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 -6/1 -5/1 -4/1 -25/7 -8/3 -5/2 -12/5 -2/1 -5/3 -15/11 -4/3 -5/4 0/1 1/1 20/17 5/4 4/3 10/7 3/2 20/13 30/19 5/3 9/5 20/11 2/1 20/9 12/5 5/2 100/39 8/3 30/11 25/9 20/7 3/1 10/3 7/2 25/7 11/3 19/5 4/1 13/3 9/2 5/1 17/3 6/1 20/3 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 1/3 1/2 -20/3 1/2 -13/2 1/2 4/7 -6/1 2/3 -5/1 1/1 -14/3 0/1 -23/5 1/1 1/0 -9/2 1/2 1/1 -13/3 2/1 1/0 -4/1 0/1 -19/5 1/2 3/5 -15/4 1/2 -11/3 2/3 1/1 -18/5 2/1 -43/12 -1/1 1/0 -25/7 -1/1 1/1 -7/2 0/1 1/2 -17/5 1/2 1/1 -10/3 1/1 -3/1 0/1 1/1 -20/7 1/1 -17/6 1/1 1/0 -14/5 0/1 -11/4 1/1 1/0 -41/15 2/1 1/0 -30/11 1/0 -19/7 1/1 1/0 -8/3 0/1 -21/8 2/1 1/0 -34/13 0/1 -13/5 0/1 1/2 -18/7 2/1 -5/2 0/1 -22/9 2/5 -61/25 9/19 1/2 -100/41 1/2 -39/16 1/2 6/11 -17/7 1/2 1/1 -29/12 1/1 1/0 -12/5 0/1 -31/13 0/1 1/3 -19/8 0/1 1/2 -7/3 1/2 1/1 -9/4 3/4 1/1 -20/9 1/1 -11/5 1/1 4/3 -2/1 0/1 -13/7 2/3 3/4 -11/6 5/6 1/1 -20/11 1/1 -9/5 1/1 4/3 -7/4 2/1 1/0 -5/3 -1/1 1/1 -13/8 2/1 1/0 -34/21 0/1 -21/13 -1/1 1/0 -8/5 0/1 -19/12 1/2 2/3 -49/31 2/3 1/1 -30/19 1/1 -11/7 0/1 1/1 -14/9 0/1 -17/11 1/2 1/1 -20/13 1/1 -3/2 1/1 1/0 -22/15 4/1 -19/13 -3/1 1/0 -16/11 0/1 -13/9 2/1 1/0 -10/7 1/0 -7/5 -1/1 1/0 -11/8 -1/1 -1/2 -26/19 0/1 -15/11 -1/1 -34/25 -2/3 -19/14 -4/7 -1/2 -4/3 0/1 -21/16 4/9 1/2 -17/13 1/2 1/1 -13/10 0/1 1/2 -9/7 1/1 2/1 -32/25 2/1 -23/18 1/1 1/0 -14/11 2/1 -19/15 3/1 1/0 -5/4 1/0 -11/9 -1/1 0/1 -6/5 0/1 -13/11 -1/4 0/1 -20/17 0/1 -7/6 0/1 1/2 -15/13 -1/1 1/1 -23/20 -1/1 1/0 -8/7 0/1 -9/8 1/1 1/0 -1/1 0/1 1/0 0/1 0/1 1/1 0/1 1/4 8/7 0/1 7/6 0/1 1/2 20/17 0/1 13/11 0/1 1/8 6/5 0/1 11/9 0/1 1/5 5/4 1/4 19/15 1/4 3/11 14/11 2/7 23/18 1/4 1/3 9/7 2/7 1/3 4/3 0/1 19/14 1/6 4/23 15/11 1/5 11/8 1/6 1/5 7/5 1/5 1/4 10/7 1/4 13/9 1/4 2/7 3/2 1/4 1/3 20/13 1/3 17/11 1/3 1/2 14/9 0/1 11/7 0/1 1/3 30/19 1/3 19/12 2/5 1/2 8/5 0/1 5/3 1/5 1/3 12/7 0/1 19/11 1/5 1/4 26/15 0/1 7/4 1/4 2/7 16/9 2/7 9/5 4/13 1/3 20/11 1/3 11/6 1/3 5/14 2/1 0/1 11/5 4/13 1/3 20/9 1/3 9/4 1/3 3/8 7/3 1/3 1/2 19/8 0/1 1/2 50/21 1/1 31/13 0/1 1/1 43/18 1/1 1/0 12/5 0/1 41/17 0/1 1/4 29/12 1/4 1/3 17/7 1/3 1/2 5/2 0/1 23/9 1/5 1/4 41/16 6/25 1/4 100/39 1/4 59/23 1/4 9/35 18/7 2/7 13/5 0/1 1/2 34/13 0/1 21/8 1/4 2/7 8/3 0/1 19/7 1/4 1/3 30/11 1/4 41/15 1/4 2/7 11/4 1/4 1/3 25/9 1/3 39/14 1/2 2/3 14/5 0/1 17/6 1/4 1/3 20/7 1/3 3/1 0/1 1/3 10/3 1/3 17/5 1/3 1/2 41/12 1/2 4/7 24/7 0/1 7/2 0/1 1/2 32/9 0/1 25/7 1/5 1/3 68/19 0/1 43/12 1/5 1/4 18/5 2/7 11/3 1/3 2/5 15/4 1/2 19/5 3/7 1/2 4/1 0/1 21/5 3/13 1/4 38/9 4/15 17/4 1/4 1/3 13/3 1/4 2/7 22/5 4/13 9/2 1/3 1/2 5/1 1/3 11/2 1/3 3/8 17/3 1/3 1/2 6/1 2/5 13/2 4/9 1/2 20/3 1/2 7/1 1/2 1/1 1/0 0/1 1/2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(19,140,8,59) (-7/1,1/0) -> (7/3,19/8) Hyperbolic Matrix(41,280,6,41) (-7/1,-20/3) -> (20/3,7/1) Hyperbolic Matrix(79,520,12,79) (-20/3,-13/2) -> (13/2,20/3) Hyperbolic Matrix(59,380,34,219) (-13/2,-6/1) -> (26/15,7/4) Hyperbolic Matrix(19,100,-4,-21) (-6/1,-5/1) -> (-5/1,-14/3) Parabolic Matrix(121,560,78,361) (-14/3,-23/5) -> (17/11,14/9) Hyperbolic Matrix(179,820,74,339) (-23/5,-9/2) -> (29/12,17/7) Hyperbolic Matrix(59,260,-32,-141) (-9/2,-13/3) -> (-13/7,-11/6) Hyperbolic Matrix(61,260,-42,-179) (-13/3,-4/1) -> (-16/11,-13/9) Hyperbolic Matrix(99,380,-68,-261) (-4/1,-19/5) -> (-19/13,-16/11) Hyperbolic Matrix(101,380,80,301) (-19/5,-15/4) -> (5/4,19/15) Hyperbolic Matrix(59,220,48,179) (-15/4,-11/3) -> (11/9,5/4) Hyperbolic Matrix(61,220,28,101) (-11/3,-18/5) -> (2/1,11/5) Hyperbolic Matrix(279,1000,-190,-681) (-18/5,-43/12) -> (-3/2,-22/15) Hyperbolic Matrix(341,1220,-296,-1059) (-43/12,-25/7) -> (-15/13,-23/20) Hyperbolic Matrix(79,280,-68,-241) (-25/7,-7/2) -> (-7/6,-15/13) Hyperbolic Matrix(99,340,-76,-261) (-7/2,-17/5) -> (-17/13,-13/10) Hyperbolic Matrix(101,340,30,101) (-17/5,-10/3) -> (10/3,17/5) Hyperbolic Matrix(19,60,6,19) (-10/3,-3/1) -> (3/1,10/3) Hyperbolic Matrix(41,120,14,41) (-3/1,-20/7) -> (20/7,3/1) Hyperbolic Matrix(239,680,84,239) (-20/7,-17/6) -> (17/6,20/7) Hyperbolic Matrix(199,560,156,439) (-17/6,-14/5) -> (14/11,23/18) Hyperbolic Matrix(159,440,-116,-321) (-14/5,-11/4) -> (-11/8,-26/19) Hyperbolic Matrix(599,1640,248,679) (-11/4,-41/15) -> (41/17,29/12) Hyperbolic Matrix(901,2460,330,901) (-41/15,-30/11) -> (30/11,41/15) Hyperbolic Matrix(419,1140,154,419) (-30/11,-19/7) -> (19/7,30/11) Hyperbolic Matrix(141,380,82,221) (-19/7,-8/3) -> (12/7,19/11) Hyperbolic Matrix(121,320,76,201) (-8/3,-21/8) -> (19/12,8/5) Hyperbolic Matrix(519,1360,-382,-1001) (-21/8,-34/13) -> (-34/25,-19/14) Hyperbolic Matrix(199,520,168,439) (-34/13,-13/5) -> (13/11,6/5) Hyperbolic Matrix(101,260,-54,-139) (-13/5,-18/7) -> (-2/1,-13/7) Hyperbolic Matrix(79,200,-32,-81) (-18/7,-5/2) -> (-5/2,-22/9) Parabolic Matrix(1139,2780,270,659) (-22/9,-61/25) -> (21/5,38/9) Hyperbolic Matrix(4919,12000,1918,4679) (-61/25,-100/41) -> (100/39,59/23) Hyperbolic Matrix(3281,8000,1280,3121) (-100/41,-39/16) -> (41/16,100/39) Hyperbolic Matrix(559,1360,164,399) (-39/16,-17/7) -> (17/5,41/12) Hyperbolic Matrix(281,680,50,121) (-17/7,-29/12) -> (11/2,17/3) Hyperbolic Matrix(141,340,-124,-299) (-29/12,-12/5) -> (-8/7,-9/8) Hyperbolic Matrix(461,1100,-360,-859) (-12/5,-31/13) -> (-9/7,-32/25) Hyperbolic Matrix(639,1520,-404,-961) (-31/13,-19/8) -> (-19/12,-49/31) Hyperbolic Matrix(59,140,8,19) (-19/8,-7/3) -> (7/1,1/0) Hyperbolic Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(161,360,72,161) (-9/4,-20/9) -> (20/9,9/4) Hyperbolic Matrix(199,440,90,199) (-20/9,-11/5) -> (11/5,20/9) Hyperbolic Matrix(101,220,28,61) (-11/5,-2/1) -> (18/5,11/3) Hyperbolic Matrix(241,440,132,241) (-11/6,-20/11) -> (20/11,11/6) Hyperbolic Matrix(199,360,110,199) (-20/11,-9/5) -> (9/5,20/11) Hyperbolic Matrix(101,180,-78,-139) (-9/5,-7/4) -> (-13/10,-9/7) Hyperbolic Matrix(59,100,-36,-61) (-7/4,-5/3) -> (-5/3,-13/8) Parabolic Matrix(321,520,50,81) (-13/8,-34/21) -> (6/1,13/2) Hyperbolic Matrix(581,940,458,741) (-34/21,-21/13) -> (19/15,14/11) Hyperbolic Matrix(199,320,74,119) (-21/13,-8/5) -> (8/3,19/7) Hyperbolic Matrix(201,320,76,121) (-8/5,-19/12) -> (21/8,8/3) Hyperbolic Matrix(2139,3380,898,1419) (-49/31,-30/19) -> (50/21,31/13) Hyperbolic Matrix(419,660,266,419) (-30/19,-11/7) -> (11/7,30/19) Hyperbolic Matrix(141,220,116,181) (-11/7,-14/9) -> (6/5,11/9) Hyperbolic Matrix(219,340,38,59) (-14/9,-17/11) -> (17/3,6/1) Hyperbolic Matrix(441,680,286,441) (-17/11,-20/13) -> (20/13,17/11) Hyperbolic Matrix(79,120,52,79) (-20/13,-3/2) -> (3/2,20/13) Hyperbolic Matrix(1119,1640,436,639) (-22/15,-19/13) -> (59/23,18/7) Hyperbolic Matrix(181,260,126,181) (-13/9,-10/7) -> (10/7,13/9) Hyperbolic Matrix(99,140,70,99) (-10/7,-7/5) -> (7/5,10/7) Hyperbolic Matrix(101,140,44,61) (-7/5,-11/8) -> (9/4,7/3) Hyperbolic Matrix(659,900,-484,-661) (-26/19,-15/11) -> (-15/11,-34/25) Parabolic Matrix(119,160,-90,-121) (-19/14,-4/3) -> (-4/3,-21/16) Parabolic Matrix(901,1180,352,461) (-21/16,-17/13) -> (23/9,41/16) Hyperbolic Matrix(719,920,-626,-801) (-32/25,-23/18) -> (-23/20,-8/7) Hyperbolic Matrix(439,560,156,199) (-23/18,-14/11) -> (14/5,17/6) Hyperbolic Matrix(599,760,346,439) (-14/11,-19/15) -> (19/11,26/15) Hyperbolic Matrix(301,380,80,101) (-19/15,-5/4) -> (15/4,19/5) Hyperbolic Matrix(179,220,48,59) (-5/4,-11/9) -> (11/3,15/4) Hyperbolic Matrix(181,220,116,141) (-11/9,-6/5) -> (14/9,11/7) Hyperbolic Matrix(439,520,168,199) (-6/5,-13/11) -> (13/5,34/13) Hyperbolic Matrix(441,520,374,441) (-13/11,-20/17) -> (20/17,13/11) Hyperbolic Matrix(239,280,204,239) (-20/17,-7/6) -> (7/6,20/17) Hyperbolic Matrix(339,380,124,139) (-9/8,-1/1) -> (41/15,11/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(299,-340,124,-141) (1/1,8/7) -> (12/5,41/17) Hyperbolic Matrix(241,-280,68,-79) (8/7,7/6) -> (7/2,32/9) Hyperbolic Matrix(859,-1100,360,-461) (23/18,9/7) -> (31/13,43/18) Hyperbolic Matrix(139,-180,78,-101) (9/7,4/3) -> (16/9,9/5) Hyperbolic Matrix(459,-620,134,-181) (4/3,19/14) -> (41/12,24/7) Hyperbolic Matrix(779,-1060,280,-381) (19/14,15/11) -> (25/9,39/14) Hyperbolic Matrix(321,-440,116,-159) (15/11,11/8) -> (11/4,25/9) Hyperbolic Matrix(179,-260,42,-61) (13/9,3/2) -> (17/4,13/3) Hyperbolic Matrix(961,-1520,404,-639) (30/19,19/12) -> (19/8,50/21) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(159,-280,46,-81) (7/4,16/9) -> (24/7,7/2) Hyperbolic Matrix(141,-260,32,-59) (11/6,2/1) -> (22/5,9/2) Hyperbolic Matrix(1439,-3440,402,-961) (43/18,12/5) -> (68/19,43/12) Hyperbolic Matrix(81,-200,32,-79) (17/7,5/2) -> (5/2,23/9) Parabolic Matrix(201,-520,46,-119) (18/7,13/5) -> (13/3,22/5) Hyperbolic Matrix(619,-1620,222,-581) (34/13,21/8) -> (39/14,14/5) Hyperbolic Matrix(701,-2500,196,-699) (32/9,25/7) -> (25/7,68/19) Parabolic Matrix(541,-1940,128,-459) (43/12,18/5) -> (38/9,17/4) Hyperbolic Matrix(41,-160,10,-39) (19/5,4/1) -> (4/1,21/5) Parabolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(19,140,8,59) -> Matrix(1,0,0,1) Matrix(41,280,6,41) -> Matrix(5,-2,8,-3) Matrix(79,520,12,79) -> Matrix(15,-8,32,-17) Matrix(59,380,34,219) -> Matrix(3,-2,14,-9) Matrix(19,100,-4,-21) -> Matrix(3,-2,2,-1) Matrix(121,560,78,361) -> Matrix(1,0,2,1) Matrix(179,820,74,339) -> Matrix(1,0,2,1) Matrix(59,260,-32,-141) -> Matrix(3,-4,4,-5) Matrix(61,260,-42,-179) -> Matrix(1,0,0,1) Matrix(99,380,-68,-261) -> Matrix(1,0,-2,1) Matrix(101,380,80,301) -> Matrix(1,0,2,1) Matrix(59,220,48,179) -> Matrix(3,-2,14,-9) Matrix(61,220,28,101) -> Matrix(1,-2,4,-7) Matrix(279,1000,-190,-681) -> Matrix(1,2,0,1) Matrix(341,1220,-296,-1059) -> Matrix(1,0,0,1) Matrix(79,280,-68,-241) -> Matrix(1,0,0,1) Matrix(99,340,-76,-261) -> Matrix(1,0,0,1) Matrix(101,340,30,101) -> Matrix(3,-2,8,-5) Matrix(19,60,6,19) -> Matrix(1,0,2,1) Matrix(41,120,14,41) -> Matrix(1,0,2,1) Matrix(239,680,84,239) -> Matrix(1,-2,4,-7) Matrix(199,560,156,439) -> Matrix(1,-2,4,-7) Matrix(159,440,-116,-321) -> Matrix(1,0,-2,1) Matrix(599,1640,248,679) -> Matrix(1,-2,4,-7) Matrix(901,2460,330,901) -> Matrix(1,-4,4,-15) Matrix(419,1140,154,419) -> Matrix(1,-2,4,-7) Matrix(141,380,82,221) -> Matrix(1,0,4,1) Matrix(121,320,76,201) -> Matrix(1,0,2,1) Matrix(519,1360,-382,-1001) -> Matrix(1,2,-2,-3) Matrix(199,520,168,439) -> Matrix(1,0,6,1) Matrix(101,260,-54,-139) -> Matrix(1,-2,2,-3) Matrix(79,200,-32,-81) -> Matrix(1,0,2,1) Matrix(1139,2780,270,659) -> Matrix(13,-6,50,-23) Matrix(4919,12000,1918,4679) -> Matrix(37,-18,146,-71) Matrix(3281,8000,1280,3121) -> Matrix(23,-12,94,-49) Matrix(559,1360,164,399) -> Matrix(3,-2,8,-5) Matrix(281,680,50,121) -> Matrix(3,-2,8,-5) Matrix(141,340,-124,-299) -> Matrix(1,0,0,1) Matrix(461,1100,-360,-859) -> Matrix(7,-2,4,-1) Matrix(639,1520,-404,-961) -> Matrix(5,-2,8,-3) Matrix(59,140,8,19) -> Matrix(1,0,0,1) Matrix(61,140,44,101) -> Matrix(3,-2,14,-9) Matrix(161,360,72,161) -> Matrix(7,-6,20,-17) Matrix(199,440,90,199) -> Matrix(7,-8,22,-25) Matrix(101,220,28,61) -> Matrix(1,-2,4,-7) Matrix(241,440,132,241) -> Matrix(11,-10,32,-29) Matrix(199,360,110,199) -> Matrix(7,-8,22,-25) Matrix(101,180,-78,-139) -> Matrix(1,-2,2,-3) Matrix(59,100,-36,-61) -> Matrix(1,0,0,1) Matrix(321,520,50,81) -> Matrix(1,2,2,5) Matrix(581,940,458,741) -> Matrix(1,-2,4,-7) Matrix(199,320,74,119) -> Matrix(1,0,4,1) Matrix(201,320,76,121) -> Matrix(1,0,2,1) Matrix(2139,3380,898,1419) -> Matrix(3,-2,2,-1) Matrix(419,660,266,419) -> Matrix(1,0,2,1) Matrix(141,220,116,181) -> Matrix(1,0,4,1) Matrix(219,340,38,59) -> Matrix(3,-2,8,-5) Matrix(441,680,286,441) -> Matrix(3,-2,8,-5) Matrix(79,120,52,79) -> Matrix(1,-2,4,-7) Matrix(1119,1640,436,639) -> Matrix(1,-6,4,-23) Matrix(181,260,126,181) -> Matrix(1,-4,4,-15) Matrix(99,140,70,99) -> Matrix(1,2,4,9) Matrix(101,140,44,61) -> Matrix(1,2,2,5) Matrix(659,900,-484,-661) -> Matrix(1,2,-2,-3) Matrix(119,160,-90,-121) -> Matrix(1,0,4,1) Matrix(901,1180,352,461) -> Matrix(3,-2,14,-9) Matrix(719,920,-626,-801) -> Matrix(1,-2,0,1) Matrix(439,560,156,199) -> Matrix(1,-2,4,-7) Matrix(599,760,346,439) -> Matrix(1,-2,4,-7) Matrix(301,380,80,101) -> Matrix(1,0,2,1) Matrix(179,220,48,59) -> Matrix(1,2,2,5) Matrix(181,220,116,141) -> Matrix(1,0,4,1) Matrix(439,520,168,199) -> Matrix(1,0,6,1) Matrix(441,520,374,441) -> Matrix(1,0,12,1) Matrix(239,280,204,239) -> Matrix(1,0,0,1) Matrix(339,380,124,139) -> Matrix(1,-2,4,-7) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(299,-340,124,-141) -> Matrix(1,0,0,1) Matrix(241,-280,68,-79) -> Matrix(1,0,0,1) Matrix(859,-1100,360,-461) -> Matrix(7,-2,4,-1) Matrix(139,-180,78,-101) -> Matrix(5,-2,18,-7) Matrix(459,-620,134,-181) -> Matrix(1,0,-4,1) Matrix(779,-1060,280,-381) -> Matrix(11,-2,28,-5) Matrix(321,-440,116,-159) -> Matrix(1,0,-2,1) Matrix(179,-260,42,-61) -> Matrix(1,0,0,1) Matrix(961,-1520,404,-639) -> Matrix(5,-2,8,-3) Matrix(61,-100,36,-59) -> Matrix(1,0,0,1) Matrix(159,-280,46,-81) -> Matrix(7,-2,18,-5) Matrix(141,-260,32,-59) -> Matrix(11,-4,36,-13) Matrix(1439,-3440,402,-961) -> Matrix(1,0,4,1) Matrix(81,-200,32,-79) -> Matrix(1,0,2,1) Matrix(201,-520,46,-119) -> Matrix(5,-2,18,-7) Matrix(619,-1620,222,-581) -> Matrix(1,0,-2,1) Matrix(701,-2500,196,-699) -> Matrix(1,0,0,1) Matrix(541,-1940,128,-459) -> Matrix(9,-2,32,-7) Matrix(41,-160,10,-39) -> Matrix(1,0,2,1) Matrix(21,-100,4,-19) -> Matrix(7,-2,18,-5) 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): 26 Degree of the the map X: 26 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 0/1 0/1 2 1 1/1 (0/1,1/4) 0 20 8/7 0/1 2 5 7/6 (0/1,1/2) 0 20 20/17 0/1 6 1 13/11 (0/1,1/8) 0 20 6/5 0/1 1 10 11/9 (0/1,1/5) 0 20 5/4 1/4 1 4 19/15 (1/4,3/11) 0 20 14/11 2/7 1 10 23/18 (1/4,1/3) 0 20 9/7 (2/7,1/3) 0 20 4/3 0/1 2 5 19/14 (1/6,4/23) 0 20 15/11 1/5 2 4 11/8 (1/6,1/5) 0 20 7/5 (1/5,1/4) 0 20 10/7 1/4 3 2 13/9 (1/4,2/7) 0 20 3/2 (1/4,1/3) 0 20 20/13 1/3 2 1 17/11 (1/3,1/2) 0 20 14/9 0/1 1 10 11/7 (0/1,1/3) 0 20 30/19 1/3 2 2 19/12 (2/5,1/2) 0 20 8/5 0/1 1 5 5/3 0 4 12/7 0/1 1 5 19/11 (1/5,1/4) 0 20 26/15 0/1 1 10 7/4 (1/4,2/7) 0 20 16/9 2/7 2 5 9/5 (4/13,1/3) 0 20 20/11 1/3 9 1 11/6 (1/3,5/14) 0 20 2/1 0/1 1 10 11/5 (4/13,1/3) 0 20 20/9 1/3 7 1 9/4 (1/3,3/8) 0 20 7/3 (1/3,1/2) 0 20 19/8 (0/1,1/2) 0 20 50/21 1/1 2 2 31/13 (0/1,1/1) 0 20 43/18 (1/1,1/0) 0 20 12/5 0/1 2 5 41/17 (0/1,1/4) 0 20 29/12 (1/4,1/3) 0 20 17/7 (1/3,1/2) 0 20 5/2 0/1 1 4 23/9 (1/5,1/4) 0 20 41/16 (6/25,1/4) 0 20 100/39 1/4 15 1 59/23 (1/4,9/35) 0 20 18/7 2/7 1 10 13/5 (0/1,1/2) 0 20 34/13 0/1 1 10 21/8 (1/4,2/7) 0 20 8/3 0/1 1 5 19/7 (1/4,1/3) 0 20 30/11 1/4 1 2 41/15 (1/4,2/7) 0 20 11/4 (1/4,1/3) 0 20 25/9 1/3 2 4 39/14 (1/2,2/3) 0 20 14/5 0/1 1 10 17/6 (1/4,1/3) 0 20 20/7 1/3 1 1 3/1 (0/1,1/3) 0 20 10/3 1/3 1 2 17/5 (1/3,1/2) 0 20 41/12 (1/2,4/7) 0 20 24/7 0/1 2 5 7/2 (0/1,1/2) 0 20 32/9 0/1 2 5 25/7 0 4 68/19 0/1 2 5 43/12 (1/5,1/4) 0 20 18/5 2/7 1 10 11/3 (1/3,2/5) 0 20 15/4 1/2 1 4 19/5 (3/7,1/2) 0 20 4/1 0/1 1 5 21/5 (3/13,1/4) 0 20 38/9 4/15 1 10 17/4 (1/4,1/3) 0 20 13/3 (1/4,2/7) 0 20 22/5 4/13 1 10 9/2 (1/3,1/2) 0 20 5/1 1/3 2 4 11/2 (1/3,3/8) 0 20 17/3 (1/3,1/2) 0 20 6/1 2/5 1 10 13/2 (4/9,1/2) 0 20 20/3 1/2 5 1 7/1 (1/2,1/1) 0 20 1/0 (0/1,1/2) 0 20 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(299,-340,124,-141) (1/1,8/7) -> (12/5,41/17) Hyperbolic Matrix(241,-280,68,-79) (8/7,7/6) -> (7/2,32/9) Hyperbolic Matrix(239,-280,204,-239) (7/6,20/17) -> (7/6,20/17) Reflection Matrix(441,-520,374,-441) (20/17,13/11) -> (20/17,13/11) Reflection Matrix(439,-520,168,-199) (13/11,6/5) -> (13/5,34/13) Glide Reflection Matrix(181,-220,116,-141) (6/5,11/9) -> (14/9,11/7) Glide Reflection Matrix(179,-220,48,-59) (11/9,5/4) -> (11/3,15/4) Glide Reflection Matrix(301,-380,80,-101) (5/4,19/15) -> (15/4,19/5) Glide Reflection Matrix(599,-760,346,-439) (19/15,14/11) -> (19/11,26/15) Glide Reflection Matrix(439,-560,156,-199) (14/11,23/18) -> (14/5,17/6) Glide Reflection Matrix(859,-1100,360,-461) (23/18,9/7) -> (31/13,43/18) Hyperbolic Matrix(139,-180,78,-101) (9/7,4/3) -> (16/9,9/5) Hyperbolic Matrix(459,-620,134,-181) (4/3,19/14) -> (41/12,24/7) Hyperbolic Matrix(779,-1060,280,-381) (19/14,15/11) -> (25/9,39/14) Hyperbolic Matrix(321,-440,116,-159) (15/11,11/8) -> (11/4,25/9) Hyperbolic Matrix(101,-140,44,-61) (11/8,7/5) -> (9/4,7/3) Glide Reflection Matrix(99,-140,70,-99) (7/5,10/7) -> (7/5,10/7) Reflection Matrix(181,-260,126,-181) (10/7,13/9) -> (10/7,13/9) Reflection Matrix(179,-260,42,-61) (13/9,3/2) -> (17/4,13/3) Hyperbolic Matrix(79,-120,52,-79) (3/2,20/13) -> (3/2,20/13) Reflection Matrix(441,-680,286,-441) (20/13,17/11) -> (20/13,17/11) Reflection Matrix(219,-340,38,-59) (17/11,14/9) -> (17/3,6/1) Glide Reflection Matrix(419,-660,266,-419) (11/7,30/19) -> (11/7,30/19) Reflection Matrix(961,-1520,404,-639) (30/19,19/12) -> (19/8,50/21) Hyperbolic Matrix(201,-320,76,-121) (19/12,8/5) -> (21/8,8/3) Glide Reflection Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(221,-380,82,-141) (12/7,19/11) -> (8/3,19/7) Glide Reflection Matrix(219,-380,34,-59) (26/15,7/4) -> (6/1,13/2) Glide Reflection Matrix(159,-280,46,-81) (7/4,16/9) -> (24/7,7/2) Hyperbolic Matrix(199,-360,110,-199) (9/5,20/11) -> (9/5,20/11) Reflection Matrix(241,-440,132,-241) (20/11,11/6) -> (20/11,11/6) Reflection Matrix(141,-260,32,-59) (11/6,2/1) -> (22/5,9/2) Hyperbolic Matrix(101,-220,28,-61) (2/1,11/5) -> (18/5,11/3) Glide Reflection Matrix(199,-440,90,-199) (11/5,20/9) -> (11/5,20/9) Reflection Matrix(161,-360,72,-161) (20/9,9/4) -> (20/9,9/4) Reflection Matrix(59,-140,8,-19) (7/3,19/8) -> (7/1,1/0) Glide Reflection Matrix(1301,-3100,546,-1301) (50/21,31/13) -> (50/21,31/13) Reflection Matrix(1439,-3440,402,-961) (43/18,12/5) -> (68/19,43/12) Hyperbolic Matrix(679,-1640,248,-599) (41/17,29/12) -> (41/15,11/4) Glide Reflection Matrix(281,-680,50,-121) (29/12,17/7) -> (11/2,17/3) Glide Reflection Matrix(81,-200,32,-79) (17/7,5/2) -> (5/2,23/9) Parabolic Matrix(641,-1640,188,-481) (23/9,41/16) -> (17/5,41/12) Glide Reflection Matrix(3199,-8200,1248,-3199) (41/16,100/39) -> (41/16,100/39) Reflection Matrix(4601,-11800,1794,-4601) (100/39,59/23) -> (100/39,59/23) Reflection Matrix(1021,-2620,242,-621) (59/23,18/7) -> (21/5,38/9) Glide Reflection Matrix(201,-520,46,-119) (18/7,13/5) -> (13/3,22/5) Hyperbolic Matrix(619,-1620,222,-581) (34/13,21/8) -> (39/14,14/5) Hyperbolic Matrix(419,-1140,154,-419) (19/7,30/11) -> (19/7,30/11) Reflection Matrix(901,-2460,330,-901) (30/11,41/15) -> (30/11,41/15) Reflection Matrix(239,-680,84,-239) (17/6,20/7) -> (17/6,20/7) Reflection Matrix(41,-120,14,-41) (20/7,3/1) -> (20/7,3/1) Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) Reflection Matrix(101,-340,30,-101) (10/3,17/5) -> (10/3,17/5) Reflection Matrix(701,-2500,196,-699) (32/9,25/7) -> (25/7,68/19) Parabolic Matrix(541,-1940,128,-459) (43/12,18/5) -> (38/9,17/4) Hyperbolic Matrix(41,-160,10,-39) (19/5,4/1) -> (4/1,21/5) Parabolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic Matrix(79,-520,12,-79) (13/2,20/3) -> (13/2,20/3) Reflection Matrix(41,-280,6,-41) (20/3,7/1) -> (20/3,7/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,4,-1) (0/1,1/0) -> (0/1,1/2) Matrix(1,0,2,-1) -> Matrix(1,0,8,-1) (0/1,1/1) -> (0/1,1/4) Matrix(299,-340,124,-141) -> Matrix(1,0,0,1) Matrix(241,-280,68,-79) -> Matrix(1,0,0,1) Matrix(239,-280,204,-239) -> Matrix(1,0,4,-1) (7/6,20/17) -> (0/1,1/2) Matrix(441,-520,374,-441) -> Matrix(1,0,16,-1) (20/17,13/11) -> (0/1,1/8) Matrix(439,-520,168,-199) -> Matrix(1,0,10,-1) *** -> (0/1,1/5) Matrix(181,-220,116,-141) -> Matrix(1,0,8,-1) *** -> (0/1,1/4) Matrix(179,-220,48,-59) -> Matrix(9,-2,22,-5) Matrix(301,-380,80,-101) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(599,-760,346,-439) -> Matrix(7,-2,24,-7) *** -> (1/4,1/3) Matrix(439,-560,156,-199) -> Matrix(7,-2,24,-7) *** -> (1/4,1/3) Matrix(859,-1100,360,-461) -> Matrix(7,-2,4,-1) Matrix(139,-180,78,-101) -> Matrix(5,-2,18,-7) 1/3 Matrix(459,-620,134,-181) -> Matrix(1,0,-4,1) 0/1 Matrix(779,-1060,280,-381) -> Matrix(11,-2,28,-5) Matrix(321,-440,116,-159) -> Matrix(1,0,-2,1) 0/1 Matrix(101,-140,44,-61) -> Matrix(9,-2,22,-5) Matrix(99,-140,70,-99) -> Matrix(9,-2,40,-9) (7/5,10/7) -> (1/5,1/4) Matrix(181,-260,126,-181) -> Matrix(15,-4,56,-15) (10/7,13/9) -> (1/4,2/7) Matrix(179,-260,42,-61) -> Matrix(1,0,0,1) Matrix(79,-120,52,-79) -> Matrix(7,-2,24,-7) (3/2,20/13) -> (1/4,1/3) Matrix(441,-680,286,-441) -> Matrix(5,-2,12,-5) (20/13,17/11) -> (1/3,1/2) Matrix(219,-340,38,-59) -> Matrix(5,-2,12,-5) *** -> (1/3,1/2) Matrix(419,-660,266,-419) -> Matrix(1,0,6,-1) (11/7,30/19) -> (0/1,1/3) Matrix(961,-1520,404,-639) -> Matrix(5,-2,8,-3) 1/2 Matrix(201,-320,76,-121) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(61,-100,36,-59) -> Matrix(1,0,0,1) Matrix(221,-380,82,-141) -> Matrix(1,0,8,-1) *** -> (0/1,1/4) Matrix(219,-380,34,-59) -> Matrix(9,-2,22,-5) Matrix(159,-280,46,-81) -> Matrix(7,-2,18,-5) 1/3 Matrix(199,-360,110,-199) -> Matrix(25,-8,78,-25) (9/5,20/11) -> (4/13,1/3) Matrix(241,-440,132,-241) -> Matrix(29,-10,84,-29) (20/11,11/6) -> (1/3,5/14) Matrix(141,-260,32,-59) -> Matrix(11,-4,36,-13) 1/3 Matrix(101,-220,28,-61) -> Matrix(7,-2,24,-7) *** -> (1/4,1/3) Matrix(199,-440,90,-199) -> Matrix(25,-8,78,-25) (11/5,20/9) -> (4/13,1/3) Matrix(161,-360,72,-161) -> Matrix(17,-6,48,-17) (20/9,9/4) -> (1/3,3/8) Matrix(59,-140,8,-19) -> Matrix(1,0,4,-1) *** -> (0/1,1/2) Matrix(1301,-3100,546,-1301) -> Matrix(1,0,2,-1) (50/21,31/13) -> (0/1,1/1) Matrix(1439,-3440,402,-961) -> Matrix(1,0,4,1) 0/1 Matrix(679,-1640,248,-599) -> Matrix(7,-2,24,-7) *** -> (1/4,1/3) Matrix(281,-680,50,-121) -> Matrix(5,-2,12,-5) *** -> (1/3,1/2) Matrix(81,-200,32,-79) -> Matrix(1,0,2,1) 0/1 Matrix(641,-1640,188,-481) -> Matrix(9,-2,22,-5) Matrix(3199,-8200,1248,-3199) -> Matrix(49,-12,200,-49) (41/16,100/39) -> (6/25,1/4) Matrix(4601,-11800,1794,-4601) -> Matrix(71,-18,280,-71) (100/39,59/23) -> (1/4,9/35) Matrix(1021,-2620,242,-621) -> Matrix(23,-6,88,-23) *** -> (1/4,3/11) Matrix(201,-520,46,-119) -> Matrix(5,-2,18,-7) 1/3 Matrix(619,-1620,222,-581) -> Matrix(1,0,-2,1) 0/1 Matrix(419,-1140,154,-419) -> Matrix(7,-2,24,-7) (19/7,30/11) -> (1/4,1/3) Matrix(901,-2460,330,-901) -> Matrix(15,-4,56,-15) (30/11,41/15) -> (1/4,2/7) Matrix(239,-680,84,-239) -> Matrix(7,-2,24,-7) (17/6,20/7) -> (1/4,1/3) Matrix(41,-120,14,-41) -> Matrix(1,0,6,-1) (20/7,3/1) -> (0/1,1/3) Matrix(19,-60,6,-19) -> Matrix(1,0,6,-1) (3/1,10/3) -> (0/1,1/3) Matrix(101,-340,30,-101) -> Matrix(5,-2,12,-5) (10/3,17/5) -> (1/3,1/2) Matrix(701,-2500,196,-699) -> Matrix(1,0,0,1) Matrix(541,-1940,128,-459) -> Matrix(9,-2,32,-7) 1/4 Matrix(41,-160,10,-39) -> Matrix(1,0,2,1) 0/1 Matrix(21,-100,4,-19) -> Matrix(7,-2,18,-5) 1/3 Matrix(79,-520,12,-79) -> Matrix(17,-8,36,-17) (13/2,20/3) -> (4/9,1/2) Matrix(41,-280,6,-41) -> Matrix(3,-2,4,-3) (20/3,7/1) -> (1/2,1/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.