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 -11/7 -9/7 -1/1 -6/7 -5/7 -13/21 -4/7 -5/9 -1/2 -3/7 -1/3 -2/7 -3/11 -1/4 -3/13 -1/5 -1/6 -1/7 0/1 1/7 1/6 1/5 3/14 3/13 1/4 3/11 2/7 1/3 5/14 2/5 3/7 1/2 15/28 5/9 4/7 13/21 9/14 2/3 5/7 11/14 4/5 6/7 1/1 9/7 4/3 11/7 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 0/1 1/0 -9/5 -1/1 1/0 -16/9 -4/1 1/0 -7/4 -2/1 -5/3 0/1 -18/11 0/1 1/0 -13/8 0/1 -21/13 1/1 1/0 -8/5 0/1 1/0 -11/7 1/0 -14/9 -4/1 1/0 -3/2 -2/1 -7/5 -2/3 -11/8 0/1 -15/11 -1/3 0/1 -4/3 0/1 1/0 -9/7 1/0 -14/11 -2/1 1/0 -5/4 -2/1 -11/9 -2/1 -6/5 -2/1 -3/2 -7/6 -6/5 -1/1 -1/1 0/1 -7/8 -6/5 -6/7 -1/1 -5/6 -6/7 -9/11 -4/5 -13/16 -4/5 -4/5 -3/4 -2/3 -11/14 -2/3 -18/23 -2/3 -5/8 -7/9 -2/3 -17/22 -4/7 -10/13 -2/3 -1/2 -3/4 -2/3 -5/7 -1/2 -7/10 0/1 -9/13 -1/1 -1/2 -11/16 -2/5 -2/3 -1/2 0/1 -9/14 0/1 -16/25 0/1 1/4 -7/11 0/1 1/1 -12/19 0/1 1/0 -5/8 0/1 -13/21 1/0 -21/34 -4/1 -8/13 -2/1 1/0 -3/5 -2/1 -7/12 -6/5 -4/7 -1/1 -9/16 -10/11 -5/9 -1/1 -6/7 -11/20 -4/5 -6/11 -5/6 -4/5 -7/13 -10/13 -1/2 -2/3 -3/7 -1/2 -5/12 -2/5 -17/41 -1/1 -1/2 -29/70 -1/2 -12/29 -1/2 -4/9 -7/17 -2/5 -2/5 -1/2 0/1 -7/18 0/1 -12/31 -1/2 0/1 -5/13 -1/2 -1/3 -13/34 -2/5 -8/21 -1/3 -3/8 0/1 -10/27 -1/2 -2/5 -7/19 -2/5 -1/3 -11/30 -2/7 -4/11 -1/2 0/1 -5/14 0/1 -6/17 -1/2 0/1 -1/3 0/1 -3/10 -2/1 -2/7 -1/1 -5/18 -6/7 -8/29 -6/7 -5/6 -3/11 -1/1 -4/5 -7/26 -4/5 -11/41 -16/21 -15/56 -3/4 -4/15 -3/4 -2/3 -5/19 -2/3 -1/4 -2/3 -3/13 -8/13 -2/9 -4/7 -1/2 -3/14 -1/2 -4/19 -1/2 -2/5 -1/5 -1/1 -1/2 -1/6 -4/7 -1/7 -1/2 -1/8 -2/5 0/1 -1/2 0/1 1/7 -1/2 2/13 -1/2 -4/9 1/6 -2/5 1/5 0/1 3/14 -1/2 2/9 -1/2 -2/5 3/13 -1/2 -1/3 1/4 -2/5 4/15 -2/5 -3/8 7/26 -4/11 3/11 -4/11 2/7 -1/3 1/3 -1/3 0/1 5/14 0/1 4/11 -1/2 0/1 7/19 0/1 10/27 -1/2 0/1 3/8 -2/5 8/21 -1/3 5/13 -4/13 7/18 -2/7 2/5 -1/4 0/1 3/7 0/1 4/9 -1/2 0/1 5/11 -1/1 0/1 6/13 -2/3 -1/2 1/2 0/1 8/15 -2/3 -1/2 15/28 -1/2 7/13 -1/2 -3/7 6/11 -1/2 -2/5 5/9 -2/5 4/7 -1/3 3/5 -1/3 -1/4 8/13 -2/7 -1/4 13/21 -1/4 18/29 -1/4 -4/17 5/8 -2/9 12/19 -2/11 -1/6 7/11 0/1 9/14 0/1 2/3 -1/4 0/1 5/7 0/1 8/11 0/1 1/4 19/26 2/5 30/41 4/9 1/2 41/56 1/2 11/15 2/3 3/4 0/1 10/13 -1/2 0/1 17/22 -2/5 7/9 -1/3 0/1 11/14 0/1 4/5 -1/2 0/1 9/11 -1/3 0/1 5/6 0/1 6/7 -1/3 1/1 0/1 8/7 -1/1 7/6 0/1 6/5 -1/2 0/1 17/14 0/1 11/9 -1/1 0/1 16/13 -1/2 0/1 5/4 0/1 9/7 0/1 13/10 0/1 17/13 2/5 4/3 0/1 1/0 19/14 0/1 15/11 0/1 11/8 2/1 7/5 -1/1 1/0 10/7 -1/1 13/9 -2/3 16/11 -2/3 -1/2 3/2 0/1 11/7 0/1 19/12 0/1 27/17 -1/1 0/1 8/5 0/1 1/0 21/13 -4/3 34/21 -1/1 13/8 -2/3 31/19 0/1 18/11 -1/2 0/1 23/14 0/1 5/3 -1/1 0/1 12/7 -1/1 19/11 -4/5 26/15 -3/4 -2/3 7/4 -2/3 16/9 -2/3 -1/2 25/14 -1/2 9/5 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(43,78,70,127) (-2/1,-9/5) -> (3/5,8/13) Hyperbolic Matrix(29,52,-140,-251) (-9/5,-16/9) -> (-4/19,-1/5) Hyperbolic Matrix(71,126,182,323) (-16/9,-7/4) -> (7/18,2/5) Hyperbolic Matrix(13,22,-42,-71) (-7/4,-5/3) -> (-1/3,-3/10) Hyperbolic Matrix(29,48,-84,-139) (-5/3,-18/11) -> (-6/17,-1/3) Hyperbolic Matrix(169,276,-308,-503) (-18/11,-13/8) -> (-11/20,-6/11) Hyperbolic Matrix(209,338,-546,-883) (-13/8,-21/13) -> (-5/13,-13/34) Hyperbolic Matrix(41,66,182,293) (-21/13,-8/5) -> (2/9,3/13) Hyperbolic Matrix(153,242,-98,-155) (-8/5,-11/7) -> (-11/7,-14/9) Parabolic Matrix(125,194,-154,-239) (-14/9,-3/2) -> (-13/16,-4/5) Hyperbolic Matrix(41,58,-70,-99) (-3/2,-7/5) -> (-3/5,-7/12) Hyperbolic Matrix(13,18,70,97) (-7/5,-11/8) -> (1/6,1/5) Hyperbolic Matrix(127,174,154,211) (-11/8,-15/11) -> (9/11,5/6) Hyperbolic Matrix(197,268,-308,-419) (-15/11,-4/3) -> (-16/25,-7/11) Hyperbolic Matrix(125,162,-98,-127) (-4/3,-9/7) -> (-9/7,-14/11) Parabolic Matrix(125,158,-322,-407) (-14/11,-5/4) -> (-7/18,-12/31) Hyperbolic Matrix(29,36,-112,-139) (-5/4,-11/9) -> (-5/19,-1/4) Hyperbolic Matrix(197,240,-252,-307) (-11/9,-6/5) -> (-18/23,-7/9) Hyperbolic Matrix(97,114,154,181) (-6/5,-7/6) -> (5/8,12/19) Hyperbolic Matrix(13,14,-14,-15) (-7/6,-1/1) -> (-1/1,-7/8) Parabolic Matrix(363,316,224,195) (-7/8,-6/7) -> (34/21,13/8) Hyperbolic Matrix(69,58,182,153) (-6/7,-5/6) -> (3/8,8/21) Hyperbolic Matrix(211,174,154,127) (-5/6,-9/11) -> (15/11,11/8) Hyperbolic Matrix(125,102,462,377) (-9/11,-13/16) -> (7/26,3/11) 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(799,618,490,379) (-7/9,-17/22) -> (13/8,31/19) Hyperbolic Matrix(57,44,364,281) (-17/22,-10/13) -> (2/13,1/6) Hyperbolic Matrix(97,74,-350,-267) (-10/13,-3/4) -> (-5/18,-8/29) Hyperbolic Matrix(69,50,-98,-71) (-3/4,-5/7) -> (-5/7,-7/10) Parabolic Matrix(43,30,182,127) (-7/10,-9/13) -> (3/13,1/4) Hyperbolic Matrix(337,232,-812,-559) (-9/13,-11/16) -> (-5/12,-17/41) Hyperbolic Matrix(265,182,182,125) (-11/16,-2/3) -> (16/11,3/2) 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(181,114,154,97) (-12/19,-5/8) -> (7/6,6/5) Hyperbolic Matrix(545,338,-882,-547) (-5/8,-13/21) -> (-13/21,-21/34) Parabolic Matrix(379,234,1022,631) (-21/34,-8/13) -> (10/27,3/8) Hyperbolic Matrix(127,78,70,43) (-8/13,-3/5) -> (9/5,2/1) Hyperbolic Matrix(111,64,-196,-113) (-7/12,-4/7) -> (-4/7,-9/16) Parabolic Matrix(211,118,-574,-321) (-9/16,-5/9) -> (-7/19,-11/30) Hyperbolic Matrix(293,162,378,209) (-5/9,-11/20) -> (17/22,7/9) Hyperbolic Matrix(239,130,182,99) (-6/11,-7/13) -> (17/13,4/3) Hyperbolic Matrix(113,60,-420,-223) (-7/13,-1/2) -> (-7/26,-11/41) Hyperbolic Matrix(41,18,-98,-43) (-1/2,-3/7) -> (-3/7,-5/12) Parabolic Matrix(1105,458,2058,853) (-17/41,-29/70) -> (15/28,7/13) Hyperbolic Matrix(3289,1362,4494,1861) (-29/70,-12/29) -> (30/41,41/56) Hyperbolic Matrix(993,410,574,237) (-12/29,-7/17) -> (19/11,26/15) Hyperbolic Matrix(239,98,378,155) (-7/17,-2/5) -> (12/19,7/11) Hyperbolic Matrix(323,126,182,71) (-2/5,-7/18) -> (7/4,16/9) Hyperbolic Matrix(295,114,546,211) (-12/31,-5/13) -> (7/13,6/11) Hyperbolic Matrix(419,160,364,139) (-13/34,-8/21) -> (8/7,7/6) Hyperbolic Matrix(153,58,182,69) (-8/21,-3/8) -> (5/6,6/7) Hyperbolic Matrix(279,104,448,167) (-3/8,-10/27) -> (18/29,5/8) Hyperbolic Matrix(601,222,490,181) (-10/27,-7/19) -> (11/9,16/13) 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(55,16,-196,-57) (-3/10,-2/7) -> (-2/7,-5/18) Parabolic Matrix(211,58,462,127) (-8/29,-3/11) -> (5/11,6/13) Hyperbolic Matrix(911,246,574,155) (-3/11,-7/26) -> (19/12,27/17) Hyperbolic Matrix(2297,616,3136,841) (-11/41,-15/56) -> (41/56,11/15) Hyperbolic Matrix(673,180,1260,337) (-15/56,-4/15) -> (8/15,15/28) Hyperbolic Matrix(295,78,798,211) (-4/15,-5/19) -> (7/19,10/27) Hyperbolic Matrix(83,20,112,27) (-1/4,-3/13) -> (11/15,3/4) Hyperbolic Matrix(293,66,182,41) (-3/13,-2/9) -> (8/5,21/13) 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(97,18,70,13) (-1/5,-1/6) -> (11/8,7/5) Hyperbolic Matrix(13,2,-98,-15) (-1/6,-1/7) -> (-1/7,-1/8) Parabolic Matrix(97,10,126,13) (-1/8,0/1) -> (10/13,17/22) Hyperbolic Matrix(15,-2,98,-13) (0/1,1/7) -> (1/7,2/13) Parabolic Matrix(251,-52,140,-29) (1/5,3/14) -> (25/14,9/5) Hyperbolic Matrix(139,-36,112,-29) (1/4,4/15) -> (16/13,5/4) Hyperbolic Matrix(223,-60,420,-113) (4/15,7/26) -> (1/2,8/15) Hyperbolic Matrix(265,-74,154,-43) (3/11,2/7) -> (12/7,19/11) Hyperbolic Matrix(71,-22,42,-13) (2/7,1/3) -> (5/3,12/7) 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(883,-338,546,-209) (8/21,5/13) -> (21/13,34/21) Hyperbolic Matrix(475,-184,364,-141) (5/13,7/18) -> (13/10,17/13) Hyperbolic Matrix(43,-18,98,-41) (2/5,3/7) -> (3/7,4/9) Parabolic Matrix(307,-144,420,-197) (6/13,1/2) -> (19/26,30/41) Hyperbolic Matrix(503,-276,308,-169) (6/11,5/9) -> (31/19,18/11) Hyperbolic Matrix(181,-102,126,-71) (5/9,4/7) -> (10/7,13/9) Hyperbolic Matrix(99,-58,70,-41) (4/7,3/5) -> (7/5,10/7) Hyperbolic Matrix(547,-338,882,-545) (8/13,13/21) -> (13/21,18/29) Parabolic Matrix(419,-268,308,-197) (7/11,9/14) -> (19/14,15/11) Hyperbolic Matrix(71,-50,98,-69) (2/3,5/7) -> (5/7,8/11) Parabolic 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(223,-180,140,-113) (4/5,9/11) -> (27/17,8/5) Hyperbolic Matrix(15,-14,14,-13) (6/7,1/1) -> (1/1,8/7) Parabolic Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(155,-242,98,-153) (3/2,11/7) -> (11/7,19/12) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-2,1) Matrix(43,78,70,127) -> Matrix(1,2,-4,-7) Matrix(29,52,-140,-251) -> Matrix(1,2,-2,-3) Matrix(71,126,182,323) -> Matrix(1,4,-4,-15) Matrix(13,22,-42,-71) -> Matrix(1,0,0,1) Matrix(29,48,-84,-139) -> Matrix(1,0,-2,1) Matrix(169,276,-308,-503) -> Matrix(5,-4,-6,5) Matrix(209,338,-546,-883) -> Matrix(1,-2,-2,5) Matrix(41,66,182,293) -> Matrix(1,-2,-2,5) Matrix(153,242,-98,-155) -> Matrix(1,-4,0,1) Matrix(125,194,-154,-239) -> Matrix(3,10,-4,-13) Matrix(41,58,-70,-99) -> Matrix(5,4,-4,-3) Matrix(13,18,70,97) -> Matrix(3,2,-8,-5) Matrix(127,174,154,211) -> Matrix(1,0,0,1) Matrix(197,268,-308,-419) -> Matrix(1,0,4,1) Matrix(125,162,-98,-127) -> Matrix(1,-2,0,1) Matrix(125,158,-322,-407) -> Matrix(1,2,-2,-3) Matrix(29,36,-112,-139) -> Matrix(3,4,-4,-5) Matrix(197,240,-252,-307) -> Matrix(1,4,-2,-7) Matrix(97,114,154,181) -> Matrix(3,4,-16,-21) Matrix(13,14,-14,-15) -> Matrix(1,0,0,1) Matrix(363,316,224,195) -> Matrix(7,8,-8,-9) Matrix(69,58,182,153) -> Matrix(9,8,-26,-23) Matrix(211,174,154,127) -> Matrix(5,4,6,5) Matrix(125,102,462,377) -> Matrix(31,24,-84,-65) Matrix(111,88,140,111) -> Matrix(3,2,-2,-1) Matrix(475,372,392,307) -> Matrix(3,2,-14,-9) Matrix(799,618,490,379) -> Matrix(3,2,-8,-5) Matrix(57,44,364,281) -> Matrix(11,6,-24,-13) Matrix(97,74,-350,-267) -> Matrix(3,4,-4,-5) Matrix(69,50,-98,-71) -> Matrix(3,2,-8,-5) Matrix(43,30,182,127) -> Matrix(3,2,-8,-5) Matrix(337,232,-812,-559) -> Matrix(1,0,0,1) Matrix(265,182,182,125) -> Matrix(5,2,-8,-3) Matrix(55,36,84,55) -> Matrix(1,0,-2,1) Matrix(531,340,392,251) -> Matrix(1,0,-4,1) Matrix(139,88,308,195) -> Matrix(1,0,-2,1) Matrix(181,114,154,97) -> Matrix(1,0,-2,1) Matrix(545,338,-882,-547) -> Matrix(1,-4,0,1) Matrix(379,234,1022,631) -> Matrix(1,2,-2,-3) Matrix(127,78,70,43) -> Matrix(1,2,-2,-3) Matrix(111,64,-196,-113) -> Matrix(15,16,-16,-17) Matrix(211,118,-574,-321) -> Matrix(9,8,-26,-23) Matrix(293,162,378,209) -> Matrix(7,6,-20,-17) Matrix(239,130,182,99) -> Matrix(5,4,6,5) Matrix(113,60,-420,-223) -> Matrix(25,18,-32,-23) Matrix(41,18,-98,-43) -> Matrix(7,4,-16,-9) Matrix(1105,458,2058,853) -> Matrix(7,4,-16,-9) Matrix(3289,1362,4494,1861) -> Matrix(17,8,36,17) Matrix(993,410,574,237) -> Matrix(23,10,-30,-13) Matrix(239,98,378,155) -> Matrix(5,2,-28,-11) Matrix(323,126,182,71) -> Matrix(5,2,-8,-3) Matrix(295,114,546,211) -> Matrix(3,2,-8,-5) Matrix(419,160,364,139) -> Matrix(5,2,-8,-3) Matrix(153,58,182,69) -> Matrix(1,0,0,1) Matrix(279,104,448,167) -> Matrix(3,2,-14,-9) Matrix(601,222,490,181) -> Matrix(5,2,-8,-3) Matrix(449,164,616,225) -> Matrix(1,0,6,1) Matrix(111,40,308,111) -> Matrix(1,0,0,1) Matrix(643,228,392,139) -> Matrix(1,0,0,1) Matrix(55,16,-196,-57) -> Matrix(7,8,-8,-9) Matrix(211,58,462,127) -> Matrix(5,4,-4,-3) Matrix(911,246,574,155) -> Matrix(5,4,-4,-3) Matrix(2297,616,3136,841) -> Matrix(29,22,54,41) Matrix(673,180,1260,337) -> Matrix(11,8,-18,-13) Matrix(295,78,798,211) -> Matrix(3,2,-2,-1) Matrix(83,20,112,27) -> Matrix(3,2,-2,-1) Matrix(293,66,182,41) -> Matrix(7,4,-2,-1) Matrix(55,12,252,55) -> Matrix(11,6,-24,-13) Matrix(699,148,392,83) -> Matrix(9,4,-16,-7) Matrix(97,18,70,13) -> Matrix(3,2,-2,-1) Matrix(13,2,-98,-15) -> Matrix(11,6,-24,-13) Matrix(97,10,126,13) -> Matrix(1,0,0,1) Matrix(15,-2,98,-13) -> Matrix(7,4,-16,-9) Matrix(251,-52,140,-29) -> Matrix(1,0,0,1) Matrix(139,-36,112,-29) -> Matrix(5,2,-18,-7) Matrix(223,-60,420,-113) -> Matrix(11,4,-14,-5) Matrix(265,-74,154,-43) -> Matrix(23,8,-26,-9) Matrix(71,-22,42,-13) -> Matrix(1,0,2,1) Matrix(139,-48,84,-29) -> Matrix(1,0,2,1) Matrix(447,-164,308,-113) -> Matrix(5,2,-8,-3) Matrix(883,-338,546,-209) -> Matrix(25,8,-22,-7) Matrix(475,-184,364,-141) -> Matrix(7,2,24,7) Matrix(43,-18,98,-41) -> Matrix(1,0,2,1) Matrix(307,-144,420,-197) -> Matrix(5,2,12,5) Matrix(503,-276,308,-169) -> Matrix(5,2,-8,-3) Matrix(181,-102,126,-71) -> Matrix(11,4,-14,-5) Matrix(99,-58,70,-41) -> Matrix(7,2,-4,-1) Matrix(547,-338,882,-545) -> Matrix(23,6,-96,-25) Matrix(419,-268,308,-197) -> Matrix(1,0,8,1) Matrix(71,-50,98,-69) -> Matrix(1,0,8,1) Matrix(195,-148,112,-85) -> Matrix(1,2,-2,-3) Matrix(307,-240,252,-197) -> Matrix(1,0,2,1) Matrix(223,-180,140,-113) -> Matrix(1,0,2,1) Matrix(15,-14,14,-13) -> Matrix(1,0,2,1) Matrix(127,-162,98,-125) -> Matrix(1,0,8,1) Matrix(155,-242,98,-153) -> 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): 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 -1/1 (-1/1,0/1) 0 14 -6/7 -1/1 12 2 -5/6 -6/7 1 14 -4/5 0 14 -11/14 -2/3 3 2 -7/9 -2/3 2 14 -10/13 0 14 -3/4 -2/3 1 14 -5/7 -1/2 2 2 -7/10 0/1 1 14 -9/13 (-1/1,-1/2) 0 14 -2/3 0 14 -9/14 0/1 5 2 -7/11 (0/1,1/1) 0 14 -5/8 0/1 1 14 -3/5 -2/1 2 14 -4/7 -1/1 8 2 -5/9 (-1/1,-6/7) 0 14 -6/11 0 14 -1/2 -2/3 1 14 -3/7 -1/2 4 2 -5/12 -2/5 1 14 -7/17 -2/5 2 14 -2/5 0 14 -5/13 (-1/2,-1/3) 0 14 -8/21 -1/3 1 2 -3/8 0/1 1 14 -7/19 (-2/5,-1/3) 0 14 -4/11 0 14 -5/14 0/1 1 2 -1/3 0/1 2 14 -2/7 -1/1 4 2 -3/11 (-1/1,-4/5) 0 14 -4/15 0 14 -1/4 -2/3 1 14 -2/9 0 14 -3/14 -1/2 5 2 -1/5 (-1/1,-1/2) 0 14 0/1 0 14 1/7 -1/2 4 2 2/13 0 14 1/6 -2/5 1 14 1/5 0/1 2 14 3/14 -1/2 5 2 2/9 0 14 3/13 (-1/2,-1/3) 0 14 1/4 -2/5 1 14 4/15 0 14 7/26 -4/11 1 14 3/11 -4/11 2 14 2/7 -1/3 4 2 1/3 (-1/3,0/1) 0 14 5/14 0/1 1 2 4/11 0 14 7/19 0/1 2 14 10/27 0 14 3/8 -2/5 1 14 8/21 -1/3 12 2 5/13 -4/13 2 14 7/18 -2/7 1 14 2/5 0 14 3/7 0/1 2 2 4/9 0 14 5/11 (-1/1,0/1) 0 14 6/13 0 14 1/2 0/1 1 14 8/15 0 14 15/28 -1/2 11 2 7/13 (-1/2,-3/7) 0 14 6/11 0 14 5/9 -2/5 2 14 4/7 -1/3 3 2 3/5 (-1/3,-1/4) 0 14 8/13 0 14 13/21 -1/4 6 2 18/29 0 14 5/8 -2/9 1 14 12/19 0 14 7/11 0/1 2 14 9/14 0/1 5 2 2/3 0 14 5/7 0/1 8 2 8/11 0 14 19/26 2/5 1 14 30/41 0 14 41/56 1/2 11 2 11/15 2/3 2 14 3/4 0/1 1 14 10/13 0 14 17/22 -2/5 1 14 7/9 (-1/3,0/1) 0 14 11/14 0/1 3 2 4/5 0 14 9/11 (-1/3,0/1) 0 14 5/6 0/1 1 14 6/7 -1/3 1 2 1/1 0/1 2 14 1/0 0/1 1 2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(13,12,-14,-13) (-1/1,-6/7) -> (-1/1,-6/7) Reflection Matrix(69,58,182,153) (-6/7,-5/6) -> (3/8,8/21) Hyperbolic Matrix(97,80,154,127) (-5/6,-4/5) -> (5/8,12/19) Glide Reflection Matrix(111,88,140,111) (-4/5,-11/14) -> (11/14,4/5) Hyperbolic Matrix(197,154,-252,-197) (-11/14,-7/9) -> (-11/14,-7/9) Reflection Matrix(181,140,490,379) (-7/9,-10/13) -> (7/19,10/27) Glide Reflection Matrix(29,22,-112,-85) (-10/13,-3/4) -> (-4/15,-1/4) Glide Reflection Matrix(69,50,-98,-71) (-3/4,-5/7) -> (-5/7,-7/10) Parabolic Matrix(43,30,182,127) (-7/10,-9/13) -> (3/13,1/4) Hyperbolic Matrix(99,68,182,125) (-9/13,-2/3) -> (7/13,6/11) Glide Reflection Matrix(55,36,84,55) (-2/3,-9/14) -> (9/14,2/3) Hyperbolic Matrix(197,126,-308,-197) (-9/14,-7/11) -> (-9/14,-7/11) Reflection Matrix(127,80,154,97) (-7/11,-5/8) -> (9/11,5/6) Glide Reflection Matrix(13,8,70,43) (-5/8,-3/5) -> (1/6,1/5) Glide Reflection Matrix(41,24,-70,-41) (-3/5,-4/7) -> (-3/5,-4/7) Reflection Matrix(71,40,-126,-71) (-4/7,-5/9) -> (-4/7,-5/9) Reflection Matrix(113,62,-308,-169) (-5/9,-6/11) -> (-7/19,-4/11) Glide Reflection Matrix(225,122,308,167) (-6/11,-1/2) -> (8/11,19/26) Glide Reflection Matrix(41,18,-98,-43) (-1/2,-3/7) -> (-3/7,-5/12) Parabolic Matrix(155,64,574,237) (-5/12,-7/17) -> (7/26,3/11) Glide Reflection Matrix(239,98,378,155) (-7/17,-2/5) -> (12/19,7/11) Hyperbolic Matrix(41,16,182,71) (-2/5,-5/13) -> (2/9,3/13) Glide Reflection Matrix(209,80,-546,-209) (-5/13,-8/21) -> (-5/13,-8/21) Reflection Matrix(153,58,182,69) (-8/21,-3/8) -> (5/6,6/7) Hyperbolic Matrix(379,140,490,181) (-3/8,-7/19) -> (17/22,7/9) Glide Reflection Matrix(111,40,308,111) (-4/11,-5/14) -> (5/14,4/11) Hyperbolic Matrix(29,10,-84,-29) (-5/14,-1/3) -> (-5/14,-1/3) Reflection Matrix(13,4,-42,-13) (-1/3,-2/7) -> (-1/3,-2/7) Reflection Matrix(43,12,-154,-43) (-2/7,-3/11) -> (-2/7,-3/11) Reflection Matrix(141,38,308,83) (-3/11,-4/15) -> (5/11,6/13) Glide Reflection Matrix(71,16,182,41) (-1/4,-2/9) -> (7/18,2/5) Glide Reflection Matrix(55,12,252,55) (-2/9,-3/14) -> (3/14,2/9) Hyperbolic Matrix(29,6,-140,-29) (-3/14,-1/5) -> (-3/14,-1/5) Reflection Matrix(43,8,70,13) (-1/5,0/1) -> (3/5,8/13) Glide Reflection Matrix(15,-2,98,-13) (0/1,1/7) -> (1/7,2/13) Parabolic Matrix(99,-16,266,-43) (2/13,1/6) -> (10/27,3/8) Glide Reflection Matrix(29,-6,140,-29) (1/5,3/14) -> (1/5,3/14) Reflection Matrix(85,-22,112,-29) (1/4,4/15) -> (3/4,10/13) Glide Reflection Matrix(223,-60,420,-113) (4/15,7/26) -> (1/2,8/15) Hyperbolic Matrix(43,-12,154,-43) (3/11,2/7) -> (3/11,2/7) Reflection Matrix(13,-4,42,-13) (2/7,1/3) -> (2/7,1/3) Reflection Matrix(29,-10,84,-29) (1/3,5/14) -> (1/3,5/14) Reflection Matrix(169,-62,308,-113) (4/11,7/19) -> (6/11,5/9) Glide Reflection Matrix(209,-80,546,-209) (8/21,5/13) -> (8/21,5/13) Reflection Matrix(237,-92,322,-125) (5/13,7/18) -> (11/15,3/4) Glide Reflection Matrix(43,-18,98,-41) (2/5,3/7) -> (3/7,4/9) Parabolic Matrix(125,-56,154,-69) (4/9,5/11) -> (4/5,9/11) Glide Reflection Matrix(307,-144,420,-197) (6/13,1/2) -> (19/26,30/41) Hyperbolic Matrix(1455,-778,1988,-1063) (8/15,15/28) -> (30/41,41/56) Glide Reflection Matrix(391,-210,728,-391) (15/28,7/13) -> (15/28,7/13) Reflection Matrix(71,-40,126,-71) (5/9,4/7) -> (5/9,4/7) Reflection Matrix(41,-24,70,-41) (4/7,3/5) -> (4/7,3/5) Reflection Matrix(547,-338,882,-545) (8/13,13/21) -> (13/21,18/29) Parabolic Matrix(573,-356,742,-461) (18/29,5/8) -> (10/13,17/22) Glide Reflection Matrix(197,-126,308,-197) (7/11,9/14) -> (7/11,9/14) Reflection Matrix(71,-50,98,-69) (2/3,5/7) -> (5/7,8/11) Parabolic Matrix(1231,-902,1680,-1231) (41/56,11/15) -> (41/56,11/15) Reflection Matrix(197,-154,252,-197) (7/9,11/14) -> (7/9,11/14) Reflection Matrix(13,-12,14,-13) (6/7,1/1) -> (6/7,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(13,12,-14,-13) -> Matrix(-1,0,2,1) (-1/1,-6/7) -> (-1/1,0/1) Matrix(69,58,182,153) -> Matrix(9,8,-26,-23) Matrix(97,80,154,127) -> Matrix(5,4,-26,-21) Matrix(111,88,140,111) -> Matrix(3,2,-2,-1) -1/1 Matrix(197,154,-252,-197) -> Matrix(7,4,-12,-7) (-11/14,-7/9) -> (-2/3,-1/2) Matrix(181,140,490,379) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(29,22,-112,-85) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(69,50,-98,-71) -> Matrix(3,2,-8,-5) -1/2 Matrix(43,30,182,127) -> Matrix(3,2,-8,-5) -1/2 Matrix(99,68,182,125) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(55,36,84,55) -> Matrix(1,0,-2,1) 0/1 Matrix(197,126,-308,-197) -> Matrix(1,0,2,-1) (-9/14,-7/11) -> (0/1,1/1) Matrix(127,80,154,97) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(13,8,70,43) -> Matrix(1,2,-2,-5) Matrix(41,24,-70,-41) -> Matrix(3,4,-2,-3) (-3/5,-4/7) -> (-2/1,-1/1) Matrix(71,40,-126,-71) -> Matrix(13,12,-14,-13) (-4/7,-5/9) -> (-1/1,-6/7) Matrix(113,62,-308,-169) -> Matrix(5,4,-16,-13) Matrix(225,122,308,167) -> Matrix(5,4,14,11) Matrix(41,18,-98,-43) -> Matrix(7,4,-16,-9) -1/2 Matrix(155,64,574,237) -> Matrix(23,10,-62,-27) Matrix(239,98,378,155) -> Matrix(5,2,-28,-11) Matrix(41,16,182,71) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(209,80,-546,-209) -> Matrix(5,2,-12,-5) (-5/13,-8/21) -> (-1/2,-1/3) Matrix(153,58,182,69) -> Matrix(1,0,0,1) Matrix(379,140,490,181) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(111,40,308,111) -> Matrix(1,0,0,1) Matrix(29,10,-84,-29) -> Matrix(-1,0,4,1) (-5/14,-1/3) -> (-1/2,0/1) Matrix(13,4,-42,-13) -> Matrix(-1,0,2,1) (-1/3,-2/7) -> (-1/1,0/1) Matrix(43,12,-154,-43) -> Matrix(9,8,-10,-9) (-2/7,-3/11) -> (-1/1,-4/5) Matrix(141,38,308,83) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(71,16,182,41) -> Matrix(7,4,-26,-15) Matrix(55,12,252,55) -> Matrix(11,6,-24,-13) -1/2 Matrix(29,6,-140,-29) -> Matrix(3,2,-4,-3) (-3/14,-1/5) -> (-1/1,-1/2) Matrix(43,8,70,13) -> Matrix(3,2,-10,-7) Matrix(15,-2,98,-13) -> Matrix(7,4,-16,-9) -1/2 Matrix(99,-16,266,-43) -> Matrix(9,4,-20,-9) *** -> (-1/2,-2/5) Matrix(29,-6,140,-29) -> Matrix(-1,0,4,1) (1/5,3/14) -> (-1/2,0/1) Matrix(85,-22,112,-29) -> Matrix(5,2,-2,-1) Matrix(223,-60,420,-113) -> Matrix(11,4,-14,-5) Matrix(43,-12,154,-43) -> Matrix(23,8,-66,-23) (3/11,2/7) -> (-4/11,-1/3) Matrix(13,-4,42,-13) -> Matrix(-1,0,6,1) (2/7,1/3) -> (-1/3,0/1) Matrix(29,-10,84,-29) -> Matrix(-1,0,6,1) (1/3,5/14) -> (-1/3,0/1) Matrix(169,-62,308,-113) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(209,-80,546,-209) -> Matrix(25,8,-78,-25) (8/21,5/13) -> (-1/3,-4/13) Matrix(237,-92,322,-125) -> Matrix(7,2,4,1) Matrix(43,-18,98,-41) -> Matrix(1,0,2,1) 0/1 Matrix(125,-56,154,-69) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(307,-144,420,-197) -> Matrix(5,2,12,5) Matrix(1455,-778,1988,-1063) -> Matrix(11,6,24,13) Matrix(391,-210,728,-391) -> Matrix(13,6,-28,-13) (15/28,7/13) -> (-1/2,-3/7) Matrix(71,-40,126,-71) -> Matrix(11,4,-30,-11) (5/9,4/7) -> (-2/5,-1/3) Matrix(41,-24,70,-41) -> Matrix(7,2,-24,-7) (4/7,3/5) -> (-1/3,-1/4) Matrix(547,-338,882,-545) -> Matrix(23,6,-96,-25) -1/4 Matrix(573,-356,742,-461) -> Matrix(17,4,-38,-9) Matrix(197,-126,308,-197) -> Matrix(-1,0,12,1) (7/11,9/14) -> (-1/6,0/1) Matrix(71,-50,98,-69) -> Matrix(1,0,8,1) 0/1 Matrix(1231,-902,1680,-1231) -> Matrix(7,-4,12,-7) (41/56,11/15) -> (1/2,2/3) Matrix(197,-154,252,-197) -> Matrix(-1,0,6,1) (7/9,11/14) -> (-1/3,0/1) Matrix(13,-12,14,-13) -> Matrix(-1,0,6,1) (6/7,1/1) -> (-1/3,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,4,1) (1/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.