33) -> (76/21,29/8) Hyperbolic Matrix(175,-76,76,-33) (3/7,4/9) -> (16/7,7/3) Hyperbolic Matrix(25,-12,48,-23) (5/11,1/2) -> (1/2,7/13) Parabolic Matrix(2033,-1100,560,-303) (7/13,13/24) -> (29/8,69/19) Hyperbolic Matrix(2777,-1506,1938,-1051) (13/24,19/35) -> (53/37,43/30) Hyperbolic Matrix(933,-508,652,-355) (19/35,6/11) -> (10/7,53/37) Hyperbolic Matrix(303,-166,418,-229) (6/11,5/9) -> (21/29,8/11) Hyperbolic Matrix(207,-116,116,-65) (5/9,4/7) -> (16/9,9/5) Hyperbolic Matrix(523,-304,160,-93) (11/19,7/12) -> (13/4,23/7) Hyperbolic Matrix(499,-292,364,-213) (7/12,10/17) -> (26/19,11/8) Hyperbolic Matrix(159,-94,22,-13) (10/17,3/5) -> (7/1,8/1) Hyperbolic Matrix(963,-584,404,-245) (3/5,14/23) -> (50/21,31/13) Hyperbolic Matrix(2077,-1272,1272,-779) (11/18,19/31) -> (31/19,49/30) Hyperbolic Matrix(1383,-848,380,-233) (19/31,8/13) -> (40/11,11/3) Hyperbolic Matrix(1669,-1032,600,-371) (21/34,13/21) -> (25/9,39/14) Hyperbolic Matrix(4175,-2594,1154,-717) (18/29,23/37) -> (47/13,76/21) Hyperbolic Matrix(1087,-676,1288,-801) (23/37,5/8) -> (27/32,11/13) Hyperbolic Matrix(155,-98,242,-153) (5/8,7/11) -> (7/11,9/14) Parabolic Matrix(85,-58,22,-15) (2/3,7/10) -> (15/4,4/1) Hyperbolic Matrix(499,-364,292,-213) (8/11,11/15) -> (29/17,12/7) Hyperbolic Matrix(933,-722,650,-503) (17/22,7/9) -> (33/23,23/16) Hyperbolic Matrix(81,-64,100,-79) (7/9,4/5) -> (4/5,9/11) Parabolic Matrix(1027,-844,376,-309) (9/11,14/17) -> (30/11,41/15) Hyperbolic Matrix(563,-480,156,-133) (11/13,6/7) -> (18/5,47/13) Hyperbolic Matrix(303,-274,94,-85) (8/9,1/1) -> (29/9,42/13) Hyperbolic Matrix(219,-248,68,-77) (1/1,8/7) -> (16/5,29/9) Hyperbolic Matrix(1289,-1518,546,-643) (20/17,13/11) -> (59/25,26/11) Hyperbolic Matrix(263,-314,98,-117) (13/11,6/5) -> (8/3,35/13) Hyperbolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(2005,-2568,552,-707) (32/25,9/7) -> (69/19,40/11) Hyperbolic Matrix(879,-1136,236,-305) (9/7,22/17) -> (26/7,41/11) Hyperbolic Matrix(531,-692,188,-245) (13/10,17/13) -> (31/11,17/6) Hyperbolic Matrix(259,-354,30,-41) (15/11,26/19) -> (8/1,9/1) Hyperbolic Matrix(303,-418,166,-229) (11/8,18/13) -> (20/11,11/6) Hyperbolic Matrix(179,-254,74,-105) (7/5,10/7) -> (12/5,17/7) Hyperbolic Matrix(155,-242,98,-153) (3/2,11/7) -> (11/7,19/12) Parabolic Matrix(1601,-2574,594,-955) (8/5,37/23) -> (35/13,62/23) Hyperbolic Matrix(1821,-2932,772,-1243) (37/23,29/18) -> (33/14,59/25) Hyperbolic Matrix(343,-556,124,-201) (21/13,13/8) -> (11/4,25/9) Hyperbolic Matrix(713,-1160,260,-423) (13/8,44/27) -> (52/19,11/4) Hyperbolic Matrix(1955,-3194,314,-513) (49/30,67/41) -> (31/5,25/4) Hyperbolic Matrix(587,-960,96,-157) (67/41,18/11) -> (6/1,31/5) Hyperbolic Matrix(681,-1156,400,-679) (22/13,17/10) -> (17/10,46/27) Parabolic Matrix(119,-206,26,-45) (12/7,7/4) -> (9/2,14/3) Hyperbolic Matrix(599,-1058,338,-597) (7/4,23/13) -> (23/13,39/22) Parabolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(1169,-2540,272,-591) (13/6,37/17) -> (73/17,43/10) Hyperbolic Matrix(1313,-2862,306,-667) (37/17,24/11) -> (30/7,73/17) Hyperbolic Matrix(711,-1556,260,-569) (24/11,11/5) -> (41/15,52/19) Hyperbolic Matrix(123,-274,22,-49) (11/5,9/4) -> (11/2,17/3) Hyperbolic Matrix(551,-1250,242,-549) (9/4,25/11) -> (25/11,41/18) Parabolic Matrix(609,-1444,256,-607) (26/11,19/8) -> (19/8,50/21) Parabolic Matrix(131,-338,50,-129) (5/2,13/5) -> (13/5,21/8) Parabolic Matrix(97,-262,10,-27) (27/10,19/7) -> (9/1,1/0) Hyperbolic Matrix(1247,-3396,300,-817) (49/18,79/29) -> (29/7,25/6) Hyperbolic Matrix(435,-1186,106,-289) (79/29,30/11) -> (4/1,29/7) Hyperbolic Matrix(1017,-2834,314,-875) (39/14,53/19) -> (55/17,13/4) Hyperbolic Matrix(1073,-2996,332,-927) (53/19,14/5) -> (42/13,55/17) Hyperbolic Matrix(57,-196,16,-55) (10/3,7/2) -> (7/2,18/5) Parabolic Matrix(525,-1958,122,-455) (41/11,15/4) -> (43/10,13/3) Hyperbolic Matrix(219,-934,34,-145) (17/4,30/7) -> (32/5,13/2) Hyperbolic Matrix(31,-150,6,-29) (14/3,5/1) -> (5/1,16/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(335,156,204,95) -> Matrix(3,-4,-2,3) Matrix(265,122,202,93) -> Matrix(1,-2,2,-3) Matrix(457,206,-1178,-531) -> Matrix(1,2,0,1) Matrix(325,146,394,177) -> Matrix(1,0,2,1) Matrix(581,256,320,141) -> Matrix(1,0,-2,1) Matrix(447,196,-1024,-449) -> Matrix(7,-6,6,-5) Matrix(955,416,4084,1779) -> Matrix(3,-4,-2,3) Matrix(4465,1944,1656,721) -> Matrix(13,-16,-4,5) Matrix(65,28,188,81) -> Matrix(3,-4,-2,3) Matrix(61,26,190,81) -> Matrix(1,-4,0,1) Matrix(1311,554,310,131) -> Matrix(1,-2,0,1) Matrix(687,290,1618,683) -> Matrix(3,-8,-4,11) Matrix(495,208,188,79) -> Matrix(1,-8,0,1) Matrix(183,76,248,103) -> Matrix(1,0,-2,1) Matrix(185,76,-796,-327) -> Matrix(1,0,2,1) Matrix(549,224,424,173) -> Matrix(1,0,2,1) Matrix(653,256,176,69) -> Matrix(1,0,-2,1) Matrix(297,116,-1124,-439) -> Matrix(1,0,0,1) Matrix(1647,638,586,227) -> Matrix(1,-4,0,1) Matrix(1117,432,468,181) -> Matrix(1,-4,0,1) Matrix(407,156,60,23) -> Matrix(3,-4,-2,3) Matrix(351,134,406,155) -> Matrix(1,-2,0,1) Matrix(463,176,292,111) -> Matrix(1,-6,0,1) Matrix(287,106,398,147) -> Matrix(1,2,-2,-3) Matrix(283,104,400,147) -> Matrix(1,2,-2,-3) Matrix(337,122,58,21) -> Matrix(3,2,-2,-1) Matrix(167,60,732,263) -> Matrix(1,0,2,1) Matrix(113,40,500,177) -> Matrix(1,0,-4,1) Matrix(275,96,444,155) -> Matrix(1,-2,0,1) Matrix(423,134,262,83) -> Matrix(1,-2,0,1) Matrix(469,146,106,33) -> Matrix(1,0,2,1) Matrix(675,206,154,47) -> Matrix(1,0,-4,1) Matrix(257,78,570,173) -> Matrix(1,0,-4,1) Matrix(255,76,104,31) -> Matrix(1,0,-2,1) Matrix(355,104,256,75) -> Matrix(3,-2,2,-1) Matrix(99,28,152,43) -> Matrix(3,-2,-4,3) Matrix(347,96,300,83) -> Matrix(1,-2,0,1) Matrix(51,14,346,95) -> Matrix(1,-4,0,1) Matrix(393,106,938,253) -> Matrix(3,-4,-2,3) Matrix(97,26,-638,-171) -> Matrix(1,-2,2,-3) Matrix(1129,300,636,169) -> Matrix(1,-2,0,1) Matrix(489,128,340,89) -> Matrix(1,0,0,1) Matrix(329,78,426,101) -> Matrix(1,0,-4,1) Matrix(1177,276,516,121) -> Matrix(1,0,2,1) Matrix(187,42,138,31) -> Matrix(1,0,0,1) Matrix(91,20,232,51) -> Matrix(3,-2,-4,3) Matrix(1187,252,504,107) -> Matrix(5,-8,2,-3) Matrix(227,48,960,203) -> Matrix(1,-2,0,1) Matrix(137,28,44,9) -> Matrix(3,-4,-2,3) Matrix(133,26,46,9) -> Matrix(1,-4,0,1) Matrix(87,16,-484,-89) -> Matrix(1,-6,0,1) Matrix(1663,296,1972,351) -> Matrix(1,2,0,1) Matrix(1399,248,220,39) -> Matrix(3,8,-2,-5) Matrix(347,60,480,83) -> Matrix(1,2,-2,-3) Matrix(383,60,300,47) -> Matrix(1,0,2,1) Matrix(421,62,550,81) -> Matrix(1,0,-4,1) Matrix(125,16,164,21) -> Matrix(1,-2,-2,5) Matrix(123,14,202,23) -> Matrix(1,-4,0,1) Matrix(159,-22,94,-13) -> Matrix(1,4,0,1) Matrix(219,-34,934,-145) -> Matrix(1,2,0,1) Matrix(589,-94,94,-15) -> Matrix(7,16,-4,-9) Matrix(123,-22,274,-49) -> Matrix(1,2,-2,-3) Matrix(303,-56,92,-17) -> Matrix(1,0,0,1) Matrix(119,-26,206,-45) -> Matrix(3,2,-2,-1) Matrix(1549,-360,1080,-251) -> Matrix(1,0,0,1) Matrix(609,-146,146,-35) -> Matrix(1,0,4,1) Matrix(85,-22,58,-15) -> Matrix(1,0,0,1) Matrix(57,-16,196,-55) -> Matrix(1,-6,0,1) Matrix(1585,-468,972,-287) -> Matrix(1,2,0,1) Matrix(389,-116,332,-99) -> Matrix(1,4,-2,-7) Matrix(303,-92,56,-17) -> Matrix(1,0,0,1) Matrix(219,-68,248,-77) -> Matrix(1,2,0,1) Matrix(531,-188,692,-245) -> Matrix(1,2,-4,-7) Matrix(343,-124,556,-201) -> Matrix(1,0,0,1) Matrix(1213,-444,948,-347) -> Matrix(1,2,2,5) Matrix(1501,-552,552,-203) -> Matrix(9,16,-4,-7) Matrix(1657,-612,972,-359) -> Matrix(13,20,-2,-3) Matrix(263,-98,314,-117) -> Matrix(3,4,2,3) Matrix(131,-50,338,-129) -> Matrix(9,10,-10,-11) Matrix(179,-74,254,-105) -> Matrix(1,2,-2,-3) Matrix(963,-404,584,-245) -> Matrix(3,4,-4,-5) Matrix(1165,-492,708,-299) -> Matrix(9,8,-8,-7) Matrix(2933,-1242,810,-343) -> Matrix(3,2,-2,-1) Matrix(175,-76,76,-33) -> Matrix(1,0,2,1) Matrix(25,-12,48,-23) -> Matrix(1,0,0,1) Matrix(2033,-1100,560,-303) -> Matrix(3,-2,-4,3) Matrix(2777,-1506,1938,-1051) -> Matrix(1,-2,0,1) Matrix(933,-508,652,-355) -> Matrix(1,0,0,1) Matrix(303,-166,418,-229) -> Matrix(1,2,-2,-3) Matrix(207,-116,116,-65) -> Matrix(1,0,0,1) Matrix(523,-304,160,-93) -> Matrix(1,0,0,1) Matrix(499,-292,364,-213) -> Matrix(3,4,2,3) Matrix(159,-94,22,-13) -> Matrix(5,4,-4,-3) Matrix(963,-584,404,-245) -> Matrix(1,-4,0,1) Matrix(2077,-1272,1272,-779) -> Matrix(5,8,-2,-3) Matrix(1383,-848,380,-233) -> Matrix(3,4,-4,-5) Matrix(1669,-1032,600,-371) -> Matrix(1,0,0,1) Matrix(4175,-2594,1154,-717) -> Matrix(3,4,-4,-5) Matrix(1087,-676,1288,-801) -> Matrix(1,2,-2,-3) Matrix(155,-98,242,-153) -> Matrix(7,8,-8,-9) Matrix(85,-58,22,-15) -> Matrix(1,0,0,1) Matrix(499,-364,292,-213) -> Matrix(7,4,-2,-1) Matrix(933,-722,650,-503) -> Matrix(1,0,4,1) Matrix(81,-64,100,-79) -> Matrix(1,0,6,1) Matrix(1027,-844,376,-309) -> Matrix(5,-2,-2,1) Matrix(563,-480,156,-133) -> Matrix(3,-2,-4,3) Matrix(303,-274,94,-85) -> Matrix(3,2,-2,-1) Matrix(219,-248,68,-77) -> Matrix(3,2,-2,-1) Matrix(1289,-1518,546,-643) -> Matrix(9,4,2,1) Matrix(263,-314,98,-117) -> Matrix(7,4,-2,-1) Matrix(81,-100,64,-79) -> Matrix(1,0,6,1) Matrix(2005,-2568,552,-707) -> Matrix(7,-2,-10,3) Matrix(879,-1136,236,-305) -> Matrix(1,0,-4,1) Matrix(531,-692,188,-245) -> Matrix(5,-2,-2,1) Matrix(259,-354,30,-41) -> Matrix(5,-4,-6,5) Matrix(303,-418,166,-229) -> Matrix(1,-2,0,1) Matrix(179,-254,74,-105) -> Matrix(1,-2,0,1) Matrix(155,-242,98,-153) -> Matrix(1,-8,0,1) Matrix(1601,-2574,594,-955) -> Matrix(7,18,-2,-5) Matrix(1821,-2932,772,-1243) -> Matrix(1,6,0,1) Matrix(343,-556,124,-201) -> Matrix(1,0,0,1) Matrix(713,-1160,260,-423) -> Matrix(1,0,0,1) Matrix(1955,-3194,314,-513) -> Matrix(15,32,-8,-17) Matrix(587,-960,96,-157) -> Matrix(5,8,-2,-3) Matrix(681,-1156,400,-679) -> Matrix(1,-12,0,1) Matrix(119,-206,26,-45) -> Matrix(1,2,0,1) Matrix(599,-1058,338,-597) -> Matrix(1,0,0,1) Matrix(25,-48,12,-23) -> Matrix(1,0,0,1) Matrix(1169,-2540,272,-591) -> Matrix(1,0,0,1) Matrix(1313,-2862,306,-667) -> Matrix(3,2,-2,-1) Matrix(711,-1556,260,-569) -> Matrix(3,2,-2,-1) Matrix(123,-274,22,-49) -> Matrix(1,-2,0,1) Matrix(551,-1250,242,-549) -> Matrix(1,0,6,1) Matrix(609,-1444,256,-607) -> Matrix(1,-12,0,1) Matrix(131,-338,50,-129) -> Matrix(1,-10,0,1) Matrix(97,-262,10,-27) -> Matrix(1,4,-2,-7) Matrix(1247,-3396,300,-817) -> Matrix(1,2,10,21) Matrix(435,-1186,106,-289) -> Matrix(1,2,-4,-7) Matrix(1017,-2834,314,-875) -> Matrix(3,2,-2,-1) Matrix(1073,-2996,332,-927) -> Matrix(3,2,-2,-1) Matrix(57,-196,16,-55) -> Matrix(5,6,-6,-7) Matrix(525,-1958,122,-455) -> Matrix(1,0,0,1) Matrix(219,-934,34,-145) -> Matrix(3,2,-2,-1) Matrix(31,-150,6,-29) -> Matrix(1,-2,0,1) 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): 36 Degree of the the map X: 36 Degree of the the map Y: 144 ----------------------------------------------------------------------- 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): 432 Minimal number of generators: 73 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: 40 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/5 2/7 1/3 1/2 5/7 4/5 11/13 1/1 13/11 5/4 7/5 11/7 67/41 5/3 17/10 23/13 2/1 37/17 11/5 25/11 7/3 13/5 79/29 11/4 53/19 3/1 7/2 11/3 4/1 17/4 13/3 9/2 14/3 5/1 11/2 6/1 7/1 8/1 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 1/7 -4/1 1/6 -1/1 2/11 -3/2 1/5 -1/1 2/9 -1/2 3/13 0/1 7/30 -1/1 4/17 -1/2 1/0 1/4 1/1 3/11 0/1 2/1 2/7 1/0 5/17 -6/1 -4/1 8/27 1/0 3/10 -3/1 4/13 1/0 1/3 -2/1 5/14 -1/1 4/11 -3/2 1/0 11/30 -3/1 7/19 -2/1 -8/5 10/27 -3/2 3/8 -1/1 5/13 -1/1 2/5 -1/2 5/12 -1/1 8/19 -1/1 11/26 -1/1 3/7 -2/3 0/1 4/9 1/0 5/11 0/1 1/2 -1/1 1/1 7/13 0/1 13/24 1/1 19/35 1/0 6/11 1/0 5/9 -2/1 0/1 4/7 1/0 11/19 -1/1 7/12 -1/1 10/17 -1/1 3/5 0/1 11/18 -3/1 19/31 -2/1 -4/3 8/13 -3/2 1/0 21/34 -1/1 13/21 -2/1 0/1 5/8 -1/1 7/11 -1/1 2/3 -1/2 7/10 -1/1 5/7 -2/3 0/1 8/11 -1/2 3/4 -1/3 10/13 -1/4 7/9 0/1 4/5 0/1 9/11 0/1 5/6 1/1 16/19 1/0 11/13 0/1 6/7 1/0 1/1 -2/1 0/1 7/6 -1/1 20/17 -1/2 13/11 0/1 6/5 -1/2 5/4 0/1 14/11 1/4 9/7 0/1 13/10 1/3 4/3 1/2 11/8 1/1 7/5 0/1 2/1 10/7 1/0 13/9 0/1 2/1 3/2 1/1 11/7 1/0 8/5 1/0 21/13 -2/1 0/1 13/8 -3/1 -1/1 44/27 1/0 31/19 -4/1 -2/1 49/30 -7/3 67/41 -2/1 18/11 -3/2 5/3 0/1 22/13 1/0 17/10 1/0 12/7 1/0 7/4 -1/1 23/13 -2/1 0/1 16/9 1/0 9/5 -2/1 0/1 11/6 -1/1 2/1 -1/2 1/0 13/6 -1/1 37/17 -1/1 24/11 -1/2 11/5 0/1 9/4 -1/1 25/11 0/1 16/7 1/2 7/3 0/1 2/1 26/11 1/0 19/8 1/0 12/5 1/0 5/2 1/1 13/5 1/0 8/3 1/0 35/13 -4/1 27/10 -3/1 19/7 -8/3 -2/1 49/18 -11/5 79/29 -2/1 30/11 -3/2 11/4 -3/1 -1/1 25/9 -2/1 0/1 39/14 -1/1 53/19 -2/1 0/1 14/5 1/0 3/1 -2/1 13/4 -1/1 23/7 -1/1 10/3 -3/2 7/2 -1/1 18/5 -3/4 47/13 -2/3 29/8 -1/1 40/11 -3/4 -1/2 11/3 -2/3 0/1 15/4 -1/1 4/1 -1/2 17/4 -1/1 1/1 30/7 1/0 13/3 0/1 9/2 1/1 14/3 1/0 5/1 1/0 16/3 1/0 11/2 -3/1 6/1 1/0 7/1 -4/3 8/1 -1/1 1/0 -1/1 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(159,-22,94,-13) (0/1,1/7) -> (5/3,22/13) Hyperbolic Matrix(156,-23,95,-14) (1/7,1/6) -> (18/11,5/3) Hyperbolic Matrix(122,-21,93,-16) (1/6,2/11) -> (13/10,4/3) Hyperbolic Matrix(303,-56,92,-17) (2/11,1/5) -> (23/7,10/3) Hyperbolic Matrix(119,-26,206,-45) (1/5,2/9) -> (4/7,11/19) Hyperbolic Matrix(146,-33,177,-40) (2/9,3/13) -> (9/11,5/6) Hyperbolic Matrix(642,-149,293,-68) (3/13,7/30) -> (24/11,11/5) Hyperbolic Matrix(1138,-267,699,-164) (7/30,4/17) -> (13/8,44/27) Hyperbolic Matrix(554,-131,203,-48) (4/17,1/4) -> (30/11,11/4) Hyperbolic Matrix(85,-22,58,-15) (1/4,3/11) -> (13/9,3/2) Hyperbolic Matrix(57,-16,196,-55) (3/11,2/7) -> (2/7,5/17) Parabolic Matrix(1585,-468,972,-287) (5/17,8/27) -> (44/27,31/19) Hyperbolic Matrix(389,-116,332,-99) (8/27,3/10) -> (7/6,20/17) Hyperbolic Matrix(303,-92,56,-17) (3/10,4/13) -> (16/3,11/2) Hyperbolic Matrix(28,-9,81,-26) (4/13,1/3) -> (1/3,5/14) Parabolic Matrix(343,-124,556,-201) (5/14,4/11) -> (8/13,21/34) Hyperbolic Matrix(554,-203,131,-48) (4/11,11/30) -> (4/1,17/4) Hyperbolic Matrix(1501,-552,552,-203) (11/30,7/19) -> (19/7,49/18) Hyperbolic Matrix(290,-107,683,-252) (7/19,10/27) -> (11/26,3/7) Hyperbolic Matrix(263,-98,314,-117) (10/27,3/8) -> (5/6,16/19) Hyperbolic Matrix(208,-79,79,-30) (3/8,5/13) -> (13/5,8/3) Hyperbolic Matrix(130,-51,51,-20) (5/13,2/5) -> (5/2,13/5) Hyperbolic Matrix(76,-31,103,-42) (2/5,5/12) -> (8/11,3/4) Hyperbolic Matrix(432,-181,253,-106) (5/12,8/19) -> (17/10,12/7) Hyperbolic Matrix(860,-363,507,-214) (8/19,11/26) -> (22/13,17/10) Hyperbolic Matrix(175,-76,76,-33) (3/7,4/9) -> (16/7,7/3) Hyperbolic Matrix(224,-101,173,-78) (4/9,5/11) -> (9/7,13/10) Hyperbolic Matrix(25,-12,48,-23) (5/11,1/2) -> (1/2,7/13) Parabolic Matrix(702,-379,163,-88) (7/13,13/24) -> (30/7,13/3) Hyperbolic Matrix(1728,-937,793,-430) (13/24,19/35) -> (37/17,24/11) Hyperbolic Matrix(862,-469,397,-216) (19/35,6/11) -> (13/6,37/17) Hyperbolic Matrix(256,-141,69,-38) (6/11,5/9) -> (11/3,15/4) Hyperbolic Matrix(207,-116,116,-65) (5/9,4/7) -> (16/9,9/5) Hyperbolic Matrix(523,-304,160,-93) (11/19,7/12) -> (13/4,23/7) Hyperbolic Matrix(432,-253,181,-106) (7/12,10/17) -> (19/8,12/5) Hyperbolic Matrix(159,-94,22,-13) (10/17,3/5) -> (7/1,8/1) Hyperbolic Matrix(156,-95,23,-14) (3/5,11/18) -> (6/1,7/1) Hyperbolic Matrix(2077,-1272,1272,-779) (11/18,19/31) -> (31/19,49/30) Hyperbolic Matrix(1383,-848,380,-233) (19/31,8/13) -> (40/11,11/3) Hyperbolic Matrix(1669,-1032,600,-371) (21/34,13/21) -> (25/9,39/14) Hyperbolic Matrix(134,-83,155,-96) (13/21,5/8) -> (6/7,1/1) Hyperbolic Matrix(176,-111,111,-70) (5/8,7/11) -> (11/7,8/5) Hyperbolic Matrix(66,-43,43,-28) (7/11,2/3) -> (3/2,11/7) Hyperbolic Matrix(85,-58,22,-15) (2/3,7/10) -> (15/4,4/1) Hyperbolic Matrix(106,-75,147,-104) (7/10,5/7) -> (5/7,8/11) Parabolic Matrix(122,-93,21,-16) (3/4,10/13) -> (11/2,6/1) Hyperbolic Matrix(224,-173,101,-78) (10/13,7/9) -> (11/5,9/4) Hyperbolic Matrix(81,-64,100,-79) (7/9,4/5) -> (4/5,9/11) Parabolic Matrix(1270,-1071,351,-296) (16/19,11/13) -> (47/13,29/8) Hyperbolic Matrix(563,-480,156,-133) (11/13,6/7) -> (18/5,47/13) Hyperbolic Matrix(134,-155,83,-96) (1/1,7/6) -> (8/5,21/13) Hyperbolic Matrix(892,-1051,331,-390) (20/17,13/11) -> (35/13,27/10) Hyperbolic Matrix(263,-314,98,-117) (13/11,6/5) -> (8/3,35/13) Hyperbolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(206,-263,47,-60) (14/11,9/7) -> (13/3,9/2) Hyperbolic Matrix(76,-103,31,-42) (4/3,11/8) -> (12/5,5/2) Hyperbolic Matrix(106,-147,75,-104) (11/8,7/5) -> (7/5,10/7) Parabolic Matrix(162,-233,89,-128) (10/7,13/9) -> (9/5,11/6) Hyperbolic Matrix(343,-556,124,-201) (21/13,13/8) -> (11/4,25/9) Hyperbolic Matrix(2748,-4489,1681,-2746) (49/30,67/41) -> (67/41,18/11) Parabolic Matrix(119,-206,26,-45) (12/7,7/4) -> (9/2,14/3) Hyperbolic Matrix(300,-529,169,-298) (7/4,23/13) -> (23/13,16/9) Parabolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(276,-625,121,-274) (9/4,25/11) -> (25/11,16/7) Parabolic Matrix(290,-683,107,-252) (7/3,26/11) -> (27/10,19/7) Hyperbolic Matrix(96,-227,11,-26) (26/11,19/8) -> (8/1,1/0) Hyperbolic Matrix(2292,-6241,841,-2290) (49/18,79/29) -> (79/29,30/11) Parabolic Matrix(1008,-2809,361,-1006) (39/14,53/19) -> (53/19,14/5) Parabolic Matrix(28,-81,9,-26) (14/5,3/1) -> (3/1,13/4) Parabolic Matrix(57,-196,16,-55) (10/3,7/2) -> (7/2,18/5) Parabolic Matrix(466,-1693,109,-396) (29/8,40/11) -> (17/4,30/7) Hyperbolic Matrix(31,-150,6,-29) (14/3,5/1) -> (5/1,16/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,1,1) Matrix(159,-22,94,-13) -> Matrix(1,4,0,1) Matrix(156,-23,95,-14) -> Matrix(1,4,-1,-3) Matrix(122,-21,93,-16) -> Matrix(1,2,1,3) Matrix(303,-56,92,-17) -> Matrix(1,0,0,1) Matrix(119,-26,206,-45) -> Matrix(3,2,-2,-1) Matrix(146,-33,177,-40) -> Matrix(1,0,3,1) Matrix(642,-149,293,-68) -> Matrix(1,0,-1,1) Matrix(1138,-267,699,-164) -> Matrix(1,2,-1,-1) Matrix(554,-131,203,-48) -> Matrix(1,2,-1,-1) Matrix(85,-22,58,-15) -> Matrix(1,0,0,1) Matrix(57,-16,196,-55) -> Matrix(1,-6,0,1) Matrix(1585,-468,972,-287) -> Matrix(1,2,0,1) Matrix(389,-116,332,-99) -> Matrix(1,4,-2,-7) Matrix(303,-92,56,-17) -> Matrix(1,0,0,1) Matrix(28,-9,81,-26) -> Matrix(1,4,-1,-3) Matrix(343,-124,556,-201) -> Matrix(1,0,0,1) Matrix(554,-203,131,-48) -> Matrix(1,2,-1,-1) Matrix(1501,-552,552,-203) -> Matrix(9,16,-4,-7) Matrix(290,-107,683,-252) -> Matrix(5,8,-7,-11) Matrix(263,-98,314,-117) -> Matrix(3,4,2,3) Matrix(208,-79,79,-30) -> Matrix(7,8,-1,-1) Matrix(130,-51,51,-20) -> Matrix(3,2,1,1) Matrix(76,-31,103,-42) -> Matrix(1,0,-1,1) Matrix(432,-181,253,-106) -> Matrix(3,4,-1,-1) Matrix(860,-363,507,-214) -> Matrix(9,8,1,1) Matrix(175,-76,76,-33) -> Matrix(1,0,2,1) Matrix(224,-101,173,-78) -> Matrix(1,0,3,1) Matrix(25,-12,48,-23) -> Matrix(1,0,0,1) Matrix(702,-379,163,-88) -> Matrix(1,0,-1,1) Matrix(1728,-937,793,-430) -> Matrix(1,-2,-1,3) Matrix(862,-469,397,-216) -> Matrix(1,0,-1,1) Matrix(256,-141,69,-38) -> Matrix(1,0,-1,1) Matrix(207,-116,116,-65) -> Matrix(1,0,0,1) Matrix(523,-304,160,-93) -> Matrix(1,0,0,1) Matrix(432,-253,181,-106) -> Matrix(3,4,-1,-1) Matrix(159,-94,22,-13) -> Matrix(5,4,-4,-3) Matrix(156,-95,23,-14) -> Matrix(1,4,-1,-3) Matrix(2077,-1272,1272,-779) -> Matrix(5,8,-2,-3) Matrix(1383,-848,380,-233) -> Matrix(3,4,-4,-5) Matrix(1669,-1032,600,-371) -> Matrix(1,0,0,1) Matrix(134,-83,155,-96) -> Matrix(1,2,-1,-1) Matrix(176,-111,111,-70) -> Matrix(5,6,-1,-1) Matrix(66,-43,43,-28) -> Matrix(3,2,1,1) Matrix(85,-58,22,-15) -> Matrix(1,0,0,1) Matrix(106,-75,147,-104) -> Matrix(3,2,-5,-3) Matrix(122,-93,21,-16) -> Matrix(5,2,-3,-1) Matrix(224,-173,101,-78) -> Matrix(1,0,3,1) Matrix(81,-64,100,-79) -> Matrix(1,0,6,1) Matrix(1270,-1071,351,-296) -> Matrix(1,-2,-1,3) Matrix(563,-480,156,-133) -> Matrix(3,-2,-4,3) Matrix(134,-155,83,-96) -> Matrix(1,2,-1,-1) Matrix(892,-1051,331,-390) -> Matrix(11,4,-3,-1) Matrix(263,-314,98,-117) -> Matrix(7,4,-2,-1) Matrix(81,-100,64,-79) -> Matrix(1,0,6,1) Matrix(206,-263,47,-60) -> Matrix(1,0,-3,1) Matrix(76,-103,31,-42) -> Matrix(1,0,-1,1) Matrix(106,-147,75,-104) -> Matrix(1,-2,1,-1) Matrix(162,-233,89,-128) -> Matrix(1,0,-1,1) Matrix(343,-556,124,-201) -> Matrix(1,0,0,1) Matrix(2748,-4489,1681,-2746) -> Matrix(9,20,-5,-11) Matrix(119,-206,26,-45) -> Matrix(1,2,0,1) Matrix(300,-529,169,-298) -> Matrix(1,2,-1,-1) Matrix(25,-48,12,-23) -> Matrix(1,0,0,1) Matrix(276,-625,121,-274) -> Matrix(1,0,3,1) Matrix(290,-683,107,-252) -> Matrix(3,-8,-1,3) Matrix(96,-227,11,-26) -> Matrix(1,-4,-1,5) Matrix(2292,-6241,841,-2290) -> Matrix(13,28,-7,-15) Matrix(1008,-2809,361,-1006) -> Matrix(1,2,-1,-1) Matrix(28,-81,9,-26) -> Matrix(1,4,-1,-3) Matrix(57,-196,16,-55) -> Matrix(5,6,-6,-7) Matrix(466,-1693,109,-396) -> Matrix(3,2,1,1) Matrix(31,-150,6,-29) -> Matrix(1,-2,0,1) 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): 36 ----------------------------------------------------------------------- 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 1/1 (-2/1,0/1) 0 9 7/6 -1/1 1 18 13/11 0/1 1 3 6/5 -1/2 1 18 5/4 0/1 3 2 14/11 1/4 1 18 9/7 0/1 1 9 13/10 1/3 1 18 4/3 1/2 1 18 11/8 1/1 1 18 7/5 (0/1,2/1).(1/1,1/0) 0 3 10/7 1/0 1 18 3/2 1/1 1 18 11/7 1/0 4 1 8/5 1/0 1 18 21/13 (-2/1,0/1) 0 9 13/8 0 6 31/19 (-4/1,-2/1) 0 9 67/41 -2/1 5 1 18/11 -3/2 1 18 5/3 0/1 1 9 17/10 1/0 6 2 12/7 1/0 1 18 7/4 -1/1 1 18 23/13 (-2/1,0/1).(-1/1,1/0) 0 1 9/5 (-2/1,0/1) 0 9 11/6 -1/1 1 18 2/1 0 6 13/6 -1/1 1 18 37/17 -1/1 1 1 24/11 -1/2 1 18 11/5 0/1 1 9 9/4 -1/1 1 18 25/11 0/1 3 1 7/3 (0/1,2/1) 0 9 26/11 1/0 1 18 19/8 1/0 6 2 12/5 1/0 1 18 5/2 1/1 1 18 13/5 1/0 5 1 8/3 1/0 1 18 35/13 -4/1 1 3 27/10 -3/1 1 18 19/7 (-8/3,-2/1) 0 9 79/29 -2/1 7 1 30/11 -3/2 1 18 11/4 0 6 25/9 (-2/1,0/1) 0 9 53/19 (-2/1,0/1).(-1/1,1/0) 0 1 14/5 1/0 1 18 3/1 -2/1 1 9 13/4 -1/1 1 18 10/3 -3/2 1 18 7/2 -1/1 3 2 18/5 -3/4 1 18 47/13 -2/3 1 3 29/8 -1/1 1 18 40/11 0 6 11/3 (-2/3,0/1) 0 9 15/4 -1/1 1 18 4/1 -1/2 1 18 17/4 0 6 30/7 1/0 1 18 13/3 0/1 1 9 9/2 1/1 1 18 14/3 1/0 1 18 5/1 1/0 1 3 16/3 1/0 1 18 11/2 -3/1 1 18 6/1 1/0 1 18 7/1 -4/3 1 9 8/1 -1/1 6 2 1/0 -1/1 1 18 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(134,-155,83,-96) (1/1,7/6) -> (8/5,21/13) Hyperbolic Matrix(480,-563,133,-156) (7/6,13/11) -> (18/5,47/13) Glide Reflection Matrix(263,-314,98,-117) (13/11,6/5) -> (8/3,35/13) Hyperbolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(206,-263,47,-60) (14/11,9/7) -> (13/3,9/2) Hyperbolic Matrix(173,-224,78,-101) (9/7,13/10) -> (11/5,9/4) Glide Reflection Matrix(93,-122,16,-21) (13/10,4/3) -> (11/2,6/1) Glide Reflection Matrix(76,-103,31,-42) (4/3,11/8) -> (12/5,5/2) Hyperbolic Matrix(106,-147,75,-104) (11/8,7/5) -> (7/5,10/7) Parabolic Matrix(58,-85,15,-22) (10/7,3/2) -> (15/4,4/1) Glide Reflection Matrix(43,-66,28,-43) (3/2,11/7) -> (3/2,11/7) Reflection Matrix(111,-176,70,-111) (11/7,8/5) -> (11/7,8/5) Reflection Matrix(343,-556,124,-201) (21/13,13/8) -> (11/4,25/9) Hyperbolic Matrix(848,-1383,233,-380) (13/8,31/19) -> (40/11,11/3) Glide Reflection Matrix(1272,-2077,779,-1272) (31/19,67/41) -> (31/19,67/41) Reflection Matrix(1475,-2412,902,-1475) (67/41,18/11) -> (67/41,18/11) Reflection Matrix(95,-156,14,-23) (18/11,5/3) -> (6/1,7/1) Glide Reflection Matrix(94,-159,13,-22) (5/3,17/10) -> (7/1,8/1) Glide Reflection Matrix(253,-432,106,-181) (17/10,12/7) -> (19/8,12/5) Glide Reflection Matrix(119,-206,26,-45) (12/7,7/4) -> (9/2,14/3) Hyperbolic Matrix(183,-322,104,-183) (7/4,23/13) -> (7/4,23/13) Reflection Matrix(116,-207,65,-116) (23/13,9/5) -> (23/13,9/5) Reflection Matrix(141,-256,38,-69) (9/5,11/6) -> (11/3,15/4) Glide Reflection Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(443,-962,204,-443) (13/6,37/17) -> (13/6,37/17) Reflection Matrix(815,-1776,374,-815) (37/17,24/11) -> (37/17,24/11) Reflection Matrix(293,-642,68,-149) (24/11,11/5) -> (30/7,13/3) Glide Reflection Matrix(199,-450,88,-199) (9/4,25/11) -> (9/4,25/11) Reflection Matrix(76,-175,33,-76) (25/11,7/3) -> (25/11,7/3) Reflection Matrix(290,-683,107,-252) (7/3,26/11) -> (27/10,19/7) Hyperbolic Matrix(96,-227,11,-26) (26/11,19/8) -> (8/1,1/0) Hyperbolic Matrix(51,-130,20,-51) (5/2,13/5) -> (5/2,13/5) Reflection Matrix(79,-208,30,-79) (13/5,8/3) -> (13/5,8/3) Reflection Matrix(847,-2284,234,-631) (35/13,27/10) -> (47/13,29/8) Glide Reflection Matrix(552,-1501,203,-552) (19/7,79/29) -> (19/7,79/29) Reflection Matrix(1739,-4740,638,-1739) (79/29,30/11) -> (79/29,30/11) Reflection Matrix(203,-554,48,-131) (30/11,11/4) -> (4/1,17/4) Glide Reflection Matrix(476,-1325,171,-476) (25/9,53/19) -> (25/9,53/19) Reflection Matrix(531,-1484,190,-531) (53/19,14/5) -> (53/19,14/5) Reflection Matrix(28,-81,9,-26) (14/5,3/1) -> (3/1,13/4) Parabolic Matrix(92,-303,17,-56) (13/4,10/3) -> (16/3,11/2) Glide Reflection Matrix(57,-196,16,-55) (10/3,7/2) -> (7/2,18/5) Parabolic Matrix(466,-1693,109,-396) (29/8,40/11) -> (17/4,30/7) Hyperbolic Matrix(31,-150,6,-29) (14/3,5/1) -> (5/1,16/3) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(0,1,1,0) -> Matrix(-1,0,1,1) (-1/1,1/1) -> (-2/1,0/1) Matrix(134,-155,83,-96) -> Matrix(1,2,-1,-1) (-2/1,0/1).(-1/1,1/0) Matrix(480,-563,133,-156) -> Matrix(5,2,-7,-3) Matrix(263,-314,98,-117) -> Matrix(7,4,-2,-1) Matrix(81,-100,64,-79) -> Matrix(1,0,6,1) 0/1 Matrix(206,-263,47,-60) -> Matrix(1,0,-3,1) 0/1 Matrix(173,-224,78,-101) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(93,-122,16,-21) -> Matrix(3,-2,-2,1) Matrix(76,-103,31,-42) -> Matrix(1,0,-1,1) 0/1 Matrix(106,-147,75,-104) -> Matrix(1,-2,1,-1) (0/1,2/1).(1/1,1/0) Matrix(58,-85,15,-22) -> Matrix(-1,0,1,1) *** -> (-2/1,0/1) Matrix(43,-66,28,-43) -> Matrix(-1,2,0,1) (3/2,11/7) -> (1/1,1/0) Matrix(111,-176,70,-111) -> Matrix(1,6,0,-1) (11/7,8/5) -> (-3/1,1/0) Matrix(343,-556,124,-201) -> Matrix(1,0,0,1) Matrix(848,-1383,233,-380) -> Matrix(1,4,-1,-5) Matrix(1272,-2077,779,-1272) -> Matrix(3,8,-1,-3) (31/19,67/41) -> (-4/1,-2/1) Matrix(1475,-2412,902,-1475) -> Matrix(7,12,-4,-7) (67/41,18/11) -> (-2/1,-3/2) Matrix(95,-156,14,-23) -> Matrix(3,4,-2,-3) *** -> (-2/1,-1/1) Matrix(94,-159,13,-22) -> Matrix(1,-4,-1,3) Matrix(253,-432,106,-181) -> Matrix(1,4,0,-1) *** -> (-2/1,1/0) Matrix(119,-206,26,-45) -> Matrix(1,2,0,1) 1/0 Matrix(183,-322,104,-183) -> Matrix(1,2,0,-1) (7/4,23/13) -> (-1/1,1/0) Matrix(116,-207,65,-116) -> Matrix(-1,0,1,1) (23/13,9/5) -> (-2/1,0/1) Matrix(141,-256,38,-69) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(25,-48,12,-23) -> Matrix(1,0,0,1) Matrix(443,-962,204,-443) -> Matrix(-1,0,2,1) (13/6,37/17) -> (-1/1,0/1) Matrix(815,-1776,374,-815) -> Matrix(3,2,-4,-3) (37/17,24/11) -> (-1/1,-1/2) Matrix(293,-642,68,-149) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(199,-450,88,-199) -> Matrix(-1,0,2,1) (9/4,25/11) -> (-1/1,0/1) Matrix(76,-175,33,-76) -> Matrix(1,0,1,-1) (25/11,7/3) -> (0/1,2/1) Matrix(290,-683,107,-252) -> Matrix(3,-8,-1,3) Matrix(96,-227,11,-26) -> Matrix(1,-4,-1,5) Matrix(51,-130,20,-51) -> Matrix(-1,2,0,1) (5/2,13/5) -> (1/1,1/0) Matrix(79,-208,30,-79) -> Matrix(1,8,0,-1) (13/5,8/3) -> (-4/1,1/0) Matrix(847,-2284,234,-631) -> Matrix(1,2,-2,-5) Matrix(552,-1501,203,-552) -> Matrix(7,16,-3,-7) (19/7,79/29) -> (-8/3,-2/1) Matrix(1739,-4740,638,-1739) -> Matrix(7,12,-4,-7) (79/29,30/11) -> (-2/1,-3/2) Matrix(203,-554,48,-131) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(476,-1325,171,-476) -> Matrix(-1,0,1,1) (25/9,53/19) -> (-2/1,0/1) Matrix(531,-1484,190,-531) -> Matrix(1,2,0,-1) (53/19,14/5) -> (-1/1,1/0) Matrix(28,-81,9,-26) -> Matrix(1,4,-1,-3) -2/1 Matrix(92,-303,17,-56) -> Matrix(-1,0,1,1) *** -> (-2/1,0/1) Matrix(57,-196,16,-55) -> Matrix(5,6,-6,-7) -1/1 Matrix(466,-1693,109,-396) -> Matrix(3,2,1,1) Matrix(31,-150,6,-29) -> Matrix(1,-2,0,1) 1/0 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.