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 -1/1 1/1 -9/5 0/1 1/0 -16/9 -1/1 1/1 -7/4 -1/1 -1/3 -5/3 1/1 -18/11 -5/1 -3/1 -13/8 -5/3 -1/1 -21/13 -1/1 -2/3 -8/5 -1/1 -1/3 -11/7 0/1 -14/9 1/3 1/1 -3/2 -1/1 1/1 -7/5 1/1 -11/8 -3/1 -1/1 -15/11 -1/2 0/1 -4/3 -1/1 1/1 -9/7 0/1 -14/11 1/3 1/1 -5/4 -1/1 1/1 -11/9 -1/1 -6/5 -1/1 1/1 -7/6 -1/1 1/1 -1/1 0/1 1/0 -7/8 -1/1 1/1 -6/7 0/1 -5/6 -1/1 1/1 -9/11 1/1 -13/16 -1/1 -1/3 -4/5 -1/1 1/1 -11/14 0/1 -18/23 1/3 1/1 -7/9 1/1 -17/22 -1/1 1/1 -10/13 -3/1 -1/1 -3/4 -1/1 1/1 -5/7 0/1 -7/10 1/3 1/1 -9/13 0/1 1/1 -11/16 1/1 3/1 -2/3 -1/1 1/1 -9/14 0/1 -16/25 1/5 1/3 -7/11 0/1 1/2 -12/19 3/5 1/1 -5/8 1/1 3/1 -13/21 1/0 -21/34 -3/1 -1/1 -8/13 -1/1 1/1 -3/5 -1/1 -7/12 -1/1 -1/3 -4/7 0/1 -9/16 1/5 1/3 -5/9 0/1 1/2 -11/20 1/5 1/3 -6/11 3/5 1/1 -7/13 1/1 -1/2 -1/1 1/1 -3/7 0/1 -5/12 1/5 1/3 -17/41 2/7 1/3 -29/70 1/3 -12/29 1/3 5/13 -7/17 1/1 -2/5 1/3 1/1 -7/18 3/5 1/1 -12/31 5/7 1/1 -5/13 2/3 1/1 -13/34 13/15 1/1 -8/21 1/1 -3/8 1/1 5/3 -10/27 7/3 3/1 -7/19 2/1 1/0 -11/30 7/3 3/1 -4/11 3/1 5/1 -5/14 1/0 -6/17 -7/1 -5/1 -1/3 -1/1 -3/10 -1/3 -1/5 -2/7 0/1 -5/18 1/7 1/5 -8/29 3/17 1/5 -3/11 0/1 1/4 -7/26 3/11 1/3 -11/41 1/3 -15/56 1/3 -4/15 1/3 3/7 -5/19 1/1 -1/4 1/3 1/1 -3/13 1/1 -2/9 -1/1 1/1 -3/14 1/0 -4/19 -1/1 1/1 -1/5 0/1 1/0 -1/6 -1/1 1/1 -1/7 0/1 -1/8 1/5 1/3 0/1 -1/1 1/1 1/7 1/0 2/13 -5/1 -3/1 1/6 -3/1 -1/1 1/5 1/1 3/14 1/0 2/9 -1/1 1/1 3/13 1/1 2/1 1/4 1/1 3/1 4/15 13/3 5/1 7/26 5/1 17/3 3/11 7/1 2/7 1/0 1/3 -1/1 1/0 5/14 1/0 4/11 -5/1 -3/1 7/19 -3/1 10/27 -1/1 1/1 3/8 -3/1 -1/1 8/21 -2/1 5/13 -1/1 7/18 -1/1 -1/3 2/5 -1/1 1/1 3/7 1/0 4/9 -7/1 -5/1 5/11 -4/1 1/0 6/13 -17/5 -3/1 1/2 -3/1 -1/1 8/15 -9/7 -1/1 15/28 -1/1 7/13 -1/1 0/1 6/11 -1/1 1/1 5/9 -1/1 4/7 1/0 3/5 -2/1 1/0 8/13 -3/1 -1/1 13/21 -2/1 18/29 -9/5 -5/3 5/8 -3/1 -1/1 12/19 -3/1 -1/1 7/11 -3/1 9/14 -2/1 2/3 -5/3 -1/1 5/7 -1/1 8/11 -1/1 -3/5 19/26 -1/1 1/1 30/41 -7/5 -1/1 41/56 -1/1 11/15 -1/1 3/4 -1/1 1/1 10/13 -3/1 -7/3 17/22 -5/3 -1/1 7/9 -2/1 1/0 11/14 -2/1 4/5 -5/3 -1/1 9/11 -3/2 -4/3 5/6 -11/9 -1/1 6/7 -1/1 1/1 -1/1 8/7 -1/1 7/6 -1/1 -11/13 6/5 -1/1 -5/7 17/14 -2/3 11/9 -2/3 -1/2 16/13 -7/11 -3/5 5/4 -1/1 -1/3 9/7 -1/1 13/10 -1/1 -9/11 17/13 -1/1 4/3 -1/1 -5/7 19/14 -2/3 15/11 -3/5 11/8 -1/1 -3/5 7/5 -2/3 -1/2 10/7 -1/2 13/9 -1/1 16/11 -1/1 -1/3 3/2 -1/1 -3/5 11/7 -1/2 19/12 -5/11 -3/7 27/17 -1/2 -2/5 8/5 -1/1 -1/3 21/13 -1/1 34/21 -2/3 13/8 -1/1 -3/5 31/19 -3/5 18/11 -3/5 -5/9 23/14 -1/2 5/3 -1/1 -1/2 12/7 -1/2 19/11 -7/15 26/15 -5/11 -13/29 7/4 -3/7 -1/3 16/9 -1/1 -1/3 25/14 -1/2 9/5 -1/3 2/1 -1/1 -1/3 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,-2,0,1) Matrix(29,52,-140,-251) -> Matrix(1,0,0,1) Matrix(71,126,182,323) -> Matrix(1,0,0,1) Matrix(13,22,-42,-71) -> Matrix(1,0,-2,1) Matrix(29,48,-84,-139) -> Matrix(1,-2,0,1) Matrix(169,276,-308,-503) -> Matrix(1,2,2,5) Matrix(209,338,-546,-883) -> Matrix(5,4,6,5) Matrix(41,66,182,293) -> Matrix(1,0,2,1) Matrix(153,242,-98,-155) -> Matrix(1,0,4,1) Matrix(125,194,-154,-239) -> Matrix(1,0,-2,1) Matrix(41,58,-70,-99) -> Matrix(1,0,-2,1) Matrix(13,18,70,97) -> Matrix(1,0,0,1) Matrix(127,174,154,211) -> Matrix(5,4,-4,-3) Matrix(197,268,-308,-419) -> Matrix(1,0,4,1) Matrix(125,162,-98,-127) -> Matrix(1,0,2,1) Matrix(125,158,-322,-407) -> Matrix(1,-2,2,-3) Matrix(29,36,-112,-139) -> Matrix(1,0,2,1) Matrix(197,240,-252,-307) -> Matrix(1,0,2,1) Matrix(97,114,154,181) -> Matrix(1,-2,0,1) Matrix(13,14,-14,-15) -> Matrix(1,0,0,1) Matrix(363,316,224,195) -> Matrix(1,2,-2,-3) Matrix(69,58,182,153) -> Matrix(1,-2,0,1) Matrix(211,174,154,127) -> Matrix(1,2,-2,-3) Matrix(125,102,462,377) -> Matrix(1,6,0,1) Matrix(111,88,140,111) -> Matrix(3,2,-2,-1) Matrix(475,372,392,307) -> Matrix(7,-2,-10,3) Matrix(799,618,490,379) -> Matrix(1,2,-2,-3) Matrix(57,44,364,281) -> Matrix(1,-2,0,1) Matrix(97,74,-350,-267) -> Matrix(1,0,6,1) Matrix(69,50,-98,-71) -> Matrix(1,0,2,1) Matrix(43,30,182,127) -> Matrix(3,-2,2,-1) Matrix(337,232,-812,-559) -> Matrix(1,-2,4,-7) Matrix(265,182,182,125) -> Matrix(1,0,-2,1) Matrix(55,36,84,55) -> Matrix(3,2,-2,-1) Matrix(531,340,392,251) -> Matrix(11,-2,-16,3) Matrix(139,88,308,195) -> Matrix(9,-4,-2,1) Matrix(181,114,154,97) -> Matrix(5,-4,-6,5) Matrix(545,338,-882,-547) -> Matrix(1,-4,0,1) Matrix(379,234,1022,631) -> Matrix(1,0,0,1) Matrix(127,78,70,43) -> Matrix(1,0,-2,1) Matrix(111,64,-196,-113) -> Matrix(1,0,6,1) Matrix(211,118,-574,-321) -> Matrix(3,-2,2,-1) Matrix(293,162,378,209) -> Matrix(5,-2,-2,1) Matrix(239,130,182,99) -> Matrix(5,-4,-6,5) Matrix(113,60,-420,-223) -> Matrix(1,-2,4,-7) Matrix(41,18,-98,-43) -> Matrix(1,0,4,1) Matrix(1105,458,2058,853) -> Matrix(7,-2,-10,3) Matrix(3289,1362,4494,1861) -> Matrix(17,-6,-14,5) Matrix(993,410,574,237) -> Matrix(13,-6,-28,13) Matrix(239,98,378,155) -> Matrix(5,-2,-2,1) Matrix(323,126,182,71) -> Matrix(1,0,-4,1) Matrix(295,114,546,211) -> Matrix(3,-2,-4,3) Matrix(419,160,364,139) -> Matrix(13,-12,-14,13) Matrix(153,58,182,69) -> Matrix(7,-8,-6,7) Matrix(279,104,448,167) -> Matrix(3,-4,-2,3) Matrix(601,222,490,181) -> Matrix(1,0,-2,1) Matrix(449,164,616,225) -> Matrix(1,-2,-2,5) Matrix(111,40,308,111) -> Matrix(1,-8,0,1) Matrix(643,228,392,139) -> Matrix(1,10,-2,-19) Matrix(55,16,-196,-57) -> Matrix(1,0,10,1) Matrix(211,58,462,127) -> Matrix(17,-4,-4,1) Matrix(911,246,574,155) -> Matrix(9,-2,-22,5) Matrix(2297,616,3136,841) -> Matrix(13,-4,-16,5) Matrix(673,180,1260,337) -> Matrix(17,-6,-14,5) Matrix(295,78,798,211) -> Matrix(5,-2,-2,1) Matrix(83,20,112,27) -> Matrix(1,0,-2,1) Matrix(293,66,182,41) -> Matrix(1,0,-2,1) Matrix(55,12,252,55) -> Matrix(1,0,0,1) Matrix(699,148,392,83) -> Matrix(1,0,-2,1) Matrix(97,18,70,13) -> Matrix(1,2,-2,-3) Matrix(13,2,-98,-15) -> Matrix(1,0,4,1) Matrix(97,10,126,13) -> Matrix(5,-2,-2,1) Matrix(15,-2,98,-13) -> Matrix(1,-4,0,1) Matrix(251,-52,140,-29) -> Matrix(1,-2,-2,5) Matrix(139,-36,112,-29) -> Matrix(1,-2,-2,5) Matrix(223,-60,420,-113) -> Matrix(3,-16,-2,11) Matrix(265,-74,154,-43) -> Matrix(1,-14,-2,29) Matrix(71,-22,42,-13) -> Matrix(1,2,-2,-3) Matrix(139,-48,84,-29) -> Matrix(1,2,-2,-3) Matrix(447,-164,308,-113) -> Matrix(1,4,-2,-7) Matrix(883,-338,546,-209) -> Matrix(3,4,-4,-5) Matrix(475,-184,364,-141) -> Matrix(3,4,-4,-5) Matrix(43,-18,98,-41) -> Matrix(1,-6,0,1) Matrix(307,-144,420,-197) -> Matrix(1,2,0,1) Matrix(503,-276,308,-169) -> Matrix(1,4,-2,-7) Matrix(181,-102,126,-71) -> Matrix(1,2,-2,-3) Matrix(99,-58,70,-41) -> Matrix(1,4,-2,-7) Matrix(547,-338,882,-545) -> Matrix(7,16,-4,-9) Matrix(419,-268,308,-197) -> Matrix(5,12,-8,-19) Matrix(71,-50,98,-69) -> Matrix(3,4,-4,-5) Matrix(195,-148,112,-85) -> Matrix(1,-2,-2,5) Matrix(307,-240,252,-197) -> Matrix(1,4,-2,-7) Matrix(223,-180,140,-113) -> Matrix(1,2,-4,-7) Matrix(15,-14,14,-13) -> Matrix(1,2,-2,-3) Matrix(127,-162,98,-125) -> Matrix(3,4,-4,-5) Matrix(155,-242,98,-153) -> Matrix(11,6,-24,-13) 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): 28 Degree of the the map X: 28 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 -11/7 -1/1 -5/7 -13/21 -4/7 -5/9 -1/2 -3/7 -1/3 -2/7 -3/11 0/1 1/7 1/5 3/13 2/7 1/3 3/7 1/2 4/7 3/5 13/21 2/3 5/7 6/7 1/1 9/7 4/3 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 1/1 -7/4 -1/1 -1/3 -5/3 1/1 -13/8 -5/3 -1/1 -21/13 -1/1 -2/3 -8/5 -1/1 -1/3 -11/7 0/1 -3/2 -1/1 1/1 -7/5 1/1 -11/8 -3/1 -1/1 -15/11 -1/2 0/1 -4/3 -1/1 1/1 -1/1 0/1 1/0 -3/4 -1/1 1/1 -5/7 0/1 -7/10 1/3 1/1 -9/13 0/1 1/1 -2/3 -1/1 1/1 -7/11 0/1 1/2 -12/19 3/5 1/1 -5/8 1/1 3/1 -13/21 1/0 -8/13 -1/1 1/1 -3/5 -1/1 -7/12 -1/1 -1/3 -4/7 0/1 -5/9 0/1 1/2 -1/2 -1/1 1/1 -3/7 0/1 -2/5 1/3 1/1 -7/18 3/5 1/1 -5/13 2/3 1/1 -13/34 13/15 1/1 -8/21 1/1 -3/8 1/1 5/3 -4/11 3/1 5/1 -1/3 -1/1 -3/10 -1/3 -1/5 -2/7 0/1 -5/18 1/7 1/5 -3/11 0/1 1/4 -1/4 1/3 1/1 0/1 -1/1 1/1 1/7 1/0 1/6 -3/1 -1/1 1/5 1/1 2/9 -1/1 1/1 3/13 1/1 2/1 1/4 1/1 3/1 4/15 13/3 5/1 3/11 7/1 2/7 1/0 1/3 -1/1 1/0 2/5 -1/1 1/1 3/7 1/0 4/9 -7/1 -5/1 5/11 -4/1 1/0 1/2 -3/1 -1/1 5/9 -1/1 4/7 1/0 3/5 -2/1 1/0 8/13 -3/1 -1/1 13/21 -2/1 18/29 -9/5 -5/3 5/8 -3/1 -1/1 2/3 -5/3 -1/1 5/7 -1/1 3/4 -1/1 1/1 10/13 -3/1 -7/3 17/22 -5/3 -1/1 7/9 -2/1 1/0 11/14 -2/1 4/5 -5/3 -1/1 9/11 -3/2 -4/3 5/6 -11/9 -1/1 6/7 -1/1 1/1 -1/1 8/7 -1/1 7/6 -1/1 -11/13 6/5 -1/1 -5/7 11/9 -2/3 -1/2 16/13 -7/11 -3/5 5/4 -1/1 -1/3 9/7 -1/1 4/3 -1/1 -5/7 11/8 -1/1 -3/5 7/5 -2/3 -1/2 10/7 -1/2 13/9 -1/1 3/2 -1/1 -3/5 5/3 -1/1 -1/2 12/7 -1/2 19/11 -7/15 7/4 -3/7 -1/3 16/9 -1/1 -1/3 9/5 -1/3 2/1 -1/1 -1/3 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(48,85,35,62) (-2/1,-7/4) -> (4/3,11/8) Hyperbolic Matrix(13,22,-42,-71) (-7/4,-5/3) -> (-1/3,-3/10) Hyperbolic Matrix(36,59,133,218) (-5/3,-13/8) -> (4/15,3/11) 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(76,121,-49,-78) (-8/5,-11/7) -> (-11/7,-3/2) Parabolic 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(64,87,-231,-314) (-15/11,-4/3) -> (-5/18,-3/11) Hyperbolic Matrix(6,7,-7,-8) (-4/3,-1/1) -> (-1/1,-3/4) Parabolic 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(106,73,-273,-188) (-9/13,-2/3) -> (-7/18,-5/13) Hyperbolic Matrix(20,13,-77,-50) (-2/3,-7/11) -> (-3/11,-1/4) 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(272,169,-441,-274) (-5/8,-13/21) -> (-13/21,-8/13) Parabolic Matrix(127,78,70,43) (-8/13,-3/5) -> (9/5,2/1) Hyperbolic Matrix(160,93,203,118) (-7/12,-4/7) -> (11/14,4/5) Hyperbolic Matrix(148,83,189,106) (-4/7,-5/9) -> (7/9,11/14) Hyperbolic Matrix(76,41,63,34) (-5/9,-1/2) -> (6/5,11/9) Hyperbolic Matrix(20,9,-49,-22) (-1/2,-3/7) -> (-3/7,-2/5) Parabolic Matrix(323,126,182,71) (-2/5,-7/18) -> (7/4,16/9) 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(134,49,175,64) (-3/8,-4/11) -> (3/4,10/13) Hyperbolic Matrix(230,83,133,48) (-4/11,-1/3) -> (19/11,7/4) Hyperbolic Matrix(55,16,-196,-57) (-3/10,-2/7) -> (-2/7,-5/18) Parabolic Matrix(22,5,35,8) (-1/4,0/1) -> (5/8,2/3) Hyperbolic Matrix(8,-1,49,-6) (0/1,1/7) -> (1/7,1/6) Parabolic Matrix(50,-11,91,-20) (1/5,2/9) -> (1/2,5/9) Hyperbolic Matrix(139,-36,112,-29) (1/4,4/15) -> (16/13,5/4) 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(34,-13,21,-8) (1/3,2/5) -> (3/2,5/3) Hyperbolic Matrix(43,-18,98,-41) (2/5,3/7) -> (3/7,4/9) Parabolic Matrix(62,-29,77,-36) (5/11,1/2) -> (4/5,9/11) 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(146,-89,105,-64) (3/5,8/13) -> (11/8,7/5) Hyperbolic Matrix(547,-338,882,-545) (8/13,13/21) -> (13/21,18/29) Parabolic Matrix(426,-265,553,-344) (18/29,5/8) -> (10/13,17/22) Hyperbolic Matrix(36,-25,49,-34) (2/3,5/7) -> (5/7,3/4) Parabolic Matrix(386,-299,315,-244) (17/22,7/9) -> (11/9,16/13) Hyperbolic Matrix(15,-14,14,-13) (6/7,1/1) -> (1/1,8/7) Parabolic Matrix(64,-81,49,-62) (5/4,9/7) -> (9/7,4/3) Parabolic Matrix(162,-235,91,-132) (13/9,3/2) -> (16/9,9/5) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-2,1) Matrix(48,85,35,62) -> Matrix(1,2,-2,-3) Matrix(13,22,-42,-71) -> Matrix(1,0,-2,1) Matrix(36,59,133,218) -> Matrix(1,6,0,1) Matrix(209,338,-546,-883) -> Matrix(5,4,6,5) Matrix(41,66,182,293) -> Matrix(1,0,2,1) Matrix(76,121,-49,-78) -> Matrix(1,0,2,1) Matrix(41,58,-70,-99) -> Matrix(1,0,-2,1) Matrix(13,18,70,97) -> Matrix(1,0,0,1) Matrix(127,174,154,211) -> Matrix(5,4,-4,-3) Matrix(64,87,-231,-314) -> Matrix(1,0,6,1) Matrix(6,7,-7,-8) -> Matrix(1,0,0,1) Matrix(69,50,-98,-71) -> Matrix(1,0,2,1) Matrix(43,30,182,127) -> Matrix(3,-2,2,-1) Matrix(106,73,-273,-188) -> Matrix(1,-2,2,-3) Matrix(20,13,-77,-50) -> Matrix(1,0,2,1) Matrix(139,88,308,195) -> Matrix(9,-4,-2,1) Matrix(181,114,154,97) -> Matrix(5,-4,-6,5) Matrix(272,169,-441,-274) -> Matrix(1,-2,0,1) Matrix(127,78,70,43) -> Matrix(1,0,-2,1) Matrix(160,93,203,118) -> Matrix(7,2,-4,-1) Matrix(148,83,189,106) -> Matrix(5,-2,-2,1) Matrix(76,41,63,34) -> Matrix(3,-2,-4,3) Matrix(20,9,-49,-22) -> Matrix(1,0,2,1) Matrix(323,126,182,71) -> Matrix(1,0,-4,1) Matrix(419,160,364,139) -> Matrix(13,-12,-14,13) Matrix(153,58,182,69) -> Matrix(7,-8,-6,7) Matrix(134,49,175,64) -> Matrix(1,-4,0,1) Matrix(230,83,133,48) -> Matrix(1,-6,-2,13) Matrix(55,16,-196,-57) -> Matrix(1,0,10,1) Matrix(22,5,35,8) -> Matrix(1,-2,0,1) Matrix(8,-1,49,-6) -> Matrix(1,-2,0,1) Matrix(50,-11,91,-20) -> Matrix(1,-2,0,1) Matrix(139,-36,112,-29) -> Matrix(1,-2,-2,5) Matrix(265,-74,154,-43) -> Matrix(1,-14,-2,29) Matrix(71,-22,42,-13) -> Matrix(1,2,-2,-3) Matrix(34,-13,21,-8) -> Matrix(1,2,-2,-3) Matrix(43,-18,98,-41) -> Matrix(1,-6,0,1) Matrix(62,-29,77,-36) -> Matrix(3,8,-2,-5) Matrix(181,-102,126,-71) -> Matrix(1,2,-2,-3) Matrix(99,-58,70,-41) -> Matrix(1,4,-2,-7) Matrix(146,-89,105,-64) -> Matrix(1,4,-2,-7) Matrix(547,-338,882,-545) -> Matrix(7,16,-4,-9) Matrix(426,-265,553,-344) -> Matrix(5,8,-2,-3) Matrix(36,-25,49,-34) -> Matrix(1,2,-2,-3) Matrix(386,-299,315,-244) -> Matrix(1,4,-2,-7) Matrix(15,-14,14,-13) -> Matrix(1,2,-2,-3) Matrix(64,-81,49,-62) -> Matrix(1,2,-2,-3) Matrix(162,-235,91,-132) -> Matrix(3,2,-8,-5) 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): 14 ----------------------------------------------------------------------- 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 (0/1,1/0) 0 14 -5/7 0/1 2 2 -9/13 (0/1,1/1) 0 14 -2/3 0 14 -7/11 (0/1,1/2) 0 14 -5/8 0 14 -3/5 -1/1 2 14 -4/7 0/1 3 2 -5/9 (0/1,1/2) 0 14 -1/2 0 14 -3/7 0/1 4 2 -2/5 0 14 -5/13 (2/3,1/1) 0 14 -8/21 1/1 9 2 -3/8 0 14 -4/11 0 14 -1/3 -1/1 2 14 -2/7 0/1 5 2 -3/11 (0/1,1/4) 0 14 -1/4 0 14 0/1 0 14 1/7 1/0 4 2 1/6 0 14 1/5 1/1 2 14 2/9 0 14 3/13 (1/1,2/1) 0 14 1/4 0 14 3/11 7/1 2 14 2/7 1/0 8 2 1/3 (-1/1,1/0) 0 14 3/7 1/0 6 2 5/11 (-4/1,1/0) 0 14 1/2 0 14 5/9 -1/1 2 14 4/7 1/0 1 2 3/5 (-2/1,1/0) 0 14 13/21 -2/1 4 2 5/8 0 14 2/3 0 14 5/7 -1/1 4 2 3/4 0 14 10/13 0 14 7/9 (-2/1,1/0) 0 14 4/5 0 14 9/11 (-3/2,-4/3) 0 14 5/6 0 14 6/7 -1/1 9 2 1/1 -1/1 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(6,5,-7,-6) (-1/1,-5/7) -> (-1/1,-5/7) Reflection Matrix(64,45,-91,-64) (-5/7,-9/13) -> (-5/7,-9/13) Reflection Matrix(22,15,91,62) (-9/13,-2/3) -> (3/13,1/4) Glide Reflection Matrix(20,13,-77,-50) (-2/3,-7/11) -> (-3/11,-1/4) Hyperbolic 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(50,27,63,34) (-5/9,-1/2) -> (7/9,4/5) Glide Reflection Matrix(20,9,-49,-22) (-1/2,-3/7) -> (-3/7,-2/5) Parabolic 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(134,49,175,64) (-3/8,-4/11) -> (3/4,10/13) Hyperbolic Matrix(36,13,133,48) (-4/11,-1/3) -> (1/4,3/11) Glide 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(22,5,35,8) (-1/4,0/1) -> (5/8,2/3) Hyperbolic Matrix(8,-1,49,-6) (0/1,1/7) -> (1/7,1/6) Parabolic Matrix(50,-11,91,-20) (1/5,2/9) -> (1/2,5/9) 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(8,-3,21,-8) (1/3,3/7) -> (1/3,3/7) Reflection Matrix(34,-15,77,-34) (3/7,5/11) -> (3/7,5/11) Reflection Matrix(62,-29,77,-36) (5/11,1/2) -> (4/5,9/11) Hyperbolic 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(64,-39,105,-64) (3/5,13/21) -> (3/5,13/21) Reflection Matrix(482,-299,777,-482) (13/21,23/37) -> (13/21,23/37) Reflection Matrix(426,-265,553,-344) (18/29,5/8) -> (10/13,17/22) Hyperbolic Matrix(36,-25,49,-34) (2/3,5/7) -> (5/7,3/4) Parabolic Matrix(244,-189,315,-244) (27/35,7/9) -> (27/35,7/9) 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(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(6,5,-7,-6) -> Matrix(1,0,0,-1) (-1/1,-5/7) -> (0/1,1/0) Matrix(64,45,-91,-64) -> Matrix(1,0,2,-1) (-5/7,-9/13) -> (0/1,1/1) Matrix(22,15,91,62) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(20,13,-77,-50) -> Matrix(1,0,2,1) 0/1 Matrix(127,80,154,97) -> Matrix(5,-4,-4,3) Matrix(13,8,70,43) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(41,24,-70,-41) -> Matrix(-1,0,2,1) (-3/5,-4/7) -> (-1/1,0/1) Matrix(71,40,-126,-71) -> Matrix(1,0,4,-1) (-4/7,-5/9) -> (0/1,1/2) Matrix(50,27,63,34) -> Matrix(3,-2,-2,1) Matrix(20,9,-49,-22) -> Matrix(1,0,2,1) 0/1 Matrix(41,16,182,71) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(209,80,-546,-209) -> Matrix(5,-4,6,-5) (-5/13,-8/21) -> (2/3,1/1) Matrix(153,58,182,69) -> Matrix(7,-8,-6,7) Matrix(134,49,175,64) -> Matrix(1,-4,0,1) 1/0 Matrix(36,13,133,48) -> Matrix(-1,6,0,1) *** -> (3/1,1/0) Matrix(13,4,-42,-13) -> Matrix(-1,0,2,1) (-1/3,-2/7) -> (-1/1,0/1) Matrix(43,12,-154,-43) -> Matrix(1,0,8,-1) (-2/7,-3/11) -> (0/1,1/4) Matrix(22,5,35,8) -> Matrix(1,-2,0,1) 1/0 Matrix(8,-1,49,-6) -> Matrix(1,-2,0,1) 1/0 Matrix(50,-11,91,-20) -> Matrix(1,-2,0,1) 1/0 Matrix(43,-12,154,-43) -> Matrix(-1,14,0,1) (3/11,2/7) -> (7/1,1/0) Matrix(13,-4,42,-13) -> Matrix(1,2,0,-1) (2/7,1/3) -> (-1/1,1/0) Matrix(8,-3,21,-8) -> Matrix(1,2,0,-1) (1/3,3/7) -> (-1/1,1/0) Matrix(34,-15,77,-34) -> Matrix(1,8,0,-1) (3/7,5/11) -> (-4/1,1/0) Matrix(62,-29,77,-36) -> Matrix(3,8,-2,-5) -2/1 Matrix(71,-40,126,-71) -> Matrix(1,2,0,-1) (5/9,4/7) -> (-1/1,1/0) Matrix(41,-24,70,-41) -> Matrix(1,4,0,-1) (4/7,3/5) -> (-2/1,1/0) Matrix(64,-39,105,-64) -> Matrix(1,4,0,-1) (3/5,13/21) -> (-2/1,1/0) Matrix(482,-299,777,-482) -> Matrix(7,12,-4,-7) (13/21,23/37) -> (-2/1,-3/2) Matrix(426,-265,553,-344) -> Matrix(5,8,-2,-3) -2/1 Matrix(36,-25,49,-34) -> Matrix(1,2,-2,-3) -1/1 Matrix(244,-189,315,-244) -> Matrix(1,4,0,-1) (27/35,7/9) -> (-2/1,1/0) Matrix(13,-12,14,-13) -> Matrix(1,2,0,-1) (6/7,1/1) -> (-1/1,1/0) Matrix(-1,2,0,1) -> Matrix(-1,0,2,1) (1/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.