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): 360 Minimal number of generators: 61 Number of equivalence classes of cusps: 36 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -15/4 -40/11 -10/3 -5/2 -20/9 -15/7 -5/3 -15/11 -5/4 -15/13 0/1 1/1 5/4 15/11 25/18 3/2 5/3 25/14 15/8 2/1 5/2 30/11 25/9 20/7 3/1 10/3 25/7 40/11 15/4 50/13 4/1 5/1 20/3 15/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -1/1 -9/2 -1/2 -22/5 0/1 -35/8 -1/2 -13/3 -1/1 -4/1 -2/3 -15/4 -1/2 -26/7 0/1 -11/3 -1/1 -40/11 -2/3 -29/8 -1/2 -18/5 -4/7 -25/7 -1/2 -7/2 -1/2 -10/3 0/1 -3/1 -1/1 -5/2 -1/2 -7/3 -1/3 -16/7 -2/3 -25/11 -1/2 -9/4 -1/2 -20/9 -2/5 -11/5 -1/3 -24/11 -2/5 -13/6 -3/8 -15/7 -1/3 -2/1 0/1 -5/3 -1/3 -8/5 -2/7 -35/22 -1/4 -27/17 -1/5 -19/12 -1/4 -30/19 0/1 -11/7 -1/3 -25/16 -1/3 -14/9 -4/13 -17/11 -1/3 -20/13 -2/7 -3/2 -1/4 -10/7 0/1 -7/5 -1/3 -25/18 -1/3 -18/13 -4/13 -29/21 -1/3 -40/29 -2/7 -11/8 -1/4 -15/11 -1/3 -19/14 -3/10 -23/17 -5/17 -50/37 -2/7 -27/20 -5/18 -4/3 -2/7 -5/4 -1/4 -6/5 0/1 -13/11 -3/13 -20/17 -2/9 -7/6 -1/4 -15/13 -1/5 -8/7 -2/9 -1/1 -1/5 0/1 0/1 1/1 1/5 5/4 1/4 9/7 1/3 22/17 0/1 35/27 1/3 13/10 1/4 4/3 2/7 15/11 1/3 26/19 0/1 11/8 1/4 40/29 2/7 29/21 1/3 18/13 4/13 25/18 1/3 7/5 1/3 10/7 0/1 3/2 1/4 5/3 1/3 7/4 1/2 16/9 2/7 25/14 1/3 9/5 1/3 20/11 2/5 11/6 1/2 24/13 2/5 13/7 3/7 15/8 1/2 2/1 0/1 5/2 1/2 8/3 2/3 35/13 1/1 27/10 1/0 19/7 1/1 30/11 0/1 11/4 1/2 25/9 1/2 14/5 4/7 17/6 1/2 20/7 2/3 3/1 1/1 10/3 0/1 7/2 1/2 25/7 1/2 18/5 4/7 29/8 1/2 40/11 2/3 11/3 1/1 15/4 1/2 19/5 3/5 23/6 5/8 50/13 2/3 27/7 5/7 4/1 2/3 5/1 1/1 6/1 0/1 13/2 3/2 20/3 2/1 7/1 1/1 15/2 1/0 8/1 2/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(9,50,-2,-11) (-5/1,1/0) -> (-5/1,-9/2) Parabolic Matrix(181,800,50,221) (-9/2,-22/5) -> (18/5,29/8) Hyperbolic Matrix(91,400,48,211) (-22/5,-35/8) -> (15/8,2/1) Hyperbolic Matrix(149,650,80,349) (-35/8,-13/3) -> (13/7,15/8) Hyperbolic Matrix(59,250,-38,-161) (-13/3,-4/1) -> (-14/9,-17/11) Hyperbolic Matrix(119,450,-32,-121) (-4/1,-15/4) -> (-15/4,-26/7) Parabolic Matrix(149,550,-68,-251) (-26/7,-11/3) -> (-11/5,-24/11) Hyperbolic Matrix(439,1600,318,1159) (-11/3,-40/11) -> (40/29,29/21) Hyperbolic Matrix(441,1600,320,1161) (-40/11,-29/8) -> (11/8,40/29) Hyperbolic Matrix(69,250,8,29) (-29/8,-18/5) -> (8/1,1/0) Hyperbolic Matrix(251,900,70,251) (-18/5,-25/7) -> (25/7,18/5) Hyperbolic Matrix(99,350,28,99) (-25/7,-7/2) -> (7/2,25/7) Hyperbolic Matrix(29,100,20,69) (-7/2,-10/3) -> (10/7,3/2) Hyperbolic Matrix(31,100,22,71) (-10/3,-3/1) -> (7/5,10/7) Hyperbolic Matrix(19,50,-8,-21) (-3/1,-5/2) -> (-5/2,-7/3) Parabolic Matrix(109,250,-92,-211) (-7/3,-16/7) -> (-6/5,-13/11) Hyperbolic Matrix(329,750,118,269) (-16/7,-25/11) -> (25/9,14/5) Hyperbolic Matrix(221,500,80,181) (-25/11,-9/4) -> (11/4,25/9) Hyperbolic Matrix(179,400,98,219) (-9/4,-20/9) -> (20/11,11/6) Hyperbolic Matrix(181,400,100,221) (-20/9,-11/5) -> (9/5,20/11) Hyperbolic Matrix(321,700,-238,-519) (-24/11,-13/6) -> (-27/20,-4/3) Hyperbolic Matrix(301,650,232,501) (-13/6,-15/7) -> (35/27,13/10) Hyperbolic Matrix(189,400,146,309) (-15/7,-2/1) -> (22/17,35/27) Hyperbolic Matrix(29,50,-18,-31) (-2/1,-5/3) -> (-5/3,-8/5) Parabolic Matrix(251,400,32,51) (-8/5,-35/22) -> (15/2,8/1) Hyperbolic Matrix(409,650,56,89) (-35/22,-27/17) -> (7/1,15/2) Hyperbolic Matrix(599,950,-442,-701) (-27/17,-19/12) -> (-19/14,-23/17) Hyperbolic Matrix(569,900,208,329) (-19/12,-30/19) -> (30/11,11/4) Hyperbolic Matrix(571,900,210,331) (-30/19,-11/7) -> (19/7,30/11) Hyperbolic Matrix(319,500,178,279) (-11/7,-25/16) -> (25/14,9/5) Hyperbolic Matrix(481,750,270,421) (-25/16,-14/9) -> (16/9,25/14) Hyperbolic Matrix(259,400,90,139) (-17/11,-20/13) -> (20/7,3/1) Hyperbolic Matrix(261,400,92,141) (-20/13,-3/2) -> (17/6,20/7) Hyperbolic Matrix(69,100,20,29) (-3/2,-10/7) -> (10/3,7/2) Hyperbolic Matrix(71,100,22,31) (-10/7,-7/5) -> (3/1,10/3) Hyperbolic Matrix(251,350,180,251) (-7/5,-25/18) -> (25/18,7/5) Hyperbolic Matrix(649,900,468,649) (-25/18,-18/13) -> (18/13,25/18) Hyperbolic Matrix(579,800,448,619) (-18/13,-29/21) -> (9/7,22/17) 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(329,450,-242,-331) (-11/8,-15/11) -> (-15/11,-19/14) Parabolic Matrix(1849,2500,480,649) (-23/17,-50/37) -> (50/13,27/7) Hyperbolic Matrix(1851,2500,482,651) (-50/37,-27/20) -> (23/6,50/13) Hyperbolic Matrix(39,50,-32,-41) (-4/3,-5/4) -> (-5/4,-6/5) Parabolic Matrix(339,400,50,59) (-13/11,-20/17) -> (20/3,7/1) Hyperbolic Matrix(341,400,52,61) (-20/17,-7/6) -> (13/2,20/3) Hyperbolic Matrix(561,650,208,241) (-7/6,-15/13) -> (35/13,27/10) Hyperbolic Matrix(349,400,130,149) (-15/13,-8/7) -> (8/3,35/13) Hyperbolic Matrix(221,250,160,181) (-8/7,-1/1) -> (29/21,18/13) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(41,-50,32,-39) (1/1,5/4) -> (5/4,9/7) Parabolic Matrix(191,-250,68,-89) (13/10,4/3) -> (14/5,17/6) Hyperbolic Matrix(331,-450,242,-329) (4/3,15/11) -> (15/11,26/19) Parabolic Matrix(401,-550,218,-299) (26/19,11/8) -> (11/6,24/13) Hyperbolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(141,-250,22,-39) (7/4,16/9) -> (6/1,13/2) Hyperbolic Matrix(379,-700,98,-181) (24/13,13/7) -> (27/7,4/1) Hyperbolic Matrix(21,-50,8,-19) (2/1,5/2) -> (5/2,8/3) Parabolic Matrix(351,-950,92,-249) (27/10,19/7) -> (19/5,23/6) Hyperbolic Matrix(121,-450,32,-119) (11/3,15/4) -> (15/4,19/5) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(9,50,-2,-11) -> Matrix(1,2,-2,-3) Matrix(181,800,50,221) -> Matrix(7,4,12,7) Matrix(91,400,48,211) -> Matrix(1,0,4,1) Matrix(149,650,80,349) -> Matrix(5,2,12,5) Matrix(59,250,-38,-161) -> Matrix(1,2,-4,-7) Matrix(119,450,-32,-121) -> Matrix(3,2,-8,-5) Matrix(149,550,-68,-251) -> Matrix(3,2,-8,-5) Matrix(439,1600,318,1159) -> Matrix(5,4,16,13) Matrix(441,1600,320,1161) -> Matrix(7,4,26,15) Matrix(69,250,8,29) -> Matrix(3,2,-2,-1) Matrix(251,900,70,251) -> Matrix(15,8,28,15) Matrix(99,350,28,99) -> Matrix(5,2,12,5) Matrix(29,100,20,69) -> Matrix(1,0,6,1) Matrix(31,100,22,71) -> Matrix(1,0,4,1) Matrix(19,50,-8,-21) -> Matrix(3,2,-8,-5) Matrix(109,250,-92,-211) -> Matrix(3,2,-14,-9) Matrix(329,750,118,269) -> Matrix(11,6,20,11) Matrix(221,500,80,181) -> Matrix(9,4,20,9) Matrix(179,400,98,219) -> Matrix(9,4,20,9) Matrix(181,400,100,221) -> Matrix(11,4,30,11) Matrix(321,700,-238,-519) -> Matrix(9,4,-34,-15) Matrix(301,650,232,501) -> Matrix(5,2,12,5) Matrix(189,400,146,309) -> Matrix(1,0,6,1) Matrix(29,50,-18,-31) -> Matrix(5,2,-18,-7) Matrix(251,400,32,51) -> Matrix(1,0,4,1) Matrix(409,650,56,89) -> Matrix(9,2,4,1) Matrix(599,950,-442,-701) -> Matrix(5,2,-18,-7) Matrix(569,900,208,329) -> Matrix(1,0,6,1) Matrix(571,900,210,331) -> Matrix(1,0,4,1) Matrix(319,500,178,279) -> Matrix(11,4,30,11) Matrix(481,750,270,421) -> Matrix(19,6,60,19) Matrix(259,400,90,139) -> Matrix(13,4,16,5) Matrix(261,400,92,141) -> Matrix(15,4,26,7) Matrix(69,100,20,29) -> Matrix(1,0,6,1) Matrix(71,100,22,31) -> Matrix(1,0,4,1) Matrix(251,350,180,251) -> Matrix(5,2,12,5) Matrix(649,900,468,649) -> Matrix(25,8,78,25) Matrix(579,800,448,619) -> Matrix(13,4,42,13) Matrix(1159,1600,318,439) -> Matrix(13,4,16,5) Matrix(1161,1600,320,441) -> Matrix(15,4,26,7) Matrix(329,450,-242,-331) -> Matrix(5,2,-18,-7) Matrix(1849,2500,480,649) -> Matrix(69,20,100,29) Matrix(1851,2500,482,651) -> Matrix(71,20,110,31) Matrix(39,50,-32,-41) -> Matrix(7,2,-32,-9) Matrix(339,400,50,59) -> Matrix(17,4,4,1) Matrix(341,400,52,61) -> Matrix(19,4,14,3) Matrix(561,650,208,241) -> Matrix(9,2,4,1) Matrix(349,400,130,149) -> Matrix(1,0,6,1) Matrix(221,250,160,181) -> Matrix(11,2,38,7) Matrix(1,0,2,1) -> Matrix(1,0,10,1) Matrix(41,-50,32,-39) -> Matrix(9,-2,32,-7) Matrix(191,-250,68,-89) -> Matrix(9,-2,14,-3) Matrix(331,-450,242,-329) -> Matrix(7,-2,18,-5) Matrix(401,-550,218,-299) -> Matrix(7,-2,18,-5) Matrix(31,-50,18,-29) -> Matrix(7,-2,18,-5) Matrix(141,-250,22,-39) -> Matrix(7,-2,4,-1) Matrix(379,-700,98,-181) -> Matrix(11,-4,14,-5) Matrix(21,-50,8,-19) -> Matrix(5,-2,8,-3) Matrix(351,-950,92,-249) -> Matrix(5,-2,8,-3) Matrix(121,-450,32,-119) -> Matrix(5,-2,8,-3) Matrix(11,-50,2,-9) -> Matrix(3,-2,2,-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): 20 Degree of the the map X: 20 Degree of the the map Y: 60 Permutation triple for Y: ((2,6,18,41,49,46,45,25,48,55,28,12,4,3,11,27,54,33,30,37,36,58,42,19,7)(5,15,34,44,26,32,14,13,31,56,51,24,10,9,23,29,39,38,22,8,21,47,59,35,16); (1,4,14,32,36,52,25,24,51,27,40,18,39,29,28,43,42,47,21,46,57,33,15,5,2)(3,10,17,16,37,30,13,20,8,7,19,34,50,23,49,41,59,60,56,55,48,38,53,26,11); (1,2,8,22,48,52,36,16,35,41,40,27,26,44,19,43,28,56,31,30,57,46,23,9,3)(4,12,29,50,34,33,54,51,60,59,42,58,32,53,38,18,6,5,17,10,25,45,21,20,13)) ----------------------------------------------------------------------- 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): 90 Minimal number of generators: 17 Number of equivalence classes of elliptic points of order 2: 2 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 12 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 5/4 10/7 5/3 2/1 5/2 10/3 25/7 40/11 5/1 20/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/5 5/4 1/4 9/7 1/3 13/10 1/4 4/3 2/7 7/5 1/3 10/7 0/1 3/2 1/4 5/3 1/3 7/4 1/2 16/9 2/7 25/14 1/3 9/5 1/3 2/1 0/1 5/2 1/2 8/3 2/3 3/1 1/1 10/3 0/1 7/2 1/2 25/7 1/2 18/5 4/7 29/8 1/2 40/11 2/3 11/3 1/1 4/1 2/3 5/1 1/1 6/1 0/1 13/2 3/2 20/3 2/1 7/1 1/1 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(41,-50,32,-39) (1/1,5/4) -> (5/4,9/7) Parabolic Matrix(193,-250,149,-193) (9/7,13/10) -> (9/7,13/10) Elliptic Matrix(267,-350,74,-97) (13/10,4/3) -> (18/5,29/8) Hyperbolic Matrix(37,-50,20,-27) (4/3,7/5) -> (9/5,2/1) Hyperbolic Matrix(71,-100,49,-69) (7/5,10/7) -> (10/7,3/2) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(141,-250,22,-39) (7/4,16/9) -> (6/1,13/2) Hyperbolic Matrix(477,-850,133,-237) (16/9,25/14) -> (25/7,18/5) Hyperbolic Matrix(223,-400,63,-113) (25/14,9/5) -> (7/2,25/7) Hyperbolic Matrix(21,-50,8,-19) (2/1,5/2) -> (5/2,8/3) Parabolic Matrix(37,-100,10,-27) (8/3,3/1) -> (11/3,4/1) Hyperbolic Matrix(31,-100,9,-29) (3/1,10/3) -> (10/3,7/2) Parabolic Matrix(441,-1600,121,-439) (29/8,40/11) -> (40/11,11/3) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic Matrix(61,-400,9,-59) (13/2,20/3) -> (20/3,7/1) Parabolic Matrix(7,-50,1,-7) (7/1,1/0) -> (7/1,1/0) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,5,1) Matrix(41,-50,32,-39) -> Matrix(9,-2,32,-7) Matrix(193,-250,149,-193) -> Matrix(7,-2,25,-7) Matrix(267,-350,74,-97) -> Matrix(9,-2,14,-3) Matrix(37,-50,20,-27) -> Matrix(7,-2,18,-5) Matrix(71,-100,49,-69) -> Matrix(1,0,1,1) Matrix(31,-50,18,-29) -> Matrix(7,-2,18,-5) Matrix(141,-250,22,-39) -> Matrix(7,-2,4,-1) Matrix(477,-850,133,-237) -> Matrix(19,-6,35,-11) Matrix(223,-400,63,-113) -> Matrix(11,-4,25,-9) Matrix(21,-50,8,-19) -> Matrix(5,-2,8,-3) Matrix(37,-100,10,-27) -> Matrix(1,0,0,1) Matrix(31,-100,9,-29) -> Matrix(1,0,1,1) Matrix(441,-1600,121,-439) -> Matrix(7,-4,9,-5) Matrix(11,-50,2,-9) -> Matrix(3,-2,2,-1) Matrix(61,-400,9,-59) -> Matrix(3,-4,1,-1) Matrix(7,-50,1,-7) -> 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): 10 ----------------------------------------------------------------------- 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 5 1 2/1 0/1 1 25 5/2 1/2 5 5 8/3 2/3 1 25 3/1 1/1 1 25 10/3 0/1 5 5 7/2 1/2 1 25 25/7 1/2 5 1 18/5 4/7 1 25 11/3 1/1 1 25 4/1 2/3 1 25 5/1 1/1 5 5 6/1 0/1 1 25 13/2 3/2 1 25 20/3 2/1 5 5 7/1 1/1 1 25 1/0 1/0 1 25 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(21,-50,8,-19) (2/1,5/2) -> (5/2,8/3) Parabolic Matrix(37,-100,10,-27) (8/3,3/1) -> (11/3,4/1) Hyperbolic Matrix(31,-100,9,-29) (3/1,10/3) -> (10/3,7/2) Parabolic Matrix(99,-350,28,-99) (7/2,25/7) -> (7/2,25/7) Reflection Matrix(251,-900,70,-251) (25/7,18/5) -> (25/7,18/5) Reflection Matrix(83,-300,13,-47) (18/5,11/3) -> (6/1,13/2) Glide Reflection Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic Matrix(61,-400,9,-59) (13/2,20/3) -> (20/3,7/1) Parabolic Matrix(7,-50,1,-7) (7/1,1/0) -> (7/1,1/0) Elliptic 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(1,0,5,-1) (0/1,2/1) -> (0/1,2/5) Matrix(21,-50,8,-19) -> Matrix(5,-2,8,-3) 1/2 Matrix(37,-100,10,-27) -> Matrix(1,0,0,1) Matrix(31,-100,9,-29) -> Matrix(1,0,1,1) 0/1 Matrix(99,-350,28,-99) -> Matrix(5,-2,12,-5) (7/2,25/7) -> (1/3,1/2) Matrix(251,-900,70,-251) -> Matrix(15,-8,28,-15) (25/7,18/5) -> (1/2,4/7) Matrix(83,-300,13,-47) -> Matrix(7,-4,5,-3) Matrix(11,-50,2,-9) -> Matrix(3,-2,2,-1) 1/1 Matrix(61,-400,9,-59) -> Matrix(3,-4,1,-1) 2/1 Matrix(7,-50,1,-7) -> Matrix(1,-2,1,-1) (0/1,2/1).(1/1,1/0)