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 -1/1 -11/2 -5/6 -5/1 -2/3 -14/3 -1/1 -9/2 -3/4 -22/5 -1/1 -13/3 -2/3 -4/1 -1/1 -7/2 -1/2 -10/3 -1/3 -13/4 -1/3 0/1 -16/5 -1/1 -3/1 0/1 -14/5 -1/1 -11/4 -1/1 -2/3 -19/7 0/1 -8/3 -1/1 -21/8 -1/2 -34/13 -1/3 -13/5 0/1 -5/2 1/0 -7/3 -1/1 -9/4 -1/1 -4/5 -11/5 -4/5 -13/6 -7/10 -2/1 -1/1 -7/4 -1/2 -12/7 -3/7 -41/24 -7/17 -2/5 -70/41 -2/5 -29/17 -2/5 -17/10 -3/8 -5/3 0/1 -18/11 -1/1 -13/8 -1/1 0/1 -21/13 0/1 -8/5 -1/1 -27/17 0/1 -19/12 -1/1 0/1 -11/7 0/1 -14/9 -1/1 -3/2 -1/2 -7/5 0/1 -11/8 -1/1 0/1 -26/19 -1/3 -41/30 -1/6 -56/41 0/1 -15/11 0/1 -4/3 -1/1 -13/10 -3/8 -9/7 0/1 -14/11 0/1 -5/4 -1/1 0/1 -6/5 -1/1 -7/6 -1/2 -8/7 -1/3 -1/1 0/1 -5/6 -1/2 -14/17 0/1 -9/11 0/1 -4/5 -1/1 -15/19 0/1 -11/14 -1/4 -7/9 0/1 -3/4 0/1 1/1 -14/19 1/1 -11/15 0/1 -19/26 1/2 -8/11 1/1 -21/29 1/1 -13/18 5/4 -5/7 2/1 -7/10 1/0 -9/13 -4/1 -11/16 -3/1 -2/1 -2/3 -1/1 -11/17 0/1 -9/14 -1/2 -7/11 0/1 -5/8 0/1 1/1 -8/13 1/1 -11/18 1/2 -14/23 1/1 -3/5 2/1 -7/12 1/0 -11/19 -8/1 -15/26 -11/2 -4/7 -3/1 -13/23 -2/1 -9/16 -2/1 -1/1 -14/25 -1/1 -5/9 0/1 -6/11 -1/1 -7/13 0/1 -1/2 1/0 0/1 -1/1 1/2 -1/2 6/11 -1/1 5/9 0/1 9/16 -1/1 -2/3 13/23 -2/3 4/7 -3/5 7/12 -1/2 10/17 -7/15 3/5 -2/5 11/18 -1/4 8/13 -1/3 5/8 -1/3 0/1 7/11 0/1 9/14 1/0 11/17 0/1 2/3 -1/1 7/10 -1/2 12/17 -5/11 5/7 -2/5 13/18 -5/14 8/11 -1/3 19/26 -1/4 11/15 0/1 3/4 -1/3 0/1 7/9 0/1 11/14 1/2 15/19 0/1 4/5 -1/1 9/11 0/1 5/6 1/0 1/1 0/1 7/6 1/0 13/11 -2/1 6/5 -1/1 5/4 -1/1 0/1 14/11 0/1 23/18 1/4 9/7 0/1 13/10 3/2 17/13 4/1 4/3 -1/1 15/11 0/1 26/19 1/1 11/8 -1/1 0/1 7/5 0/1 3/2 1/0 14/9 -1/1 25/16 -1/1 0/1 11/7 0/1 19/12 -1/1 0/1 27/17 0/1 8/5 -1/1 29/18 -1/2 21/13 0/1 13/8 -1/1 0/1 18/11 -1/1 5/3 0/1 7/4 1/0 9/5 -4/1 11/6 -5/2 13/7 -2/1 2/1 -1/1 15/7 -2/1 28/13 -2/1 13/6 -7/4 11/5 -4/3 9/4 -4/3 -1/1 7/3 -1/1 5/2 -1/2 13/5 0/1 21/8 1/0 29/11 -2/1 8/3 -1/1 19/7 0/1 30/11 -3/1 11/4 -2/1 -1/1 14/5 -1/1 17/6 -3/4 3/1 0/1 7/2 1/0 11/3 -2/1 26/7 -3/1 41/11 -2/1 56/15 -2/1 15/4 -2/1 -1/1 4/1 -1/1 13/3 -2/1 22/5 -1/1 9/2 -3/2 14/3 -1/1 19/4 -1/1 0/1 5/1 -2/1 11/2 -5/4 6/1 -1/1 13/2 -11/10 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(7,6,8,7) Matrix(27,154,44,251) -> Matrix(5,4,-14,-11) Matrix(29,154,16,85) -> Matrix(13,10,-4,-3) Matrix(29,140,-52,-251) -> Matrix(3,2,-2,-1) Matrix(55,252,12,55) -> Matrix(7,6,-6,-5) Matrix(111,490,152,671) -> Matrix(3,2,-8,-5) Matrix(169,742,64,281) -> Matrix(5,4,-4,-3) Matrix(27,112,20,83) -> Matrix(3,2,-2,-1) Matrix(27,98,-8,-29) -> Matrix(3,2,-8,-5) Matrix(111,364,68,223) -> Matrix(1,0,2,1) Matrix(253,812,-148,-475) -> Matrix(1,-2,-2,5) Matrix(57,182,88,281) -> Matrix(1,0,0,1) Matrix(29,84,-48,-139) -> Matrix(1,2,0,1) Matrix(111,308,40,111) -> Matrix(5,4,-4,-3) Matrix(113,308,-164,-447) -> Matrix(7,4,-2,-1) Matrix(57,154,104,281) -> Matrix(1,0,0,1) Matrix(335,882,-128,-337) -> Matrix(3,2,-8,-5) Matrix(279,728,64,167) -> Matrix(7,2,-4,-1) Matrix(27,70,32,83) -> Matrix(1,0,0,1) Matrix(29,70,12,29) -> Matrix(1,2,-2,-3) Matrix(55,126,24,55) -> Matrix(9,8,-8,-7) Matrix(139,308,88,195) -> Matrix(5,4,-4,-3) Matrix(167,364,128,279) -> Matrix(11,8,4,3) Matrix(197,420,-144,-307) -> Matrix(3,2,-8,-5) Matrix(55,98,-32,-57) -> Matrix(7,4,-16,-9) Matrix(1959,3346,524,895) -> Matrix(39,16,-22,-9) Matrix(1091,1862,508,867) -> Matrix(41,16,-18,-7) Matrix(337,574,428,729) -> Matrix(5,2,2,1) Matrix(83,140,16,27) -> Matrix(7,2,-4,-1) Matrix(111,182,136,223) -> Matrix(1,0,0,1) Matrix(197,322,52,85) -> Matrix(3,2,-2,-1) Matrix(337,546,208,337) -> Matrix(1,0,0,1) Matrix(113,182,-208,-335) -> Matrix(1,0,0,1) Matrix(167,266,140,223) -> Matrix(3,2,-2,-1) Matrix(309,490,548,869) -> Matrix(1,2,-2,-3) Matrix(195,308,88,139) -> Matrix(5,4,-4,-3) Matrix(197,308,-268,-419) -> Matrix(1,0,2,1) Matrix(55,84,36,55) -> Matrix(3,2,-2,-1) Matrix(29,42,20,29) -> Matrix(1,0,2,1) Matrix(111,154,80,111) -> Matrix(1,0,0,1) Matrix(449,616,164,225) -> Matrix(3,2,-2,-1) Matrix(1373,1876,636,869) -> Matrix(19,2,-10,-1) Matrix(2297,3136,616,841) -> Matrix(9,-2,-4,1) Matrix(83,112,20,27) -> Matrix(3,2,-2,-1) Matrix(85,112,-148,-195) -> Matrix(7,4,-2,-1) Matrix(141,182,196,253) -> Matrix(7,2,-18,-5) Matrix(197,252,-240,-307) -> Matrix(1,0,4,1) Matrix(111,140,88,111) -> Matrix(1,0,0,1) Matrix(57,70,92,113) -> Matrix(1,0,-2,1) Matrix(83,98,-72,-85) -> Matrix(3,2,-8,-5) Matrix(197,224,124,141) -> Matrix(1,0,2,1) Matrix(83,70,32,27) -> Matrix(1,0,0,1) Matrix(475,392,372,307) -> Matrix(1,0,6,1) Matrix(223,182,136,111) -> Matrix(1,0,0,1) Matrix(141,112,248,197) -> Matrix(5,2,-8,-3) Matrix(391,308,212,167) -> Matrix(3,2,-2,-1) Matrix(197,154,252,197) -> Matrix(1,0,6,1) Matrix(55,42,72,55) -> Matrix(1,0,-4,1) Matrix(531,392,340,251) -> Matrix(1,0,-2,1) Matrix(421,308,652,477) -> Matrix(1,0,-2,1) Matrix(671,490,152,111) -> Matrix(1,-2,0,1) Matrix(503,364,76,55) -> Matrix(15,-16,-14,15) Matrix(253,182,196,141) -> Matrix(1,-2,2,-3) Matrix(139,98,-200,-141) -> Matrix(1,-6,0,1) Matrix(449,308,328,225) -> Matrix(1,2,0,1) Matrix(475,308,128,83) -> Matrix(1,-2,0,1) Matrix(477,308,652,421) -> Matrix(1,0,-2,1) Matrix(197,126,308,197) -> Matrix(1,0,2,1) Matrix(111,70,176,111) -> Matrix(1,0,-4,1) Matrix(113,70,92,57) -> Matrix(1,0,-2,1) Matrix(251,154,44,27) -> Matrix(3,-4,-2,3) Matrix(643,392,228,139) -> Matrix(5,-4,-6,5) Matrix(167,98,-288,-169) -> Matrix(1,-10,0,1) Matrix(533,308,244,141) -> Matrix(3,20,-2,-13) Matrix(197,112,248,141) -> Matrix(1,2,0,1) Matrix(869,490,548,309) -> Matrix(1,2,-2,-3) Matrix(699,392,148,83) -> Matrix(1,2,-2,-3) Matrix(281,154,104,57) -> Matrix(1,0,0,1) Matrix(419,224,260,139) -> Matrix(1,0,-2,1) Matrix(1,0,4,1) -> Matrix(1,2,-2,-3) Matrix(335,-182,208,-113) -> Matrix(1,0,0,1) Matrix(251,-140,52,-29) -> Matrix(3,2,-2,-1) Matrix(169,-98,288,-167) -> Matrix(19,10,-40,-21) Matrix(309,-182,236,-139) -> Matrix(13,6,2,1) Matrix(139,-84,48,-29) -> Matrix(5,2,-8,-3) Matrix(141,-98,200,-139) -> Matrix(11,6,-24,-13) Matrix(533,-378,196,-139) -> Matrix(5,2,2,1) Matrix(251,-182,40,-29) -> Matrix(19,6,-16,-5) Matrix(419,-308,268,-197) -> Matrix(1,0,2,1) Matrix(307,-252,240,-197) -> Matrix(1,0,4,1) Matrix(85,-98,72,-83) -> Matrix(1,-2,0,1) Matrix(307,-420,144,-197) -> Matrix(1,-2,0,1) Matrix(57,-98,32,-55) -> Matrix(1,-4,0,1) Matrix(223,-420,60,-113) -> Matrix(5,8,-2,-3) Matrix(337,-882,128,-335) -> Matrix(1,-2,0,1) 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): 32 Degree of the the map X: 32 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 -7/2 -3/1 -2/1 -7/6 -1/1 -7/10 0/1 1/2 7/12 7/10 3/4 7/9 1/1 7/6 5/4 7/5 3/2 7/4 2/1 7/3 5/2 21/8 3/1 7/2 9/2 14/3 11/2 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 -1/1 -11/2 -5/6 -5/1 -2/3 -9/2 -3/4 -13/3 -2/3 -4/1 -1/1 -7/2 -1/2 -3/1 0/1 -8/3 -1/1 -5/2 1/0 -7/3 -1/1 -9/4 -1/1 -4/5 -2/1 -1/1 -3/2 -1/2 -7/5 0/1 -11/8 -1/1 0/1 -4/3 -1/1 -13/10 -3/8 -9/7 0/1 -5/4 -1/1 0/1 -6/5 -1/1 -7/6 -1/2 -1/1 0/1 -4/5 -1/1 -11/14 -1/4 -7/9 0/1 -3/4 0/1 1/1 -8/11 1/1 -21/29 1/1 -13/18 5/4 -5/7 2/1 -7/10 1/0 -2/3 -1/1 -7/11 0/1 -5/8 0/1 1/1 -8/13 1/1 -11/18 1/2 -3/5 2/1 -1/2 1/0 0/1 -1/1 1/2 -1/2 4/7 -3/5 7/12 -1/2 10/17 -7/15 3/5 -2/5 11/18 -1/4 8/13 -1/3 5/8 -1/3 0/1 7/11 0/1 9/14 1/0 2/3 -1/1 7/10 -1/2 5/7 -2/5 13/18 -5/14 8/11 -1/3 11/15 0/1 3/4 -1/3 0/1 7/9 0/1 11/14 1/2 4/5 -1/1 1/1 0/1 7/6 1/0 6/5 -1/1 5/4 -1/1 0/1 9/7 0/1 13/10 3/2 17/13 4/1 4/3 -1/1 15/11 0/1 11/8 -1/1 0/1 7/5 0/1 3/2 1/0 5/3 0/1 7/4 1/0 9/5 -4/1 11/6 -5/2 2/1 -1/1 9/4 -4/3 -1/1 7/3 -1/1 5/2 -1/2 13/5 0/1 21/8 1/0 29/11 -2/1 8/3 -1/1 3/1 0/1 7/2 1/0 4/1 -1/1 13/3 -2/1 22/5 -1/1 9/2 -3/2 14/3 -1/1 5/1 -2/1 11/2 -5/4 6/1 -1/1 13/2 -11/10 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(13,63,20,97) (-5/1,-9/2) -> (9/14,2/3) Hyperbolic Matrix(71,315,16,71) (-9/2,-13/3) -> (22/5,9/2) Hyperbolic Matrix(27,112,20,83) (-13/3,-4/1) -> (4/3,15/11) Hyperbolic Matrix(13,49,-4,-15) (-4/1,-7/2) -> (-7/2,-3/1) Parabolic Matrix(13,35,-16,-43) (-3/1,-8/3) -> (-1/1,-4/5) Hyperbolic Matrix(41,105,16,41) (-8/3,-5/2) -> (5/2,13/5) 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(41,91,-32,-71) (-9/4,-2/1) -> (-9/7,-5/4) Hyperbolic Matrix(13,21,8,13) (-2/1,-3/2) -> (3/2,5/3) 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(97,133,132,181) (-11/8,-4/3) -> (11/15,3/4) Hyperbolic Matrix(209,273,160,209) (-4/3,-13/10) -> (13/10,17/13) Hyperbolic Matrix(141,182,196,253) (-13/10,-9/7) -> (5/7,13/18) Hyperbolic Matrix(57,70,92,113) (-5/4,-6/5) -> (8/13,5/8) Hyperbolic Matrix(41,49,-36,-43) (-6/5,-7/6) -> (-7/6,-1/1) Parabolic Matrix(97,77,-160,-127) (-4/5,-11/14) -> (-11/18,-3/5) 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(181,133,132,97) (-3/4,-8/11) -> (15/11,11/8) 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(69,49,-100,-71) (-5/7,-7/10) -> (-7/10,-2/3) Parabolic Matrix(97,63,20,13) (-2/3,-7/11) -> (14/3,5/1) 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(13,7,24,13) (-3/5,-1/2) -> (1/2,4/7) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic 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(127,-77,160,-97) (3/5,11/18) -> (11/14,4/5) Hyperbolic Matrix(295,-189,64,-41) (7/11,9/14) -> (9/2,14/3) Hyperbolic Matrix(71,-49,100,-69) (2/3,7/10) -> (7/10,5/7) Parabolic Matrix(251,-182,40,-29) (13/18,8/11) -> (6/1,13/2) Hyperbolic Matrix(239,-175,56,-41) (8/11,11/15) -> (4/1,13/3) Hyperbolic Matrix(43,-35,16,-13) (4/5,1/1) -> (8/3,3/1) Hyperbolic Matrix(43,-49,36,-41) (1/1,7/6) -> (7/6,6/5) Parabolic Matrix(71,-91,32,-41) (5/4,9/7) -> (2/1,9/4) Hyperbolic Matrix(57,-98,32,-55) (5/3,7/4) -> (7/4,9/5) Parabolic Matrix(41,-77,8,-15) (11/6,2/1) -> (5/1,11/2) Hyperbolic Matrix(337,-882,128,-335) (13/5,21/8) -> (21/8,29/11) Parabolic Matrix(209,-553,48,-127) (29/11,8/3) -> (13/3,22/5) Hyperbolic Matrix(15,-49,4,-13) (3/1,7/2) -> (7/2,4/1) 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(7,6,8,7) Matrix(27,154,44,251) -> Matrix(5,4,-14,-11) Matrix(29,154,16,85) -> Matrix(13,10,-4,-3) Matrix(13,63,20,97) -> Matrix(1,1,-4,-3) Matrix(71,315,16,71) -> Matrix(5,3,-2,-1) Matrix(27,112,20,83) -> Matrix(3,2,-2,-1) Matrix(13,49,-4,-15) -> Matrix(1,1,-4,-3) Matrix(13,35,-16,-43) -> Matrix(1,1,-2,-1) Matrix(41,105,16,41) -> Matrix(1,1,-2,-1) Matrix(29,70,12,29) -> Matrix(1,2,-2,-3) Matrix(55,126,24,55) -> Matrix(9,8,-8,-7) Matrix(41,91,-32,-71) -> Matrix(1,1,-6,-5) Matrix(13,21,8,13) -> Matrix(1,1,-2,-1) Matrix(29,42,20,29) -> Matrix(1,0,2,1) Matrix(111,154,80,111) -> Matrix(1,0,0,1) Matrix(97,133,132,181) -> Matrix(1,1,-4,-3) Matrix(209,273,160,209) -> Matrix(7,3,2,1) Matrix(141,182,196,253) -> Matrix(7,2,-18,-5) Matrix(57,70,92,113) -> Matrix(1,0,-2,1) Matrix(41,49,-36,-43) -> Matrix(1,1,-4,-3) Matrix(97,77,-160,-127) -> Matrix(3,1,2,1) Matrix(197,154,252,197) -> Matrix(1,0,6,1) Matrix(55,42,72,55) -> Matrix(1,0,-4,1) Matrix(181,133,132,97) -> Matrix(1,-1,0,1) Matrix(503,364,76,55) -> Matrix(15,-16,-14,15) Matrix(253,182,196,141) -> Matrix(1,-2,2,-3) Matrix(69,49,-100,-71) -> Matrix(1,-3,0,1) Matrix(97,63,20,13) -> Matrix(1,-1,0,1) Matrix(111,70,176,111) -> Matrix(1,0,-4,1) Matrix(113,70,92,57) -> Matrix(1,0,-2,1) Matrix(251,154,44,27) -> Matrix(3,-4,-2,3) Matrix(13,7,24,13) -> Matrix(1,1,-2,-1) Matrix(1,0,4,1) -> Matrix(1,2,-2,-3) Matrix(169,-98,288,-167) -> Matrix(19,10,-40,-21) Matrix(309,-182,236,-139) -> Matrix(13,6,2,1) Matrix(127,-77,160,-97) -> Matrix(3,1,2,1) Matrix(295,-189,64,-41) -> Matrix(3,-1,-2,1) Matrix(71,-49,100,-69) -> Matrix(5,3,-12,-7) Matrix(251,-182,40,-29) -> Matrix(19,6,-16,-5) Matrix(239,-175,56,-41) -> Matrix(1,1,-2,-1) Matrix(43,-35,16,-13) -> Matrix(1,1,-2,-1) Matrix(43,-49,36,-41) -> Matrix(1,-1,0,1) Matrix(71,-91,32,-41) -> Matrix(3,-1,-2,1) Matrix(57,-98,32,-55) -> Matrix(1,-4,0,1) Matrix(41,-77,8,-15) -> Matrix(3,5,-2,-3) Matrix(337,-882,128,-335) -> Matrix(1,-2,0,1) Matrix(209,-553,48,-127) -> Matrix(3,5,-2,-3) Matrix(15,-49,4,-13) -> Matrix(1,-1,0,1) 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): 32 ----------------------------------------------------------------------- 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 1 2 1/2 -1/2 2 14 7/12 -1/2 10 2 3/5 -2/5 1 14 11/18 -1/4 2 14 8/13 -1/3 1 14 5/8 (-1/3,0/1) 0 14 7/11 0/1 3 2 2/3 -1/1 1 14 7/10 -1/2 6 2 5/7 -2/5 1 14 13/18 -5/14 2 14 8/11 -1/3 1 14 3/4 (-1/3,0/1) 0 14 7/9 0/1 5 2 11/14 1/2 2 14 4/5 -1/1 1 14 1/1 0/1 1 14 7/6 1/0 2 2 6/5 -1/1 1 14 5/4 (-1/1,0/1) 0 14 9/7 0/1 1 14 13/10 3/2 2 14 4/3 -1/1 1 14 15/11 0/1 1 14 11/8 (-1/1,0/1) 0 14 7/5 0/1 1 2 3/2 1/0 2 14 7/4 1/0 4 2 11/6 -5/2 2 14 2/1 -1/1 1 14 9/4 (-4/3,-1/1) 0 14 7/3 -1/1 5 2 5/2 -1/2 2 14 21/8 1/0 2 2 8/3 -1/1 1 14 3/1 0/1 1 14 7/2 1/0 2 2 4/1 -1/1 1 14 13/3 -2/1 1 14 9/2 -3/2 2 14 14/3 -1/1 3 2 5/1 -2/1 1 14 11/2 -5/4 2 14 6/1 -1/1 1 14 13/2 -11/10 2 14 7/1 -1/1 11 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(13,-7,24,-13) (1/2,7/12) -> (1/2,7/12) Reflection Matrix(155,-91,264,-155) (7/12,13/22) -> (7/12,13/22) Reflection Matrix(153,-91,116,-69) (10/17,3/5) -> (17/13,4/3) Glide Reflection Matrix(127,-77,160,-97) (3/5,11/18) -> (11/14,4/5) Hyperbolic Matrix(251,-154,44,-27) (11/18,8/13) -> (11/2,6/1) Glide Reflection Matrix(113,-70,92,-57) (8/13,5/8) -> (6/5,5/4) Glide Reflection Matrix(111,-70,176,-111) (5/8,7/11) -> (5/8,7/11) Reflection Matrix(97,-63,20,-13) (7/11,2/3) -> (14/3,5/1) Glide Reflection Matrix(71,-49,100,-69) (2/3,7/10) -> (7/10,5/7) Parabolic Matrix(253,-182,196,-141) (5/7,13/18) -> (9/7,13/10) Glide Reflection Matrix(251,-182,40,-29) (13/18,8/11) -> (6/1,13/2) Hyperbolic Matrix(181,-133,132,-97) (8/11,3/4) -> (15/11,11/8) Glide Reflection Matrix(55,-42,72,-55) (3/4,7/9) -> (3/4,7/9) Reflection Matrix(197,-154,252,-197) (7/9,11/14) -> (7/9,11/14) Reflection Matrix(43,-35,16,-13) (4/5,1/1) -> (8/3,3/1) Hyperbolic Matrix(43,-49,36,-41) (1/1,7/6) -> (7/6,6/5) Parabolic Matrix(71,-91,32,-41) (5/4,9/7) -> (2/1,9/4) Hyperbolic Matrix(209,-273,160,-209) (13/10,21/16) -> (13/10,21/16) Reflection Matrix(83,-112,20,-27) (4/3,15/11) -> (4/1,13/3) Glide Reflection 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(13,-21,8,-13) (3/2,7/4) -> (3/2,7/4) Reflection Matrix(43,-77,24,-43) (7/4,11/6) -> (7/4,11/6) Reflection Matrix(41,-77,8,-15) (11/6,2/1) -> (5/1,11/2) Hyperbolic 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(41,-105,16,-41) (5/2,21/8) -> (5/2,21/8) Reflection Matrix(295,-777,112,-295) (21/8,37/14) -> (21/8,37/14) Reflection Matrix(209,-553,48,-127) (29/11,8/3) -> (13/3,22/5) Hyperbolic Matrix(15,-49,4,-13) (3/1,7/2) -> (7/2,4/1) Parabolic Matrix(71,-315,16,-71) (35/8,9/2) -> (35/8,9/2) Reflection Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) Reflection Matrix(27,-182,4,-27) (13/2,7/1) -> (13/2,7/1) 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(1,0,4,-1) -> Matrix(3,2,-4,-3) (0/1,1/2) -> (-1/1,-1/2) Matrix(13,-7,24,-13) -> Matrix(1,1,0,-1) (1/2,7/12) -> (-1/2,1/0) Matrix(155,-91,264,-155) -> Matrix(19,9,-40,-19) (7/12,13/22) -> (-1/2,-9/20) Matrix(153,-91,116,-69) -> Matrix(7,3,-2,-1) Matrix(127,-77,160,-97) -> Matrix(3,1,2,1) Matrix(251,-154,44,-27) -> Matrix(11,4,-8,-3) Matrix(113,-70,92,-57) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(111,-70,176,-111) -> Matrix(-1,0,6,1) (5/8,7/11) -> (-1/3,0/1) Matrix(97,-63,20,-13) -> Matrix(3,1,-2,-1) Matrix(71,-49,100,-69) -> Matrix(5,3,-12,-7) -1/2 Matrix(253,-182,196,-141) -> Matrix(5,2,8,3) Matrix(251,-182,40,-29) -> Matrix(19,6,-16,-5) Matrix(181,-133,132,-97) -> Matrix(3,1,-2,-1) Matrix(55,-42,72,-55) -> Matrix(-1,0,6,1) (3/4,7/9) -> (-1/3,0/1) Matrix(197,-154,252,-197) -> Matrix(1,0,4,-1) (7/9,11/14) -> (0/1,1/2) Matrix(43,-35,16,-13) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(43,-49,36,-41) -> Matrix(1,-1,0,1) 1/0 Matrix(71,-91,32,-41) -> Matrix(3,-1,-2,1) Matrix(209,-273,160,-209) -> Matrix(-1,3,0,1) (13/10,21/16) -> (3/2,1/0) Matrix(83,-112,20,-27) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(111,-154,80,-111) -> Matrix(-1,0,2,1) (11/8,7/5) -> (-1/1,0/1) Matrix(29,-42,20,-29) -> Matrix(1,0,0,-1) (7/5,3/2) -> (0/1,1/0) Matrix(13,-21,8,-13) -> Matrix(1,1,0,-1) (3/2,7/4) -> (-1/2,1/0) Matrix(43,-77,24,-43) -> Matrix(1,5,0,-1) (7/4,11/6) -> (-5/2,1/0) Matrix(41,-77,8,-15) -> Matrix(3,5,-2,-3) (-2/1,-1/1).(-3/2,1/0) Matrix(55,-126,24,-55) -> Matrix(7,8,-6,-7) (9/4,7/3) -> (-4/3,-1/1) Matrix(29,-70,12,-29) -> Matrix(3,2,-4,-3) (7/3,5/2) -> (-1/1,-1/2) Matrix(41,-105,16,-41) -> Matrix(1,1,0,-1) (5/2,21/8) -> (-1/2,1/0) Matrix(295,-777,112,-295) -> Matrix(1,3,0,-1) (21/8,37/14) -> (-3/2,1/0) Matrix(209,-553,48,-127) -> Matrix(3,5,-2,-3) (-2/1,-1/1).(-3/2,1/0) Matrix(15,-49,4,-13) -> Matrix(1,-1,0,1) 1/0 Matrix(71,-315,16,-71) -> Matrix(1,3,0,-1) (35/8,9/2) -> (-3/2,1/0) Matrix(55,-252,12,-55) -> Matrix(5,6,-4,-5) (9/2,14/3) -> (-3/2,-1/1) Matrix(27,-182,4,-27) -> Matrix(21,22,-20,-21) (13/2,7/1) -> (-11/10,-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.