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 -12/7 -10/7 -1/1 -6/7 -4/7 -1/2 -13/28 -8/21 -1/3 -2/7 -1/4 -1/5 -1/6 -2/13 0/1 1/7 1/6 2/11 1/5 3/14 2/9 1/4 2/7 4/13 1/3 5/14 4/11 8/21 2/5 3/7 4/9 1/2 4/7 9/14 2/3 5/7 11/14 4/5 6/7 1/1 9/7 4/3 10/7 11/7 12/7 13/7 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 0/1 1/2 -11/6 1/4 1/2 -9/5 1/2 1/0 -16/9 1/4 -23/13 1/2 1/0 -7/4 1/4 1/2 -12/7 1/2 -17/10 1/2 3/4 -5/3 1/2 1/0 -18/11 0/1 1/4 -13/8 1/4 1/2 -8/5 1/0 -11/7 0/1 -14/9 0/1 1/6 -17/11 3/14 1/4 -3/2 1/4 1/2 -10/7 1/2 -17/12 1/2 7/12 -7/5 1/2 3/4 -18/13 1/2 1/1 -11/8 1/2 1/0 -26/19 1/2 2/3 -15/11 3/4 5/6 -4/3 1/0 -9/7 0/1 -14/11 0/1 1/8 -19/15 1/8 1/6 -5/4 1/6 1/4 -11/9 1/4 3/10 -6/5 1/3 1/2 -1/1 1/2 1/0 -6/7 1/2 -11/13 1/2 9/16 -5/6 1/2 5/8 -4/5 3/4 -11/14 1/1 -18/23 1/1 3/2 -7/9 1/2 1/0 -10/13 1/2 1/1 -13/17 1/2 1/0 -3/4 1/2 3/4 -11/15 5/6 7/8 -19/26 3/4 5/6 -8/11 7/8 -5/7 1/1 -2/3 1/1 1/0 -9/14 1/1 -16/25 9/8 -7/11 5/4 3/2 -12/19 5/4 -17/27 11/8 3/2 -5/8 3/2 1/0 -18/29 3/2 2/1 -13/21 2/1 -8/13 1/0 -11/18 -1/2 1/0 -3/5 3/2 1/0 -4/7 1/0 -5/9 -1/2 1/0 -6/11 0/1 1/0 -7/13 1/2 1/0 -1/2 1/2 1/0 -7/15 1/2 3/4 -13/28 1/1 -6/13 1/2 1/1 -5/11 1/2 3/4 -4/9 3/4 -3/7 1/1 -2/5 1/1 3/2 -5/13 11/8 3/2 -8/21 3/2 -11/29 3/2 25/16 -3/8 3/2 7/4 -7/19 3/2 1/0 -11/30 3/2 13/8 -4/11 7/4 -5/14 2/1 -6/17 2/1 9/4 -1/3 5/2 1/0 -2/7 1/0 -3/11 -11/2 1/0 -7/26 -7/2 1/0 -11/41 -7/2 1/0 -15/56 -3/1 -4/15 1/0 -1/4 -3/2 1/0 -3/13 -3/4 -1/2 -5/22 -1/2 -3/8 -2/9 -1/2 0/1 -3/14 0/1 -4/19 1/4 -1/5 1/2 1/0 -2/11 0/1 1/0 -1/6 1/2 1/0 -2/13 1/2 1/1 -1/7 1/1 0/1 1/0 1/7 -1/1 1/6 -1/2 1/0 2/11 0/1 1/0 1/5 -1/2 1/0 3/14 0/1 2/9 0/1 1/2 5/22 3/8 1/2 3/13 1/2 3/4 1/4 3/2 1/0 2/7 1/0 3/10 -13/2 1/0 4/13 1/0 5/16 -9/2 1/0 1/3 -5/2 1/0 5/14 -2/1 4/11 -7/4 7/19 -3/2 1/0 3/8 -7/4 -3/2 8/21 -3/2 13/34 -3/2 -29/20 5/13 -3/2 -11/8 2/5 -3/2 -1/1 3/7 -1/1 4/9 -3/4 5/11 -3/4 -1/2 6/13 -1/1 -1/2 1/2 -1/2 1/0 4/7 1/0 7/12 -5/2 1/0 24/41 1/0 41/70 -3/1 17/29 -5/2 1/0 10/17 -2/1 1/0 3/5 -3/2 1/0 11/18 1/2 1/0 8/13 1/0 13/21 -2/1 5/8 -3/2 1/0 17/27 -3/2 -11/8 12/19 -5/4 7/11 -3/2 -5/4 9/14 -1/1 2/3 -1/1 1/0 5/7 -1/1 8/11 -7/8 19/26 -5/6 -3/4 30/41 -1/1 -5/6 41/56 -1/1 11/15 -7/8 -5/6 3/4 -3/4 -1/2 10/13 -1/1 -1/2 7/9 -1/2 1/0 11/14 -1/1 4/5 -3/4 5/6 -5/8 -1/2 6/7 -1/2 7/8 -1/2 -3/8 1/1 -1/2 1/0 6/5 -1/2 -1/3 17/14 -1/3 11/9 -3/10 -1/4 5/4 -1/4 -1/6 19/15 -1/6 -1/8 14/11 -1/8 0/1 9/7 0/1 4/3 1/0 19/14 -1/1 15/11 -5/6 -3/4 26/19 -2/3 -1/2 11/8 -1/2 1/0 29/21 -1/1 18/13 -1/1 -1/2 7/5 -3/4 -1/2 10/7 -1/2 13/9 -1/2 -5/12 16/11 -3/8 3/2 -1/2 -1/4 17/11 -1/4 -3/14 14/9 -1/6 0/1 11/7 0/1 8/5 1/0 13/8 -1/2 -1/4 18/11 -1/4 0/1 23/14 0/1 5/3 -1/2 1/0 12/7 -1/2 19/11 -1/2 -3/8 26/15 -1/2 -1/3 7/4 -1/2 -1/4 23/13 -1/2 1/0 16/9 -1/4 25/14 0/1 9/5 -1/2 1/0 11/6 -1/2 -1/4 13/7 0/1 2/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(113,208,-182,-335) (-2/1,-11/6) -> (-5/8,-18/29) Hyperbolic Matrix(57,104,154,281) (-11/6,-9/5) -> (7/19,3/8) Hyperbolic Matrix(29,52,-140,-251) (-9/5,-16/9) -> (-4/19,-1/5) Hyperbolic Matrix(309,548,490,869) (-16/9,-23/13) -> (17/27,12/19) Hyperbolic Matrix(141,248,112,197) (-23/13,-7/4) -> (5/4,19/15) Hyperbolic Matrix(167,288,-98,-169) (-7/4,-12/7) -> (-12/7,-17/10) Parabolic Matrix(139,236,-182,-309) (-17/10,-5/3) -> (-13/17,-3/4) Hyperbolic Matrix(29,48,-84,-139) (-5/3,-18/11) -> (-6/17,-1/3) Hyperbolic Matrix(27,44,154,251) (-18/11,-13/8) -> (1/6,2/11) Hyperbolic Matrix(57,92,70,113) (-13/8,-8/5) -> (4/5,5/6) Hyperbolic Matrix(111,176,70,111) (-8/5,-11/7) -> (11/7,8/5) Hyperbolic Matrix(197,308,126,197) (-11/7,-14/9) -> (14/9,11/7) Hyperbolic Matrix(421,652,308,477) (-14/9,-17/11) -> (15/11,26/19) Hyperbolic Matrix(57,88,182,281) (-17/11,-3/2) -> (5/16,1/3) Hyperbolic Matrix(139,200,-98,-141) (-3/2,-10/7) -> (-10/7,-17/12) Parabolic Matrix(139,196,-378,-533) (-17/12,-7/5) -> (-7/19,-11/30) Hyperbolic Matrix(141,196,182,253) (-7/5,-18/13) -> (10/13,7/9) Hyperbolic Matrix(29,40,-182,-251) (-18/13,-11/8) -> (-1/6,-2/13) Hyperbolic Matrix(111,152,490,671) (-11/8,-26/19) -> (2/9,5/22) Hyperbolic Matrix(477,652,308,421) (-26/19,-15/11) -> (17/11,14/9) Hyperbolic Matrix(197,268,-308,-419) (-15/11,-4/3) -> (-16/25,-7/11) Hyperbolic Matrix(55,72,42,55) (-4/3,-9/7) -> (9/7,4/3) Hyperbolic Matrix(197,252,154,197) (-9/7,-14/11) -> (14/11,9/7) Hyperbolic Matrix(337,428,574,729) (-14/11,-19/15) -> (17/29,10/17) Hyperbolic Matrix(197,248,112,141) (-19/15,-5/4) -> (7/4,23/13) Hyperbolic Matrix(111,136,182,223) (-5/4,-11/9) -> (3/5,11/18) Hyperbolic Matrix(197,240,-252,-307) (-11/9,-6/5) -> (-18/23,-7/9) Hyperbolic Matrix(27,32,70,83) (-6/5,-1/1) -> (5/13,2/5) Hyperbolic Matrix(83,72,-98,-85) (-1/1,-6/7) -> (-6/7,-11/13) Parabolic Matrix(167,140,266,223) (-11/13,-5/6) -> (5/8,17/27) Hyperbolic Matrix(113,92,70,57) (-5/6,-4/5) -> (8/5,13/8) Hyperbolic Matrix(111,88,140,111) (-4/5,-11/14) -> (11/14,4/5) Hyperbolic Matrix(475,372,392,307) (-11/14,-18/23) -> (6/5,17/14) Hyperbolic Matrix(253,196,182,141) (-7/9,-10/13) -> (18/13,7/5) Hyperbolic Matrix(167,128,364,279) (-10/13,-13/17) -> (5/11,6/13) Hyperbolic Matrix(27,20,112,83) (-3/4,-11/15) -> (3/13,1/4) Hyperbolic Matrix(197,144,-420,-307) (-11/15,-19/26) -> (-1/2,-7/15) Hyperbolic Matrix(449,328,308,225) (-19/26,-8/11) -> (16/11,3/2) Hyperbolic Matrix(111,80,154,111) (-8/11,-5/7) -> (5/7,8/11) Hyperbolic Matrix(29,20,42,29) (-5/7,-2/3) -> (2/3,5/7) Hyperbolic Matrix(55,36,84,55) (-2/3,-9/14) -> (9/14,2/3) Hyperbolic Matrix(531,340,392,251) (-9/14,-16/25) -> (4/3,19/14) Hyperbolic Matrix(139,88,308,195) (-7/11,-12/19) -> (4/9,5/11) Hyperbolic Matrix(869,548,490,309) (-12/19,-17/27) -> (23/13,16/9) Hyperbolic Matrix(197,124,224,141) (-17/27,-5/8) -> (7/8,1/1) Hyperbolic Matrix(419,260,224,139) (-18/29,-13/21) -> (13/7,2/1) Hyperbolic Matrix(337,208,546,337) (-13/21,-8/13) -> (8/13,13/21) Hyperbolic Matrix(111,68,364,223) (-8/13,-11/18) -> (3/10,4/13) Hyperbolic Matrix(223,136,182,111) (-11/18,-3/5) -> (11/9,5/4) Hyperbolic Matrix(55,32,-98,-57) (-3/5,-4/7) -> (-4/7,-5/9) Parabolic Matrix(29,16,154,85) (-5/9,-6/11) -> (2/11,1/5) Hyperbolic Matrix(391,212,308,167) (-6/11,-7/13) -> (19/15,14/11) Hyperbolic Matrix(113,60,-420,-223) (-7/13,-1/2) -> (-7/26,-11/41) Hyperbolic Matrix(1091,508,1862,867) (-7/15,-13/28) -> (41/70,17/29) Hyperbolic Matrix(1373,636,1876,869) (-13/28,-6/13) -> (30/41,41/56) Hyperbolic Matrix(533,244,308,141) (-6/13,-5/11) -> (19/11,26/15) Hyperbolic Matrix(195,88,308,139) (-5/11,-4/9) -> (12/19,7/11) Hyperbolic Matrix(55,24,126,55) (-4/9,-3/7) -> (3/7,4/9) Hyperbolic Matrix(29,12,70,29) (-3/7,-2/5) -> (2/5,3/7) Hyperbolic Matrix(83,32,70,27) (-2/5,-5/13) -> (1/1,6/5) Hyperbolic Matrix(335,128,-882,-337) (-5/13,-8/21) -> (-8/21,-11/29) Parabolic Matrix(169,64,742,281) (-11/29,-3/8) -> (5/22,3/13) Hyperbolic Matrix(281,104,154,57) (-3/8,-7/19) -> (9/5,11/6) Hyperbolic Matrix(449,164,616,225) (-11/30,-4/11) -> (8/11,19/26) Hyperbolic Matrix(111,40,308,111) (-4/11,-5/14) -> (5/14,4/11) Hyperbolic Matrix(643,228,392,139) (-5/14,-6/17) -> (18/11,23/14) Hyperbolic Matrix(27,8,-98,-29) (-1/3,-2/7) -> (-2/7,-3/11) Parabolic Matrix(475,128,308,83) (-3/11,-7/26) -> (3/2,17/11) Hyperbolic Matrix(2297,616,3136,841) (-11/41,-15/56) -> (41/56,11/15) Hyperbolic Matrix(1959,524,3346,895) (-15/56,-4/15) -> (24/41,41/70) Hyperbolic Matrix(197,52,322,85) (-4/15,-1/4) -> (11/18,8/13) Hyperbolic Matrix(83,20,112,27) (-1/4,-3/13) -> (11/15,3/4) Hyperbolic Matrix(279,64,728,167) (-3/13,-5/22) -> (13/34,5/13) Hyperbolic Matrix(671,152,490,111) (-5/22,-2/9) -> (26/19,11/8) Hyperbolic Matrix(55,12,252,55) (-2/9,-3/14) -> (3/14,2/9) Hyperbolic Matrix(699,148,392,83) (-3/14,-4/19) -> (16/9,25/14) Hyperbolic Matrix(83,16,140,27) (-1/5,-2/11) -> (10/17,3/5) Hyperbolic Matrix(251,44,154,27) (-2/11,-1/6) -> (13/8,18/11) Hyperbolic Matrix(503,76,364,55) (-2/13,-1/7) -> (29/21,18/13) Hyperbolic Matrix(1,0,14,1) (-1/7,0/1) -> (0/1,1/7) Parabolic Matrix(251,-40,182,-29) (1/7,1/6) -> (11/8,29/21) Hyperbolic Matrix(251,-52,140,-29) (1/5,3/14) -> (25/14,9/5) Hyperbolic Matrix(29,-8,98,-27) (1/4,2/7) -> (2/7,3/10) Parabolic Matrix(475,-148,812,-253) (4/13,5/16) -> (7/12,24/41) Hyperbolic Matrix(139,-48,84,-29) (1/3,5/14) -> (23/14,5/3) Hyperbolic Matrix(447,-164,308,-113) (4/11,7/19) -> (13/9,16/11) Hyperbolic Matrix(337,-128,882,-335) (3/8,8/21) -> (8/21,13/34) Parabolic Matrix(307,-144,420,-197) (6/13,1/2) -> (19/26,30/41) Hyperbolic Matrix(57,-32,98,-55) (1/2,4/7) -> (4/7,7/12) Parabolic Matrix(335,-208,182,-113) (13/21,5/8) -> (11/6,13/7) Hyperbolic Matrix(419,-268,308,-197) (7/11,9/14) -> (19/14,15/11) Hyperbolic Matrix(195,-148,112,-85) (3/4,10/13) -> (26/15,7/4) Hyperbolic Matrix(307,-240,252,-197) (7/9,11/14) -> (17/14,11/9) Hyperbolic Matrix(85,-72,98,-83) (5/6,6/7) -> (6/7,7/8) Parabolic Matrix(141,-200,98,-139) (7/5,10/7) -> (10/7,13/9) Parabolic Matrix(169,-288,98,-167) (5/3,12/7) -> (12/7,19/11) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-4,1) Matrix(113,208,-182,-335) -> Matrix(7,-2,4,-1) Matrix(57,104,154,281) -> Matrix(1,-2,0,1) Matrix(29,52,-140,-251) -> Matrix(1,0,0,1) Matrix(309,548,490,869) -> Matrix(11,-4,-8,3) Matrix(141,248,112,197) -> Matrix(1,0,-8,1) Matrix(167,288,-98,-169) -> Matrix(5,-2,8,-3) Matrix(139,236,-182,-309) -> Matrix(1,0,0,1) Matrix(29,48,-84,-139) -> Matrix(1,2,0,1) Matrix(27,44,154,251) -> Matrix(1,0,-4,1) Matrix(57,92,70,113) -> Matrix(3,-2,-4,3) Matrix(111,176,70,111) -> Matrix(1,0,0,1) Matrix(197,308,126,197) -> Matrix(1,0,-12,1) Matrix(421,652,308,477) -> Matrix(11,-2,-16,3) Matrix(57,88,182,281) -> Matrix(17,-4,-4,1) Matrix(139,200,-98,-141) -> Matrix(9,-4,16,-7) Matrix(139,196,-378,-533) -> Matrix(5,-4,4,-3) Matrix(141,196,182,253) -> Matrix(3,-2,-4,3) Matrix(29,40,-182,-251) -> Matrix(1,0,0,1) Matrix(111,152,490,671) -> Matrix(3,-2,8,-5) Matrix(477,652,308,421) -> Matrix(3,-2,-16,11) Matrix(197,268,-308,-419) -> Matrix(9,-8,8,-7) Matrix(55,72,42,55) -> Matrix(1,0,0,1) Matrix(197,252,154,197) -> Matrix(1,0,-16,1) Matrix(337,428,574,729) -> Matrix(17,-2,-8,1) Matrix(197,248,112,141) -> Matrix(1,0,-8,1) Matrix(111,136,182,223) -> Matrix(1,0,-4,1) Matrix(197,240,-252,-307) -> Matrix(7,-2,4,-1) Matrix(27,32,70,83) -> Matrix(11,-4,-8,3) Matrix(83,72,-98,-85) -> Matrix(9,-4,16,-7) Matrix(167,140,266,223) -> Matrix(13,-8,-8,5) Matrix(113,92,70,57) -> Matrix(3,-2,-4,3) Matrix(111,88,140,111) -> Matrix(7,-6,-8,7) Matrix(475,372,392,307) -> Matrix(3,-4,-8,11) Matrix(253,196,182,141) -> Matrix(3,-2,-4,3) Matrix(167,128,364,279) -> Matrix(3,-2,-4,3) Matrix(27,20,112,83) -> Matrix(5,-4,4,-3) Matrix(197,144,-420,-307) -> Matrix(5,-4,4,-3) Matrix(449,328,308,225) -> Matrix(5,-4,-16,13) Matrix(111,80,154,111) -> Matrix(15,-14,-16,15) Matrix(29,20,42,29) -> Matrix(1,-2,0,1) Matrix(55,36,84,55) -> Matrix(1,-2,0,1) Matrix(531,340,392,251) -> Matrix(9,-10,-8,9) Matrix(139,88,308,195) -> Matrix(1,-2,0,1) Matrix(869,548,490,309) -> Matrix(3,-4,-8,11) Matrix(197,124,224,141) -> Matrix(3,-4,-8,11) Matrix(419,260,224,139) -> Matrix(1,-2,0,1) Matrix(337,208,546,337) -> Matrix(1,-4,0,1) Matrix(111,68,364,223) -> Matrix(1,-6,0,1) Matrix(223,136,182,111) -> Matrix(1,0,-4,1) Matrix(55,32,-98,-57) -> Matrix(1,-2,0,1) Matrix(29,16,154,85) -> Matrix(1,0,0,1) Matrix(391,212,308,167) -> Matrix(1,0,-8,1) Matrix(113,60,-420,-223) -> Matrix(1,-4,0,1) Matrix(1091,508,1862,867) -> Matrix(11,-8,-4,3) Matrix(1373,636,1876,869) -> Matrix(7,-6,-8,7) Matrix(533,244,308,141) -> Matrix(1,0,-4,1) Matrix(195,88,308,139) -> Matrix(1,-2,0,1) Matrix(55,24,126,55) -> Matrix(7,-6,-8,7) Matrix(29,12,70,29) -> Matrix(5,-6,-4,5) Matrix(83,32,70,27) -> Matrix(3,-4,-8,11) Matrix(335,128,-882,-337) -> Matrix(37,-54,24,-35) Matrix(169,64,742,281) -> Matrix(5,-8,12,-19) Matrix(281,104,154,57) -> Matrix(1,-2,0,1) Matrix(449,164,616,225) -> Matrix(17,-28,-20,33) Matrix(111,40,308,111) -> Matrix(15,-28,-8,15) Matrix(643,228,392,139) -> Matrix(1,-2,-8,17) Matrix(27,8,-98,-29) -> Matrix(1,-8,0,1) Matrix(475,128,308,83) -> Matrix(1,4,-4,-15) Matrix(2297,616,3136,841) -> Matrix(7,22,-8,-25) Matrix(1959,524,3346,895) -> Matrix(1,0,0,1) Matrix(197,52,322,85) -> Matrix(1,2,0,1) Matrix(83,20,112,27) -> Matrix(3,4,-4,-5) Matrix(279,64,728,167) -> Matrix(17,10,-12,-7) Matrix(671,152,490,111) -> Matrix(5,2,-8,-3) Matrix(55,12,252,55) -> Matrix(1,0,4,1) Matrix(699,148,392,83) -> Matrix(1,0,-8,1) Matrix(83,16,140,27) -> Matrix(1,-2,0,1) Matrix(251,44,154,27) -> Matrix(1,0,-4,1) Matrix(503,76,364,55) -> Matrix(3,-2,-4,3) Matrix(1,0,14,1) -> Matrix(1,-2,0,1) Matrix(251,-40,182,-29) -> Matrix(1,0,0,1) Matrix(251,-52,140,-29) -> Matrix(1,0,0,1) Matrix(29,-8,98,-27) -> Matrix(1,-8,0,1) Matrix(475,-148,812,-253) -> Matrix(1,2,0,1) Matrix(139,-48,84,-29) -> Matrix(1,2,0,1) Matrix(447,-164,308,-113) -> Matrix(5,8,-12,-19) Matrix(337,-128,882,-335) -> Matrix(35,54,-24,-37) Matrix(307,-144,420,-197) -> Matrix(3,4,-4,-5) Matrix(57,-32,98,-55) -> Matrix(1,-2,0,1) Matrix(335,-208,182,-113) -> Matrix(1,2,-4,-7) Matrix(419,-268,308,-197) -> Matrix(7,8,-8,-9) Matrix(195,-148,112,-85) -> Matrix(3,2,-8,-5) Matrix(307,-240,252,-197) -> Matrix(1,2,-4,-7) Matrix(85,-72,98,-83) -> Matrix(7,4,-16,-9) Matrix(141,-200,98,-139) -> Matrix(7,4,-16,-9) Matrix(169,-288,98,-167) -> Matrix(3,2,-8,-5) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 16 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): 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 0/1 2/7 1/3 8/21 2/5 3/7 1/2 4/7 2/3 5/7 4/5 6/7 1/1 4/3 10/7 12/7 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 0/1 1/2 -5/3 1/2 1/0 -8/5 1/0 -3/2 1/4 1/2 -4/3 1/0 -5/4 1/6 1/4 -1/1 1/2 1/0 -4/5 3/4 -3/4 1/2 3/4 -8/11 7/8 -5/7 1/1 -2/3 1/1 1/0 -1/2 1/2 1/0 -4/9 3/4 -3/7 1/1 -2/5 1/1 3/2 -3/8 3/2 7/4 -1/3 5/2 1/0 0/1 1/0 1/4 3/2 1/0 2/7 1/0 1/3 -5/2 1/0 3/8 -7/4 -3/2 8/21 -3/2 5/13 -3/2 -11/8 2/5 -3/2 -1/1 3/7 -1/1 1/2 -1/2 1/0 4/7 1/0 3/5 -3/2 1/0 2/3 -1/1 1/0 5/7 -1/1 3/4 -3/4 -1/2 4/5 -3/4 5/6 -5/8 -1/2 6/7 -1/2 1/1 -1/2 1/0 5/4 -1/4 -1/6 9/7 0/1 4/3 1/0 7/5 -3/4 -1/2 10/7 -1/2 3/2 -1/2 -1/4 11/7 0/1 8/5 1/0 5/3 -1/2 1/0 12/7 -1/2 7/4 -1/2 -1/4 2/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(13,24,7,13) (-2/1,-5/3) -> (7/4,2/1) Hyperbolic Matrix(27,44,-35,-57) (-5/3,-8/5) -> (-4/5,-3/4) Hyperbolic Matrix(13,20,-28,-43) (-8/5,-3/2) -> (-1/2,-4/9) Hyperbolic Matrix(29,40,21,29) (-3/2,-4/3) -> (4/3,7/5) Hyperbolic Matrix(41,52,-56,-71) (-4/3,-5/4) -> (-3/4,-8/11) Hyperbolic Matrix(13,16,-35,-43) (-5/4,-1/1) -> (-3/8,-1/3) Hyperbolic Matrix(29,24,35,29) (-1/1,-4/5) -> (4/5,5/6) Hyperbolic Matrix(127,92,98,71) (-8/11,-5/7) -> (9/7,4/3) Hyperbolic Matrix(29,20,42,29) (-5/7,-2/3) -> (2/3,5/7) Hyperbolic Matrix(13,8,21,13) (-2/3,-1/2) -> (3/5,2/3) Hyperbolic Matrix(155,68,98,43) (-4/9,-3/7) -> (11/7,8/5) Hyperbolic Matrix(29,12,70,29) (-3/7,-2/5) -> (2/5,3/7) Hyperbolic Matrix(41,16,105,41) (-2/5,-3/8) -> (5/13,2/5) Hyperbolic Matrix(1,0,7,1) (-1/3,0/1) -> (0/1,1/4) Parabolic Matrix(15,-4,49,-13) (1/4,2/7) -> (2/7,1/3) Parabolic Matrix(43,-16,35,-13) (1/3,3/8) -> (1/1,5/4) Hyperbolic Matrix(169,-64,441,-167) (3/8,8/21) -> (8/21,5/13) Parabolic Matrix(43,-20,28,-13) (3/7,1/2) -> (3/2,11/7) Hyperbolic Matrix(29,-16,49,-27) (1/2,4/7) -> (4/7,3/5) Parabolic Matrix(71,-52,56,-41) (5/7,3/4) -> (5/4,9/7) Hyperbolic Matrix(57,-44,35,-27) (3/4,4/5) -> (8/5,5/3) Hyperbolic Matrix(43,-36,49,-41) (5/6,6/7) -> (6/7,1/1) Parabolic Matrix(71,-100,49,-69) (7/5,10/7) -> (10/7,3/2) Parabolic Matrix(85,-144,49,-83) (5/3,12/7) -> (12/7,7/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-4,1) Matrix(13,24,7,13) -> Matrix(1,0,-4,1) Matrix(27,44,-35,-57) -> Matrix(3,-1,4,-1) Matrix(13,20,-28,-43) -> Matrix(3,-1,4,-1) Matrix(29,40,21,29) -> Matrix(1,-1,0,1) Matrix(41,52,-56,-71) -> Matrix(7,-1,8,-1) Matrix(13,16,-35,-43) -> Matrix(7,-2,4,-1) Matrix(29,24,35,29) -> Matrix(5,-3,-8,5) Matrix(127,92,98,71) -> Matrix(1,-1,8,-7) Matrix(29,20,42,29) -> Matrix(1,-2,0,1) Matrix(13,8,21,13) -> Matrix(1,-2,0,1) Matrix(155,68,98,43) -> Matrix(1,-1,4,-3) Matrix(29,12,70,29) -> Matrix(5,-6,-4,5) Matrix(41,16,105,41) -> Matrix(5,-6,-4,5) Matrix(1,0,7,1) -> Matrix(1,-1,0,1) Matrix(15,-4,49,-13) -> Matrix(1,-4,0,1) Matrix(43,-16,35,-13) -> Matrix(1,2,-4,-7) Matrix(169,-64,441,-167) -> Matrix(17,27,-12,-19) Matrix(43,-20,28,-13) -> Matrix(1,1,-4,-3) Matrix(29,-16,49,-27) -> Matrix(1,-1,0,1) Matrix(71,-52,56,-41) -> Matrix(1,1,-8,-7) Matrix(57,-44,35,-27) -> Matrix(1,1,-4,-3) Matrix(43,-36,49,-41) -> Matrix(3,2,-8,-5) Matrix(71,-100,49,-69) -> Matrix(3,2,-8,-5) Matrix(85,-144,49,-83) -> Matrix(1,1,-4,-3) 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): 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 1/0 2 14 2/7 1/0 8 2 1/3 (-5/2,1/0) 0 14 3/8 (-7/4,-3/2) 0 14 8/21 -3/2 6 2 2/5 (-3/2,-1/1) 0 14 3/7 -1/1 6 2 1/2 (-1/2,1/0) 0 14 4/7 1/0 2 2 2/3 (-1/1,1/0) 0 14 5/7 -1/1 8 2 3/4 (-3/4,-1/2) 0 14 4/5 -3/4 2 14 6/7 -1/2 4 2 1/1 (-1/2,1/0) 0 14 5/4 (-1/4,-1/6) 0 14 9/7 0/1 8 2 4/3 1/0 2 14 10/7 -1/2 4 2 3/2 (-1/2,-1/4) 0 14 11/7 0/1 6 2 8/5 1/0 2 14 5/3 (-1/2,1/0) 0 14 12/7 -1/2 2 2 2/1 (-1/2,0/1) 0 14 1/0 0/1 2 2 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(1,0,7,-1) (0/1,2/7) -> (0/1,2/7) Reflection Matrix(13,-4,42,-13) (2/7,1/3) -> (2/7,1/3) Reflection Matrix(43,-16,35,-13) (1/3,3/8) -> (1/1,5/4) Hyperbolic Matrix(127,-48,336,-127) (3/8,8/21) -> (3/8,8/21) Reflection Matrix(41,-16,105,-41) (8/21,2/5) -> (8/21,2/5) Reflection Matrix(29,-12,70,-29) (2/5,3/7) -> (2/5,3/7) Reflection Matrix(43,-20,28,-13) (3/7,1/2) -> (3/2,11/7) Hyperbolic Matrix(15,-8,28,-15) (1/2,4/7) -> (1/2,4/7) Reflection Matrix(13,-8,21,-13) (4/7,2/3) -> (4/7,2/3) Reflection Matrix(29,-20,42,-29) (2/3,5/7) -> (2/3,5/7) Reflection Matrix(71,-52,56,-41) (5/7,3/4) -> (5/4,9/7) Hyperbolic Matrix(57,-44,35,-27) (3/4,4/5) -> (8/5,5/3) Hyperbolic Matrix(29,-24,35,-29) (4/5,6/7) -> (4/5,6/7) Reflection Matrix(13,-12,14,-13) (6/7,1/1) -> (6/7,1/1) Reflection Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(29,-40,21,-29) (4/3,10/7) -> (4/3,10/7) Reflection Matrix(41,-60,28,-41) (10/7,3/2) -> (10/7,3/2) Reflection Matrix(111,-176,70,-111) (11/7,8/5) -> (11/7,8/5) Reflection Matrix(71,-120,42,-71) (5/3,12/7) -> (5/3,12/7) Reflection Matrix(13,-24,7,-13) (12/7,2/1) -> (12/7,2/1) Reflection Matrix(-1,4,0,1) (2/1,1/0) -> (2/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,7,-1) -> Matrix(1,1,0,-1) (0/1,2/7) -> (-1/2,1/0) Matrix(13,-4,42,-13) -> Matrix(1,5,0,-1) (2/7,1/3) -> (-5/2,1/0) Matrix(43,-16,35,-13) -> Matrix(1,2,-4,-7) Matrix(127,-48,336,-127) -> Matrix(13,21,-8,-13) (3/8,8/21) -> (-7/4,-3/2) Matrix(41,-16,105,-41) -> Matrix(5,6,-4,-5) (8/21,2/5) -> (-3/2,-1/1) Matrix(29,-12,70,-29) -> Matrix(5,6,-4,-5) (2/5,3/7) -> (-3/2,-1/1) Matrix(43,-20,28,-13) -> Matrix(1,1,-4,-3) -1/2 Matrix(15,-8,28,-15) -> Matrix(1,1,0,-1) (1/2,4/7) -> (-1/2,1/0) Matrix(13,-8,21,-13) -> Matrix(1,2,0,-1) (4/7,2/3) -> (-1/1,1/0) Matrix(29,-20,42,-29) -> Matrix(1,2,0,-1) (2/3,5/7) -> (-1/1,1/0) Matrix(71,-52,56,-41) -> Matrix(1,1,-8,-7) Matrix(57,-44,35,-27) -> Matrix(1,1,-4,-3) -1/2 Matrix(29,-24,35,-29) -> Matrix(5,3,-8,-5) (4/5,6/7) -> (-3/4,-1/2) Matrix(13,-12,14,-13) -> Matrix(1,1,0,-1) (6/7,1/1) -> (-1/2,1/0) Matrix(55,-72,42,-55) -> Matrix(1,0,0,-1) (9/7,4/3) -> (0/1,1/0) Matrix(29,-40,21,-29) -> Matrix(1,1,0,-1) (4/3,10/7) -> (-1/2,1/0) Matrix(41,-60,28,-41) -> Matrix(3,1,-8,-3) (10/7,3/2) -> (-1/2,-1/4) Matrix(111,-176,70,-111) -> Matrix(1,0,0,-1) (11/7,8/5) -> (0/1,1/0) Matrix(71,-120,42,-71) -> Matrix(1,1,0,-1) (5/3,12/7) -> (-1/2,1/0) Matrix(13,-24,7,-13) -> Matrix(-1,0,4,1) (12/7,2/1) -> (-1/2,0/1) Matrix(-1,4,0,1) -> Matrix(-1,0,4,1) (2/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.