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: 40 Genus: 29 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -10/3 -15/7 -2/1 -10/9 0/1 1/1 5/4 4/3 3/2 5/3 12/7 20/11 2/1 20/9 16/7 12/5 5/2 8/3 25/9 20/7 3/1 10/3 7/2 40/11 11/3 160/43 15/4 4/1 5/1 16/3 11/2 40/7 6/1 20/3 7/1 15/2 8/1 10/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 -2/1 0/1 -7/1 -1/1 -6/1 -2/1 0/1 -11/2 -3/2 -16/3 -2/1 0/1 -5/1 -1/1 -4/1 0/1 -15/4 1/0 -11/3 -1/3 -18/5 0/1 2/5 -25/7 1/1 -7/2 1/0 -10/3 0/1 -3/1 1/1 -14/5 -2/1 0/1 -11/4 1/2 -8/3 0/1 2/1 -5/2 1/0 -12/5 -2/1 -43/18 1/0 -31/13 -1/1 -19/8 -3/2 -26/11 -2/1 0/1 -7/3 -1/1 -16/7 0/1 2/1 -25/11 1/1 -9/4 1/0 -20/9 1/0 -11/5 -7/1 -13/6 1/0 -28/13 -4/1 -15/7 -3/1 -2/1 -2/1 0/1 -13/7 -1/1 -11/6 -7/6 -20/11 -1/1 -9/5 -1/1 -25/14 -1/2 -16/9 -2/3 0/1 -7/4 1/0 -19/11 -3/1 -31/18 1/0 -43/25 -1/1 -12/7 -2/1 -5/3 -1/1 -8/5 -2/3 0/1 -19/12 1/0 -49/31 -1/1 -30/19 -2/3 0/1 -41/26 -1/2 -11/7 -1/3 -25/16 1/0 -39/25 -1/1 -53/34 -1/2 -14/9 -2/1 0/1 -17/11 -1/1 -20/13 -1/1 -3/2 -1/2 -16/11 -2/5 0/1 -29/20 -3/8 -13/9 -1/3 -10/7 0/1 -17/12 1/2 -24/17 0/1 2/1 -7/5 -1/1 -32/23 -2/3 0/1 -25/18 -1/2 -68/49 -2/5 -43/31 -1/3 -18/13 -2/7 0/1 -29/21 -1/9 -40/29 0/1 -11/8 1/2 -26/19 -2/1 0/1 -93/68 -1/2 -160/117 0/1 -67/49 1/1 -41/30 1/0 -15/11 -1/1 -4/3 0/1 -5/4 1/0 -16/13 -2/1 0/1 -27/22 1/0 -11/9 -3/1 -17/14 1/0 -40/33 -2/1 0/1 -23/19 -1/1 -6/5 -2/1 0/1 -13/11 1/1 -20/17 1/0 -7/6 1/0 -15/13 -1/1 -8/7 -2/1 0/1 -9/8 1/0 -10/9 -2/1 0/1 -1/1 -1/1 0/1 0/1 1/1 1/1 8/7 0/1 2/1 7/6 1/0 6/5 0/1 2/1 11/9 3/1 16/13 0/1 2/1 5/4 1/0 4/3 0/1 15/11 1/1 11/8 -1/2 18/13 0/1 2/7 25/18 1/2 7/5 1/1 10/7 0/1 3/2 1/2 14/9 0/1 2/1 11/7 1/3 8/5 0/1 2/3 5/3 1/1 12/7 2/1 43/25 1/1 31/18 1/0 19/11 3/1 26/15 0/1 2/1 7/4 1/0 16/9 0/1 2/3 25/14 1/2 9/5 1/1 20/11 1/1 11/6 7/6 13/7 1/1 28/15 4/3 15/8 3/2 2/1 0/1 2/1 13/6 1/0 11/5 7/1 20/9 1/0 9/4 1/0 25/11 -1/1 16/7 -2/1 0/1 7/3 1/1 19/8 3/2 31/13 1/1 43/18 1/0 12/5 2/1 5/2 1/0 8/3 -2/1 0/1 19/7 1/1 49/18 1/0 30/11 -2/1 0/1 41/15 -1/1 11/4 -1/2 25/9 1/1 39/14 1/0 53/19 -1/1 14/5 0/1 2/1 17/6 1/0 20/7 1/0 3/1 -1/1 16/5 -2/3 0/1 29/9 -3/5 13/4 -1/2 10/3 0/1 17/5 1/3 24/7 0/1 2/3 7/2 1/0 32/9 -2/1 0/1 25/7 -1/1 68/19 -2/3 43/12 -1/2 18/5 -2/5 0/1 29/8 -1/8 40/11 0/1 11/3 1/3 26/7 0/1 2/1 93/25 -1/1 160/43 0/1 67/18 1/2 41/11 1/1 15/4 1/0 4/1 0/1 5/1 1/1 16/3 0/1 2/1 27/5 1/1 11/2 3/2 17/3 1/1 40/7 0/1 2/1 23/4 1/0 6/1 0/1 2/1 13/2 1/2 20/3 1/1 7/1 1/1 15/2 1/0 8/1 0/1 2/1 9/1 1/1 10/1 0/1 2/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(9,80,-8,-71) (-8/1,1/0) -> (-8/7,-9/8) Hyperbolic Matrix(39,280,-28,-201) (-8/1,-7/1) -> (-7/5,-32/23) Hyperbolic Matrix(31,200,-20,-129) (-7/1,-6/1) -> (-14/9,-17/11) Hyperbolic Matrix(71,400,-30,-169) (-6/1,-11/2) -> (-19/8,-26/11) Hyperbolic Matrix(119,640,-82,-441) (-11/2,-16/3) -> (-16/11,-29/20) Hyperbolic Matrix(31,160,6,31) (-16/3,-5/1) -> (5/1,16/3) Hyperbolic Matrix(9,40,2,9) (-5/1,-4/1) -> (4/1,5/1) Hyperbolic Matrix(31,120,8,31) (-4/1,-15/4) -> (15/4,4/1) Hyperbolic Matrix(119,440,-76,-281) (-15/4,-11/3) -> (-11/7,-25/16) Hyperbolic Matrix(199,720,-144,-521) (-11/3,-18/5) -> (-18/13,-29/21) Hyperbolic Matrix(89,320,-42,-151) (-18/5,-25/7) -> (-15/7,-2/1) Hyperbolic Matrix(79,280,-68,-241) (-25/7,-7/2) -> (-7/6,-15/13) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(49,160,-34,-111) (-10/3,-3/1) -> (-13/9,-10/7) Hyperbolic Matrix(71,200,-60,-169) (-3/1,-14/5) -> (-6/5,-13/11) Hyperbolic Matrix(159,440,-116,-321) (-14/5,-11/4) -> (-11/8,-26/19) Hyperbolic Matrix(89,240,-56,-151) (-11/4,-8/3) -> (-8/5,-19/12) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(569,1360,-264,-631) (-12/5,-43/18) -> (-13/6,-28/13) Hyperbolic Matrix(1391,3320,-892,-2129) (-43/18,-31/13) -> (-39/25,-53/34) Hyperbolic Matrix(639,1520,-404,-961) (-31/13,-19/8) -> (-19/12,-49/31) Hyperbolic Matrix(271,640,-224,-529) (-26/11,-7/3) -> (-23/19,-6/5) Hyperbolic Matrix(121,280,-86,-199) (-7/3,-16/7) -> (-24/17,-7/5) Hyperbolic Matrix(351,800,154,351) (-16/7,-25/11) -> (25/11,16/7) Hyperbolic Matrix(511,1160,-374,-849) (-25/11,-9/4) -> (-41/30,-15/11) 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(201,440,-164,-359) (-11/5,-13/6) -> (-27/22,-11/9) Hyperbolic Matrix(801,1720,224,481) (-28/13,-15/7) -> (25/7,68/19) Hyperbolic Matrix(169,320,-122,-231) (-2/1,-13/7) -> (-43/31,-18/13) Hyperbolic Matrix(281,520,-194,-359) (-13/7,-11/6) -> (-29/20,-13/9) 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(671,1200,-430,-769) (-9/5,-25/14) -> (-25/16,-39/25) Hyperbolic Matrix(449,800,252,449) (-25/14,-16/9) -> (16/9,25/14) Hyperbolic Matrix(249,440,-176,-311) (-16/9,-7/4) -> (-17/12,-24/17) Hyperbolic Matrix(231,400,-190,-329) (-7/4,-19/11) -> (-11/9,-17/14) Hyperbolic Matrix(649,1120,-412,-711) (-19/11,-31/18) -> (-41/26,-11/7) Hyperbolic Matrix(2231,3840,-1632,-2809) (-31/18,-43/25) -> (-67/49,-41/30) Hyperbolic Matrix(2001,3440,-1442,-2479) (-43/25,-12/7) -> (-68/49,-43/31) Hyperbolic Matrix(71,120,42,71) (-12/7,-5/3) -> (5/3,12/7) Hyperbolic Matrix(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(329,520,-298,-471) (-49/31,-30/19) -> (-10/9,-1/1) Hyperbolic Matrix(431,680,-386,-609) (-30/19,-41/26) -> (-9/8,-10/9) Hyperbolic Matrix(1721,2680,-1258,-1959) (-53/34,-14/9) -> (-26/19,-93/68) Hyperbolic Matrix(311,480,46,71) (-17/11,-20/13) -> (20/3,7/1) Hyperbolic Matrix(209,320,32,49) (-20/13,-3/2) -> (13/2,20/3) Hyperbolic Matrix(329,480,-268,-391) (-3/2,-16/11) -> (-16/13,-27/22) Hyperbolic Matrix(489,680,64,89) (-32/23,-25/18) -> (15/2,8/1) Hyperbolic Matrix(1239,1720,662,919) (-25/18,-68/49) -> (28/15,15/8) Hyperbolic Matrix(1159,1600,318,439) (-29/21,-40/29) -> (40/11,11/3) Hyperbolic Matrix(1161,1600,320,441) (-40/29,-11/8) -> (29/8,40/11) Hyperbolic Matrix(18719,25600,5030,6879) (-93/68,-160/117) -> (160/43,67/18) Hyperbolic Matrix(18721,25600,5032,6881) (-160/117,-67/49) -> (93/25,160/43) Hyperbolic Matrix(89,120,66,89) (-15/11,-4/3) -> (4/3,15/11) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(129,160,104,129) (-5/4,-16/13) -> (16/13,5/4) Hyperbolic Matrix(1319,1600,230,279) (-17/14,-40/33) -> (40/7,23/4) Hyperbolic Matrix(1321,1600,232,281) (-40/33,-23/19) -> (17/3,40/7) Hyperbolic Matrix(271,320,94,111) (-13/11,-20/17) -> (20/7,3/1) Hyperbolic Matrix(409,480,144,169) (-20/17,-7/6) -> (17/6,20/7) Hyperbolic Matrix(591,680,166,191) (-15/13,-8/7) -> (32/9,25/7) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(71,-80,8,-9) (1/1,8/7) -> (8/1,9/1) Hyperbolic Matrix(241,-280,68,-79) (8/7,7/6) -> (7/2,32/9) Hyperbolic Matrix(169,-200,60,-71) (7/6,6/5) -> (14/5,17/6) Hyperbolic Matrix(329,-400,190,-231) (6/5,11/9) -> (19/11,26/15) Hyperbolic Matrix(521,-640,162,-199) (11/9,16/13) -> (16/5,29/9) Hyperbolic Matrix(321,-440,116,-159) (15/11,11/8) -> (11/4,25/9) Hyperbolic Matrix(521,-720,144,-199) (11/8,18/13) -> (18/5,29/8) Hyperbolic Matrix(231,-320,122,-169) (18/13,25/18) -> (15/8,2/1) Hyperbolic Matrix(201,-280,28,-39) (25/18,7/5) -> (7/1,15/2) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(129,-200,20,-31) (3/2,14/9) -> (6/1,13/2) Hyperbolic Matrix(281,-440,76,-119) (14/9,11/7) -> (11/3,26/7) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(791,-1360,424,-729) (12/7,43/25) -> (13/7,28/15) Hyperbolic Matrix(1929,-3320,692,-1191) (43/25,31/18) -> (39/14,53/19) Hyperbolic Matrix(881,-1520,324,-559) (31/18,19/11) -> (19/7,49/18) Hyperbolic Matrix(369,-640,64,-111) (26/15,7/4) -> (23/4,6/1) Hyperbolic Matrix(159,-280,46,-81) (7/4,16/9) -> (24/7,7/2) Hyperbolic Matrix(649,-1160,174,-311) (25/14,9/5) -> (41/11,15/4) Hyperbolic Matrix(239,-440,44,-81) (11/6,13/7) -> (27/5,11/2) Hyperbolic Matrix(151,-320,42,-89) (2/1,13/6) -> (43/12,18/5) Hyperbolic Matrix(239,-520,74,-161) (13/6,11/5) -> (29/9,13/4) Hyperbolic Matrix(529,-1200,190,-431) (9/4,25/11) -> (25/9,39/14) Hyperbolic Matrix(191,-440,56,-129) (16/7,7/3) -> (17/5,24/7) Hyperbolic Matrix(169,-400,30,-71) (7/3,19/8) -> (11/2,17/3) Hyperbolic Matrix(471,-1120,172,-409) (19/8,31/13) -> (41/15,11/4) Hyperbolic Matrix(1609,-3840,432,-1031) (31/13,43/18) -> (67/18,41/11) Hyperbolic Matrix(1439,-3440,402,-961) (43/18,12/5) -> (68/19,43/12) Hyperbolic Matrix(191,-520,18,-49) (49/18,30/11) -> (10/1,1/0) Hyperbolic Matrix(249,-680,26,-71) (30/11,41/15) -> (9/1,10/1) Hyperbolic Matrix(959,-2680,258,-721) (53/19,14/5) -> (26/7,93/25) Hyperbolic Matrix(151,-480,28,-89) (3/1,16/5) -> (16/3,27/5) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(9,80,-8,-71) -> Matrix(1,0,0,1) Matrix(39,280,-28,-201) -> Matrix(1,2,-2,-3) Matrix(31,200,-20,-129) -> Matrix(1,0,0,1) Matrix(71,400,-30,-169) -> Matrix(1,0,0,1) Matrix(119,640,-82,-441) -> Matrix(1,0,-2,1) Matrix(31,160,6,31) -> Matrix(1,2,0,1) Matrix(9,40,2,9) -> Matrix(1,0,2,1) Matrix(31,120,8,31) -> Matrix(1,0,0,1) Matrix(119,440,-76,-281) -> Matrix(1,0,0,1) Matrix(199,720,-144,-521) -> Matrix(1,0,-6,1) Matrix(89,320,-42,-151) -> Matrix(5,-2,-2,1) Matrix(79,280,-68,-241) -> Matrix(1,-2,0,1) Matrix(71,240,-50,-169) -> Matrix(1,0,2,1) Matrix(49,160,-34,-111) -> Matrix(1,0,-4,1) Matrix(71,200,-60,-169) -> Matrix(1,0,0,1) Matrix(159,440,-116,-321) -> Matrix(1,0,0,1) Matrix(89,240,-56,-151) -> Matrix(1,0,-2,1) Matrix(31,80,12,31) -> Matrix(1,-2,0,1) Matrix(49,120,20,49) -> Matrix(1,4,0,1) Matrix(569,1360,-264,-631) -> Matrix(1,-2,0,1) Matrix(1391,3320,-892,-2129) -> Matrix(1,2,-2,-3) Matrix(639,1520,-404,-961) -> Matrix(1,2,-2,-3) Matrix(271,640,-224,-529) -> Matrix(1,0,0,1) Matrix(121,280,-86,-199) -> Matrix(1,0,0,1) Matrix(351,800,154,351) -> Matrix(1,-2,0,1) Matrix(511,1160,-374,-849) -> Matrix(1,-2,0,1) Matrix(161,360,72,161) -> Matrix(1,-6,0,1) Matrix(199,440,90,199) -> Matrix(1,14,0,1) Matrix(201,440,-164,-359) -> Matrix(1,4,0,1) Matrix(801,1720,224,481) -> Matrix(3,10,-4,-13) Matrix(169,320,-122,-231) -> Matrix(1,2,-4,-7) Matrix(281,520,-194,-359) -> Matrix(3,4,-10,-13) Matrix(241,440,132,241) -> Matrix(13,14,12,13) Matrix(199,360,110,199) -> Matrix(7,6,8,7) Matrix(671,1200,-430,-769) -> Matrix(3,2,-2,-1) Matrix(449,800,252,449) -> Matrix(3,2,4,3) Matrix(249,440,-176,-311) -> Matrix(1,0,2,1) Matrix(231,400,-190,-329) -> Matrix(1,0,0,1) Matrix(649,1120,-412,-711) -> Matrix(1,2,-2,-3) Matrix(2231,3840,-1632,-2809) -> Matrix(1,2,0,1) Matrix(2001,3440,-1442,-2479) -> Matrix(1,0,-2,1) Matrix(71,120,42,71) -> Matrix(3,4,2,3) Matrix(49,80,30,49) -> Matrix(3,2,4,3) Matrix(329,520,-298,-471) -> Matrix(3,2,-2,-1) Matrix(431,680,-386,-609) -> Matrix(3,2,-2,-1) Matrix(1721,2680,-1258,-1959) -> Matrix(1,0,0,1) Matrix(311,480,46,71) -> Matrix(3,4,2,3) Matrix(209,320,32,49) -> Matrix(3,2,4,3) Matrix(329,480,-268,-391) -> Matrix(1,0,2,1) Matrix(489,680,64,89) -> Matrix(1,0,2,1) Matrix(1239,1720,662,919) -> Matrix(23,10,16,7) Matrix(1159,1600,318,439) -> Matrix(1,0,12,1) Matrix(1161,1600,320,441) -> Matrix(1,0,-10,1) Matrix(18719,25600,5030,6879) -> Matrix(1,0,4,1) Matrix(18721,25600,5032,6881) -> Matrix(1,0,-2,1) Matrix(89,120,66,89) -> Matrix(1,0,2,1) Matrix(31,40,24,31) -> Matrix(1,0,0,1) Matrix(129,160,104,129) -> Matrix(1,2,0,1) Matrix(1319,1600,230,279) -> Matrix(1,2,0,1) Matrix(1321,1600,232,281) -> Matrix(1,2,0,1) Matrix(271,320,94,111) -> Matrix(1,-2,0,1) Matrix(409,480,144,169) -> Matrix(1,4,0,1) Matrix(591,680,166,191) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(1,0,2,1) Matrix(71,-80,8,-9) -> Matrix(1,0,0,1) Matrix(241,-280,68,-79) -> Matrix(1,-2,0,1) Matrix(169,-200,60,-71) -> Matrix(1,0,0,1) Matrix(329,-400,190,-231) -> Matrix(1,0,0,1) Matrix(521,-640,162,-199) -> Matrix(1,0,-2,1) Matrix(321,-440,116,-159) -> Matrix(1,0,0,1) Matrix(521,-720,144,-199) -> Matrix(1,0,-6,1) Matrix(231,-320,122,-169) -> Matrix(7,-2,4,-1) Matrix(201,-280,28,-39) -> Matrix(3,-2,2,-1) Matrix(169,-240,50,-71) -> Matrix(1,0,2,1) Matrix(111,-160,34,-49) -> Matrix(1,0,-4,1) Matrix(129,-200,20,-31) -> Matrix(1,0,0,1) Matrix(281,-440,76,-119) -> Matrix(1,0,0,1) Matrix(151,-240,56,-89) -> Matrix(1,0,-2,1) Matrix(791,-1360,424,-729) -> Matrix(3,-2,2,-1) Matrix(1929,-3320,692,-1191) -> Matrix(1,-2,0,1) Matrix(881,-1520,324,-559) -> Matrix(1,-2,0,1) Matrix(369,-640,64,-111) -> Matrix(1,0,0,1) Matrix(159,-280,46,-81) -> Matrix(1,0,0,1) Matrix(649,-1160,174,-311) -> Matrix(3,-2,2,-1) Matrix(239,-440,44,-81) -> Matrix(3,-4,4,-5) Matrix(151,-320,42,-89) -> Matrix(1,-2,-2,5) Matrix(239,-520,74,-161) -> Matrix(1,-4,-2,9) Matrix(529,-1200,190,-431) -> Matrix(1,2,0,1) Matrix(191,-440,56,-129) -> Matrix(1,0,2,1) Matrix(169,-400,30,-71) -> Matrix(1,0,0,1) Matrix(471,-1120,172,-409) -> Matrix(1,-2,0,1) Matrix(1609,-3840,432,-1031) -> Matrix(1,-2,2,-3) Matrix(1439,-3440,402,-961) -> Matrix(1,0,-2,1) Matrix(191,-520,18,-49) -> Matrix(1,2,0,1) Matrix(249,-680,26,-71) -> Matrix(1,2,0,1) Matrix(959,-2680,258,-721) -> Matrix(1,0,0,1) Matrix(151,-480,28,-89) -> Matrix(1,0,2,1) 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): 25 Degree of the the map X: 25 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. ----------------------------------------------------------------------- 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): 288 Minimal number of generators: 49 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: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 20/9 16/7 12/5 5/2 8/3 3/1 10/3 40/11 160/43 15/4 4/1 5/1 16/3 11/2 40/7 6/1 20/3 7/1 15/2 8/1 10/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/1 8/7 0/1 2/1 7/6 1/0 6/5 0/1 2/1 11/9 3/1 16/13 0/1 2/1 5/4 1/0 4/3 0/1 15/11 1/1 11/8 -1/2 18/13 0/1 2/7 25/18 1/2 7/5 1/1 10/7 0/1 3/2 1/2 14/9 0/1 2/1 11/7 1/3 8/5 0/1 2/3 5/3 1/1 12/7 2/1 43/25 1/1 31/18 1/0 19/11 3/1 26/15 0/1 2/1 7/4 1/0 16/9 0/1 2/3 25/14 1/2 9/5 1/1 20/11 1/1 11/6 7/6 13/7 1/1 28/15 4/3 15/8 3/2 2/1 0/1 2/1 13/6 1/0 11/5 7/1 20/9 1/0 9/4 1/0 25/11 -1/1 16/7 -2/1 0/1 7/3 1/1 19/8 3/2 31/13 1/1 43/18 1/0 12/5 2/1 5/2 1/0 8/3 -2/1 0/1 19/7 1/1 49/18 1/0 30/11 -2/1 0/1 41/15 -1/1 11/4 -1/2 25/9 1/1 39/14 1/0 53/19 -1/1 14/5 0/1 2/1 17/6 1/0 20/7 1/0 3/1 -1/1 16/5 -2/3 0/1 29/9 -3/5 13/4 -1/2 10/3 0/1 17/5 1/3 24/7 0/1 2/3 7/2 1/0 32/9 -2/1 0/1 25/7 -1/1 68/19 -2/3 43/12 -1/2 18/5 -2/5 0/1 29/8 -1/8 40/11 0/1 11/3 1/3 26/7 0/1 2/1 93/25 -1/1 160/43 0/1 67/18 1/2 41/11 1/1 15/4 1/0 4/1 0/1 5/1 1/1 16/3 0/1 2/1 27/5 1/1 11/2 3/2 17/3 1/1 40/7 0/1 2/1 23/4 1/0 6/1 0/1 2/1 13/2 1/2 20/3 1/1 7/1 1/1 15/2 1/0 8/1 0/1 2/1 9/1 1/1 10/1 0/1 2/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(71,-80,8,-9) (1/1,8/7) -> (8/1,9/1) Hyperbolic Matrix(241,-280,68,-79) (8/7,7/6) -> (7/2,32/9) Hyperbolic Matrix(169,-200,60,-71) (7/6,6/5) -> (14/5,17/6) Hyperbolic Matrix(329,-400,190,-231) (6/5,11/9) -> (19/11,26/15) Hyperbolic Matrix(521,-640,162,-199) (11/9,16/13) -> (16/5,29/9) Hyperbolic Matrix(129,-160,25,-31) (16/13,5/4) -> (5/1,16/3) Hyperbolic Matrix(31,-40,7,-9) (5/4,4/3) -> (4/1,5/1) Hyperbolic Matrix(89,-120,23,-31) (4/3,15/11) -> (15/4,4/1) Hyperbolic Matrix(321,-440,116,-159) (15/11,11/8) -> (11/4,25/9) Hyperbolic Matrix(521,-720,144,-199) (11/8,18/13) -> (18/5,29/8) Hyperbolic Matrix(231,-320,122,-169) (18/13,25/18) -> (15/8,2/1) Hyperbolic Matrix(201,-280,28,-39) (25/18,7/5) -> (7/1,15/2) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(129,-200,20,-31) (3/2,14/9) -> (6/1,13/2) Hyperbolic Matrix(281,-440,76,-119) (14/9,11/7) -> (11/3,26/7) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(49,-80,19,-31) (8/5,5/3) -> (5/2,8/3) Hyperbolic Matrix(71,-120,29,-49) (5/3,12/7) -> (12/5,5/2) Hyperbolic Matrix(791,-1360,424,-729) (12/7,43/25) -> (13/7,28/15) Hyperbolic Matrix(1929,-3320,692,-1191) (43/25,31/18) -> (39/14,53/19) Hyperbolic Matrix(881,-1520,324,-559) (31/18,19/11) -> (19/7,49/18) Hyperbolic Matrix(369,-640,64,-111) (26/15,7/4) -> (23/4,6/1) Hyperbolic Matrix(159,-280,46,-81) (7/4,16/9) -> (24/7,7/2) Hyperbolic Matrix(449,-800,197,-351) (16/9,25/14) -> (25/11,16/7) Hyperbolic Matrix(649,-1160,174,-311) (25/14,9/5) -> (41/11,15/4) Hyperbolic Matrix(199,-360,89,-161) (9/5,20/11) -> (20/9,9/4) Hyperbolic Matrix(241,-440,109,-199) (20/11,11/6) -> (11/5,20/9) Hyperbolic Matrix(239,-440,44,-81) (11/6,13/7) -> (27/5,11/2) Hyperbolic Matrix(919,-1720,257,-481) (28/15,15/8) -> (25/7,68/19) Hyperbolic Matrix(151,-320,42,-89) (2/1,13/6) -> (43/12,18/5) Hyperbolic Matrix(239,-520,74,-161) (13/6,11/5) -> (29/9,13/4) Hyperbolic Matrix(529,-1200,190,-431) (9/4,25/11) -> (25/9,39/14) Hyperbolic Matrix(191,-440,56,-129) (16/7,7/3) -> (17/5,24/7) Hyperbolic Matrix(169,-400,30,-71) (7/3,19/8) -> (11/2,17/3) Hyperbolic Matrix(471,-1120,172,-409) (19/8,31/13) -> (41/15,11/4) Hyperbolic Matrix(1609,-3840,432,-1031) (31/13,43/18) -> (67/18,41/11) Hyperbolic Matrix(1439,-3440,402,-961) (43/18,12/5) -> (68/19,43/12) Hyperbolic Matrix(191,-520,18,-49) (49/18,30/11) -> (10/1,1/0) Hyperbolic Matrix(249,-680,26,-71) (30/11,41/15) -> (9/1,10/1) Hyperbolic Matrix(959,-2680,258,-721) (53/19,14/5) -> (26/7,93/25) Hyperbolic Matrix(169,-480,25,-71) (17/6,20/7) -> (20/3,7/1) Hyperbolic Matrix(111,-320,17,-49) (20/7,3/1) -> (13/2,20/3) Hyperbolic Matrix(151,-480,28,-89) (3/1,16/5) -> (16/3,27/5) Hyperbolic Matrix(191,-680,25,-89) (32/9,25/7) -> (15/2,8/1) Hyperbolic Matrix(441,-1600,121,-439) (29/8,40/11) -> (40/11,11/3) Parabolic Matrix(6881,-25600,1849,-6879) (93/25,160/43) -> (160/43,67/18) Parabolic Matrix(281,-1600,49,-279) (17/3,40/7) -> (40/7,23/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,1,1) Matrix(71,-80,8,-9) -> Matrix(1,0,0,1) Matrix(241,-280,68,-79) -> Matrix(1,-2,0,1) Matrix(169,-200,60,-71) -> Matrix(1,0,0,1) Matrix(329,-400,190,-231) -> Matrix(1,0,0,1) Matrix(521,-640,162,-199) -> Matrix(1,0,-2,1) Matrix(129,-160,25,-31) -> Matrix(1,-2,1,-1) Matrix(31,-40,7,-9) -> Matrix(1,0,1,1) Matrix(89,-120,23,-31) -> Matrix(1,0,-1,1) Matrix(321,-440,116,-159) -> Matrix(1,0,0,1) Matrix(521,-720,144,-199) -> Matrix(1,0,-6,1) Matrix(231,-320,122,-169) -> Matrix(7,-2,4,-1) Matrix(201,-280,28,-39) -> Matrix(3,-2,2,-1) Matrix(169,-240,50,-71) -> Matrix(1,0,2,1) Matrix(111,-160,34,-49) -> Matrix(1,0,-4,1) Matrix(129,-200,20,-31) -> Matrix(1,0,0,1) Matrix(281,-440,76,-119) -> Matrix(1,0,0,1) Matrix(151,-240,56,-89) -> Matrix(1,0,-2,1) Matrix(49,-80,19,-31) -> Matrix(3,-2,-1,1) Matrix(71,-120,29,-49) -> Matrix(3,-4,1,-1) Matrix(791,-1360,424,-729) -> Matrix(3,-2,2,-1) Matrix(1929,-3320,692,-1191) -> Matrix(1,-2,0,1) Matrix(881,-1520,324,-559) -> Matrix(1,-2,0,1) Matrix(369,-640,64,-111) -> Matrix(1,0,0,1) Matrix(159,-280,46,-81) -> Matrix(1,0,0,1) Matrix(449,-800,197,-351) -> Matrix(3,-2,-1,1) Matrix(649,-1160,174,-311) -> Matrix(3,-2,2,-1) Matrix(199,-360,89,-161) -> Matrix(7,-6,-1,1) Matrix(241,-440,109,-199) -> Matrix(13,-14,1,-1) Matrix(239,-440,44,-81) -> Matrix(3,-4,4,-5) Matrix(919,-1720,257,-481) -> Matrix(7,-10,-9,13) Matrix(151,-320,42,-89) -> Matrix(1,-2,-2,5) Matrix(239,-520,74,-161) -> Matrix(1,-4,-2,9) Matrix(529,-1200,190,-431) -> Matrix(1,2,0,1) Matrix(191,-440,56,-129) -> Matrix(1,0,2,1) Matrix(169,-400,30,-71) -> Matrix(1,0,0,1) Matrix(471,-1120,172,-409) -> Matrix(1,-2,0,1) Matrix(1609,-3840,432,-1031) -> Matrix(1,-2,2,-3) Matrix(1439,-3440,402,-961) -> Matrix(1,0,-2,1) Matrix(191,-520,18,-49) -> Matrix(1,2,0,1) Matrix(249,-680,26,-71) -> Matrix(1,2,0,1) Matrix(959,-2680,258,-721) -> Matrix(1,0,0,1) Matrix(169,-480,25,-71) -> Matrix(1,-4,1,-3) Matrix(111,-320,17,-49) -> Matrix(1,2,1,3) Matrix(151,-480,28,-89) -> Matrix(1,0,2,1) Matrix(191,-680,25,-89) -> Matrix(1,0,1,1) Matrix(441,-1600,121,-439) -> Matrix(1,0,11,1) Matrix(6881,-25600,1849,-6879) -> Matrix(1,0,3,1) Matrix(281,-1600,49,-279) -> Matrix(1,-2,1,-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): 25 ----------------------------------------------------------------------- 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 1 1 2/1 (0/1,2/1) 0 10 13/6 1/0 1 20 11/5 7/1 1 20 20/9 1/0 10 1 9/4 1/0 1 20 25/11 -1/1 1 4 16/7 (-2/1,0/1).(-1/1,1/0) 0 5 7/3 1/1 1 20 19/8 3/2 1 20 31/13 1/1 1 20 43/18 1/0 1 20 12/5 2/1 1 5 5/2 1/0 3 4 8/3 (-2/1,0/1).(-1/1,1/0) 0 5 30/11 (-2/1,0/1) 0 2 41/15 -1/1 1 20 11/4 -1/2 1 20 14/5 (0/1,2/1) 0 10 3/1 -1/1 1 20 16/5 (-1/1,-1/2).(-2/3,0/1) 0 5 10/3 0/1 3 2 24/7 (0/1,2/3).(1/2,1/1) 0 5 7/2 1/0 1 20 25/7 -1/1 3 4 68/19 -2/3 1 5 43/12 -1/2 1 20 18/5 (-2/5,0/1) 0 10 40/11 0/1 11 1 11/3 1/3 1 20 26/7 (0/1,2/1) 0 10 160/43 0/1 3 1 67/18 1/2 1 20 41/11 1/1 1 20 15/4 1/0 1 4 4/1 0/1 1 5 5/1 1/1 1 4 16/3 (0/1,2/1).(1/1,1/0) 0 5 27/5 1/1 1 20 11/2 3/2 1 20 17/3 1/1 1 20 40/7 (0/1,2/1).(1/1,1/0) 0 1 6/1 (0/1,2/1) 0 10 13/2 1/2 1 20 20/3 1/1 3 1 7/1 1/1 1 20 15/2 1/0 3 4 8/1 (0/1,2/1).(1/1,1/0) 0 5 10/1 (0/1,2/1) 0 2 1/0 1/0 1 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(151,-320,42,-89) (2/1,13/6) -> (43/12,18/5) Hyperbolic Matrix(201,-440,37,-81) (13/6,11/5) -> (27/5,11/2) 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(511,-1160,137,-311) (9/4,25/11) -> (41/11,15/4) Glide Reflection Matrix(351,-800,154,-351) (25/11,16/7) -> (25/11,16/7) Reflection Matrix(121,-280,35,-81) (16/7,7/3) -> (24/7,7/2) Glide Reflection Matrix(169,-400,30,-71) (7/3,19/8) -> (11/2,17/3) Hyperbolic Matrix(471,-1120,172,-409) (19/8,31/13) -> (41/15,11/4) Hyperbolic Matrix(1609,-3840,432,-1031) (31/13,43/18) -> (67/18,41/11) Hyperbolic Matrix(1439,-3440,402,-961) (43/18,12/5) -> (68/19,43/12) Hyperbolic Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(89,-240,33,-89) (8/3,30/11) -> (8/3,30/11) Reflection Matrix(161,-440,15,-41) (30/11,41/15) -> (10/1,1/0) Glide Reflection Matrix(159,-440,43,-119) (11/4,14/5) -> (11/3,26/7) Glide Reflection Matrix(71,-200,11,-31) (14/5,3/1) -> (6/1,13/2) Glide Reflection Matrix(151,-480,28,-89) (3/1,16/5) -> (16/3,27/5) Hyperbolic Matrix(49,-160,15,-49) (16/5,10/3) -> (16/5,10/3) Reflection Matrix(71,-240,21,-71) (10/3,24/7) -> (10/3,24/7) Reflection Matrix(79,-280,11,-39) (7/2,25/7) -> (7/1,15/2) Glide Reflection Matrix(951,-3400,266,-951) (25/7,68/19) -> (25/7,68/19) Reflection Matrix(199,-720,55,-199) (18/5,40/11) -> (18/5,40/11) Reflection Matrix(241,-880,66,-241) (40/11,11/3) -> (40/11,11/3) Reflection Matrix(1119,-4160,301,-1119) (26/7,160/43) -> (26/7,160/43) Reflection Matrix(5761,-21440,1548,-5761) (160/43,67/18) -> (160/43,67/18) Reflection Matrix(31,-120,8,-31) (15/4,4/1) -> (15/4,4/1) Reflection Matrix(9,-40,2,-9) (4/1,5/1) -> (4/1,5/1) Reflection Matrix(31,-160,6,-31) (5/1,16/3) -> (5/1,16/3) Reflection Matrix(239,-1360,42,-239) (17/3,40/7) -> (17/3,40/7) Reflection Matrix(41,-240,7,-41) (40/7,6/1) -> (40/7,6/1) Reflection 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 Matrix(31,-240,4,-31) (15/2,8/1) -> (15/2,8/1) Reflection Matrix(9,-80,1,-9) (8/1,10/1) -> (8/1,10/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Matrix(151,-320,42,-89) -> Matrix(1,-2,-2,5) Matrix(201,-440,37,-81) -> Matrix(1,-4,1,-5) Matrix(199,-440,90,-199) -> Matrix(-1,14,0,1) (11/5,20/9) -> (7/1,1/0) Matrix(161,-360,72,-161) -> Matrix(1,6,0,-1) (20/9,9/4) -> (-3/1,1/0) Matrix(511,-1160,137,-311) -> Matrix(1,2,1,1) Matrix(351,-800,154,-351) -> Matrix(1,2,0,-1) (25/11,16/7) -> (-1/1,1/0) Matrix(121,-280,35,-81) -> Matrix(1,0,1,-1) *** -> (0/1,2/1) Matrix(169,-400,30,-71) -> Matrix(1,0,0,1) Matrix(471,-1120,172,-409) -> Matrix(1,-2,0,1) 1/0 Matrix(1609,-3840,432,-1031) -> Matrix(1,-2,2,-3) 1/1 Matrix(1439,-3440,402,-961) -> Matrix(1,0,-2,1) 0/1 Matrix(49,-120,20,-49) -> Matrix(-1,4,0,1) (12/5,5/2) -> (2/1,1/0) Matrix(31,-80,12,-31) -> Matrix(1,2,0,-1) (5/2,8/3) -> (-1/1,1/0) Matrix(89,-240,33,-89) -> Matrix(-1,0,1,1) (8/3,30/11) -> (-2/1,0/1) Matrix(161,-440,15,-41) -> Matrix(1,2,1,1) Matrix(159,-440,43,-119) -> Matrix(1,0,1,-1) *** -> (0/1,2/1) Matrix(71,-200,11,-31) -> Matrix(1,0,1,-1) *** -> (0/1,2/1) Matrix(151,-480,28,-89) -> Matrix(1,0,2,1) 0/1 Matrix(49,-160,15,-49) -> Matrix(-1,0,3,1) (16/5,10/3) -> (-2/3,0/1) Matrix(71,-240,21,-71) -> Matrix(1,0,3,-1) (10/3,24/7) -> (0/1,2/3) Matrix(79,-280,11,-39) -> Matrix(1,2,1,1) Matrix(951,-3400,266,-951) -> Matrix(5,4,-6,-5) (25/7,68/19) -> (-1/1,-2/3) Matrix(199,-720,55,-199) -> Matrix(-1,0,5,1) (18/5,40/11) -> (-2/5,0/1) Matrix(241,-880,66,-241) -> Matrix(1,0,6,-1) (40/11,11/3) -> (0/1,1/3) Matrix(1119,-4160,301,-1119) -> Matrix(1,0,1,-1) (26/7,160/43) -> (0/1,2/1) Matrix(5761,-21440,1548,-5761) -> Matrix(1,0,4,-1) (160/43,67/18) -> (0/1,1/2) Matrix(31,-120,8,-31) -> Matrix(1,0,0,-1) (15/4,4/1) -> (0/1,1/0) Matrix(9,-40,2,-9) -> Matrix(1,0,2,-1) (4/1,5/1) -> (0/1,1/1) Matrix(31,-160,6,-31) -> Matrix(-1,2,0,1) (5/1,16/3) -> (1/1,1/0) Matrix(239,-1360,42,-239) -> Matrix(-1,2,0,1) (17/3,40/7) -> (1/1,1/0) Matrix(41,-240,7,-41) -> Matrix(1,0,1,-1) (40/7,6/1) -> (0/1,2/1) Matrix(79,-520,12,-79) -> Matrix(3,-2,4,-3) (13/2,20/3) -> (1/2,1/1) Matrix(41,-280,6,-41) -> Matrix(3,-4,2,-3) (20/3,7/1) -> (1/1,2/1) Matrix(31,-240,4,-31) -> Matrix(-1,2,0,1) (15/2,8/1) -> (1/1,1/0) Matrix(9,-80,1,-9) -> Matrix(1,0,1,-1) (8/1,10/1) -> (0/1,2/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.