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 -4/1 -3/1 -7/3 -2/1 -3/2 -4/3 -1/1 -3/4 -2/3 -1/2 -1/3 -1/4 0/1 1/5 1/4 3/10 1/3 2/5 1/2 3/5 2/3 3/4 4/5 1/1 5/4 4/3 3/2 5/3 2/1 7/3 5/2 3/1 10/3 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -4/1 -2/1 -9/2 -3/1 -2/1 1/0 -4/1 -5/2 1/0 -7/2 1/0 -10/3 -4/1 -2/1 -3/1 -2/1 -8/3 -2/1 -5/2 -2/1 -1/1 1/0 -7/3 -1/1 -16/7 -1/1 -9/4 1/0 -2/1 -2/1 -1/1 1/0 -9/5 -1/1 -16/9 -1/2 1/0 -7/4 1/0 -5/3 -2/1 -8/5 -2/1 -3/2 -3/2 1/0 -10/7 -2/1 -4/3 -7/5 -1/1 -4/3 -1/1 -9/7 -2/1 -5/4 -2/1 -3/2 -1/1 -1/1 -1/1 -5/6 -1/1 -2/3 -1/2 -4/5 -1/1 -7/9 -2/3 -10/13 -2/3 0/1 -3/4 -1/1 -2/3 -1/2 -5/7 -2/3 -7/10 -1/2 -2/3 -1/2 -5/8 -1/1 -1/2 0/1 -3/5 -1/1 -4/7 -1/1 -2/3 -1/2 -9/16 -1/2 -5/9 -2/3 0/1 -1/2 -1/2 -4/9 -1/3 -7/16 -1/2 -1/3 0/1 -10/23 -2/5 0/1 -3/7 -1/3 -5/12 -1/3 -1/4 0/1 -2/5 0/1 -3/8 -1/2 -4/11 -1/2 -1/4 -9/25 -1/3 -5/14 -1/2 -1/3 0/1 -1/3 0/1 -4/13 -1/1 -7/23 -1/1 -10/33 -2/3 0/1 -3/10 -1/2 -2/7 0/1 -3/11 0/1 -4/15 -1/1 -5/19 -2/3 0/1 -1/4 -1/2 -2/9 -1/3 -1/5 -1/3 0/1 0/1 1/5 1/3 2/9 0/1 1/3 1/2 1/4 1/2 2/7 1/2 3/10 1/2 1/3 0/1 2/3 3/8 1/2 1/0 2/5 0/1 3/7 1/2 7/16 1/2 4/9 1/2 1/0 1/2 0/1 1/2 1/1 5/9 0/1 2/3 9/16 1/2 4/7 1/1 3/5 1/1 5/8 0/1 1/1 1/0 2/3 0/1 7/10 1/2 5/7 0/1 3/4 1/2 7/9 2/3 4/5 1/1 1/1 1/1 6/5 1/1 5/4 1/1 2/1 1/0 9/7 0/1 2/1 13/10 1/0 4/3 1/1 2/1 1/0 7/5 1/1 10/7 0/1 2/1 3/2 1/0 8/5 0/1 5/3 0/1 7/4 0/1 1/2 1/1 16/9 1/1 9/5 1/1 2/1 1/1 9/4 3/2 16/7 1/1 4/3 3/2 23/10 3/2 7/3 3/2 12/5 2/1 5/2 1/1 3/2 2/1 8/3 3/2 11/4 3/2 25/9 8/5 2/1 14/5 5/3 3/1 2/1 13/4 5/2 23/7 5/2 33/10 5/2 10/3 2/1 8/3 7/2 5/2 1/0 11/3 2/1 8/3 15/4 2/1 5/2 3/1 19/5 3/1 4/1 3/1 9/2 1/0 5/1 2/1 4/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,10,0,1) (-5/1,1/0) -> (5/1,1/0) Parabolic Matrix(11,50,20,91) (-5/1,-9/2) -> (1/2,5/9) Hyperbolic Matrix(9,40,20,89) (-9/2,-4/1) -> (4/9,1/2) Hyperbolic Matrix(11,40,-30,-109) (-4/1,-7/2) -> (-3/8,-4/11) Hyperbolic Matrix(29,100,20,69) (-7/2,-10/3) -> (10/7,3/2) Hyperbolic Matrix(31,100,-40,-129) (-10/3,-3/1) -> (-7/9,-10/13) Hyperbolic Matrix(11,30,-40,-109) (-3/1,-8/3) -> (-2/7,-3/11) Hyperbolic Matrix(19,50,30,79) (-8/3,-5/2) -> (5/8,2/3) Hyperbolic Matrix(21,50,-50,-119) (-5/2,-7/3) -> (-3/7,-5/12) Hyperbolic Matrix(61,140,-200,-459) (-7/3,-16/7) -> (-4/13,-7/23) Hyperbolic Matrix(79,180,140,319) (-16/7,-9/4) -> (9/16,4/7) Hyperbolic Matrix(9,20,40,89) (-9/4,-2/1) -> (2/9,1/4) Hyperbolic Matrix(11,20,50,91) (-2/1,-9/5) -> (1/5,2/9) Hyperbolic Matrix(101,180,-280,-499) (-9/5,-16/9) -> (-4/11,-9/25) Hyperbolic Matrix(79,140,180,319) (-16/9,-7/4) -> (7/16,4/9) Hyperbolic Matrix(29,50,40,69) (-7/4,-5/3) -> (5/7,3/4) Hyperbolic Matrix(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(19,30,50,79) (-8/5,-3/2) -> (3/8,2/5) Hyperbolic Matrix(69,100,20,29) (-3/2,-10/7) -> (10/3,7/2) Hyperbolic Matrix(99,140,70,99) (-10/7,-7/5) -> (7/5,10/7) Hyperbolic Matrix(29,40,50,69) (-7/5,-4/3) -> (4/7,3/5) Hyperbolic Matrix(31,40,-100,-129) (-4/3,-9/7) -> (-1/3,-4/13) Hyperbolic Matrix(39,50,-110,-141) (-9/7,-5/4) -> (-5/14,-1/3) Hyperbolic Matrix(9,10,-10,-11) (-5/4,-1/1) -> (-1/1,-5/6) Parabolic Matrix(61,50,50,41) (-5/6,-4/5) -> (6/5,5/4) Hyperbolic Matrix(51,40,-190,-149) (-4/5,-7/9) -> (-3/11,-4/15) Hyperbolic Matrix(131,100,-300,-229) (-10/13,-3/4) -> (-7/16,-10/23) Hyperbolic Matrix(69,50,40,29) (-3/4,-5/7) -> (5/3,7/4) Hyperbolic Matrix(99,70,140,99) (-5/7,-7/10) -> (7/10,5/7) Hyperbolic Matrix(29,20,100,69) (-7/10,-2/3) -> (2/7,3/10) Hyperbolic Matrix(79,50,30,19) (-2/3,-5/8) -> (5/2,8/3) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(69,40,50,29) (-3/5,-4/7) -> (4/3,7/5) Hyperbolic Matrix(319,180,140,79) (-4/7,-9/16) -> (9/4,16/7) Hyperbolic Matrix(89,50,-340,-191) (-9/16,-5/9) -> (-5/19,-1/4) Hyperbolic Matrix(91,50,20,11) (-5/9,-1/2) -> (9/2,5/1) Hyperbolic Matrix(89,40,20,9) (-1/2,-4/9) -> (4/1,9/2) Hyperbolic Matrix(319,140,180,79) (-4/9,-7/16) -> (7/4,16/9) Hyperbolic Matrix(231,100,-760,-329) (-10/23,-3/7) -> (-7/23,-10/33) Hyperbolic Matrix(169,70,70,29) (-5/12,-2/5) -> (12/5,5/2) Hyperbolic Matrix(79,30,50,19) (-2/5,-3/8) -> (3/2,8/5) Hyperbolic Matrix(641,230,170,61) (-9/25,-5/14) -> (15/4,19/5) Hyperbolic Matrix(1189,360,360,109) (-10/33,-3/10) -> (33/10,10/3) Hyperbolic Matrix(69,20,100,29) (-3/10,-2/7) -> (2/3,7/10) Hyperbolic Matrix(641,170,230,61) (-4/15,-5/19) -> (25/9,14/5) Hyperbolic Matrix(89,20,40,9) (-1/4,-2/9) -> (2/1,9/4) Hyperbolic Matrix(91,20,50,11) (-2/9,-1/5) -> (9/5,2/1) Hyperbolic Matrix(1,0,10,1) (-1/5,0/1) -> (0/1,1/5) Parabolic Matrix(109,-30,40,-11) (1/4,2/7) -> (8/3,11/4) Hyperbolic Matrix(129,-40,100,-31) (3/10,1/3) -> (9/7,13/10) Hyperbolic Matrix(109,-40,30,-11) (1/3,3/8) -> (7/2,11/3) Hyperbolic Matrix(119,-50,50,-21) (2/5,3/7) -> (7/3,12/5) Hyperbolic Matrix(459,-200,140,-61) (3/7,7/16) -> (13/4,23/7) Hyperbolic Matrix(499,-280,180,-101) (5/9,9/16) -> (11/4,25/9) Hyperbolic Matrix(129,-100,40,-31) (3/4,7/9) -> (3/1,13/4) Hyperbolic Matrix(141,-110,50,-39) (7/9,4/5) -> (14/5,3/1) Hyperbolic Matrix(11,-10,10,-9) (4/5,1/1) -> (1/1,6/5) Parabolic Matrix(149,-190,40,-51) (5/4,9/7) -> (11/3,15/4) Hyperbolic Matrix(229,-300,100,-131) (13/10,4/3) -> (16/7,23/10) Hyperbolic Matrix(191,-340,50,-89) (16/9,9/5) -> (19/5,4/1) Hyperbolic Matrix(329,-760,100,-231) (23/10,7/3) -> (23/7,33/10) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,10,0,1) -> Matrix(1,6,0,1) Matrix(11,50,20,91) -> Matrix(1,2,2,5) Matrix(9,40,20,89) -> Matrix(1,2,2,5) Matrix(11,40,-30,-109) -> Matrix(1,2,-2,-3) Matrix(29,100,20,69) -> Matrix(1,4,0,1) Matrix(31,100,-40,-129) -> Matrix(1,4,-2,-7) Matrix(11,30,-40,-109) -> Matrix(1,2,-2,-3) Matrix(19,50,30,79) -> Matrix(1,2,0,1) Matrix(21,50,-50,-119) -> Matrix(1,2,-4,-7) Matrix(61,140,-200,-459) -> Matrix(1,2,-2,-3) Matrix(79,180,140,319) -> Matrix(1,0,2,1) Matrix(9,20,40,89) -> Matrix(1,2,2,5) Matrix(11,20,50,91) -> Matrix(1,2,2,5) Matrix(101,180,-280,-499) -> Matrix(1,0,-2,1) Matrix(79,140,180,319) -> Matrix(1,0,2,1) Matrix(29,50,40,69) -> Matrix(1,2,2,5) Matrix(49,80,30,49) -> Matrix(1,2,0,1) Matrix(19,30,50,79) -> Matrix(1,2,0,1) Matrix(69,100,20,29) -> Matrix(1,4,0,1) Matrix(99,140,70,99) -> Matrix(3,4,2,3) Matrix(29,40,50,69) -> Matrix(1,2,0,1) Matrix(31,40,-100,-129) -> Matrix(1,2,-2,-3) Matrix(39,50,-110,-141) -> Matrix(1,2,-4,-7) Matrix(9,10,-10,-11) -> Matrix(3,4,-4,-5) Matrix(61,50,50,41) -> Matrix(1,0,2,1) Matrix(51,40,-190,-149) -> Matrix(3,2,-2,-1) Matrix(131,100,-300,-229) -> Matrix(3,2,-8,-5) Matrix(69,50,40,29) -> Matrix(3,2,4,3) Matrix(99,70,140,99) -> Matrix(3,2,4,3) Matrix(29,20,100,69) -> Matrix(3,2,4,3) Matrix(79,50,30,19) -> Matrix(1,2,0,1) Matrix(49,30,80,49) -> Matrix(1,0,2,1) Matrix(69,40,50,29) -> Matrix(1,0,2,1) Matrix(319,180,140,79) -> Matrix(1,2,0,1) Matrix(89,50,-340,-191) -> Matrix(1,0,0,1) Matrix(91,50,20,11) -> Matrix(5,2,2,1) Matrix(89,40,20,9) -> Matrix(9,4,2,1) Matrix(319,140,180,79) -> Matrix(1,0,4,1) Matrix(231,100,-760,-329) -> Matrix(5,2,-8,-3) Matrix(169,70,70,29) -> Matrix(5,2,2,1) Matrix(79,30,50,19) -> Matrix(1,0,2,1) Matrix(641,230,170,61) -> Matrix(9,2,4,1) Matrix(1189,360,360,109) -> Matrix(11,8,4,3) Matrix(69,20,100,29) -> Matrix(1,0,4,1) Matrix(641,170,230,61) -> Matrix(13,8,8,5) Matrix(89,20,40,9) -> Matrix(11,4,8,3) Matrix(91,20,50,11) -> Matrix(1,0,4,1) Matrix(1,0,10,1) -> Matrix(1,0,6,1) Matrix(109,-30,40,-11) -> Matrix(7,-2,4,-1) Matrix(129,-40,100,-31) -> Matrix(3,-2,2,-1) Matrix(109,-40,30,-11) -> Matrix(1,2,0,1) Matrix(119,-50,50,-21) -> Matrix(7,-2,4,-1) Matrix(459,-200,140,-61) -> Matrix(1,2,0,1) Matrix(499,-280,180,-101) -> Matrix(7,-2,4,-1) Matrix(129,-100,40,-31) -> Matrix(19,-12,8,-5) Matrix(141,-110,50,-39) -> Matrix(17,-12,10,-7) Matrix(11,-10,10,-9) -> Matrix(3,-2,2,-1) Matrix(149,-190,40,-51) -> Matrix(5,-8,2,-3) Matrix(229,-300,100,-131) -> Matrix(3,-2,2,-1) Matrix(191,-340,50,-89) -> Matrix(1,2,0,1) Matrix(329,-760,100,-231) -> Matrix(11,-14,4,-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): 12 Degree of the the map X: 12 Degree of the the map Y: 60 Permutation triple for Y: ((1,7,31,8,2)(3,13,44,45,14)(4,19,32,20,5)(6,25,34,33,26)(9,21,29,28,36)(10,18,30,39,11)(12,42,57,54,27)(15,35,58,55,47)(16,37,52,48,17)(22,38,51,53,23)(24,41,40,56,50)(43,46,59,60,49); (1,5,23,24,6)(2,11,41,12,3)(4,17,49,50,18)(7,29,48,53,30)(8,34,42,35,9)(10,22,43,27,26)(13,21,20,39,25)(14,33,40,46,15)(16,47,54,56,38)(19,28,55,60,51)(31,44,58,52,32)(36,45,57,59,37); (1,3,15,16,4)(2,9,37,38,10)(5,21,35,46,22)(6,27,47,28,7)(8,32,51,56,33)(11,20,52,59,40)(12,43,17,29,13)(14,36,19,18,26)(23,48,58,42,41)(24,49,55,44,25)(30,50,54,45,31)(34,39,53,60,57)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- The image of the extended modular group liftables in PGL(2,Z) equals the image of the modular liftables. ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.