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 -7/8 -4/5 -3/4 -11/15 -3/5 -1/2 -2/5 -39/100 -3/8 -7/25 -1/4 -1/5 -3/17 -1/6 0/1 1/7 3/20 1/6 1/5 2/9 3/13 1/4 5/19 3/11 2/7 3/10 1/3 7/20 11/30 3/8 2/5 9/20 1/2 11/20 5/9 3/5 19/30 13/20 2/3 7/10 11/15 3/4 4/5 17/20 13/15 7/8 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/0 -7/8 0/1 -6/7 -1/1 0/1 -17/20 0/1 -11/13 0/1 1/2 -5/6 0/1 -9/11 1/1 1/0 -4/5 1/0 -15/19 -1/1 1/0 -11/14 0/1 -18/23 -3/1 1/0 -7/9 -1/1 1/0 -3/4 0/1 -14/19 0/1 1/1 -11/15 1/1 -8/11 1/1 1/0 -5/7 1/1 1/0 -7/10 1/0 -9/13 -2/1 1/0 -2/3 -1/1 1/0 -13/20 -1/1 -11/17 -1/1 0/1 -9/14 0/1 -7/11 -1/1 1/0 -19/30 -1/1 -12/19 -1/1 0/1 -5/8 -2/1 0/1 -3/5 -1/1 1/1 -7/12 -2/1 0/1 -11/19 -1/1 1/0 -15/26 -2/1 -4/7 -1/1 0/1 -9/16 0/1 -5/9 -1/1 1/0 -11/20 -1/1 -6/11 -1/1 -1/2 -1/2 0/1 -5/11 1/1 1/0 -9/20 1/0 -4/9 -1/1 1/0 -3/7 -1/1 1/0 -8/19 -1/1 0/1 -21/50 -1/1 -13/31 -1/1 -1/2 -18/43 -1/2 -1/3 -5/12 0/1 -17/41 0/1 1/0 -12/29 -1/1 1/0 -7/17 -1/1 0/1 -2/5 0/1 -9/23 0/1 1/1 -16/41 0/1 1/1 -39/100 1/1 -23/59 1/1 1/0 -7/18 0/1 -5/13 0/1 1/0 -13/34 0/1 -8/21 0/1 1/1 -3/8 0/1 2/1 -7/19 1/1 1/0 -11/30 1/0 -15/41 0/1 1/0 -4/11 1/1 1/0 -9/25 1/1 -14/39 1/1 2/1 -5/14 2/1 -6/17 5/1 1/0 -7/20 1/0 -1/3 -1/1 1/0 -3/10 -1/1 -5/17 -1/1 0/1 -12/41 -1/1 0/1 -7/24 0/1 -2/7 -1/1 0/1 -9/32 0/1 -7/25 -1/1 1/1 -19/68 0/1 -12/43 -1/1 1/0 -5/18 0/1 -3/11 -1/1 1/0 -4/15 1/0 -5/19 -1/1 1/0 -1/4 -2/1 0/1 -5/21 -1/1 1/0 -9/38 0/1 -4/17 -1/1 1/0 -3/13 -2/1 1/0 -5/22 -2/1 -2/9 -1/1 1/0 -1/5 -1/1 -2/11 -1/1 -1/2 -3/17 -1/1 0/1 -1/6 0/1 -2/13 -2/1 -1/1 -3/20 -1/1 -1/7 -1/1 -1/2 0/1 -1/1 0/1 1/7 -1/1 1/0 3/20 -1/1 2/13 -1/1 -2/3 1/6 0/1 1/5 -1/1 3/14 -2/3 5/23 -1/1 -2/3 2/9 -1/1 -1/2 3/13 -2/3 -1/2 1/4 -2/3 0/1 5/19 -1/1 -1/2 4/15 -1/2 3/11 -1/1 -1/2 5/18 0/1 12/43 -1/1 -1/2 7/25 -1/1 -1/3 2/7 -1/1 0/1 5/17 -1/1 0/1 3/10 -1/1 1/3 -1/1 -1/2 7/20 -1/2 6/17 -1/2 -5/11 5/14 -2/5 4/11 -1/2 -1/3 15/41 -1/2 0/1 11/30 -1/2 7/19 -1/2 -1/3 3/8 -2/5 0/1 8/21 -1/3 0/1 13/34 0/1 5/13 -1/2 0/1 7/18 0/1 2/5 0/1 9/22 0/1 25/61 -1/1 1/0 41/100 -1/1 16/39 -1/1 0/1 7/17 -1/1 0/1 12/29 -1/1 -1/2 5/12 0/1 13/31 -1/1 1/0 8/19 -1/1 0/1 3/7 -1/1 -1/2 4/9 -1/1 -1/2 9/20 -1/2 5/11 -1/2 -1/3 1/2 0/1 7/13 0/1 1/0 6/11 -1/1 1/0 11/20 -1/1 5/9 -1/1 -1/2 4/7 -1/1 0/1 3/5 -1/1 -1/3 8/13 -1/1 0/1 21/34 -2/3 13/21 -1/1 -1/2 5/8 -2/3 0/1 12/19 -1/1 0/1 31/49 -1/1 1/0 19/30 -1/1 7/11 -1/1 -1/2 9/14 0/1 11/17 -1/1 0/1 13/20 -1/1 2/3 -1/1 -1/2 15/22 0/1 13/19 -1/1 -1/2 11/16 -2/3 0/1 9/13 -2/3 -1/2 7/10 -1/2 5/7 -1/2 -1/3 8/11 -1/2 -1/3 19/26 -2/5 11/15 -1/3 25/34 0/1 14/19 -1/3 0/1 3/4 0/1 16/21 -1/1 0/1 13/17 -1/1 0/1 10/13 -1/1 0/1 7/9 -1/1 -1/2 25/32 -2/3 18/23 -3/5 -1/2 11/14 0/1 15/19 -1/1 -1/2 4/5 -1/2 9/11 -1/2 -1/3 5/6 0/1 11/13 -1/4 0/1 17/20 0/1 6/7 -1/1 0/1 13/15 -1/1 -1/3 20/23 -1/1 -1/2 7/8 0/1 8/9 -1/1 -1/2 1/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(141,124,-340,-299) (-1/1,-7/8) -> (-5/12,-17/41) Hyperbolic Matrix(79,68,-280,-241) (-7/8,-6/7) -> (-2/7,-9/32) Hyperbolic Matrix(239,204,280,239) (-6/7,-17/20) -> (17/20,6/7) Hyperbolic Matrix(441,374,520,441) (-17/20,-11/13) -> (11/13,17/20) Hyperbolic Matrix(199,168,520,439) (-11/13,-5/6) -> (13/34,5/13) Hyperbolic Matrix(141,116,220,181) (-5/6,-9/11) -> (7/11,9/14) Hyperbolic Matrix(59,48,220,179) (-9/11,-4/5) -> (4/15,3/11) Hyperbolic Matrix(101,80,380,301) (-4/5,-15/19) -> (5/19,4/15) Hyperbolic Matrix(581,458,940,741) (-15/19,-11/14) -> (21/34,13/21) Hyperbolic Matrix(199,156,560,439) (-11/14,-18/23) -> (6/17,5/14) Hyperbolic Matrix(461,360,-1100,-859) (-18/23,-7/9) -> (-13/31,-18/43) Hyperbolic Matrix(101,78,-180,-139) (-7/9,-3/4) -> (-9/16,-5/9) Hyperbolic Matrix(181,134,-620,-459) (-3/4,-14/19) -> (-12/41,-7/24) Hyperbolic Matrix(381,280,-1060,-779) (-14/19,-11/15) -> (-9/25,-14/39) Hyperbolic Matrix(159,116,-440,-321) (-11/15,-8/11) -> (-4/11,-9/25) Hyperbolic Matrix(61,44,140,101) (-8/11,-5/7) -> (3/7,4/9) Hyperbolic Matrix(99,70,140,99) (-5/7,-7/10) -> (7/10,5/7) Hyperbolic Matrix(181,126,260,181) (-7/10,-9/13) -> (9/13,7/10) Hyperbolic Matrix(61,42,-260,-179) (-9/13,-2/3) -> (-4/17,-3/13) Hyperbolic Matrix(79,52,120,79) (-2/3,-13/20) -> (13/20,2/3) Hyperbolic Matrix(441,286,680,441) (-13/20,-11/17) -> (11/17,13/20) Hyperbolic Matrix(121,78,560,361) (-11/17,-9/14) -> (3/14,5/23) Hyperbolic Matrix(181,116,220,141) (-9/14,-7/11) -> (9/11,5/6) Hyperbolic Matrix(419,266,660,419) (-7/11,-19/30) -> (19/30,7/11) Hyperbolic Matrix(639,404,-1520,-961) (-19/30,-12/19) -> (-8/19,-21/50) Hyperbolic Matrix(121,76,320,201) (-12/19,-5/8) -> (3/8,8/21) Hyperbolic Matrix(59,36,-100,-61) (-5/8,-3/5) -> (-3/5,-7/12) Parabolic Matrix(141,82,380,221) (-7/12,-11/19) -> (7/19,3/8) Hyperbolic Matrix(599,346,760,439) (-11/19,-15/26) -> (11/14,15/19) Hyperbolic Matrix(59,34,380,219) (-15/26,-4/7) -> (2/13,1/6) Hyperbolic Matrix(81,46,-280,-159) (-4/7,-9/16) -> (-7/24,-2/7) Hyperbolic Matrix(199,110,360,199) (-5/9,-11/20) -> (11/20,5/9) Hyperbolic Matrix(241,132,440,241) (-11/20,-6/11) -> (6/11,11/20) Hyperbolic Matrix(59,32,-260,-141) (-6/11,-1/2) -> (-5/22,-2/9) Hyperbolic Matrix(61,28,220,101) (-1/2,-5/11) -> (3/11,5/18) Hyperbolic Matrix(199,90,440,199) (-5/11,-9/20) -> (9/20,5/11) Hyperbolic Matrix(161,72,360,161) (-9/20,-4/9) -> (4/9,9/20) Hyperbolic Matrix(101,44,140,61) (-4/9,-3/7) -> (5/7,8/11) Hyperbolic Matrix(19,8,140,59) (-3/7,-8/19) -> (0/1,1/7) Hyperbolic Matrix(2139,898,3380,1419) (-21/50,-13/31) -> (31/49,19/30) Hyperbolic Matrix(961,402,-3440,-1439) (-18/43,-5/12) -> (-19/68,-12/43) Hyperbolic Matrix(599,248,1640,679) (-17/41,-12/29) -> (4/11,15/41) Hyperbolic Matrix(179,74,820,339) (-12/29,-7/17) -> (5/23,2/9) Hyperbolic Matrix(79,32,-200,-81) (-7/17,-2/5) -> (-2/5,-9/23) Parabolic Matrix(901,352,1180,461) (-9/23,-16/41) -> (16/21,13/17) Hyperbolic Matrix(3281,1280,8000,3121) (-16/41,-39/100) -> (41/100,16/39) Hyperbolic Matrix(4919,1918,12000,4679) (-39/100,-23/59) -> (25/61,41/100) Hyperbolic Matrix(1119,436,1640,639) (-23/59,-7/18) -> (15/22,13/19) Hyperbolic Matrix(119,46,-520,-201) (-7/18,-5/13) -> (-3/13,-5/22) Hyperbolic Matrix(439,168,520,199) (-5/13,-13/34) -> (5/6,11/13) Hyperbolic Matrix(581,222,-1620,-619) (-13/34,-8/21) -> (-14/39,-5/14) Hyperbolic Matrix(201,76,320,121) (-8/21,-3/8) -> (5/8,12/19) Hyperbolic Matrix(199,74,320,119) (-3/8,-7/19) -> (13/21,5/8) Hyperbolic Matrix(419,154,1140,419) (-7/19,-11/30) -> (11/30,7/19) Hyperbolic Matrix(901,330,2460,901) (-11/30,-15/41) -> (15/41,11/30) Hyperbolic Matrix(339,124,380,139) (-15/41,-4/11) -> (8/9,1/1) Hyperbolic Matrix(439,156,560,199) (-5/14,-6/17) -> (18/23,11/14) Hyperbolic Matrix(239,84,680,239) (-6/17,-7/20) -> (7/20,6/17) Hyperbolic Matrix(41,14,120,41) (-7/20,-1/3) -> (1/3,7/20) Hyperbolic Matrix(19,6,60,19) (-1/3,-3/10) -> (3/10,1/3) Hyperbolic Matrix(101,30,340,101) (-3/10,-5/17) -> (5/17,3/10) Hyperbolic Matrix(559,164,1360,399) (-5/17,-12/41) -> (16/39,7/17) Hyperbolic Matrix(699,196,-2500,-701) (-9/32,-7/25) -> (-7/25,-19/68) Parabolic Matrix(459,128,-1940,-541) (-12/43,-5/18) -> (-9/38,-4/17) Hyperbolic Matrix(101,28,220,61) (-5/18,-3/11) -> (5/11,1/2) Hyperbolic Matrix(179,48,220,59) (-3/11,-4/15) -> (4/5,9/11) Hyperbolic Matrix(301,80,380,101) (-4/15,-5/19) -> (15/19,4/5) Hyperbolic Matrix(39,10,-160,-41) (-5/19,-1/4) -> (-1/4,-5/21) Parabolic Matrix(1139,270,2780,659) (-5/21,-9/38) -> (9/22,25/61) Hyperbolic Matrix(19,4,-100,-21) (-2/9,-1/5) -> (-1/5,-2/11) Parabolic Matrix(281,50,680,121) (-2/11,-3/17) -> (7/17,12/29) Hyperbolic Matrix(219,38,340,59) (-3/17,-1/6) -> (9/14,11/17) Hyperbolic Matrix(321,50,520,81) (-1/6,-2/13) -> (8/13,21/34) Hyperbolic Matrix(79,12,520,79) (-2/13,-3/20) -> (3/20,2/13) Hyperbolic Matrix(41,6,280,41) (-3/20,-1/7) -> (1/7,3/20) Hyperbolic Matrix(59,8,140,19) (-1/7,0/1) -> (8/19,3/7) Hyperbolic Matrix(21,-4,100,-19) (1/6,1/5) -> (1/5,3/14) Parabolic Matrix(141,-32,260,-59) (2/9,3/13) -> (7/13,6/11) Hyperbolic Matrix(179,-42,260,-61) (3/13,1/4) -> (11/16,9/13) Hyperbolic Matrix(261,-68,380,-99) (1/4,5/19) -> (13/19,11/16) Hyperbolic Matrix(681,-190,1000,-279) (5/18,12/43) -> (2/3,15/22) Hyperbolic Matrix(1059,-296,1220,-341) (12/43,7/25) -> (13/15,20/23) Hyperbolic Matrix(241,-68,280,-79) (7/25,2/7) -> (6/7,13/15) Hyperbolic Matrix(261,-76,340,-99) (2/7,5/17) -> (13/17,10/13) Hyperbolic Matrix(321,-116,440,-159) (5/14,4/11) -> (8/11,19/26) Hyperbolic Matrix(1001,-382,1360,-519) (8/21,13/34) -> (25/34,14/19) Hyperbolic Matrix(139,-54,260,-101) (5/13,7/18) -> (1/2,7/13) Hyperbolic Matrix(81,-32,200,-79) (7/18,2/5) -> (2/5,9/22) Parabolic Matrix(299,-124,340,-141) (12/29,5/12) -> (7/8,8/9) Hyperbolic Matrix(859,-360,1100,-461) (5/12,13/31) -> (7/9,25/32) Hyperbolic Matrix(961,-404,1520,-639) (13/31,8/19) -> (12/19,31/49) Hyperbolic Matrix(139,-78,180,-101) (5/9,4/7) -> (10/13,7/9) Hyperbolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(661,-484,900,-659) (19/26,11/15) -> (11/15,25/34) Parabolic Matrix(121,-90,160,-119) (14/19,3/4) -> (3/4,16/21) Parabolic Matrix(801,-626,920,-719) (25/32,18/23) -> (20/23,7/8) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(141,124,-340,-299) -> Matrix(1,0,0,1) Matrix(79,68,-280,-241) -> Matrix(1,0,0,1) Matrix(239,204,280,239) -> Matrix(1,0,0,1) Matrix(441,374,520,441) -> Matrix(1,0,-6,1) Matrix(199,168,520,439) -> Matrix(1,0,-4,1) Matrix(141,116,220,181) -> Matrix(1,0,-2,1) Matrix(59,48,220,179) -> Matrix(1,0,-2,1) Matrix(101,80,380,301) -> Matrix(1,2,-2,-3) Matrix(581,458,940,741) -> Matrix(1,2,-2,-3) Matrix(199,156,560,439) -> Matrix(1,-2,-2,5) Matrix(461,360,-1100,-859) -> Matrix(1,2,-2,-3) Matrix(101,78,-180,-139) -> Matrix(1,0,0,1) Matrix(181,134,-620,-459) -> Matrix(1,0,-2,1) Matrix(381,280,-1060,-779) -> Matrix(3,-2,2,-1) Matrix(159,116,-440,-321) -> Matrix(1,0,0,1) Matrix(61,44,140,101) -> Matrix(1,0,-2,1) Matrix(99,70,140,99) -> Matrix(1,-2,-2,5) Matrix(181,126,260,181) -> Matrix(1,4,-2,-7) Matrix(61,42,-260,-179) -> Matrix(1,0,0,1) Matrix(79,52,120,79) -> Matrix(1,2,-2,-3) Matrix(441,286,680,441) -> Matrix(1,0,0,1) Matrix(121,78,560,361) -> Matrix(1,2,-2,-3) Matrix(181,116,220,141) -> Matrix(1,0,-2,1) Matrix(419,266,660,419) -> Matrix(1,2,-2,-3) Matrix(639,404,-1520,-961) -> Matrix(1,0,0,1) Matrix(121,76,320,201) -> Matrix(1,0,-2,1) Matrix(59,36,-100,-61) -> Matrix(1,0,0,1) Matrix(141,82,380,221) -> Matrix(1,0,-2,1) Matrix(599,346,760,439) -> Matrix(1,2,-2,-3) Matrix(59,34,380,219) -> Matrix(1,2,-2,-3) Matrix(81,46,-280,-159) -> Matrix(1,0,0,1) Matrix(199,110,360,199) -> Matrix(1,2,-2,-3) Matrix(241,132,440,241) -> Matrix(3,2,-2,-1) Matrix(59,32,-260,-141) -> Matrix(3,2,-2,-1) Matrix(61,28,220,101) -> Matrix(1,0,-2,1) Matrix(199,90,440,199) -> Matrix(1,-2,-2,5) Matrix(161,72,360,161) -> Matrix(1,2,-2,-3) Matrix(101,44,140,61) -> Matrix(1,0,-2,1) Matrix(19,8,140,59) -> Matrix(1,0,0,1) Matrix(2139,898,3380,1419) -> Matrix(3,2,-2,-1) Matrix(961,402,-3440,-1439) -> Matrix(1,0,2,1) Matrix(599,248,1640,679) -> Matrix(1,0,-2,1) Matrix(179,74,820,339) -> Matrix(1,2,-2,-3) Matrix(79,32,-200,-81) -> Matrix(1,0,2,1) Matrix(901,352,1180,461) -> Matrix(1,0,-2,1) Matrix(3281,1280,8000,3121) -> Matrix(1,0,-2,1) Matrix(4919,1918,12000,4679) -> Matrix(1,-2,0,1) Matrix(1119,436,1640,639) -> Matrix(1,0,-2,1) Matrix(119,46,-520,-201) -> Matrix(1,-2,0,1) Matrix(439,168,520,199) -> Matrix(1,0,-4,1) Matrix(581,222,-1620,-619) -> Matrix(3,-2,2,-1) Matrix(201,76,320,121) -> Matrix(1,0,-2,1) Matrix(199,74,320,119) -> Matrix(1,0,-2,1) Matrix(419,154,1140,419) -> Matrix(1,-2,-2,5) Matrix(901,330,2460,901) -> Matrix(1,0,-2,1) Matrix(339,124,380,139) -> Matrix(1,0,-2,1) Matrix(439,156,560,199) -> Matrix(1,-2,-2,5) Matrix(239,84,680,239) -> Matrix(1,-10,-2,21) Matrix(41,14,120,41) -> Matrix(1,2,-2,-3) Matrix(19,6,60,19) -> Matrix(1,2,-2,-3) Matrix(101,30,340,101) -> Matrix(1,0,0,1) Matrix(559,164,1360,399) -> Matrix(1,0,0,1) Matrix(699,196,-2500,-701) -> Matrix(1,0,0,1) Matrix(459,128,-1940,-541) -> Matrix(1,0,0,1) Matrix(101,28,220,61) -> Matrix(1,0,-2,1) Matrix(179,48,220,59) -> Matrix(1,0,-2,1) Matrix(301,80,380,101) -> Matrix(1,2,-2,-3) Matrix(39,10,-160,-41) -> Matrix(1,0,0,1) Matrix(1139,270,2780,659) -> Matrix(1,0,0,1) Matrix(19,4,-100,-21) -> Matrix(1,2,-2,-3) Matrix(281,50,680,121) -> Matrix(1,0,0,1) Matrix(219,38,340,59) -> Matrix(1,0,0,1) Matrix(321,50,520,81) -> Matrix(1,2,-2,-3) Matrix(79,12,520,79) -> Matrix(3,4,-4,-5) Matrix(41,6,280,41) -> Matrix(3,2,-2,-1) Matrix(59,8,140,19) -> Matrix(1,0,0,1) Matrix(21,-4,100,-19) -> Matrix(1,2,-2,-3) Matrix(141,-32,260,-59) -> Matrix(3,2,-2,-1) Matrix(179,-42,260,-61) -> Matrix(1,0,0,1) Matrix(261,-68,380,-99) -> Matrix(1,0,0,1) Matrix(681,-190,1000,-279) -> Matrix(1,0,0,1) Matrix(1059,-296,1220,-341) -> Matrix(1,0,0,1) Matrix(241,-68,280,-79) -> Matrix(1,0,0,1) Matrix(261,-76,340,-99) -> Matrix(1,0,0,1) Matrix(321,-116,440,-159) -> Matrix(1,0,0,1) Matrix(1001,-382,1360,-519) -> Matrix(1,0,0,1) Matrix(139,-54,260,-101) -> Matrix(1,0,2,1) Matrix(81,-32,200,-79) -> Matrix(1,0,2,1) Matrix(299,-124,340,-141) -> Matrix(1,0,0,1) Matrix(859,-360,1100,-461) -> Matrix(1,2,-2,-3) Matrix(961,-404,1520,-639) -> Matrix(1,0,0,1) Matrix(139,-78,180,-101) -> Matrix(1,0,0,1) Matrix(61,-36,100,-59) -> Matrix(1,0,0,1) Matrix(661,-484,900,-659) -> Matrix(5,2,-18,-7) Matrix(121,-90,160,-119) -> Matrix(1,0,2,1) Matrix(801,-626,920,-719) -> Matrix(3,2,-8,-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): 14 Degree of the the map X: 14 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 (-1/1,0/1) 0 20 1/7 (-1/1,1/0) 0 20 3/20 -1/1 6 2 2/13 (-1/1,-2/3) 0 20 1/6 0/1 1 10 1/5 -1/1 2 4 3/14 -2/3 1 10 5/23 (-1/1,-2/3) 0 20 2/9 (-1/1,-1/2) 0 20 3/13 (-2/3,-1/2) 0 20 1/4 0 10 5/19 (-1/1,-1/2) 0 20 4/15 -1/2 1 4 3/11 (-1/1,-1/2) 0 20 5/18 0/1 1 10 12/43 (-1/1,-1/2) 0 20 7/25 0 4 2/7 (-1/1,0/1) 0 20 5/17 (-1/1,0/1) 0 20 3/10 -1/1 1 2 1/3 (-1/1,-1/2) 0 20 7/20 -1/2 12 2 6/17 (-1/2,-5/11) 0 20 5/14 -2/5 1 10 4/11 (-1/2,-1/3) 0 20 15/41 (-1/2,0/1) 0 20 11/30 -1/2 1 2 7/19 (-1/2,-1/3) 0 20 3/8 0 10 8/21 (-1/3,0/1) 0 20 13/34 0/1 1 10 5/13 (-1/2,0/1) 0 20 7/18 0/1 1 10 2/5 0/1 1 4 9/22 0/1 1 10 25/61 (-1/1,1/0) 0 20 41/100 -1/1 2 2 16/39 (-1/1,0/1) 0 20 7/17 (-1/1,0/1) 0 20 12/29 (-1/1,-1/2) 0 20 5/12 0/1 2 10 13/31 (-1/1,1/0) 0 20 8/19 (-1/1,0/1) 0 20 3/7 (-1/1,-1/2) 0 20 4/9 (-1/1,-1/2) 0 20 9/20 -1/2 4 2 5/11 (-1/2,-1/3) 0 20 1/2 0/1 1 10 7/13 (0/1,1/0) 0 20 6/11 (-1/1,1/0) 0 20 11/20 -1/1 4 2 5/9 (-1/1,-1/2) 0 20 4/7 (-1/1,0/1) 0 20 3/5 0 4 8/13 (-1/1,0/1) 0 20 21/34 -2/3 1 10 13/21 (-1/1,-1/2) 0 20 5/8 0 10 12/19 (-1/1,0/1) 0 20 31/49 (-1/1,1/0) 0 20 19/30 -1/1 2 2 7/11 (-1/1,-1/2) 0 20 9/14 0/1 1 10 11/17 (-1/1,0/1) 0 20 13/20 -1/1 2 2 2/3 (-1/1,-1/2) 0 20 15/22 0/1 1 10 13/19 (-1/1,-1/2) 0 20 11/16 0 10 9/13 (-2/3,-1/2) 0 20 7/10 -1/2 3 2 5/7 (-1/2,-1/3) 0 20 8/11 (-1/2,-1/3) 0 20 19/26 -2/5 1 10 11/15 -1/3 2 4 25/34 0/1 1 10 14/19 (-1/3,0/1) 0 20 3/4 0/1 2 10 16/21 (-1/1,0/1) 0 20 13/17 (-1/1,0/1) 0 20 10/13 (-1/1,0/1) 0 20 7/9 (-1/1,-1/2) 0 20 25/32 -2/3 2 10 18/23 (-3/5,-1/2) 0 20 11/14 0/1 1 10 15/19 (-1/1,-1/2) 0 20 4/5 -1/2 1 4 9/11 (-1/2,-1/3) 0 20 5/6 0/1 1 10 11/13 (-1/4,0/1) 0 20 17/20 0/1 6 2 6/7 (-1/1,0/1) 0 20 13/15 0 4 20/23 (-1/1,-1/2) 0 20 7/8 0/1 2 10 8/9 (-1/1,-1/2) 0 20 1/1 (-1/2,0/1) 0 20 1/0 0/1 2 2 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(59,-8,140,-19) (0/1,1/7) -> (8/19,3/7) Glide Reflection Matrix(41,-6,280,-41) (1/7,3/20) -> (1/7,3/20) Reflection Matrix(79,-12,520,-79) (3/20,2/13) -> (3/20,2/13) Reflection Matrix(321,-50,520,-81) (2/13,1/6) -> (8/13,21/34) Glide Reflection Matrix(21,-4,100,-19) (1/6,1/5) -> (1/5,3/14) Parabolic Matrix(361,-78,560,-121) (3/14,5/23) -> (9/14,11/17) Glide Reflection Matrix(339,-74,820,-179) (5/23,2/9) -> (7/17,12/29) Glide Reflection Matrix(141,-32,260,-59) (2/9,3/13) -> (7/13,6/11) Hyperbolic Matrix(179,-42,260,-61) (3/13,1/4) -> (11/16,9/13) Hyperbolic Matrix(261,-68,380,-99) (1/4,5/19) -> (13/19,11/16) Hyperbolic Matrix(301,-80,380,-101) (5/19,4/15) -> (15/19,4/5) Glide Reflection Matrix(179,-48,220,-59) (4/15,3/11) -> (4/5,9/11) Glide Reflection Matrix(101,-28,220,-61) (3/11,5/18) -> (5/11,1/2) Glide Reflection Matrix(681,-190,1000,-279) (5/18,12/43) -> (2/3,15/22) Hyperbolic Matrix(1059,-296,1220,-341) (12/43,7/25) -> (13/15,20/23) Hyperbolic Matrix(241,-68,280,-79) (7/25,2/7) -> (6/7,13/15) Hyperbolic Matrix(261,-76,340,-99) (2/7,5/17) -> (13/17,10/13) Hyperbolic Matrix(101,-30,340,-101) (5/17,3/10) -> (5/17,3/10) Reflection Matrix(19,-6,60,-19) (3/10,1/3) -> (3/10,1/3) Reflection Matrix(41,-14,120,-41) (1/3,7/20) -> (1/3,7/20) Reflection Matrix(239,-84,680,-239) (7/20,6/17) -> (7/20,6/17) Reflection Matrix(439,-156,560,-199) (6/17,5/14) -> (18/23,11/14) Glide Reflection Matrix(321,-116,440,-159) (5/14,4/11) -> (8/11,19/26) Hyperbolic Matrix(339,-124,380,-139) (4/11,15/41) -> (8/9,1/1) Glide Reflection Matrix(901,-330,2460,-901) (15/41,11/30) -> (15/41,11/30) Reflection Matrix(419,-154,1140,-419) (11/30,7/19) -> (11/30,7/19) Reflection Matrix(199,-74,320,-119) (7/19,3/8) -> (13/21,5/8) Glide Reflection Matrix(201,-76,320,-121) (3/8,8/21) -> (5/8,12/19) Glide Reflection Matrix(1001,-382,1360,-519) (8/21,13/34) -> (25/34,14/19) Hyperbolic Matrix(439,-168,520,-199) (13/34,5/13) -> (5/6,11/13) Glide Reflection Matrix(139,-54,260,-101) (5/13,7/18) -> (1/2,7/13) Hyperbolic Matrix(81,-32,200,-79) (7/18,2/5) -> (2/5,9/22) Parabolic Matrix(1201,-492,1760,-721) (9/22,25/61) -> (15/22,13/19) Glide Reflection Matrix(5001,-2050,12200,-5001) (25/61,41/100) -> (25/61,41/100) Reflection Matrix(3199,-1312,7800,-3199) (41/100,16/39) -> (41/100,16/39) Reflection Matrix(779,-320,1020,-419) (16/39,7/17) -> (16/21,13/17) Glide Reflection Matrix(299,-124,340,-141) (12/29,5/12) -> (7/8,8/9) Hyperbolic Matrix(859,-360,1100,-461) (5/12,13/31) -> (7/9,25/32) Hyperbolic Matrix(961,-404,1520,-639) (13/31,8/19) -> (12/19,31/49) Hyperbolic Matrix(101,-44,140,-61) (3/7,4/9) -> (5/7,8/11) Glide Reflection Matrix(161,-72,360,-161) (4/9,9/20) -> (4/9,9/20) Reflection Matrix(199,-90,440,-199) (9/20,5/11) -> (9/20,5/11) Reflection Matrix(241,-132,440,-241) (6/11,11/20) -> (6/11,11/20) Reflection Matrix(199,-110,360,-199) (11/20,5/9) -> (11/20,5/9) Reflection Matrix(139,-78,180,-101) (5/9,4/7) -> (10/13,7/9) Hyperbolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(741,-458,940,-581) (21/34,13/21) -> (11/14,15/19) Glide Reflection Matrix(1861,-1178,2940,-1861) (31/49,19/30) -> (31/49,19/30) Reflection Matrix(419,-266,660,-419) (19/30,7/11) -> (19/30,7/11) Reflection Matrix(181,-116,220,-141) (7/11,9/14) -> (9/11,5/6) Glide Reflection Matrix(441,-286,680,-441) (11/17,13/20) -> (11/17,13/20) Reflection Matrix(79,-52,120,-79) (13/20,2/3) -> (13/20,2/3) Reflection Matrix(181,-126,260,-181) (9/13,7/10) -> (9/13,7/10) Reflection Matrix(99,-70,140,-99) (7/10,5/7) -> (7/10,5/7) Reflection Matrix(661,-484,900,-659) (19/26,11/15) -> (11/15,25/34) Parabolic Matrix(121,-90,160,-119) (14/19,3/4) -> (3/4,16/21) Parabolic Matrix(801,-626,920,-719) (25/32,18/23) -> (20/23,7/8) Hyperbolic Matrix(441,-374,520,-441) (11/13,17/20) -> (11/13,17/20) Reflection Matrix(239,-204,280,-239) (17/20,6/7) -> (17/20,6/7) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(59,-8,140,-19) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(41,-6,280,-41) -> Matrix(1,2,0,-1) (1/7,3/20) -> (-1/1,1/0) Matrix(79,-12,520,-79) -> Matrix(5,4,-6,-5) (3/20,2/13) -> (-1/1,-2/3) Matrix(321,-50,520,-81) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(21,-4,100,-19) -> Matrix(1,2,-2,-3) -1/1 Matrix(361,-78,560,-121) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(339,-74,820,-179) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(141,-32,260,-59) -> Matrix(3,2,-2,-1) -1/1 Matrix(179,-42,260,-61) -> Matrix(1,0,0,1) Matrix(261,-68,380,-99) -> Matrix(1,0,0,1) Matrix(301,-80,380,-101) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(179,-48,220,-59) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(101,-28,220,-61) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(681,-190,1000,-279) -> Matrix(1,0,0,1) Matrix(1059,-296,1220,-341) -> Matrix(1,0,0,1) Matrix(241,-68,280,-79) -> Matrix(1,0,0,1) Matrix(261,-76,340,-99) -> Matrix(1,0,0,1) Matrix(101,-30,340,-101) -> Matrix(-1,0,2,1) (5/17,3/10) -> (-1/1,0/1) Matrix(19,-6,60,-19) -> Matrix(3,2,-4,-3) (3/10,1/3) -> (-1/1,-1/2) Matrix(41,-14,120,-41) -> Matrix(3,2,-4,-3) (1/3,7/20) -> (-1/1,-1/2) Matrix(239,-84,680,-239) -> Matrix(21,10,-44,-21) (7/20,6/17) -> (-1/2,-5/11) Matrix(439,-156,560,-199) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(321,-116,440,-159) -> Matrix(1,0,0,1) Matrix(339,-124,380,-139) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(901,-330,2460,-901) -> Matrix(-1,0,4,1) (15/41,11/30) -> (-1/2,0/1) Matrix(419,-154,1140,-419) -> Matrix(5,2,-12,-5) (11/30,7/19) -> (-1/2,-1/3) Matrix(199,-74,320,-119) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(201,-76,320,-121) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(1001,-382,1360,-519) -> Matrix(1,0,0,1) Matrix(439,-168,520,-199) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(139,-54,260,-101) -> Matrix(1,0,2,1) 0/1 Matrix(81,-32,200,-79) -> Matrix(1,0,2,1) 0/1 Matrix(1201,-492,1760,-721) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(5001,-2050,12200,-5001) -> Matrix(1,2,0,-1) (25/61,41/100) -> (-1/1,1/0) Matrix(3199,-1312,7800,-3199) -> Matrix(-1,0,2,1) (41/100,16/39) -> (-1/1,0/1) Matrix(779,-320,1020,-419) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(299,-124,340,-141) -> Matrix(1,0,0,1) Matrix(859,-360,1100,-461) -> Matrix(1,2,-2,-3) -1/1 Matrix(961,-404,1520,-639) -> Matrix(1,0,0,1) Matrix(101,-44,140,-61) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(161,-72,360,-161) -> Matrix(3,2,-4,-3) (4/9,9/20) -> (-1/1,-1/2) Matrix(199,-90,440,-199) -> Matrix(5,2,-12,-5) (9/20,5/11) -> (-1/2,-1/3) Matrix(241,-132,440,-241) -> Matrix(1,2,0,-1) (6/11,11/20) -> (-1/1,1/0) Matrix(199,-110,360,-199) -> Matrix(3,2,-4,-3) (11/20,5/9) -> (-1/1,-1/2) Matrix(139,-78,180,-101) -> Matrix(1,0,0,1) Matrix(61,-36,100,-59) -> Matrix(1,0,0,1) Matrix(741,-458,940,-581) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(1861,-1178,2940,-1861) -> Matrix(1,2,0,-1) (31/49,19/30) -> (-1/1,1/0) Matrix(419,-266,660,-419) -> Matrix(3,2,-4,-3) (19/30,7/11) -> (-1/1,-1/2) Matrix(181,-116,220,-141) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(441,-286,680,-441) -> Matrix(-1,0,2,1) (11/17,13/20) -> (-1/1,0/1) Matrix(79,-52,120,-79) -> Matrix(3,2,-4,-3) (13/20,2/3) -> (-1/1,-1/2) Matrix(181,-126,260,-181) -> Matrix(7,4,-12,-7) (9/13,7/10) -> (-2/3,-1/2) Matrix(99,-70,140,-99) -> Matrix(5,2,-12,-5) (7/10,5/7) -> (-1/2,-1/3) Matrix(661,-484,900,-659) -> Matrix(5,2,-18,-7) -1/3 Matrix(121,-90,160,-119) -> Matrix(1,0,2,1) 0/1 Matrix(801,-626,920,-719) -> Matrix(3,2,-8,-5) -1/2 Matrix(441,-374,520,-441) -> Matrix(-1,0,8,1) (11/13,17/20) -> (-1/4,0/1) Matrix(239,-204,280,-239) -> Matrix(-1,0,2,1) (17/20,6/7) -> (-1/1,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,4,1) (1/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.