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 -2/3 -11/2 -1/1 -1/2 -5/1 -2/3 -14/3 -1/2 -9/2 -1/2 0/1 -22/5 -2/3 -13/3 0/1 -4/1 0/1 -7/2 0/1 -10/3 0/1 -13/4 0/1 1/0 -16/5 0/1 -3/1 0/1 -14/5 1/0 -11/4 0/1 1/0 -19/7 0/1 -8/3 0/1 -21/8 1/0 -34/13 -4/1 -13/5 -2/1 -5/2 0/1 1/0 -7/3 1/0 -9/4 -3/1 1/0 -11/5 -2/1 -13/6 -1/1 1/0 -2/1 -2/1 -7/4 -1/1 -12/7 -2/3 -41/24 -2/3 -1/2 -70/41 -1/2 -29/17 0/1 -17/10 -1/1 -1/2 -5/3 0/1 -18/11 0/1 -13/8 0/1 1/0 -21/13 1/0 -8/5 -2/1 -27/17 0/1 -19/12 -1/1 1/0 -11/7 -2/1 -14/9 -1/1 -3/2 -1/1 1/0 -7/5 1/0 -11/8 -2/1 1/0 -26/19 -2/1 -41/30 -1/1 1/0 -56/41 1/0 -15/11 -2/1 -4/3 -2/1 -13/10 -3/2 -1/1 -9/7 -2/1 -14/11 -2/1 -5/4 -2/1 -1/1 -6/5 -2/1 -7/6 -1/1 -8/7 0/1 -1/1 -2/1 -5/6 -3/2 -4/3 -14/17 -4/3 -9/11 -4/3 -4/5 -4/3 -15/19 -4/3 -11/14 -5/4 -1/1 -7/9 -1/1 -3/4 -3/2 -1/1 -14/19 -1/1 -11/15 -2/1 -19/26 -2/1 -3/2 -8/11 -4/3 -21/29 -1/1 -13/18 -1/1 1/0 -5/7 -2/1 -7/10 -3/2 -9/13 -4/3 -11/16 -3/2 -4/3 -2/3 -4/3 -11/17 -4/3 -9/14 -3/2 -4/3 -7/11 -4/3 -5/8 -4/3 -1/1 -8/13 -4/3 -11/18 -3/2 -1/1 -14/23 -3/2 -3/5 -4/3 -7/12 -4/3 -11/19 -4/3 -15/26 -13/10 -9/7 -4/7 -4/3 -13/23 -4/3 -9/16 -9/7 -5/4 -14/25 -5/4 -5/9 -4/3 -6/11 -4/3 -7/13 -5/4 -1/2 -5/4 -1/1 0/1 -1/1 1/2 -1/1 -5/6 6/11 -4/5 5/9 -4/5 9/16 -5/6 -9/11 13/23 -4/5 4/7 -4/5 7/12 -4/5 10/17 -4/5 3/5 -4/5 11/18 -1/1 -3/4 8/13 -4/5 5/8 -1/1 -4/5 7/11 -4/5 9/14 -4/5 -3/4 11/17 -4/5 2/3 -4/5 7/10 -3/4 12/17 -2/3 5/7 -2/3 13/18 -1/1 -1/2 8/11 -4/5 19/26 -3/4 -2/3 11/15 -2/3 3/4 -1/1 -3/4 7/9 -1/1 11/14 -1/1 -5/6 15/19 -4/5 4/5 -4/5 9/11 -4/5 5/6 -4/5 -3/4 1/1 -2/3 7/6 -1/1 13/11 -4/5 6/5 -2/3 5/4 -1/1 -2/3 14/11 -2/3 23/18 -2/3 -1/2 9/7 -2/3 13/10 -1/1 -3/4 17/13 -2/3 4/3 -2/3 15/11 -2/3 26/19 -2/3 11/8 -2/3 -1/2 7/5 -1/2 3/2 -1/1 -1/2 14/9 -1/1 25/16 -1/1 -3/4 11/7 -2/3 19/12 -1/1 -1/2 27/17 0/1 8/5 -2/3 29/18 -3/5 -1/2 21/13 -1/2 13/8 -1/2 0/1 18/11 0/1 5/3 0/1 7/4 -1/1 9/5 -4/5 11/6 -1/1 -3/4 13/7 -4/5 2/1 -2/3 15/7 -2/3 28/13 -1/2 13/6 -1/1 -1/2 11/5 -2/3 9/4 -3/5 -1/2 7/3 -1/2 5/2 -1/2 0/1 13/5 -2/3 21/8 -1/2 29/11 -2/5 8/3 0/1 19/7 0/1 30/11 0/1 11/4 -1/2 0/1 14/5 -1/2 17/6 -1/2 -1/3 3/1 0/1 7/2 0/1 11/3 0/1 26/7 0/1 41/11 2/1 56/15 1/0 15/4 0/1 1/0 4/1 0/1 13/3 0/1 22/5 -2/1 9/2 0/1 1/0 14/3 1/0 19/4 -3/1 1/0 5/1 -2/1 11/2 -1/1 1/0 6/1 -2/1 13/2 -5/4 -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(1,2,-2,-3) Matrix(29,154,16,85) -> Matrix(1,2,-2,-3) Matrix(29,140,-52,-251) -> Matrix(7,6,-6,-5) Matrix(55,252,12,55) -> Matrix(1,0,2,1) Matrix(111,490,152,671) -> Matrix(1,2,-2,-3) Matrix(169,742,64,281) -> Matrix(3,2,-8,-5) Matrix(27,112,20,83) -> Matrix(5,2,-8,-3) Matrix(27,98,-8,-29) -> Matrix(1,0,2,1) Matrix(111,364,68,223) -> Matrix(1,0,-2,1) Matrix(253,812,-148,-475) -> Matrix(1,2,-2,-3) Matrix(57,182,88,281) -> Matrix(3,4,-4,-5) Matrix(29,84,-48,-139) -> Matrix(3,-4,-2,3) Matrix(111,308,40,111) -> Matrix(1,0,-2,1) Matrix(113,308,-164,-447) -> Matrix(3,-4,-2,3) Matrix(57,154,104,281) -> Matrix(5,-4,-6,5) Matrix(335,882,-128,-337) -> Matrix(1,-4,0,1) Matrix(279,728,64,167) -> Matrix(1,2,0,1) Matrix(27,70,32,83) -> Matrix(3,4,-4,-5) Matrix(29,70,12,29) -> Matrix(1,0,-2,1) Matrix(55,126,24,55) -> Matrix(1,6,-2,-11) Matrix(139,308,88,195) -> Matrix(1,4,-2,-7) Matrix(167,364,128,279) -> Matrix(3,4,-4,-5) Matrix(197,420,-144,-307) -> Matrix(1,0,0,1) Matrix(55,98,-32,-57) -> Matrix(3,4,-4,-5) Matrix(1959,3346,524,895) -> Matrix(3,2,-2,-1) Matrix(1091,1862,508,867) -> Matrix(5,2,-8,-3) Matrix(337,574,428,729) -> Matrix(3,4,-4,-5) Matrix(83,140,16,27) -> Matrix(3,2,-2,-1) Matrix(111,182,136,223) -> Matrix(3,4,-4,-5) Matrix(197,322,52,85) -> Matrix(1,0,0,1) Matrix(337,546,208,337) -> Matrix(1,0,-2,1) Matrix(113,182,-208,-335) -> Matrix(5,14,-4,-11) Matrix(167,266,140,223) -> Matrix(3,4,-4,-5) Matrix(309,490,548,869) -> Matrix(5,-4,-6,5) Matrix(195,308,88,139) -> Matrix(1,4,-2,-7) Matrix(197,308,-268,-419) -> Matrix(1,0,0,1) Matrix(55,84,36,55) -> Matrix(1,2,-2,-3) Matrix(29,42,20,29) -> Matrix(1,2,-2,-3) Matrix(111,154,80,111) -> Matrix(1,4,-2,-7) Matrix(449,616,164,225) -> Matrix(1,2,-2,-3) Matrix(1373,1876,636,869) -> Matrix(1,2,-2,-3) Matrix(2297,3136,616,841) -> Matrix(1,4,0,1) Matrix(83,112,20,27) -> Matrix(1,2,0,1) Matrix(85,112,-148,-195) -> Matrix(5,14,-4,-11) Matrix(141,182,196,253) -> Matrix(3,4,-4,-5) Matrix(197,252,-240,-307) -> Matrix(3,2,-2,-1) Matrix(111,140,88,111) -> Matrix(3,4,-4,-5) Matrix(57,70,92,113) -> Matrix(5,6,-6,-7) Matrix(83,98,-72,-85) -> Matrix(1,2,-2,-3) Matrix(197,224,124,141) -> Matrix(1,2,-2,-3) Matrix(83,70,32,27) -> Matrix(3,4,-4,-5) Matrix(475,392,372,307) -> Matrix(7,10,-12,-17) Matrix(223,182,136,111) -> Matrix(3,4,-4,-5) Matrix(141,112,248,197) -> Matrix(31,40,-38,-49) Matrix(391,308,212,167) -> Matrix(7,8,-8,-9) Matrix(197,154,252,197) -> Matrix(9,10,-10,-11) Matrix(55,42,72,55) -> Matrix(5,6,-6,-7) Matrix(531,392,340,251) -> Matrix(5,6,-6,-7) Matrix(421,308,652,477) -> Matrix(5,6,-6,-7) Matrix(671,490,152,111) -> Matrix(1,2,-2,-3) Matrix(503,364,76,55) -> Matrix(5,4,-4,-3) Matrix(253,182,196,141) -> Matrix(3,4,-4,-5) Matrix(139,98,-200,-141) -> Matrix(11,18,-8,-13) Matrix(449,308,328,225) -> Matrix(7,10,-12,-17) Matrix(475,308,128,83) -> Matrix(3,4,2,3) Matrix(477,308,652,421) -> Matrix(5,6,-6,-7) Matrix(197,126,308,197) -> Matrix(17,24,-22,-31) Matrix(111,70,176,111) -> Matrix(7,8,-8,-9) Matrix(113,70,92,57) -> Matrix(5,6,-6,-7) Matrix(251,154,44,27) -> Matrix(1,2,-2,-3) Matrix(643,392,228,139) -> Matrix(1,2,-4,-7) Matrix(167,98,-288,-169) -> Matrix(47,64,-36,-49) Matrix(533,308,244,141) -> Matrix(17,22,-24,-31) Matrix(197,112,248,141) -> Matrix(31,40,-38,-49) Matrix(869,490,548,309) -> Matrix(3,4,-10,-13) Matrix(699,392,148,83) -> Matrix(19,24,-4,-5) Matrix(281,154,104,57) -> Matrix(3,4,-10,-13) Matrix(419,224,260,139) -> Matrix(13,16,-22,-27) Matrix(1,0,4,1) -> Matrix(9,10,-10,-11) Matrix(335,-182,208,-113) -> Matrix(17,14,-28,-23) Matrix(251,-140,52,-29) -> Matrix(7,6,-6,-5) Matrix(169,-98,288,-167) -> Matrix(79,64,-100,-81) Matrix(309,-182,236,-139) -> Matrix(23,18,-32,-25) Matrix(139,-84,48,-29) -> Matrix(5,4,-14,-11) Matrix(141,-98,200,-139) -> Matrix(23,18,-32,-25) Matrix(533,-378,196,-139) -> Matrix(3,2,-2,-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(3,2,-2,-1) Matrix(85,-98,72,-83) -> Matrix(1,2,-2,-3) Matrix(307,-420,144,-197) -> Matrix(1,0,0,1) Matrix(57,-98,32,-55) -> Matrix(3,4,-4,-5) Matrix(223,-420,60,-113) -> Matrix(3,2,4,3) Matrix(337,-882,128,-335) -> Matrix(7,4,-16,-9) Matrix(29,-98,8,-27) -> 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): 20 Degree of the the map X: 20 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: 30 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/1 -21/8 -2/1 -7/4 -1/1 -7/12 -7/13 0/1 1/2 7/12 7/11 7/10 3/4 7/9 1/1 7/6 5/4 7/5 3/2 7/4 2/1 9/4 7/3 5/2 21/8 11/4 3/1 7/2 15/4 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 0/1 -11/4 0/1 1/0 -19/7 0/1 -8/3 0/1 -21/8 1/0 -13/5 -2/1 -5/2 0/1 1/0 -7/3 1/0 -9/4 -3/1 1/0 -2/1 -2/1 -7/4 -1/1 -5/3 0/1 -18/11 0/1 -13/8 0/1 1/0 -21/13 1/0 -8/5 -2/1 -19/12 -1/1 1/0 -11/7 -2/1 -3/2 -1/1 1/0 -7/5 1/0 -11/8 -2/1 1/0 -4/3 -2/1 -5/4 -2/1 -1/1 -1/1 -2/1 -5/6 -3/2 -4/3 -9/11 -4/3 -4/5 -4/3 -7/9 -1/1 -3/4 -3/2 -1/1 -2/3 -4/3 -9/14 -3/2 -4/3 -7/11 -4/3 -5/8 -4/3 -1/1 -3/5 -4/3 -7/12 -4/3 -4/7 -4/3 -5/9 -4/3 -6/11 -4/3 -7/13 -5/4 -1/2 -5/4 -1/1 0/1 -1/1 1/2 -1/1 -5/6 6/11 -4/5 5/9 -4/5 9/16 -5/6 -9/11 4/7 -4/5 7/12 -4/5 3/5 -4/5 5/8 -1/1 -4/5 7/11 -4/5 9/14 -4/5 -3/4 2/3 -4/5 7/10 -3/4 12/17 -2/3 5/7 -2/3 3/4 -1/1 -3/4 7/9 -1/1 11/14 -1/1 -5/6 4/5 -4/5 9/11 -4/5 5/6 -4/5 -3/4 1/1 -2/3 7/6 -1/1 13/11 -4/5 6/5 -2/3 5/4 -1/1 -2/3 4/3 -2/3 11/8 -2/3 -1/2 7/5 -1/2 3/2 -1/1 -1/2 14/9 -1/1 11/7 -2/3 19/12 -1/1 -1/2 27/17 0/1 8/5 -2/3 29/18 -3/5 -1/2 21/13 -1/2 13/8 -1/2 0/1 18/11 0/1 5/3 0/1 7/4 -1/1 2/1 -2/3 11/5 -2/3 9/4 -3/5 -1/2 7/3 -1/2 5/2 -1/2 0/1 13/5 -2/3 21/8 -1/2 8/3 0/1 19/7 0/1 30/11 0/1 11/4 -1/2 0/1 3/1 0/1 7/2 0/1 11/3 0/1 15/4 0/1 1/0 4/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,7,0,1) (-3/1,1/0) -> (4/1,1/0) Parabolic Matrix(27,77,-20,-57) (-3/1,-11/4) -> (-11/8,-4/3) Hyperbolic Matrix(197,539,72,197) (-11/4,-19/7) -> (30/11,11/4) Hyperbolic Matrix(57,154,104,281) (-19/7,-8/3) -> (6/11,5/9) Hyperbolic Matrix(167,441,-64,-169) (-8/3,-21/8) -> (-21/8,-13/5) Parabolic 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(29,63,52,113) (-9/4,-2/1) -> (5/9,9/16) Hyperbolic Matrix(27,49,-16,-29) (-2/1,-7/4) -> (-7/4,-5/3) Parabolic 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(419,665,264,419) (-8/5,-19/12) -> (19/12,27/17) Hyperbolic Matrix(195,308,88,139) (-19/12,-11/7) -> (11/5,9/4) Hyperbolic Matrix(85,133,108,169) (-11/7,-3/2) -> (11/14,4/5) 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(27,35,-44,-57) (-4/3,-5/4) -> (-5/8,-3/5) Hyperbolic Matrix(29,35,24,29) (-5/4,-1/1) -> (6/5,5/4) Hyperbolic Matrix(83,70,32,27) (-1/1,-5/6) -> (5/2,13/5) Hyperbolic Matrix(111,91,-172,-141) (-5/6,-9/11) -> (-2/3,-9/14) Hyperbolic Matrix(223,182,136,111) (-9/11,-4/5) -> (18/11,5/3) Hyperbolic Matrix(169,133,108,85) (-4/5,-7/9) -> (14/9,11/7) Hyperbolic Matrix(55,42,72,55) (-7/9,-3/4) -> (3/4,7/9) Hyperbolic Matrix(29,21,40,29) (-3/4,-2/3) -> (5/7,3/4) 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(83,49,-144,-85) (-3/5,-7/12) -> (-7/12,-4/7) Parabolic Matrix(113,63,52,29) (-4/7,-5/9) -> (2/1,11/5) 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(309,-175,196,-111) (9/16,4/7) -> (11/7,19/12) Hyperbolic Matrix(85,-49,144,-83) (4/7,7/12) -> (7/12,3/5) Parabolic Matrix(57,-35,44,-27) (3/5,5/8) -> (5/4,4/3) Hyperbolic Matrix(141,-91,172,-111) (9/14,2/3) -> (9/11,5/6) 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(223,-175,144,-113) (7/9,11/14) -> (3/2,14/9) Hyperbolic Matrix(85,-98,72,-83) (1/1,7/6) -> (7/6,13/11) Parabolic Matrix(337,-399,212,-251) (13/11,6/5) -> (27/17,8/5) Hyperbolic Matrix(57,-77,20,-27) (4/3,11/8) -> (11/4,3/1) Hyperbolic Matrix(253,-413,68,-111) (13/8,18/11) -> (11/3,15/4) Hyperbolic Matrix(29,-49,16,-27) (5/3,7/4) -> (7/4,2/1) Parabolic Matrix(169,-441,64,-167) (13/5,21/8) -> (21/8,8/3) Parabolic Matrix(29,-98,8,-27) (3/1,7/2) -> (7/2,11/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,7,0,1) -> Matrix(1,0,0,1) Matrix(27,77,-20,-57) -> Matrix(1,-2,0,1) Matrix(197,539,72,197) -> Matrix(1,0,-2,1) Matrix(57,154,104,281) -> Matrix(5,-4,-6,5) Matrix(167,441,-64,-169) -> Matrix(1,-2,0,1) Matrix(27,70,32,83) -> Matrix(3,4,-4,-5) Matrix(29,70,12,29) -> Matrix(1,0,-2,1) Matrix(55,126,24,55) -> Matrix(1,6,-2,-11) Matrix(29,63,52,113) -> Matrix(5,6,-6,-7) Matrix(27,49,-16,-29) -> Matrix(1,2,-2,-3) Matrix(111,182,136,223) -> Matrix(3,4,-4,-5) Matrix(197,322,52,85) -> Matrix(1,0,0,1) Matrix(337,546,208,337) -> Matrix(1,0,-2,1) Matrix(113,182,-208,-335) -> Matrix(5,14,-4,-11) Matrix(419,665,264,419) -> Matrix(1,2,-2,-3) Matrix(195,308,88,139) -> Matrix(1,4,-2,-7) Matrix(85,133,108,169) -> Matrix(5,6,-6,-7) Matrix(29,42,20,29) -> Matrix(1,2,-2,-3) Matrix(111,154,80,111) -> Matrix(1,4,-2,-7) Matrix(27,35,-44,-57) -> Matrix(3,2,-2,-1) Matrix(29,35,24,29) -> Matrix(3,4,-4,-5) Matrix(83,70,32,27) -> Matrix(3,4,-4,-5) Matrix(111,91,-172,-141) -> Matrix(1,0,0,1) Matrix(223,182,136,111) -> Matrix(3,4,-4,-5) Matrix(169,133,108,85) -> Matrix(5,6,-6,-7) Matrix(55,42,72,55) -> Matrix(5,6,-6,-7) Matrix(29,21,40,29) -> Matrix(5,6,-6,-7) Matrix(197,126,308,197) -> Matrix(17,24,-22,-31) Matrix(111,70,176,111) -> Matrix(7,8,-8,-9) Matrix(83,49,-144,-85) -> Matrix(23,32,-18,-25) Matrix(113,63,52,29) -> Matrix(5,6,-6,-7) Matrix(281,154,104,57) -> Matrix(3,4,-10,-13) Matrix(419,224,260,139) -> Matrix(13,16,-22,-27) Matrix(1,0,4,1) -> Matrix(9,10,-10,-11) Matrix(335,-182,208,-113) -> Matrix(17,14,-28,-23) Matrix(309,-175,196,-111) -> Matrix(17,14,-28,-23) Matrix(85,-49,144,-83) -> Matrix(39,32,-50,-41) Matrix(57,-35,44,-27) -> Matrix(3,2,-2,-1) Matrix(141,-91,172,-111) -> Matrix(1,0,0,1) Matrix(141,-98,200,-139) -> Matrix(23,18,-32,-25) Matrix(533,-378,196,-139) -> Matrix(3,2,-2,-1) Matrix(223,-175,144,-113) -> Matrix(5,4,-4,-3) Matrix(85,-98,72,-83) -> Matrix(1,2,-2,-3) Matrix(337,-399,212,-251) -> Matrix(3,2,-2,-1) Matrix(57,-77,20,-27) -> Matrix(3,2,-8,-5) Matrix(253,-413,68,-111) -> Matrix(1,0,2,1) Matrix(29,-49,16,-27) -> Matrix(1,2,-2,-3) Matrix(169,-441,64,-167) -> Matrix(3,2,-8,-5) Matrix(29,-98,8,-27) -> 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 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 -1/1 5 2 1/2 (-1/1,-5/6) 0 14 6/11 -4/5 1 14 5/9 -4/5 1 14 4/7 -4/5 1 14 7/12 -4/5 4 2 3/5 -4/5 1 14 5/8 (-1/1,-4/5) 0 14 7/11 -4/5 1 2 9/14 (-4/5,-3/4) 0 14 2/3 -4/5 1 14 7/10 -3/4 2 2 3/4 (-1/1,-3/4) 0 14 7/9 -1/1 2 2 4/5 -4/5 1 14 9/11 -4/5 1 14 5/6 (-4/5,-3/4) 0 14 1/1 -2/3 1 14 7/6 -1/1 2 2 5/4 (-1/1,-2/3) 0 14 4/3 -2/3 1 14 11/8 (-2/3,-1/2) 0 14 7/5 -1/2 1 2 3/2 (-1/1,-1/2) 0 14 14/9 -1/1 2 2 11/7 -2/3 1 14 19/12 (-1/1,-1/2) 0 14 8/5 -2/3 1 14 29/18 (-3/5,-1/2) 0 14 21/13 -1/2 3 2 13/8 (-1/2,0/1) 0 14 18/11 0/1 1 14 5/3 0/1 1 14 7/4 -1/1 4 2 2/1 -2/3 1 14 11/5 -2/3 1 14 9/4 (-3/5,-1/2) 0 14 7/3 -1/2 3 2 5/2 (-1/2,0/1) 0 14 13/5 -2/3 1 14 21/8 -1/2 4 2 8/3 0/1 1 14 19/7 0/1 1 14 11/4 (-1/2,0/1) 0 14 3/1 0/1 1 14 13/4 (-1/2,0/1) 0 14 7/2 0/1 2 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(335,-182,208,-113) (1/2,6/11) -> (8/5,29/18) Hyperbolic Matrix(281,-154,104,-57) (6/11,5/9) -> (8/3,19/7) Glide Reflection Matrix(113,-63,52,-29) (5/9,4/7) -> (2/1,11/5) Glide Reflection Matrix(85,-49,144,-83) (4/7,7/12) -> (7/12,3/5) Parabolic Matrix(57,-35,44,-27) (3/5,5/8) -> (5/4,4/3) Hyperbolic Matrix(111,-70,176,-111) (5/8,7/11) -> (5/8,7/11) Reflection Matrix(197,-126,308,-197) (7/11,9/14) -> (7/11,9/14) Reflection Matrix(141,-91,172,-111) (9/14,2/3) -> (9/11,5/6) Hyperbolic Matrix(337,-231,124,-85) (2/3,9/13) -> (19/7,30/11) Glide Reflection Matrix(111,-77,160,-111) (11/16,7/10) -> (11/16,7/10) Reflection Matrix(29,-21,40,-29) (7/10,3/4) -> (7/10,3/4) Reflection Matrix(55,-42,72,-55) (3/4,7/9) -> (3/4,7/9) Reflection Matrix(169,-133,108,-85) (7/9,4/5) -> (14/9,11/7) Glide Reflection Matrix(223,-182,136,-111) (4/5,9/11) -> (18/11,5/3) Glide Reflection Matrix(83,-70,32,-27) (5/6,1/1) -> (5/2,13/5) Glide Reflection Matrix(83,-91,52,-57) (1/1,8/7) -> (27/17,8/5) Hyperbolic Matrix(55,-63,48,-55) (9/8,7/6) -> (9/8,7/6) Reflection Matrix(29,-35,24,-29) (7/6,5/4) -> (7/6,5/4) Reflection Matrix(57,-77,20,-27) (4/3,11/8) -> (11/4,3/1) Hyperbolic Matrix(111,-154,80,-111) (11/8,7/5) -> (11/8,7/5) Reflection Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) Reflection Matrix(55,-84,36,-55) (3/2,14/9) -> (3/2,14/9) Reflection Matrix(195,-308,88,-139) (11/7,19/12) -> (11/5,9/4) Glide Reflection Matrix(419,-665,264,-419) (19/12,35/22) -> (19/12,35/22) Reflection Matrix(755,-1218,468,-755) (29/18,21/13) -> (29/18,21/13) Reflection Matrix(337,-546,208,-337) (21/13,13/8) -> (21/13,13/8) Reflection Matrix(167,-273,52,-85) (13/8,18/11) -> (3/1,13/4) Hyperbolic Matrix(29,-49,16,-27) (5/3,7/4) -> (7/4,2/1) Parabolic Matrix(55,-126,24,-55) (9/4,7/3) -> (9/4,7/3) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(169,-441,64,-167) (13/5,21/8) -> (21/8,8/3) Parabolic Matrix(197,-539,72,-197) (49/18,11/4) -> (49/18,11/4) Reflection Matrix(27,-91,8,-27) (13/4,7/2) -> (13/4,7/2) Reflection Matrix(-1,7,0,1) (7/2,1/0) -> (7/2,1/0) 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(1,0,4,-1) -> Matrix(11,10,-12,-11) (0/1,1/2) -> (-1/1,-5/6) Matrix(335,-182,208,-113) -> Matrix(17,14,-28,-23) Matrix(281,-154,104,-57) -> Matrix(5,4,-16,-13) Matrix(113,-63,52,-29) -> Matrix(7,6,-8,-7) *** -> (-1/1,-3/4) Matrix(85,-49,144,-83) -> Matrix(39,32,-50,-41) -4/5 Matrix(57,-35,44,-27) -> Matrix(3,2,-2,-1) -1/1 Matrix(111,-70,176,-111) -> Matrix(9,8,-10,-9) (5/8,7/11) -> (-1/1,-4/5) Matrix(197,-126,308,-197) -> Matrix(31,24,-40,-31) (7/11,9/14) -> (-4/5,-3/4) Matrix(141,-91,172,-111) -> Matrix(1,0,0,1) Matrix(337,-231,124,-85) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(111,-77,160,-111) -> Matrix(31,24,-40,-31) (11/16,7/10) -> (-4/5,-3/4) Matrix(29,-21,40,-29) -> Matrix(7,6,-8,-7) (7/10,3/4) -> (-1/1,-3/4) Matrix(55,-42,72,-55) -> Matrix(7,6,-8,-7) (3/4,7/9) -> (-1/1,-3/4) Matrix(169,-133,108,-85) -> Matrix(7,6,-8,-7) *** -> (-1/1,-3/4) Matrix(223,-182,136,-111) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(83,-70,32,-27) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(83,-91,52,-57) -> Matrix(1,0,0,1) Matrix(55,-63,48,-55) -> Matrix(3,2,-4,-3) (9/8,7/6) -> (-1/1,-1/2) Matrix(29,-35,24,-29) -> Matrix(5,4,-6,-5) (7/6,5/4) -> (-1/1,-2/3) Matrix(57,-77,20,-27) -> Matrix(3,2,-8,-5) -1/2 Matrix(111,-154,80,-111) -> Matrix(7,4,-12,-7) (11/8,7/5) -> (-2/3,-1/2) Matrix(29,-42,20,-29) -> Matrix(3,2,-4,-3) (7/5,3/2) -> (-1/1,-1/2) Matrix(55,-84,36,-55) -> Matrix(3,2,-4,-3) (3/2,14/9) -> (-1/1,-1/2) Matrix(195,-308,88,-139) -> Matrix(7,4,-12,-7) *** -> (-2/3,-1/2) Matrix(419,-665,264,-419) -> Matrix(3,2,-4,-3) (19/12,35/22) -> (-1/1,-1/2) Matrix(755,-1218,468,-755) -> Matrix(11,6,-20,-11) (29/18,21/13) -> (-3/5,-1/2) Matrix(337,-546,208,-337) -> Matrix(-1,0,4,1) (21/13,13/8) -> (-1/2,0/1) Matrix(167,-273,52,-85) -> Matrix(1,0,0,1) Matrix(29,-49,16,-27) -> Matrix(1,2,-2,-3) -1/1 Matrix(55,-126,24,-55) -> Matrix(11,6,-20,-11) (9/4,7/3) -> (-3/5,-1/2) Matrix(29,-70,12,-29) -> Matrix(-1,0,4,1) (7/3,5/2) -> (-1/2,0/1) Matrix(169,-441,64,-167) -> Matrix(3,2,-8,-5) -1/2 Matrix(197,-539,72,-197) -> Matrix(-1,0,4,1) (49/18,11/4) -> (-1/2,0/1) Matrix(27,-91,8,-27) -> Matrix(-1,0,4,1) (13/4,7/2) -> (-1/2,0/1) Matrix(-1,7,0,1) -> Matrix(-1,0,2,1) (7/2,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.