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): 432 Minimal number of generators: 73 Number of equivalence classes of cusps: 32 Genus: 21 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -2/5 -1/4 0/1 1/5 2/7 1/2 5/7 4/5 1/1 5/4 7/5 3/2 11/7 23/13 2/1 37/17 25/11 7/3 19/8 5/2 13/5 3/1 7/2 11/3 4/1 13/3 5/1 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 -1/2 0/1 1/1 -5/11 1/1 -4/9 1/1 2/1 -3/7 0/1 1/0 -5/12 0/1 1/2 -7/17 1/2 -9/22 1/1 2/1 -2/5 0/1 2/1 -9/23 1/0 -7/18 -2/1 1/0 -5/13 0/1 1/0 -3/8 0/1 1/1 -7/19 0/1 -4/11 0/1 1/2 -1/3 1/0 -3/10 -1/2 0/1 -5/17 0/1 -2/7 -1/1 0/1 -5/18 0/1 1/1 -3/11 0/1 1/0 -4/15 2/1 1/0 -9/34 -4/1 1/0 -5/19 1/0 -1/4 -2/1 0/1 -4/17 -2/1 -1/1 -7/30 -1/1 -4/5 -3/13 -1/2 -2/9 -1/2 0/1 -3/14 0/1 1/2 -1/5 0/1 1/0 -1/6 -2/1 -1/1 -1/7 -1/1 0/1 -1/1 0/1 1/7 0/1 1/0 1/6 -2/1 1/0 2/11 -1/1 0/1 1/5 -1/1 2/9 -1/2 0/1 1/4 -1/1 -2/3 3/11 -1/2 2/7 -1/2 5/17 -1/2 3/10 -2/5 -1/3 1/3 -1/2 0/1 2/5 0/1 1/0 5/12 -1/1 0/1 8/19 0/1 3/7 1/0 4/9 -1/1 -2/3 5/11 -1/1 -1/2 1/2 -2/3 0/1 7/13 -1/1 -1/2 6/11 -1/1 -2/3 5/9 -1/2 4/7 -1/2 -2/5 11/19 -1/3 7/12 -1/3 0/1 10/17 0/1 3/5 -1/2 0/1 2/3 -1/1 0/1 7/10 -1/3 0/1 12/17 -1/2 0/1 5/7 0/1 13/18 0/1 1/1 21/29 1/0 8/11 -1/1 0/1 11/15 1/0 3/4 0/1 1/0 7/9 -1/1 1/0 4/5 -1/1 9/11 -1/1 -3/4 14/17 -1/1 -4/5 5/6 -3/4 -2/3 1/1 -1/2 6/5 -2/5 -3/8 5/4 -1/3 14/11 -2/7 -1/4 9/7 -1/3 -1/4 22/17 -2/7 0/1 13/10 -1/3 -2/7 4/3 -1/4 0/1 15/11 -1/4 26/19 0/1 11/8 -1/3 0/1 18/13 -1/5 0/1 7/5 0/1 10/7 -1/1 0/1 23/16 -2/3 0/1 13/9 -1/2 3/2 -1/3 0/1 11/7 0/1 19/12 0/1 1/1 8/5 -1/1 0/1 5/3 -1/2 0/1 17/10 0/1 12/7 -1/1 0/1 7/4 -2/3 -1/2 23/13 -1/2 39/22 -1/2 -10/21 16/9 -1/2 -4/9 9/5 -1/2 11/6 -2/5 -1/3 2/1 -2/5 0/1 13/6 -2/5 -1/3 37/17 -1/3 24/11 -1/3 0/1 11/5 -1/2 -1/3 9/4 -2/5 -1/3 25/11 -1/3 41/18 -1/3 -10/31 16/7 -1/3 -4/13 7/3 -1/4 26/11 -1/5 0/1 19/8 0/1 31/13 -1/2 0/1 12/5 -1/3 0/1 17/7 0/1 5/2 -1/4 0/1 13/5 0/1 21/8 0/1 1/4 8/3 0/1 1/0 3/1 -1/2 0/1 10/3 -1/1 -2/3 7/2 -1/2 18/5 -3/7 -2/5 29/8 -2/5 -1/3 11/3 -1/2 26/7 -4/9 -2/5 41/11 -1/2 -3/7 15/4 -3/7 -2/5 4/1 -2/5 -1/3 17/4 -2/5 0/1 30/7 -2/5 -1/3 73/17 -1/3 43/10 -1/3 0/1 13/3 -1/2 -1/3 9/2 -1/2 0/1 14/3 -2/5 -1/3 5/1 -1/3 11/2 -1/3 0/1 17/3 -1/3 -1/4 6/1 -2/7 -1/4 7/1 -1/4 0/1 8/1 0/1 1/0 -1/3 0/1 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(99,46,170,79) (-1/2,-5/11) -> (11/19,7/12) Hyperbolic Matrix(31,14,166,75) (-5/11,-4/9) -> (2/11,1/5) Hyperbolic Matrix(97,42,30,13) (-4/9,-3/7) -> (3/1,10/3) Hyperbolic Matrix(33,14,-158,-67) (-3/7,-5/12) -> (-3/14,-1/5) 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(31,12,204,79) (-7/18,-5/13) -> (1/7,1/6) Hyperbolic Matrix(89,34,-322,-123) (-5/13,-3/8) -> (-5/18,-3/11) 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(29,10,26,9) (-4/11,-1/3) -> (1/1,6/5) Hyperbolic Matrix(25,8,28,9) (-1/3,-3/10) -> (5/6,1/1) 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(277,78,174,49) (-2/7,-5/18) -> (19/12,8/5) Hyperbolic Matrix(171,46,26,7) (-3/11,-4/15) -> (6/1,7/1) 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(305,72,72,17) (-1/4,-4/17) -> (4/1,17/4) 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(301,66,114,25) (-2/9,-3/14) -> (21/8,8/3) Hyperbolic Matrix(21,4,68,13) (-1/5,-1/6) -> (3/10,1/3) Hyperbolic Matrix(107,16,20,3) (-1/6,-1/7) -> (5/1,11/2) Hyperbolic Matrix(103,14,22,3) (-1/7,0/1) -> (14/3,5/1) Hyperbolic Matrix(115,-12,48,-5) (0/1,1/7) -> (31/13,12/5) Hyperbolic Matrix(139,-24,168,-29) (1/6,2/11) -> (14/17,5/6) Hyperbolic Matrix(119,-26,206,-45) (1/5,2/9) -> (4/7,11/19) Hyperbolic Matrix(133,-30,102,-23) (2/9,1/4) -> (13/10,4/3) 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(431,-128,596,-177) (5/17,3/10) -> (13/18,21/29) Hyperbolic Matrix(39,-14,14,-5) (1/3,2/5) -> (8/3,3/1) Hyperbolic Matrix(179,-74,254,-105) (2/5,5/12) -> (7/10,12/17) Hyperbolic Matrix(115,-48,12,-5) (5/12,8/19) -> (8/1,1/0) Hyperbolic Matrix(467,-198,342,-145) (8/19,3/7) -> (15/11,26/19) Hyperbolic Matrix(175,-76,76,-33) (3/7,4/9) -> (16/7,7/3) Hyperbolic Matrix(235,-106,286,-129) (4/9,5/11) -> (9/11,14/17) Hyperbolic Matrix(25,-12,48,-23) (5/11,1/2) -> (1/2,7/13) Parabolic Matrix(433,-234,198,-107) (7/13,6/11) -> (24/11,11/5) 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(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(55,-34,34,-21) (3/5,2/3) -> (8/5,5/3) Hyperbolic Matrix(85,-58,22,-15) (2/3,7/10) -> (15/4,4/1) Hyperbolic Matrix(467,-342,198,-145) (8/11,11/15) -> (7/3,26/11) Hyperbolic Matrix(133,-102,30,-23) (3/4,7/9) -> (13/3,9/2) Hyperbolic Matrix(81,-64,100,-79) (7/9,4/5) -> (4/5,9/11) Parabolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(229,-292,40,-51) (14/11,9/7) -> (17/3,6/1) Hyperbolic Matrix(879,-1136,236,-305) (9/7,22/17) -> (26/7,41/11) Hyperbolic Matrix(521,-720,144,-199) (11/8,18/13) -> (18/5,29/8) Hyperbolic Matrix(179,-254,74,-105) (7/5,10/7) -> (12/5,17/7) Hyperbolic Matrix(599,-860,140,-201) (10/7,23/16) -> (17/4,30/7) Hyperbolic Matrix(155,-242,98,-153) (3/2,11/7) -> (11/7,19/12) Parabolic Matrix(367,-622,154,-261) (5/3,17/10) -> (19/8,31/13) Hyperbolic Matrix(393,-670,166,-283) (17/10,12/7) -> (26/11,19/8) Hyperbolic 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(211,-382,58,-105) (9/5,11/6) -> (29/8,11/3) Hyperbolic 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(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(131,-338,50,-129) (5/2,13/5) -> (13/5,21/8) Parabolic 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 IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(99,46,170,79) -> Matrix(1,0,-4,1) Matrix(31,14,166,75) -> Matrix(1,-2,0,1) Matrix(97,42,30,13) -> Matrix(1,0,-2,1) Matrix(33,14,-158,-67) -> Matrix(1,0,0,1) Matrix(183,76,248,103) -> Matrix(1,0,-2,1) Matrix(185,76,-796,-327) -> Matrix(3,-2,-4,3) Matrix(549,224,424,173) -> Matrix(1,0,-4,1) Matrix(653,256,176,69) -> Matrix(1,-4,-2,9) Matrix(297,116,-1124,-439) -> Matrix(1,-2,0,1) Matrix(31,12,204,79) -> Matrix(1,0,0,1) Matrix(89,34,-322,-123) -> Matrix(1,0,0,1) Matrix(287,106,398,147) -> Matrix(1,0,0,1) Matrix(283,104,400,147) -> Matrix(1,0,-4,1) Matrix(29,10,26,9) -> Matrix(1,-2,-2,5) Matrix(25,8,28,9) -> Matrix(1,2,-2,-3) Matrix(255,76,104,31) -> Matrix(1,0,-2,1) Matrix(355,104,256,75) -> Matrix(1,0,-4,1) Matrix(277,78,174,49) -> Matrix(1,0,0,1) Matrix(171,46,26,7) -> Matrix(1,0,-4,1) Matrix(1129,300,636,169) -> Matrix(1,-6,-2,13) Matrix(489,128,340,89) -> Matrix(1,2,-2,-3) Matrix(305,72,72,17) -> Matrix(1,0,-2,1) Matrix(1177,276,516,121) -> Matrix(5,6,-16,-19) Matrix(187,42,138,31) -> Matrix(1,0,-2,1) Matrix(301,66,114,25) -> Matrix(1,0,2,1) Matrix(21,4,68,13) -> Matrix(1,0,-2,1) Matrix(107,16,20,3) -> Matrix(1,2,-4,-7) Matrix(103,14,22,3) -> Matrix(3,2,-8,-5) Matrix(115,-12,48,-5) -> Matrix(1,0,-2,1) Matrix(139,-24,168,-29) -> Matrix(3,4,-4,-5) Matrix(119,-26,206,-45) -> Matrix(3,2,-8,-5) Matrix(133,-30,102,-23) -> Matrix(1,0,-2,1) Matrix(85,-22,58,-15) -> Matrix(3,2,-8,-5) Matrix(57,-16,196,-55) -> Matrix(7,4,-16,-9) Matrix(431,-128,596,-177) -> Matrix(5,2,2,1) Matrix(39,-14,14,-5) -> Matrix(1,0,0,1) Matrix(179,-74,254,-105) -> Matrix(1,0,-2,1) Matrix(115,-48,12,-5) -> Matrix(1,0,-2,1) Matrix(467,-198,342,-145) -> Matrix(1,0,-4,1) Matrix(175,-76,76,-33) -> Matrix(1,2,-4,-7) Matrix(235,-106,286,-129) -> Matrix(1,2,-2,-3) Matrix(25,-12,48,-23) -> Matrix(1,0,0,1) Matrix(433,-234,198,-107) -> Matrix(3,2,-8,-5) Matrix(303,-166,418,-229) -> Matrix(3,2,-2,-1) Matrix(207,-116,116,-65) -> Matrix(3,2,-8,-5) Matrix(499,-292,364,-213) -> Matrix(1,0,0,1) Matrix(159,-94,22,-13) -> Matrix(1,0,-2,1) Matrix(55,-34,34,-21) -> Matrix(1,0,0,1) Matrix(85,-58,22,-15) -> Matrix(3,2,-8,-5) Matrix(467,-342,198,-145) -> Matrix(1,0,-4,1) Matrix(133,-102,30,-23) -> Matrix(1,0,-2,1) Matrix(81,-64,100,-79) -> Matrix(3,4,-4,-5) Matrix(81,-100,64,-79) -> Matrix(11,4,-36,-13) Matrix(229,-292,40,-51) -> Matrix(1,0,0,1) Matrix(879,-1136,236,-305) -> Matrix(15,4,-34,-9) Matrix(521,-720,144,-199) -> Matrix(7,2,-18,-5) Matrix(179,-254,74,-105) -> Matrix(1,0,-2,1) Matrix(599,-860,140,-201) -> Matrix(3,2,-8,-5) Matrix(155,-242,98,-153) -> Matrix(1,0,4,1) Matrix(367,-622,154,-261) -> Matrix(1,0,0,1) Matrix(393,-670,166,-283) -> Matrix(1,0,-4,1) Matrix(119,-206,26,-45) -> Matrix(3,2,-8,-5) Matrix(599,-1058,338,-597) -> Matrix(23,12,-48,-25) Matrix(211,-382,58,-105) -> Matrix(1,0,0,1) Matrix(25,-48,12,-23) -> Matrix(1,0,0,1) Matrix(1169,-2540,272,-591) -> Matrix(5,2,-18,-7) Matrix(1313,-2862,306,-667) -> Matrix(7,2,-18,-5) Matrix(123,-274,22,-49) -> Matrix(5,2,-18,-7) Matrix(551,-1250,242,-549) -> Matrix(35,12,-108,-37) Matrix(131,-338,50,-129) -> Matrix(1,0,8,1) Matrix(57,-196,16,-55) -> Matrix(7,4,-16,-9) Matrix(525,-1958,122,-455) -> Matrix(5,2,-8,-3) 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): 16 Degree of the the map X: 16 Degree of the the map Y: 72 ----------------------------------------------------------------------- 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): 216 Minimal number of generators: 37 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: 24 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/5 1/2 5/7 1/1 5/4 7/5 11/7 5/3 23/13 2/1 37/17 25/11 7/3 19/8 13/5 3/1 7/2 11/3 4/1 13/3 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 -1/1 0/1 1/6 -2/1 1/0 1/5 -1/1 2/9 -1/2 0/1 1/4 -1/1 -2/3 3/11 -1/2 2/7 -1/2 1/3 -1/2 0/1 2/5 0/1 1/0 5/12 -1/1 0/1 8/19 0/1 3/7 1/0 4/9 -1/1 -2/3 5/11 -1/1 -1/2 1/2 -2/3 0/1 7/13 -1/1 -1/2 6/11 -1/1 -2/3 5/9 -1/2 4/7 -1/2 -2/5 7/12 -1/3 0/1 10/17 0/1 3/5 -1/2 0/1 2/3 -1/1 0/1 7/10 -1/3 0/1 5/7 0/1 8/11 -1/1 0/1 3/4 0/1 1/0 7/9 -1/1 1/0 4/5 -1/1 1/1 -1/2 5/4 -1/3 9/7 -1/3 -1/4 13/10 -1/3 -2/7 4/3 -1/4 0/1 11/8 -1/3 0/1 7/5 0/1 10/7 -1/1 0/1 13/9 -1/2 3/2 -1/3 0/1 11/7 0/1 8/5 -1/1 0/1 5/3 -1/2 0/1 17/10 0/1 12/7 -1/1 0/1 7/4 -2/3 -1/2 23/13 -1/2 16/9 -1/2 -4/9 9/5 -1/2 11/6 -2/5 -1/3 2/1 -2/5 0/1 13/6 -2/5 -1/3 37/17 -1/3 24/11 -1/3 0/1 11/5 -1/2 -1/3 9/4 -2/5 -1/3 25/11 -1/3 16/7 -1/3 -4/13 7/3 -1/4 19/8 0/1 12/5 -1/3 0/1 5/2 -1/4 0/1 13/5 0/1 8/3 0/1 1/0 3/1 -1/2 0/1 7/2 -1/2 11/3 -1/2 15/4 -3/7 -2/5 4/1 -2/5 -1/3 13/3 -1/2 -1/3 9/2 -1/2 0/1 5/1 -1/3 6/1 -2/7 -1/4 1/0 -1/3 0/1 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(46,-7,79,-12) (0/1,1/6) -> (4/7,7/12) Hyperbolic Matrix(16,-3,75,-14) (1/6,1/5) -> (1/5,2/9) Parabolic Matrix(133,-30,102,-23) (2/9,1/4) -> (13/10,4/3) Hyperbolic Matrix(85,-22,58,-15) (1/4,3/11) -> (13/9,3/2) Hyperbolic Matrix(154,-43,43,-12) (3/11,2/7) -> (7/2,11/3) Hyperbolic Matrix(42,-13,13,-4) (2/7,1/3) -> (3/1,7/2) Hyperbolic Matrix(39,-14,14,-5) (1/3,2/5) -> (8/3,3/1) 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(266,-113,113,-48) (8/19,3/7) -> (7/3,19/8) 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(433,-234,198,-107) (7/13,6/11) -> (24/11,11/5) 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(432,-253,181,-106) (7/12,10/17) -> (19/8,12/5) Hyperbolic Matrix(170,-101,101,-60) (10/17,3/5) -> (5/3,17/10) Hyperbolic Matrix(55,-34,34,-21) (3/5,2/3) -> (8/5,5/3) 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(133,-102,30,-23) (3/4,7/9) -> (13/3,9/2) Hyperbolic Matrix(90,-71,71,-56) (7/9,4/5) -> (5/4,9/7) Hyperbolic Matrix(10,-9,9,-8) (4/5,1/1) -> (1/1,5/4) Parabolic 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(78,-121,49,-76) (3/2,11/7) -> (11/7,8/5) Parabolic Matrix(46,-79,7,-12) (12/7,7/4) -> (6/1,1/0) 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(630,-1369,289,-628) (13/6,37/17) -> (37/17,24/11) Parabolic Matrix(72,-161,17,-38) (11/5,9/4) -> (4/1,13/3) Hyperbolic Matrix(276,-625,121,-274) (9/4,25/11) -> (25/11,16/7) Parabolic Matrix(66,-169,25,-64) (5/2,13/5) -> (13/5,8/3) Parabolic Matrix(16,-75,3,-14) (9/2,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,2,1) Matrix(46,-7,79,-12) -> Matrix(1,0,-2,1) Matrix(16,-3,75,-14) -> Matrix(1,2,-2,-3) Matrix(133,-30,102,-23) -> Matrix(1,0,-2,1) Matrix(85,-22,58,-15) -> Matrix(3,2,-8,-5) Matrix(154,-43,43,-12) -> Matrix(7,4,-16,-9) Matrix(42,-13,13,-4) -> Matrix(1,0,0,1) Matrix(39,-14,14,-5) -> Matrix(1,0,0,1) Matrix(76,-31,103,-42) -> Matrix(1,0,0,1) Matrix(432,-181,253,-106) -> Matrix(1,0,0,1) Matrix(266,-113,113,-48) -> Matrix(1,0,-4,1) Matrix(175,-76,76,-33) -> Matrix(1,2,-4,-7) Matrix(224,-101,173,-78) -> Matrix(1,0,-2,1) Matrix(25,-12,48,-23) -> Matrix(1,0,0,1) Matrix(433,-234,198,-107) -> Matrix(3,2,-8,-5) Matrix(256,-141,69,-38) -> Matrix(7,4,-16,-9) Matrix(207,-116,116,-65) -> Matrix(3,2,-8,-5) Matrix(432,-253,181,-106) -> Matrix(1,0,0,1) Matrix(170,-101,101,-60) -> Matrix(1,0,0,1) Matrix(55,-34,34,-21) -> Matrix(1,0,0,1) Matrix(85,-58,22,-15) -> Matrix(3,2,-8,-5) Matrix(106,-75,147,-104) -> Matrix(1,0,2,1) Matrix(133,-102,30,-23) -> Matrix(1,0,-2,1) Matrix(90,-71,71,-56) -> Matrix(1,2,-4,-7) Matrix(10,-9,9,-8) -> Matrix(3,2,-8,-5) Matrix(76,-103,31,-42) -> Matrix(1,0,0,1) Matrix(106,-147,75,-104) -> Matrix(1,0,2,1) Matrix(162,-233,89,-128) -> Matrix(3,2,-8,-5) Matrix(78,-121,49,-76) -> Matrix(1,0,2,1) Matrix(46,-79,7,-12) -> Matrix(1,0,-2,1) Matrix(300,-529,169,-298) -> Matrix(11,6,-24,-13) Matrix(25,-48,12,-23) -> Matrix(1,0,0,1) Matrix(630,-1369,289,-628) -> Matrix(5,2,-18,-7) Matrix(72,-161,17,-38) -> Matrix(1,0,0,1) Matrix(276,-625,121,-274) -> Matrix(17,6,-54,-19) Matrix(66,-169,25,-64) -> Matrix(1,0,4,1) Matrix(16,-75,3,-14) -> Matrix(5,2,-18,-7) 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): 8 ----------------------------------------------------------------------- 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 1 1/1 -1/2 2 9 5/4 -1/3 2 2 9/7 (-1/3,-1/4) 0 9 4/3 (-1/4,0/1) 0 18 11/8 (-1/3,0/1) 0 18 7/5 0/1 2 3 10/7 (-1/1,0/1) 0 18 3/2 (-1/3,0/1) 0 18 11/7 0/1 2 1 5/3 (-1/2,0/1) 0 9 17/10 0/1 2 2 12/7 (-1/1,0/1) 0 18 7/4 (-2/3,-1/2) 0 18 23/13 -1/2 6 1 9/5 -1/2 2 9 11/6 (-2/5,-1/3) 0 18 2/1 0 6 13/6 (-2/5,-1/3) 0 18 37/17 -1/3 2 1 11/5 (-1/2,-1/3) 0 9 9/4 (-2/5,-1/3) 0 18 25/11 -1/3 6 1 7/3 -1/4 2 9 19/8 0/1 2 2 12/5 (-1/3,0/1) 0 18 5/2 (-1/4,0/1) 0 18 13/5 0/1 4 1 3/1 (-1/2,0/1) 0 9 7/2 -1/2 2 2 11/3 -1/2 2 9 15/4 (-3/7,-2/5) 0 18 4/1 (-2/5,-1/3) 0 18 13/3 (-1/2,-1/3) 0 9 9/2 (-1/2,0/1) 0 18 5/1 -1/3 2 3 6/1 (-2/7,-1/4) 0 18 1/0 (-1/3,0/1) 0 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(9,-10,8,-9) (1/1,5/4) -> (1/1,5/4) Reflection Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(102,-133,23,-30) (9/7,4/3) -> (13/3,9/2) 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(34,-55,21,-34) (11/7,5/3) -> (11/7,5/3) Reflection Matrix(101,-170,60,-101) (5/3,17/10) -> (5/3,17/10) Reflection Matrix(253,-432,106,-181) (17/10,12/7) -> (19/8,12/5) Glide Reflection Matrix(46,-79,7,-12) (12/7,7/4) -> (6/1,1/0) 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(186,-407,85,-186) (37/17,11/5) -> (37/17,11/5) Reflection Matrix(72,-161,17,-38) (11/5,9/4) -> (4/1,13/3) Hyperbolic 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(113,-266,48,-113) (7/3,19/8) -> (7/3,19/8) Reflection Matrix(51,-130,20,-51) (5/2,13/5) -> (5/2,13/5) Reflection Matrix(14,-39,5,-14) (13/5,3/1) -> (13/5,3/1) Reflection Matrix(13,-42,4,-13) (3/1,7/2) -> (3/1,7/2) Reflection Matrix(43,-154,12,-43) (7/2,11/3) -> (7/2,11/3) Reflection Matrix(16,-75,3,-14) (9/2,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,6,1) (-1/1,1/0) -> (-1/3,0/1) Matrix(0,1,1,0) -> Matrix(-1,0,4,1) (-1/1,1/1) -> (-1/2,0/1) Matrix(9,-10,8,-9) -> Matrix(5,2,-12,-5) (1/1,5/4) -> (-1/2,-1/3) Matrix(71,-90,56,-71) -> Matrix(7,2,-24,-7) (5/4,9/7) -> (-1/3,-1/4) Matrix(102,-133,23,-30) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(76,-103,31,-42) -> Matrix(1,0,0,1) Matrix(106,-147,75,-104) -> Matrix(1,0,2,1) 0/1 Matrix(58,-85,15,-22) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(43,-66,28,-43) -> Matrix(-1,0,6,1) (3/2,11/7) -> (-1/3,0/1) Matrix(34,-55,21,-34) -> Matrix(-1,0,4,1) (11/7,5/3) -> (-1/2,0/1) Matrix(101,-170,60,-101) -> Matrix(-1,0,4,1) (5/3,17/10) -> (-1/2,0/1) Matrix(253,-432,106,-181) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(46,-79,7,-12) -> Matrix(1,0,-2,1) 0/1 Matrix(183,-322,104,-183) -> Matrix(7,4,-12,-7) (7/4,23/13) -> (-2/3,-1/2) Matrix(116,-207,65,-116) -> Matrix(5,2,-12,-5) (23/13,9/5) -> (-1/2,-1/3) Matrix(141,-256,38,-69) -> Matrix(9,4,-20,-9) *** -> (-1/2,-2/5) Matrix(25,-48,12,-23) -> Matrix(1,0,0,1) Matrix(443,-962,204,-443) -> Matrix(11,4,-30,-11) (13/6,37/17) -> (-2/5,-1/3) Matrix(186,-407,85,-186) -> Matrix(5,2,-12,-5) (37/17,11/5) -> (-1/2,-1/3) Matrix(72,-161,17,-38) -> Matrix(1,0,0,1) Matrix(199,-450,88,-199) -> Matrix(11,4,-30,-11) (9/4,25/11) -> (-2/5,-1/3) Matrix(76,-175,33,-76) -> Matrix(7,2,-24,-7) (25/11,7/3) -> (-1/3,-1/4) Matrix(113,-266,48,-113) -> Matrix(-1,0,8,1) (7/3,19/8) -> (-1/4,0/1) Matrix(51,-130,20,-51) -> Matrix(-1,0,8,1) (5/2,13/5) -> (-1/4,0/1) Matrix(14,-39,5,-14) -> Matrix(-1,0,4,1) (13/5,3/1) -> (-1/2,0/1) Matrix(13,-42,4,-13) -> Matrix(-1,0,4,1) (3/1,7/2) -> (-1/2,0/1) Matrix(43,-154,12,-43) -> Matrix(9,4,-20,-9) (7/2,11/3) -> (-1/2,-2/5) Matrix(16,-75,3,-14) -> Matrix(5,2,-18,-7) -1/3 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.