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 -6/1 -5/1 -4/1 -7/2 -13/4 -3/1 -21/8 -2/1 -7/4 -7/6 -1/1 -7/10 -7/12 -7/13 0/1 1/2 7/12 7/11 7/10 3/4 7/9 1/1 7/6 5/4 14/11 7/5 3/2 14/9 7/4 2/1 28/13 9/4 7/3 5/2 21/8 11/4 14/5 3/1 7/2 4/1 9/2 14/3 5/1 11/2 6/1 13/2 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -1/1 -6/1 -5/6 -11/2 -1/1 -3/4 -5/1 -3/4 -1/2 -14/3 -3/4 -1/2 -9/2 -1/1 -2/3 -22/5 -3/4 -13/3 -3/4 -1/2 -4/1 -1/2 -7/2 -1/2 -10/3 -1/2 -13/4 -1/1 0/1 -16/5 -1/2 -3/1 -1/2 1/0 -14/5 -1/2 -11/4 -1/2 0/1 -19/7 -1/2 -1/4 -8/3 1/0 -21/8 -1/2 1/0 -34/13 1/0 -13/5 -1/2 1/0 -5/2 -1/1 0/1 -7/3 -1/1 -9/4 -1/1 -2/3 -11/5 -3/4 -1/2 -13/6 -1/1 -2/3 -2/1 -1/2 -7/4 -1/2 -12/7 -1/2 -41/24 -2/5 -1/3 -70/41 -1/2 -3/8 -29/17 -1/2 -3/8 -17/10 -1/2 -1/3 -5/3 -1/2 -1/4 -18/11 -1/4 -13/8 -1/5 0/1 -21/13 0/1 -8/5 1/0 -27/17 -1/2 1/0 -19/12 -1/1 0/1 -11/7 -1/2 1/0 -14/9 -1/2 -3/2 -1/2 0/1 -7/5 0/1 -11/8 0/1 1/2 -26/19 1/2 -41/30 0/1 1/1 -56/41 1/2 1/0 -15/11 1/2 1/0 -4/3 1/0 -13/10 -2/1 -1/1 -9/7 -1/2 1/0 -14/11 -1/2 1/0 -5/4 -1/1 0/1 -6/5 -1/2 -7/6 -1/2 1/0 -8/7 -1/2 -1/1 -1/2 1/0 -5/6 -1/1 0/1 -14/17 -1/2 1/0 -9/11 -1/2 1/0 -4/5 1/0 -15/19 -3/2 1/0 -11/14 -3/2 -1/1 -7/9 -1/1 -3/4 -1/1 -1/2 -14/19 -1/2 -11/15 -1/2 1/0 -19/26 -1/1 0/1 -8/11 1/0 -21/29 -1/1 -13/18 -1/1 -4/5 -5/7 -3/4 -1/2 -7/10 -1/2 -9/13 -1/2 -1/4 -11/16 -1/2 0/1 -2/3 -1/2 -11/17 -1/2 -1/4 -9/14 -1/3 0/1 -7/11 0/1 -5/8 -1/1 0/1 -8/13 1/0 -11/18 -1/1 -1/2 -14/23 -1/2 -3/5 -1/2 1/0 -7/12 -1/2 -11/19 -1/2 -3/8 -15/26 -2/5 -1/3 -4/7 -1/2 -13/23 -1/2 -1/4 -9/16 -1/3 0/1 -14/25 -1/2 -1/4 -5/9 -1/2 -1/4 -6/11 -1/6 -7/13 0/1 -1/2 -1/1 0/1 0/1 -1/2 1/0 1/2 -1/1 0/1 6/11 1/4 5/9 1/2 1/0 9/16 0/1 1/1 13/23 1/2 1/0 4/7 1/0 7/12 1/0 10/17 1/0 3/5 -1/2 1/0 11/18 -1/1 1/0 8/13 -1/2 5/8 -1/1 0/1 7/11 0/1 9/14 0/1 1/1 11/17 1/2 1/0 2/3 1/0 7/10 1/0 12/17 1/0 5/7 -3/2 1/0 13/18 -4/3 -1/1 8/11 -1/2 19/26 -1/1 0/1 11/15 -1/2 1/0 3/4 -1/1 1/0 7/9 -1/1 11/14 -1/1 -3/4 15/19 -3/4 -1/2 4/5 -1/2 9/11 -1/2 1/0 5/6 -1/1 0/1 1/1 -1/2 1/0 7/6 -1/2 1/0 13/11 -1/2 1/0 6/5 1/0 5/4 -1/1 0/1 14/11 -1/2 1/0 23/18 -1/1 0/1 9/7 -1/2 1/0 13/10 -1/1 -2/3 17/13 -3/4 -1/2 4/3 -1/2 15/11 -1/2 -1/4 26/19 -1/4 11/8 -1/4 0/1 7/5 0/1 3/2 0/1 1/0 14/9 1/0 25/16 -1/1 1/0 11/7 -1/2 1/0 19/12 -1/1 0/1 27/17 -1/2 1/0 8/5 -1/2 29/18 -1/7 0/1 21/13 0/1 13/8 0/1 1/3 18/11 1/2 5/3 1/2 1/0 7/4 1/0 9/5 -3/2 1/0 11/6 -1/1 1/0 13/7 -1/2 1/0 2/1 1/0 15/7 -3/2 1/0 28/13 -3/2 1/0 13/6 -2/1 -1/1 11/5 -3/2 1/0 9/4 -2/1 -1/1 7/3 -1/1 5/2 -1/1 0/1 13/5 -1/2 1/0 21/8 -1/2 1/0 29/11 -1/2 1/0 8/3 -1/2 19/7 1/2 1/0 30/11 1/0 11/4 0/1 1/0 14/5 1/0 17/6 -1/1 1/0 3/1 -1/2 1/0 7/2 1/0 11/3 -5/2 1/0 26/7 1/0 41/11 -5/2 1/0 56/15 -5/2 1/0 15/4 -3/1 -2/1 4/1 1/0 13/3 -3/2 1/0 22/5 -3/2 9/2 -2/1 -1/1 14/3 -3/2 1/0 19/4 -2/1 -1/1 5/1 -3/2 1/0 11/2 -3/2 -1/1 6/1 -5/4 13/2 -10/9 -1/1 7/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,14,0,1) (-7/1,1/0) -> (7/1,1/0) Parabolic Matrix(29,182,-40,-251) (-7/1,-6/1) -> (-8/11,-21/29) Hyperbolic Matrix(27,154,44,251) (-6/1,-11/2) -> (11/18,8/13) Hyperbolic Matrix(29,154,16,85) (-11/2,-5/1) -> (9/5,11/6) Hyperbolic Matrix(29,140,-52,-251) (-5/1,-14/3) -> (-14/25,-5/9) Hyperbolic Matrix(55,252,12,55) (-14/3,-9/2) -> (9/2,14/3) Hyperbolic Matrix(111,490,152,671) (-9/2,-22/5) -> (8/11,19/26) Hyperbolic Matrix(169,742,64,281) (-22/5,-13/3) -> (29/11,8/3) Hyperbolic Matrix(27,112,20,83) (-13/3,-4/1) -> (4/3,15/11) Hyperbolic Matrix(27,98,-8,-29) (-4/1,-7/2) -> (-7/2,-10/3) Parabolic Matrix(111,364,68,223) (-10/3,-13/4) -> (13/8,18/11) Hyperbolic Matrix(253,812,-148,-475) (-13/4,-16/5) -> (-12/7,-41/24) Hyperbolic Matrix(57,182,88,281) (-16/5,-3/1) -> (11/17,2/3) Hyperbolic Matrix(29,84,-48,-139) (-3/1,-14/5) -> (-14/23,-3/5) Hyperbolic Matrix(111,308,40,111) (-14/5,-11/4) -> (11/4,14/5) Hyperbolic Matrix(113,308,-164,-447) (-11/4,-19/7) -> (-9/13,-11/16) Hyperbolic Matrix(57,154,104,281) (-19/7,-8/3) -> (6/11,5/9) Hyperbolic Matrix(335,882,-128,-337) (-8/3,-21/8) -> (-21/8,-34/13) Parabolic Matrix(279,728,64,167) (-34/13,-13/5) -> (13/3,22/5) Hyperbolic Matrix(27,70,32,83) (-13/5,-5/2) -> (5/6,1/1) Hyperbolic Matrix(29,70,12,29) (-5/2,-7/3) -> (7/3,5/2) Hyperbolic Matrix(55,126,24,55) (-7/3,-9/4) -> (9/4,7/3) Hyperbolic Matrix(139,308,88,195) (-9/4,-11/5) -> (11/7,19/12) Hyperbolic Matrix(167,364,128,279) (-11/5,-13/6) -> (13/10,17/13) Hyperbolic Matrix(197,420,-144,-307) (-13/6,-2/1) -> (-26/19,-41/30) Hyperbolic Matrix(55,98,-32,-57) (-2/1,-7/4) -> (-7/4,-12/7) Parabolic Matrix(1959,3346,524,895) (-41/24,-70/41) -> (56/15,15/4) Hyperbolic Matrix(1091,1862,508,867) (-70/41,-29/17) -> (15/7,28/13) Hyperbolic Matrix(337,574,428,729) (-29/17,-17/10) -> (11/14,15/19) Hyperbolic Matrix(83,140,16,27) (-17/10,-5/3) -> (5/1,11/2) Hyperbolic Matrix(111,182,136,223) (-5/3,-18/11) -> (4/5,9/11) Hyperbolic Matrix(197,322,52,85) (-18/11,-13/8) -> (15/4,4/1) Hyperbolic Matrix(337,546,208,337) (-13/8,-21/13) -> (21/13,13/8) Hyperbolic Matrix(113,182,-208,-335) (-21/13,-8/5) -> (-6/11,-7/13) Hyperbolic Matrix(167,266,140,223) (-8/5,-27/17) -> (13/11,6/5) Hyperbolic Matrix(309,490,548,869) (-27/17,-19/12) -> (9/16,13/23) Hyperbolic Matrix(195,308,88,139) (-19/12,-11/7) -> (11/5,9/4) Hyperbolic Matrix(197,308,-268,-419) (-11/7,-14/9) -> (-14/19,-11/15) Hyperbolic Matrix(55,84,36,55) (-14/9,-3/2) -> (3/2,14/9) Hyperbolic Matrix(29,42,20,29) (-3/2,-7/5) -> (7/5,3/2) Hyperbolic Matrix(111,154,80,111) (-7/5,-11/8) -> (11/8,7/5) Hyperbolic Matrix(449,616,164,225) (-11/8,-26/19) -> (30/11,11/4) Hyperbolic Matrix(1373,1876,636,869) (-41/30,-56/41) -> (28/13,13/6) Hyperbolic Matrix(2297,3136,616,841) (-56/41,-15/11) -> (41/11,56/15) Hyperbolic Matrix(83,112,20,27) (-15/11,-4/3) -> (4/1,13/3) Hyperbolic Matrix(85,112,-148,-195) (-4/3,-13/10) -> (-15/26,-4/7) Hyperbolic Matrix(141,182,196,253) (-13/10,-9/7) -> (5/7,13/18) Hyperbolic Matrix(197,252,-240,-307) (-9/7,-14/11) -> (-14/17,-9/11) Hyperbolic Matrix(111,140,88,111) (-14/11,-5/4) -> (5/4,14/11) Hyperbolic Matrix(57,70,92,113) (-5/4,-6/5) -> (8/13,5/8) Hyperbolic Matrix(83,98,-72,-85) (-6/5,-7/6) -> (-7/6,-8/7) Parabolic Matrix(197,224,124,141) (-8/7,-1/1) -> (27/17,8/5) Hyperbolic Matrix(83,70,32,27) (-1/1,-5/6) -> (5/2,13/5) Hyperbolic Matrix(475,392,372,307) (-5/6,-14/17) -> (14/11,23/18) Hyperbolic Matrix(223,182,136,111) (-9/11,-4/5) -> (18/11,5/3) Hyperbolic Matrix(141,112,248,197) (-4/5,-15/19) -> (13/23,4/7) Hyperbolic Matrix(391,308,212,167) (-15/19,-11/14) -> (11/6,13/7) Hyperbolic Matrix(197,154,252,197) (-11/14,-7/9) -> (7/9,11/14) Hyperbolic Matrix(55,42,72,55) (-7/9,-3/4) -> (3/4,7/9) Hyperbolic Matrix(531,392,340,251) (-3/4,-14/19) -> (14/9,25/16) Hyperbolic Matrix(421,308,652,477) (-11/15,-19/26) -> (9/14,11/17) Hyperbolic Matrix(671,490,152,111) (-19/26,-8/11) -> (22/5,9/2) Hyperbolic Matrix(503,364,76,55) (-21/29,-13/18) -> (13/2,7/1) Hyperbolic Matrix(253,182,196,141) (-13/18,-5/7) -> (9/7,13/10) Hyperbolic Matrix(139,98,-200,-141) (-5/7,-7/10) -> (-7/10,-9/13) Parabolic Matrix(449,308,328,225) (-11/16,-2/3) -> (26/19,11/8) Hyperbolic Matrix(475,308,128,83) (-2/3,-11/17) -> (11/3,26/7) Hyperbolic Matrix(477,308,652,421) (-11/17,-9/14) -> (19/26,11/15) Hyperbolic Matrix(197,126,308,197) (-9/14,-7/11) -> (7/11,9/14) Hyperbolic Matrix(111,70,176,111) (-7/11,-5/8) -> (5/8,7/11) Hyperbolic Matrix(113,70,92,57) (-5/8,-8/13) -> (6/5,5/4) Hyperbolic Matrix(251,154,44,27) (-8/13,-11/18) -> (11/2,6/1) Hyperbolic Matrix(643,392,228,139) (-11/18,-14/23) -> (14/5,17/6) Hyperbolic Matrix(167,98,-288,-169) (-3/5,-7/12) -> (-7/12,-11/19) Parabolic Matrix(533,308,244,141) (-11/19,-15/26) -> (13/6,11/5) Hyperbolic Matrix(197,112,248,141) (-4/7,-13/23) -> (15/19,4/5) Hyperbolic Matrix(869,490,548,309) (-13/23,-9/16) -> (19/12,27/17) Hyperbolic Matrix(699,392,148,83) (-9/16,-14/25) -> (14/3,19/4) Hyperbolic Matrix(281,154,104,57) (-5/9,-6/11) -> (8/3,19/7) Hyperbolic Matrix(419,224,260,139) (-7/13,-1/2) -> (29/18,21/13) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(335,-182,208,-113) (1/2,6/11) -> (8/5,29/18) Hyperbolic Matrix(251,-140,52,-29) (5/9,9/16) -> (19/4,5/1) Hyperbolic Matrix(169,-98,288,-167) (4/7,7/12) -> (7/12,10/17) Parabolic Matrix(309,-182,236,-139) (10/17,3/5) -> (17/13,4/3) Hyperbolic Matrix(139,-84,48,-29) (3/5,11/18) -> (17/6,3/1) Hyperbolic Matrix(141,-98,200,-139) (2/3,7/10) -> (7/10,12/17) Parabolic Matrix(533,-378,196,-139) (12/17,5/7) -> (19/7,30/11) Hyperbolic Matrix(251,-182,40,-29) (13/18,8/11) -> (6/1,13/2) Hyperbolic Matrix(419,-308,268,-197) (11/15,3/4) -> (25/16,11/7) Hyperbolic Matrix(307,-252,240,-197) (9/11,5/6) -> (23/18,9/7) Hyperbolic Matrix(85,-98,72,-83) (1/1,7/6) -> (7/6,13/11) Parabolic Matrix(307,-420,144,-197) (15/11,26/19) -> (2/1,15/7) Hyperbolic Matrix(57,-98,32,-55) (5/3,7/4) -> (7/4,9/5) Parabolic Matrix(223,-420,60,-113) (13/7,2/1) -> (26/7,41/11) Hyperbolic Matrix(337,-882,128,-335) (13/5,21/8) -> (21/8,29/11) Parabolic Matrix(29,-98,8,-27) (3/1,7/2) -> (7/2,11/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,14,0,1) -> Matrix(1,0,0,1) Matrix(29,182,-40,-251) -> Matrix(7,6,-6,-5) Matrix(27,154,44,251) -> Matrix(5,4,-4,-3) Matrix(29,154,16,85) -> Matrix(5,4,-4,-3) Matrix(29,140,-52,-251) -> Matrix(3,2,-8,-5) Matrix(55,252,12,55) -> Matrix(5,4,-4,-3) Matrix(111,490,152,671) -> Matrix(3,2,-2,-1) Matrix(169,742,64,281) -> Matrix(3,2,-2,-1) Matrix(27,112,20,83) -> Matrix(3,2,-8,-5) Matrix(27,98,-8,-29) -> Matrix(3,2,-8,-5) Matrix(111,364,68,223) -> Matrix(1,0,4,1) Matrix(253,812,-148,-475) -> Matrix(3,2,-8,-5) Matrix(57,182,88,281) -> Matrix(1,0,2,1) Matrix(29,84,-48,-139) -> Matrix(1,0,0,1) Matrix(111,308,40,111) -> Matrix(1,0,2,1) Matrix(113,308,-164,-447) -> Matrix(1,0,0,1) Matrix(57,154,104,281) -> Matrix(1,0,4,1) Matrix(335,882,-128,-337) -> Matrix(1,0,0,1) Matrix(279,728,64,167) -> Matrix(3,2,-2,-1) Matrix(27,70,32,83) -> Matrix(1,0,0,1) Matrix(29,70,12,29) -> Matrix(1,0,0,1) Matrix(55,126,24,55) -> Matrix(5,4,-4,-3) Matrix(139,308,88,195) -> Matrix(3,2,-2,-1) Matrix(167,364,128,279) -> Matrix(1,0,0,1) Matrix(197,420,-144,-307) -> Matrix(3,2,4,3) Matrix(55,98,-32,-57) -> Matrix(3,2,-8,-5) Matrix(1959,3346,524,895) -> Matrix(21,8,-8,-3) Matrix(1091,1862,508,867) -> Matrix(1,0,2,1) Matrix(337,574,428,729) -> Matrix(11,4,-14,-5) Matrix(83,140,16,27) -> Matrix(7,2,-4,-1) Matrix(111,182,136,223) -> Matrix(1,0,2,1) Matrix(197,322,52,85) -> Matrix(7,2,-4,-1) Matrix(337,546,208,337) -> Matrix(1,0,8,1) Matrix(113,182,-208,-335) -> Matrix(1,0,-6,1) Matrix(167,266,140,223) -> Matrix(1,0,0,1) Matrix(309,490,548,869) -> Matrix(1,0,2,1) Matrix(195,308,88,139) -> Matrix(3,2,-2,-1) Matrix(197,308,-268,-419) -> Matrix(1,0,0,1) Matrix(55,84,36,55) -> Matrix(1,0,2,1) Matrix(29,42,20,29) -> Matrix(1,0,2,1) Matrix(111,154,80,111) -> Matrix(1,0,-6,1) Matrix(449,616,164,225) -> Matrix(1,0,-2,1) Matrix(1373,1876,636,869) -> Matrix(1,-2,0,1) Matrix(2297,3136,616,841) -> Matrix(5,-2,-2,1) Matrix(83,112,20,27) -> Matrix(1,-2,0,1) Matrix(85,112,-148,-195) -> Matrix(1,0,-2,1) Matrix(141,182,196,253) -> Matrix(3,2,-2,-1) Matrix(197,252,-240,-307) -> Matrix(1,0,0,1) Matrix(111,140,88,111) -> Matrix(1,0,0,1) Matrix(57,70,92,113) -> Matrix(1,0,0,1) Matrix(83,98,-72,-85) -> Matrix(1,0,0,1) Matrix(197,224,124,141) -> Matrix(1,0,0,1) Matrix(83,70,32,27) -> Matrix(1,0,0,1) Matrix(475,392,372,307) -> Matrix(1,0,0,1) Matrix(223,182,136,111) -> Matrix(1,0,2,1) Matrix(141,112,248,197) -> Matrix(1,2,0,1) Matrix(391,308,212,167) -> Matrix(1,2,-2,-3) Matrix(197,154,252,197) -> Matrix(5,6,-6,-7) Matrix(55,42,72,55) -> Matrix(3,2,-2,-1) Matrix(531,392,340,251) -> Matrix(3,2,-2,-1) Matrix(421,308,652,477) -> Matrix(1,0,2,1) Matrix(671,490,152,111) -> Matrix(3,2,-2,-1) Matrix(503,364,76,55) -> Matrix(15,14,-14,-13) Matrix(253,182,196,141) -> Matrix(3,2,-2,-1) Matrix(139,98,-200,-141) -> Matrix(3,2,-8,-5) Matrix(449,308,328,225) -> Matrix(1,0,-2,1) Matrix(475,308,128,83) -> Matrix(3,2,-2,-1) Matrix(477,308,652,421) -> Matrix(1,0,2,1) Matrix(197,126,308,197) -> Matrix(1,0,4,1) Matrix(111,70,176,111) -> Matrix(1,0,0,1) Matrix(113,70,92,57) -> Matrix(1,0,0,1) Matrix(251,154,44,27) -> Matrix(5,4,-4,-3) Matrix(643,392,228,139) -> Matrix(3,2,-2,-1) Matrix(167,98,-288,-169) -> Matrix(3,2,-8,-5) Matrix(533,308,244,141) -> Matrix(1,0,2,1) Matrix(197,112,248,141) -> Matrix(5,2,-8,-3) Matrix(869,490,548,309) -> Matrix(1,0,2,1) Matrix(699,392,148,83) -> Matrix(7,2,-4,-1) Matrix(281,154,104,57) -> Matrix(1,0,4,1) Matrix(419,224,260,139) -> Matrix(1,0,-6,1) Matrix(1,0,4,1) -> Matrix(1,0,0,1) Matrix(335,-182,208,-113) -> Matrix(1,0,-6,1) Matrix(251,-140,52,-29) -> Matrix(1,-2,0,1) Matrix(169,-98,288,-167) -> Matrix(1,-2,0,1) Matrix(309,-182,236,-139) -> Matrix(1,2,-2,-3) Matrix(139,-84,48,-29) -> Matrix(1,0,0,1) Matrix(141,-98,200,-139) -> Matrix(1,-2,0,1) Matrix(533,-378,196,-139) -> Matrix(1,2,0,1) Matrix(251,-182,40,-29) -> Matrix(7,6,-6,-5) Matrix(419,-308,268,-197) -> Matrix(1,0,0,1) Matrix(307,-252,240,-197) -> Matrix(1,0,0,1) Matrix(85,-98,72,-83) -> Matrix(1,0,0,1) Matrix(307,-420,144,-197) -> Matrix(7,2,-4,-1) Matrix(57,-98,32,-55) -> Matrix(1,-2,0,1) Matrix(223,-420,60,-113) -> Matrix(1,-2,0,1) Matrix(337,-882,128,-335) -> Matrix(1,0,0,1) Matrix(29,-98,8,-27) -> Matrix(1,-2,0,1) 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: 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): 144 Minimal number of generators: 25 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: 18 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -7/6 0/1 1/1 7/5 3/2 7/4 2/1 7/3 5/2 14/5 3/1 7/2 4/1 14/3 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -3/4 -1/2 -4/1 -1/2 -7/2 -1/2 -3/1 -1/2 1/0 -8/3 1/0 -5/2 -1/1 0/1 -7/3 -1/1 -2/1 -1/2 -7/4 -1/2 -5/3 -1/2 -1/4 -13/8 -1/5 0/1 -21/13 0/1 -8/5 1/0 -11/7 -1/2 1/0 -14/9 -1/2 -3/2 -1/2 0/1 -7/5 0/1 -4/3 1/0 -9/7 -1/2 1/0 -14/11 -1/2 1/0 -5/4 -1/1 0/1 -6/5 -1/2 -7/6 -1/2 1/0 -1/1 -1/2 1/0 0/1 -1/2 1/0 1/1 -1/2 1/0 5/4 -1/1 0/1 4/3 -1/2 7/5 0/1 3/2 0/1 1/0 8/5 -1/2 5/3 1/2 1/0 7/4 1/0 2/1 1/0 7/3 -1/1 5/2 -1/1 0/1 13/5 -1/2 1/0 21/8 -1/2 1/0 8/3 -1/2 11/4 0/1 1/0 14/5 1/0 3/1 -1/2 1/0 7/2 1/0 4/1 1/0 9/2 -2/1 -1/1 14/3 -3/2 1/0 5/1 -3/2 1/0 6/1 -5/4 7/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(13,70,-8,-43) (-5/1,1/0) -> (-5/3,-13/8) Hyperbolic Matrix(13,56,-10,-43) (-5/1,-4/1) -> (-4/3,-9/7) Hyperbolic Matrix(15,56,4,15) (-4/1,-7/2) -> (7/2,4/1) Hyperbolic Matrix(13,42,4,13) (-7/2,-3/1) -> (3/1,7/2) Hyperbolic Matrix(41,112,-26,-71) (-3/1,-8/3) -> (-8/5,-11/7) Hyperbolic Matrix(27,70,-22,-57) (-8/3,-5/2) -> (-5/4,-6/5) Hyperbolic Matrix(29,70,12,29) (-5/2,-7/3) -> (7/3,5/2) Hyperbolic Matrix(13,28,6,13) (-7/3,-2/1) -> (2/1,7/3) Hyperbolic Matrix(15,28,8,15) (-2/1,-7/4) -> (7/4,2/1) Hyperbolic Matrix(41,70,24,41) (-7/4,-5/3) -> (5/3,7/4) Hyperbolic Matrix(69,112,8,13) (-13/8,-21/13) -> (7/1,1/0) Hyperbolic Matrix(113,182,18,29) (-21/13,-8/5) -> (6/1,7/1) Hyperbolic Matrix(125,196,44,69) (-11/7,-14/9) -> (14/5,3/1) Hyperbolic Matrix(127,196,46,71) (-14/9,-3/2) -> (11/4,14/5) Hyperbolic Matrix(29,42,20,29) (-3/2,-7/5) -> (7/5,3/2) Hyperbolic Matrix(41,56,30,41) (-7/5,-4/3) -> (4/3,7/5) Hyperbolic Matrix(153,196,32,41) (-9/7,-14/11) -> (14/3,5/1) Hyperbolic Matrix(155,196,34,43) (-14/11,-5/4) -> (9/2,14/3) Hyperbolic Matrix(153,182,58,69) (-6/5,-7/6) -> (21/8,8/3) Hyperbolic Matrix(99,112,38,43) (-7/6,-1/1) -> (13/5,21/8) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(43,-56,10,-13) (5/4,4/3) -> (4/1,9/2) Hyperbolic Matrix(71,-112,26,-41) (3/2,8/5) -> (8/3,11/4) Hyperbolic Matrix(43,-70,8,-13) (8/5,5/3) -> (5/1,6/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(13,70,-8,-43) -> Matrix(1,1,-6,-5) Matrix(13,56,-10,-43) -> Matrix(3,2,-2,-1) Matrix(15,56,4,15) -> Matrix(5,3,-2,-1) Matrix(13,42,4,13) -> Matrix(1,1,-2,-1) Matrix(41,112,-26,-71) -> Matrix(1,0,0,1) Matrix(27,70,-22,-57) -> Matrix(1,1,-2,-1) Matrix(29,70,12,29) -> Matrix(1,0,0,1) Matrix(13,28,6,13) -> Matrix(3,2,-2,-1) Matrix(15,28,8,15) -> Matrix(1,1,-2,-1) Matrix(41,70,24,41) -> Matrix(3,1,2,1) Matrix(69,112,8,13) -> Matrix(5,1,-6,-1) Matrix(113,182,18,29) -> Matrix(5,-1,-4,1) Matrix(125,196,44,69) -> Matrix(1,1,-2,-1) Matrix(127,196,46,71) -> Matrix(1,0,2,1) Matrix(29,42,20,29) -> Matrix(1,0,2,1) Matrix(41,56,30,41) -> Matrix(1,0,-2,1) Matrix(153,196,32,41) -> Matrix(1,-1,0,1) Matrix(155,196,34,43) -> Matrix(3,2,-2,-1) Matrix(153,182,58,69) -> Matrix(1,0,0,1) Matrix(99,112,38,43) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(1,1,-2,-1) Matrix(57,-70,22,-27) -> Matrix(1,1,-2,-1) Matrix(43,-56,10,-13) -> Matrix(3,2,-2,-1) Matrix(71,-112,26,-41) -> Matrix(1,0,0,1) Matrix(43,-70,8,-13) -> Matrix(3,-1,-2,1) 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): 8 ----------------------------------------------------------------------- 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).(-1/2,1/0) 0 2 1/1 (-1/2,1/0) 0 14 5/4 (-1/1,0/1) 0 14 4/3 -1/2 1 14 7/5 0/1 4 2 3/2 (0/1,1/0) 0 14 8/5 -1/2 1 14 5/3 (1/2,1/0) 0 14 7/4 1/0 2 2 2/1 1/0 1 14 7/3 -1/1 2 2 5/2 (-1/1,0/1) 0 14 13/5 (-1/2,1/0) 0 14 21/8 (-1/2,1/0) 0 2 8/3 -1/2 1 14 11/4 (0/1,1/0) 0 14 14/5 1/0 1 2 3/1 (-1/2,1/0) 0 14 7/2 1/0 2 2 4/1 1/0 1 14 9/2 (-2/1,-1/1) 0 14 14/3 (-2/1,-1/1).(-3/2,1/0) 0 2 5/1 (-3/2,1/0) 0 14 6/1 -5/4 1 14 7/1 -1/1 10 2 1/0 (-1/1,0/1) 0 14 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(57,-70,22,-27) (1/1,5/4) -> (5/2,13/5) Hyperbolic Matrix(43,-56,10,-13) (5/4,4/3) -> (4/1,9/2) Hyperbolic Matrix(41,-56,30,-41) (4/3,7/5) -> (4/3,7/5) Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) Reflection Matrix(71,-112,26,-41) (3/2,8/5) -> (8/3,11/4) Hyperbolic Matrix(43,-70,8,-13) (8/5,5/3) -> (5/1,6/1) Hyperbolic Matrix(41,-70,24,-41) (5/3,7/4) -> (5/3,7/4) Reflection Matrix(15,-28,8,-15) (7/4,2/1) -> (7/4,2/1) Reflection Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(209,-546,80,-209) (13/5,21/8) -> (13/5,21/8) Reflection Matrix(127,-336,48,-127) (21/8,8/3) -> (21/8,8/3) Reflection Matrix(111,-308,40,-111) (11/4,14/5) -> (11/4,14/5) Reflection Matrix(29,-84,10,-29) (14/5,3/1) -> (14/5,3/1) Reflection Matrix(13,-42,4,-13) (3/1,7/2) -> (3/1,7/2) Reflection Matrix(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) Reflection Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) Reflection Matrix(29,-140,6,-29) (14/3,5/1) -> (14/3,5/1) Reflection Matrix(13,-84,2,-13) (6/1,7/1) -> (6/1,7/1) Reflection Matrix(-1,14,0,1) (7/1,1/0) -> (7/1,1/0) 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(1,0,2,-1) -> Matrix(1,1,0,-1) (0/1,1/1) -> (-1/2,1/0) Matrix(57,-70,22,-27) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(43,-56,10,-13) -> Matrix(3,2,-2,-1) -1/1 Matrix(41,-56,30,-41) -> Matrix(-1,0,4,1) (4/3,7/5) -> (-1/2,0/1) Matrix(29,-42,20,-29) -> Matrix(1,0,0,-1) (7/5,3/2) -> (0/1,1/0) Matrix(71,-112,26,-41) -> Matrix(1,0,0,1) Matrix(43,-70,8,-13) -> Matrix(3,-1,-2,1) Matrix(41,-70,24,-41) -> Matrix(-1,1,0,1) (5/3,7/4) -> (1/2,1/0) Matrix(15,-28,8,-15) -> Matrix(1,1,0,-1) (7/4,2/1) -> (-1/2,1/0) Matrix(13,-28,6,-13) -> Matrix(1,2,0,-1) (2/1,7/3) -> (-1/1,1/0) Matrix(29,-70,12,-29) -> Matrix(-1,0,2,1) (7/3,5/2) -> (-1/1,0/1) Matrix(209,-546,80,-209) -> Matrix(1,1,0,-1) (13/5,21/8) -> (-1/2,1/0) Matrix(127,-336,48,-127) -> Matrix(1,1,0,-1) (21/8,8/3) -> (-1/2,1/0) Matrix(111,-308,40,-111) -> Matrix(1,0,0,-1) (11/4,14/5) -> (0/1,1/0) Matrix(29,-84,10,-29) -> Matrix(1,1,0,-1) (14/5,3/1) -> (-1/2,1/0) Matrix(13,-42,4,-13) -> Matrix(1,1,0,-1) (3/1,7/2) -> (-1/2,1/0) Matrix(15,-56,4,-15) -> Matrix(1,3,0,-1) (7/2,4/1) -> (-3/2,1/0) Matrix(55,-252,12,-55) -> Matrix(3,4,-2,-3) (9/2,14/3) -> (-2/1,-1/1) Matrix(29,-140,6,-29) -> Matrix(1,3,0,-1) (14/3,5/1) -> (-3/2,1/0) Matrix(13,-84,2,-13) -> Matrix(9,10,-8,-9) (6/1,7/1) -> (-5/4,-1/1) Matrix(-1,14,0,1) -> Matrix(-1,0,2,1) (7/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.