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): 384 Minimal number of generators: 65 Number of equivalence classes of cusps: 40 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -4/1 -10/3 -3/1 -2/1 -4/3 -8/7 -1/1 -4/5 -2/3 -4/7 0/1 1/2 4/7 2/3 3/4 4/5 1/1 5/4 4/3 3/2 8/5 7/4 2/1 16/7 7/3 5/2 8/3 3/1 16/5 10/3 7/2 4/1 5/1 16/3 6/1 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 -11/2 -1/2 1/0 -16/3 0/1 -5/1 -1/1 1/1 -4/1 1/0 -11/3 -3/1 -1/1 -7/2 -3/2 1/0 -10/3 1/0 -13/4 1/0 -16/5 -2/1 -3/1 -1/1 -8/3 -1/1 -5/2 -1/2 1/0 -12/5 -2/3 0/1 -7/3 -1/1 -1/3 -2/1 0/1 -9/5 -1/1 1/1 -16/9 0/1 -7/4 1/2 -12/7 1/1 -5/3 1/1 -8/5 1/0 -3/2 -1/2 1/0 -16/11 0/1 -13/9 -1/1 1/1 -10/7 -1/1 -17/12 -1/2 -24/17 -1/2 -7/5 -1/3 -4/3 0/1 -9/7 1/3 -5/4 1/2 -16/13 0/1 -11/9 1/3 -6/5 1/2 -7/6 1/2 3/4 -8/7 1/1 -1/1 -1/1 1/1 -6/7 2/1 -5/6 5/2 1/0 -4/5 1/0 -7/9 -1/1 -10/13 -2/1 0/1 -3/4 1/0 -8/11 1/0 -5/7 -1/1 -7/10 -1/2 1/0 -2/3 1/0 -9/14 -5/2 1/0 -16/25 -2/1 -7/11 -5/3 -1/1 -5/8 1/0 -8/13 -1/1 -3/5 -1/1 -10/17 0/1 -17/29 -3/1 -1/1 -24/41 -1/1 -7/12 -1/2 -4/7 0/1 -5/9 -1/1 1/1 -6/11 0/1 2/1 -7/13 1/1 -8/15 1/0 -1/2 -1/2 1/0 0/1 0/1 1/2 1/2 1/0 6/11 1/2 5/9 1/1 4/7 1/1 7/12 1/2 10/17 1/1 3/5 1/3 1/1 5/8 1/0 7/11 1/1 2/3 0/1 2/3 9/13 1/1 7/10 1/4 1/2 5/7 1/3 1/1 3/4 1/2 10/13 1/2 17/22 1/2 1/0 7/9 3/5 1/1 4/5 0/1 2/3 5/6 1/2 3/4 6/7 1/1 7/8 3/4 1/1 1/1 6/5 0/1 2/1 11/9 -1/1 1/1 16/13 0/1 5/4 1/2 4/3 1/1 11/8 5/4 7/5 1/1 7/5 10/7 2/1 13/9 1/1 16/11 2/1 3/2 3/2 1/0 8/5 1/0 13/8 1/0 5/3 -1/1 1/1 12/7 0/1 7/4 1/0 2/1 1/1 9/4 3/2 16/7 2/1 23/10 9/4 5/2 7/3 3/1 12/5 1/0 5/2 1/2 1/0 8/3 1/1 11/4 3/2 3/1 1/1 3/1 16/5 2/1 13/4 1/0 10/3 0/1 2/1 17/5 1/1 41/12 1/2 24/7 1/1 7/2 3/2 1/0 4/1 0/1 2/1 9/2 3/2 1/0 5/1 1/1 16/3 2/1 11/2 5/2 1/0 6/1 2/1 7/1 3/1 5/1 15/2 11/2 1/0 8/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(17,112,-12,-79) (-6/1,1/0) -> (-10/7,-17/12) Hyperbolic Matrix(17,96,20,113) (-6/1,-11/2) -> (5/6,6/7) Hyperbolic Matrix(65,352,12,65) (-11/2,-16/3) -> (16/3,11/2) Hyperbolic Matrix(49,256,40,209) (-16/3,-5/1) -> (11/9,16/13) Hyperbolic Matrix(15,64,-4,-17) (-5/1,-4/1) -> (-4/1,-11/3) Parabolic Matrix(31,112,44,159) (-11/3,-7/2) -> (7/10,5/7) Hyperbolic Matrix(33,112,-28,-95) (-7/2,-10/3) -> (-6/5,-7/6) Hyperbolic Matrix(49,160,64,209) (-10/3,-13/4) -> (3/4,10/13) Hyperbolic Matrix(129,416,40,129) (-13/4,-16/5) -> (16/5,13/4) Hyperbolic Matrix(81,256,56,177) (-16/5,-3/1) -> (13/9,16/11) Hyperbolic Matrix(17,48,-28,-79) (-3/1,-8/3) -> (-8/13,-3/5) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(33,80,40,97) (-5/2,-12/5) -> (4/5,5/6) Hyperbolic Matrix(47,112,60,143) (-12/5,-7/3) -> (7/9,4/5) Hyperbolic Matrix(15,32,-8,-17) (-7/3,-2/1) -> (-2/1,-9/5) Parabolic Matrix(143,256,-224,-401) (-9/5,-16/9) -> (-16/25,-7/11) Hyperbolic Matrix(145,256,64,113) (-16/9,-7/4) -> (9/4,16/7) Hyperbolic Matrix(65,112,112,193) (-7/4,-12/7) -> (4/7,7/12) Hyperbolic Matrix(47,80,84,143) (-12/7,-5/3) -> (5/9,4/7) Hyperbolic Matrix(49,80,-68,-111) (-5/3,-8/5) -> (-8/11,-5/7) Hyperbolic Matrix(31,48,20,31) (-8/5,-3/2) -> (3/2,8/5) Hyperbolic Matrix(65,96,44,65) (-3/2,-16/11) -> (16/11,3/2) Hyperbolic Matrix(177,256,56,81) (-16/11,-13/9) -> (3/1,16/5) Hyperbolic Matrix(111,160,188,271) (-13/9,-10/7) -> (10/17,3/5) Hyperbolic Matrix(113,160,12,17) (-17/12,-24/17) -> (8/1,1/0) Hyperbolic Matrix(159,224,-296,-417) (-24/17,-7/5) -> (-7/13,-8/15) Hyperbolic Matrix(47,64,-36,-49) (-7/5,-4/3) -> (-4/3,-9/7) Parabolic Matrix(63,80,100,127) (-9/7,-5/4) -> (5/8,7/11) Hyperbolic Matrix(129,160,104,129) (-5/4,-16/13) -> (16/13,5/4) Hyperbolic Matrix(209,256,40,49) (-16/13,-11/9) -> (5/1,16/3) Hyperbolic Matrix(79,96,144,175) (-11/9,-6/5) -> (6/11,5/9) Hyperbolic Matrix(193,224,56,65) (-7/6,-8/7) -> (24/7,7/2) Hyperbolic Matrix(143,160,-244,-273) (-8/7,-1/1) -> (-17/29,-24/41) Hyperbolic Matrix(129,112,-220,-191) (-1/1,-6/7) -> (-10/17,-17/29) Hyperbolic Matrix(113,96,20,17) (-6/7,-5/6) -> (11/2,6/1) Hyperbolic Matrix(97,80,40,33) (-5/6,-4/5) -> (12/5,5/2) Hyperbolic Matrix(143,112,60,47) (-4/5,-7/9) -> (7/3,12/5) Hyperbolic Matrix(145,112,-268,-207) (-7/9,-10/13) -> (-6/11,-7/13) Hyperbolic Matrix(209,160,64,49) (-10/13,-3/4) -> (13/4,10/3) Hyperbolic Matrix(175,128,108,79) (-3/4,-8/11) -> (8/5,13/8) Hyperbolic Matrix(113,80,24,17) (-5/7,-7/10) -> (9/2,5/1) Hyperbolic Matrix(47,32,-72,-49) (-7/10,-2/3) -> (-2/3,-9/14) Parabolic Matrix(799,512,348,223) (-9/14,-16/25) -> (16/7,23/10) Hyperbolic Matrix(177,112,128,81) (-7/11,-5/8) -> (11/8,7/5) Hyperbolic Matrix(207,128,76,47) (-5/8,-8/13) -> (8/3,11/4) Hyperbolic Matrix(271,160,188,111) (-3/5,-10/17) -> (10/7,13/9) Hyperbolic Matrix(1969,1152,576,337) (-24/41,-7/12) -> (41/12,24/7) Hyperbolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(143,80,84,47) (-4/7,-5/9) -> (5/3,12/7) Hyperbolic Matrix(175,96,144,79) (-5/9,-6/11) -> (6/5,11/9) Hyperbolic Matrix(241,128,32,17) (-8/15,-1/2) -> (15/2,8/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(207,-112,268,-145) (1/2,6/11) -> (10/13,17/22) Hyperbolic Matrix(191,-112,220,-129) (7/12,10/17) -> (6/7,7/8) Hyperbolic Matrix(79,-48,28,-17) (3/5,5/8) -> (11/4,3/1) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(369,-256,160,-111) (9/13,7/10) -> (23/10,7/3) Hyperbolic Matrix(111,-80,68,-49) (5/7,3/4) -> (13/8,5/3) Hyperbolic Matrix(289,-224,40,-31) (17/22,7/9) -> (7/1,15/2) Hyperbolic Matrix(177,-160,52,-47) (7/8,1/1) -> (17/5,41/12) Hyperbolic Matrix(95,-112,28,-33) (1/1,6/5) -> (10/3,17/5) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(79,-112,12,-17) (7/5,10/7) -> (6/1,7/1) Hyperbolic Matrix(17,-32,8,-15) (7/4,2/1) -> (2/1,9/4) Parabolic Matrix(17,-64,4,-15) (7/2,4/1) -> (4/1,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(17,112,-12,-79) -> Matrix(1,2,-2,-3) Matrix(17,96,20,113) -> Matrix(3,2,4,3) Matrix(65,352,12,65) -> Matrix(5,2,2,1) Matrix(49,256,40,209) -> Matrix(1,0,0,1) Matrix(15,64,-4,-17) -> Matrix(1,-2,0,1) Matrix(31,112,44,159) -> Matrix(1,2,2,5) Matrix(33,112,-28,-95) -> Matrix(1,0,2,1) Matrix(49,160,64,209) -> Matrix(1,0,2,1) Matrix(129,416,40,129) -> Matrix(1,4,0,1) Matrix(81,256,56,177) -> Matrix(3,4,2,3) Matrix(17,48,-28,-79) -> Matrix(1,0,0,1) Matrix(31,80,12,31) -> Matrix(1,0,2,1) Matrix(33,80,40,97) -> Matrix(3,2,4,3) Matrix(47,112,60,143) -> Matrix(3,2,4,3) Matrix(15,32,-8,-17) -> Matrix(1,0,2,1) Matrix(143,256,-224,-401) -> Matrix(3,2,-2,-1) Matrix(145,256,64,113) -> Matrix(7,-2,4,-1) Matrix(65,112,112,193) -> Matrix(1,0,0,1) Matrix(47,80,84,143) -> Matrix(1,-2,2,-3) Matrix(49,80,-68,-111) -> Matrix(1,-2,0,1) Matrix(31,48,20,31) -> Matrix(1,2,0,1) Matrix(65,96,44,65) -> Matrix(1,2,0,1) Matrix(177,256,56,81) -> Matrix(1,2,0,1) Matrix(111,160,188,271) -> Matrix(1,0,2,1) Matrix(113,160,12,17) -> Matrix(1,0,2,1) Matrix(159,224,-296,-417) -> Matrix(5,2,2,1) Matrix(47,64,-36,-49) -> Matrix(1,0,6,1) Matrix(63,80,100,127) -> Matrix(1,0,-2,1) Matrix(129,160,104,129) -> Matrix(1,0,0,1) Matrix(209,256,40,49) -> Matrix(7,-2,4,-1) Matrix(79,96,144,175) -> Matrix(5,-2,8,-3) Matrix(193,224,56,65) -> Matrix(5,-4,4,-3) Matrix(143,160,-244,-273) -> Matrix(1,-2,0,1) Matrix(129,112,-220,-191) -> Matrix(1,-2,0,1) Matrix(113,96,20,17) -> Matrix(1,0,0,1) Matrix(97,80,40,33) -> Matrix(1,-2,0,1) Matrix(143,112,60,47) -> Matrix(1,4,0,1) Matrix(145,112,-268,-207) -> Matrix(1,2,0,1) Matrix(209,160,64,49) -> Matrix(1,2,0,1) Matrix(175,128,108,79) -> Matrix(1,0,0,1) Matrix(113,80,24,17) -> Matrix(1,2,0,1) Matrix(47,32,-72,-49) -> Matrix(1,-2,0,1) Matrix(799,512,348,223) -> Matrix(9,20,4,9) Matrix(177,112,128,81) -> Matrix(5,6,4,5) Matrix(207,128,76,47) -> Matrix(3,4,2,3) Matrix(271,160,188,111) -> Matrix(1,2,0,1) Matrix(1969,1152,576,337) -> Matrix(3,2,4,3) Matrix(193,112,112,65) -> Matrix(1,0,2,1) Matrix(143,80,84,47) -> Matrix(1,0,0,1) Matrix(175,96,144,79) -> Matrix(1,0,0,1) Matrix(241,128,32,17) -> Matrix(1,6,0,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(207,-112,268,-145) -> Matrix(1,0,0,1) Matrix(191,-112,220,-129) -> Matrix(1,-2,2,-3) Matrix(79,-48,28,-17) -> Matrix(3,-2,2,-1) Matrix(49,-32,72,-47) -> Matrix(1,0,0,1) Matrix(369,-256,160,-111) -> Matrix(1,2,0,1) Matrix(111,-80,68,-49) -> Matrix(1,0,-2,1) Matrix(289,-224,40,-31) -> Matrix(11,-6,2,-1) Matrix(177,-160,52,-47) -> Matrix(3,-2,2,-1) Matrix(95,-112,28,-33) -> Matrix(1,0,0,1) Matrix(49,-64,36,-47) -> Matrix(7,-6,6,-5) Matrix(79,-112,12,-17) -> Matrix(5,-8,2,-3) Matrix(17,-32,8,-15) -> Matrix(3,-2,2,-1) Matrix(17,-64,4,-15) -> Matrix(1,0,0,1) 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): 16 Degree of the the map X: 16 Degree of the the map Y: 64 Permutation triple for Y: ((1,2)(3,10,37,11)(4,16,17,5)(6,22,51,23)(7,28,29,8)(9,34)(12,40)(13,14)(15,27)(18,30)(19,20)(21,38)(24,36)(25,26)(31,32)(33,39,50,53)(35,41,49,52)(42,47,55,43)(44,48,54,45)(46,56)(57,64)(58,62)(59,63)(60,61); (1,5,20,44,64,49,21,6)(2,8,32,55,57,33,9,3)(4,14,45,60,52,24,23,15)(7,26,47,61,39,12,11,27)(10,30,29,13,43,58,53,36)(16,31,54,59,35,34,51,46)(17,25,48,62,41,40,22,18)(19,42,63,50,38,37,56,28); (1,3,12,41,59,42,13,4)(2,6,24,53,63,54,25,7)(5,18,10,9,35,60,47,19)(8,30,22,21,50,61,45,31)(11,38,49,62,43,32,16,15)(14,29,56,51,40,39,57,44)(17,46,37,36,52,64,55,26)(20,28,27,23,34,33,58,48)) ----------------------------------------------------------------------- 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.