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: 36 Genus: 31 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 7/6 21/17 21/16 7/5 3/2 21/13 7/4 9/5 2/1 15/7 28/13 42/19 7/3 12/5 5/2 21/8 14/5 3/1 7/2 18/5 11/3 42/11 4/1 21/5 9/2 14/3 5/1 21/4 11/2 6/1 7/1 15/2 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -8/1 -1/3 -7/1 0/1 -6/1 1/1 -11/2 -1/2 0/1 -16/3 -1/5 -21/4 0/1 -5/1 0/1 1/2 -14/3 1/1 -9/2 1/1 -22/5 3/1 -13/3 1/1 1/0 -17/4 4/1 1/0 -21/5 1/0 -4/1 -1/1 -19/5 0/1 1/0 -15/4 1/1 -26/7 3/1 -11/3 0/1 1/0 -18/5 1/1 -7/2 1/0 -3/1 -1/1 -14/5 -1/1 -11/4 -2/3 -1/2 -19/7 -1/2 0/1 -27/10 -1/3 -35/13 0/1 -8/3 -1/1 -21/8 -1/1 -13/5 -1/1 -1/2 -18/7 -1/1 -5/2 -1/2 0/1 -17/7 -1/4 0/1 -29/12 -1/1 0/1 -12/5 -1/3 -7/3 0/1 -9/4 1/1 -20/9 -1/1 -11/5 0/1 1/0 -13/6 -1/1 0/1 -2/1 1/1 -15/8 -1/1 -28/15 -1/1 -13/7 -1/1 1/0 -11/6 0/1 1/0 -42/23 -1/1 1/1 -31/17 0/1 1/0 -20/11 -1/1 -9/5 1/1 -7/4 1/0 -12/7 -3/1 -41/24 -2/1 -1/1 -70/41 -1/1 -29/17 -1/1 1/0 -17/10 -4/1 1/0 -5/3 -2/1 1/0 -23/14 -2/1 1/0 -18/11 -1/1 -31/19 -2/1 1/0 -13/8 -2/1 -1/1 -21/13 -1/1 -8/5 -1/1 -27/17 -3/1 -19/12 -2/1 1/0 -11/7 -2/1 -3/2 -25/16 -4/3 -5/4 -14/9 -1/1 -3/2 -1/1 -7/5 0/1 -18/13 1/1 -29/21 1/1 1/0 -11/8 0/1 1/0 -26/19 1/3 -41/30 0/1 1/1 -56/41 1/1 -15/11 1/1 -19/14 0/1 1/0 -42/31 -1/1 1/1 -23/17 0/1 1/0 -4/3 -1/1 -21/16 0/1 -17/13 0/1 1/4 -13/10 0/1 1/1 -22/17 1/3 -9/7 1/1 -14/11 1/1 -19/15 2/1 1/0 -5/4 2/1 1/0 -21/17 1/0 -16/13 -5/1 -11/9 -2/1 1/0 -17/14 -1/2 0/1 -6/5 1/1 -7/6 1/0 -15/13 -5/1 -8/7 -3/1 -1/1 -1/1 1/0 0/1 -1/1 1/1 -1/1 -1/2 8/7 -3/5 7/6 -1/2 6/5 -1/3 11/9 -2/3 -1/2 16/13 -5/9 21/17 -1/2 5/4 -1/2 -2/5 14/11 -1/3 9/7 -1/3 22/17 -1/5 13/10 -1/3 0/1 17/13 -1/6 0/1 21/16 0/1 4/3 -1/1 19/14 -1/2 0/1 15/11 -1/3 26/19 -1/5 11/8 -1/2 0/1 18/13 -1/3 7/5 0/1 3/2 -1/1 14/9 -1/1 11/7 -3/4 -2/3 19/12 -2/3 -1/2 27/17 -3/5 35/22 -1/2 8/5 -1/1 21/13 -1/1 13/8 -1/1 -2/3 18/11 -1/1 5/3 -2/3 -1/2 17/10 -4/7 -1/2 29/17 -1/1 -1/2 12/7 -3/5 7/4 -1/2 9/5 -1/3 20/11 -1/1 11/6 -1/2 0/1 13/7 -1/1 -1/2 2/1 -1/3 15/7 -1/1 28/13 -1/1 13/6 -1/1 0/1 11/5 -1/2 0/1 42/19 -1/1 -1/3 31/14 -1/2 0/1 20/9 -1/1 9/4 -1/3 7/3 0/1 12/5 1/1 41/17 -1/1 1/0 70/29 -1/1 29/12 -1/1 0/1 17/7 0/1 1/2 5/2 0/1 1/0 23/9 0/1 1/0 18/7 -1/1 31/12 0/1 1/0 13/5 -1/1 1/0 21/8 -1/1 8/3 -1/1 27/10 1/1 19/7 0/1 1/0 11/4 -2/1 1/0 25/9 -3/2 -4/3 14/5 -1/1 3/1 -1/1 7/2 -1/2 18/5 -1/3 29/8 -1/3 0/1 11/3 -1/2 0/1 26/7 -3/7 41/11 -1/2 -1/3 56/15 -1/3 15/4 -1/3 19/5 -1/2 0/1 42/11 -1/1 -1/3 23/6 -1/2 0/1 4/1 -1/1 21/5 -1/2 17/4 -1/2 -4/9 13/3 -1/2 -1/3 22/5 -3/7 9/2 -1/3 14/3 -1/3 19/4 -1/4 0/1 5/1 -1/4 0/1 21/4 0/1 16/3 1/3 11/2 0/1 1/0 17/3 -2/3 -1/2 6/1 -1/3 7/1 0/1 15/2 1/3 8/1 1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(13,126,10,97) (-8/1,1/0) -> (22/17,13/10) Hyperbolic Matrix(43,336,-16,-125) (-8/1,-7/1) -> (-35/13,-8/3) Hyperbolic Matrix(13,84,2,13) (-7/1,-6/1) -> (6/1,7/1) Hyperbolic Matrix(29,168,-24,-139) (-6/1,-11/2) -> (-17/14,-6/5) Hyperbolic Matrix(85,462,62,337) (-11/2,-16/3) -> (26/19,11/8) Hyperbolic Matrix(127,672,24,127) (-16/3,-21/4) -> (21/4,16/3) Hyperbolic Matrix(41,210,8,41) (-21/4,-5/1) -> (5/1,21/4) Hyperbolic Matrix(71,336,-56,-265) (-5/1,-14/3) -> (-14/11,-19/15) Hyperbolic Matrix(55,252,12,55) (-14/3,-9/2) -> (9/2,14/3) Hyperbolic Matrix(85,378,38,169) (-9/2,-22/5) -> (20/9,9/4) Hyperbolic Matrix(29,126,26,113) (-22/5,-13/3) -> (1/1,8/7) Hyperbolic Matrix(167,714,98,419) (-13/3,-17/4) -> (17/10,29/17) Hyperbolic Matrix(169,714,40,169) (-17/4,-21/5) -> (21/5,17/4) Hyperbolic Matrix(41,168,10,41) (-21/5,-4/1) -> (4/1,21/5) Hyperbolic Matrix(43,168,-32,-125) (-4/1,-19/5) -> (-23/17,-4/3) Hyperbolic Matrix(211,798,78,295) (-19/5,-15/4) -> (27/10,19/7) Hyperbolic Matrix(113,420,-60,-223) (-15/4,-26/7) -> (-2/1,-15/8) Hyperbolic Matrix(125,462,102,377) (-26/7,-11/3) -> (11/9,16/13) Hyperbolic Matrix(209,756,-128,-463) (-11/3,-18/5) -> (-18/11,-31/19) Hyperbolic Matrix(71,252,20,71) (-18/5,-7/2) -> (7/2,18/5) Hyperbolic Matrix(13,42,4,13) (-7/2,-3/1) -> (3/1,7/2) Hyperbolic Matrix(29,84,10,29) (-3/1,-14/5) -> (14/5,3/1) Hyperbolic Matrix(181,504,-116,-323) (-14/5,-11/4) -> (-25/16,-14/9) Hyperbolic Matrix(139,378,82,223) (-11/4,-19/7) -> (5/3,17/10) Hyperbolic Matrix(295,798,78,211) (-19/7,-27/10) -> (15/4,19/5) Hyperbolic Matrix(265,714,36,97) (-27/10,-35/13) -> (7/1,15/2) Hyperbolic Matrix(127,336,48,127) (-8/3,-21/8) -> (21/8,8/3) Hyperbolic Matrix(209,546,80,209) (-21/8,-13/5) -> (13/5,21/8) Hyperbolic Matrix(293,756,-212,-547) (-13/5,-18/7) -> (-18/13,-29/21) Hyperbolic Matrix(197,504,-120,-307) (-18/7,-5/2) -> (-23/14,-18/11) Hyperbolic Matrix(155,378,98,239) (-5/2,-17/7) -> (11/7,19/12) Hyperbolic Matrix(295,714,226,547) (-17/7,-29/12) -> (13/10,17/13) Hyperbolic Matrix(349,840,-204,-491) (-29/12,-12/5) -> (-12/7,-41/24) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(55,126,24,55) (-7/3,-9/4) -> (9/4,7/3) Hyperbolic Matrix(169,378,38,85) (-9/4,-20/9) -> (22/5,9/2) Hyperbolic Matrix(379,840,-208,-461) (-20/9,-11/5) -> (-31/17,-20/11) Hyperbolic Matrix(211,462,58,127) (-11/5,-13/6) -> (29/8,11/3) Hyperbolic Matrix(197,420,-144,-307) (-13/6,-2/1) -> (-26/19,-41/30) Hyperbolic Matrix(673,1260,180,337) (-15/8,-28/15) -> (56/15,15/4) Hyperbolic Matrix(1105,2058,458,853) (-28/15,-13/7) -> (41/17,70/29) Hyperbolic Matrix(295,546,114,211) (-13/7,-11/6) -> (31/12,13/5) Hyperbolic Matrix(965,1764,436,797) (-11/6,-42/23) -> (42/19,31/14) Hyperbolic Matrix(967,1764,438,799) (-42/23,-31/17) -> (11/5,42/19) Hyperbolic Matrix(209,378,162,293) (-20/11,-9/5) -> (9/7,22/17) Hyperbolic Matrix(71,126,40,71) (-9/5,-7/4) -> (7/4,9/5) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(1205,2058,558,953) (-41/24,-70/41) -> (28/13,13/6) Hyperbolic Matrix(2633,4494,706,1205) (-70/41,-29/17) -> (41/11,56/15) Hyperbolic Matrix(419,714,98,167) (-29/17,-17/10) -> (17/4,13/3) Hyperbolic Matrix(223,378,82,139) (-17/10,-5/3) -> (19/7,11/4) Hyperbolic Matrix(127,210,26,43) (-5/3,-23/14) -> (19/4,5/1) Hyperbolic Matrix(335,546,154,251) (-31/19,-13/8) -> (13/6,11/5) Hyperbolic Matrix(337,546,208,337) (-13/8,-21/13) -> (21/13,13/8) Hyperbolic Matrix(209,336,130,209) (-21/13,-8/5) -> (8/5,21/13) Hyperbolic Matrix(211,336,-184,-293) (-8/5,-27/17) -> (-15/13,-8/7) Hyperbolic Matrix(503,798,370,587) (-27/17,-19/12) -> (19/14,15/11) Hyperbolic Matrix(239,378,98,155) (-19/12,-11/7) -> (17/7,5/2) Hyperbolic Matrix(349,546,62,97) (-11/7,-25/16) -> (11/2,17/3) 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(181,252,130,181) (-7/5,-18/13) -> (18/13,7/5) Hyperbolic Matrix(335,462,182,251) (-29/21,-11/8) -> (11/6,13/7) Hyperbolic Matrix(337,462,62,85) (-11/8,-26/19) -> (16/3,11/2) Hyperbolic Matrix(3289,4494,1362,1861) (-41/30,-56/41) -> (70/29,29/12) Hyperbolic Matrix(923,1260,430,587) (-56/41,-15/11) -> (15/7,28/13) Hyperbolic Matrix(587,798,370,503) (-15/11,-19/14) -> (19/12,27/17) Hyperbolic Matrix(1301,1764,340,461) (-19/14,-42/31) -> (42/11,23/6) Hyperbolic Matrix(1303,1764,342,463) (-42/31,-23/17) -> (19/5,42/11) Hyperbolic Matrix(127,168,96,127) (-4/3,-21/16) -> (21/16,4/3) Hyperbolic Matrix(545,714,416,545) (-21/16,-17/13) -> (17/13,21/16) Hyperbolic Matrix(547,714,226,295) (-17/13,-13/10) -> (29/12,17/7) Hyperbolic Matrix(97,126,10,13) (-13/10,-22/17) -> (8/1,1/0) Hyperbolic Matrix(293,378,162,209) (-22/17,-9/7) -> (9/5,20/11) Hyperbolic Matrix(197,252,154,197) (-9/7,-14/11) -> (14/11,9/7) Hyperbolic Matrix(167,210,66,83) (-19/15,-5/4) -> (5/2,23/9) Hyperbolic Matrix(169,210,136,169) (-5/4,-21/17) -> (21/17,5/4) Hyperbolic Matrix(545,672,442,545) (-21/17,-16/13) -> (16/13,21/17) Hyperbolic Matrix(377,462,102,125) (-16/13,-11/9) -> (11/3,26/7) Hyperbolic Matrix(449,546,162,197) (-11/9,-17/14) -> (11/4,25/9) Hyperbolic Matrix(71,84,60,71) (-6/5,-7/6) -> (7/6,6/5) Hyperbolic Matrix(617,714,388,449) (-7/6,-15/13) -> (27/17,35/22) Hyperbolic Matrix(113,126,26,29) (-8/7,-1/1) -> (13/3,22/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(293,-336,184,-211) (8/7,7/6) -> (35/22,8/5) Hyperbolic Matrix(139,-168,24,-29) (6/5,11/9) -> (17/3,6/1) Hyperbolic Matrix(265,-336,56,-71) (5/4,14/11) -> (14/3,19/4) Hyperbolic Matrix(125,-168,32,-43) (4/3,19/14) -> (23/6,4/1) Hyperbolic Matrix(307,-420,144,-197) (15/11,26/19) -> (2/1,15/7) Hyperbolic Matrix(547,-756,212,-293) (11/8,18/13) -> (18/7,31/12) Hyperbolic Matrix(323,-504,116,-181) (14/9,11/7) -> (25/9,14/5) Hyperbolic Matrix(463,-756,128,-209) (13/8,18/11) -> (18/5,29/8) Hyperbolic Matrix(307,-504,120,-197) (18/11,5/3) -> (23/9,18/7) Hyperbolic Matrix(491,-840,204,-349) (29/17,12/7) -> (12/5,41/17) Hyperbolic Matrix(461,-840,208,-379) (20/11,11/6) -> (31/14,20/9) Hyperbolic Matrix(223,-420,60,-113) (13/7,2/1) -> (26/7,41/11) Hyperbolic Matrix(125,-336,16,-43) (8/3,27/10) -> (15/2,8/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(13,126,10,97) -> Matrix(1,0,-2,1) Matrix(43,336,-16,-125) -> Matrix(1,0,2,1) Matrix(13,84,2,13) -> Matrix(1,0,-4,1) Matrix(29,168,-24,-139) -> Matrix(1,0,0,1) Matrix(85,462,62,337) -> Matrix(1,0,0,1) Matrix(127,672,24,127) -> Matrix(1,0,8,1) Matrix(41,210,8,41) -> Matrix(1,0,-6,1) Matrix(71,336,-56,-265) -> Matrix(3,-2,2,-1) Matrix(55,252,12,55) -> Matrix(1,-2,-2,5) Matrix(85,378,38,169) -> Matrix(1,-2,-2,5) Matrix(29,126,26,113) -> Matrix(1,0,-2,1) Matrix(167,714,98,419) -> Matrix(1,0,-2,1) Matrix(169,714,40,169) -> Matrix(1,-8,-2,17) Matrix(41,168,10,41) -> Matrix(1,2,-2,-3) Matrix(43,168,-32,-125) -> Matrix(1,0,0,1) Matrix(211,798,78,295) -> Matrix(1,0,0,1) Matrix(113,420,-60,-223) -> Matrix(1,-2,0,1) Matrix(125,462,102,377) -> Matrix(1,2,-2,-3) Matrix(209,756,-128,-463) -> Matrix(1,-2,0,1) Matrix(71,252,20,71) -> Matrix(1,-2,-2,5) Matrix(13,42,4,13) -> Matrix(1,2,-2,-3) Matrix(29,84,10,29) -> Matrix(1,0,0,1) Matrix(181,504,-116,-323) -> Matrix(7,6,-6,-5) Matrix(139,378,82,223) -> Matrix(5,2,-8,-3) Matrix(295,798,78,211) -> Matrix(1,0,0,1) Matrix(265,714,36,97) -> Matrix(1,0,6,1) Matrix(127,336,48,127) -> Matrix(1,0,0,1) Matrix(209,546,80,209) -> Matrix(3,2,-2,-1) Matrix(293,756,-212,-547) -> Matrix(1,0,2,1) Matrix(197,504,-120,-307) -> Matrix(3,2,-2,-1) Matrix(155,378,98,239) -> Matrix(5,2,-8,-3) Matrix(295,714,226,547) -> Matrix(1,0,-2,1) Matrix(349,840,-204,-491) -> Matrix(3,2,-2,-1) Matrix(71,168,30,71) -> Matrix(1,0,4,1) Matrix(55,126,24,55) -> Matrix(1,0,-4,1) Matrix(169,378,38,85) -> Matrix(1,-2,-2,5) Matrix(379,840,-208,-461) -> Matrix(1,0,0,1) Matrix(211,462,58,127) -> Matrix(1,0,-2,1) Matrix(197,420,-144,-307) -> Matrix(1,0,2,1) Matrix(673,1260,180,337) -> Matrix(1,0,-2,1) Matrix(1105,2058,458,853) -> Matrix(1,0,0,1) Matrix(295,546,114,211) -> Matrix(1,0,0,1) Matrix(965,1764,436,797) -> Matrix(1,0,-2,1) Matrix(967,1764,438,799) -> Matrix(1,0,-2,1) Matrix(209,378,162,293) -> Matrix(1,0,-4,1) Matrix(71,126,40,71) -> Matrix(1,-2,-2,5) Matrix(97,168,56,97) -> Matrix(1,6,-2,-11) Matrix(1205,2058,558,953) -> Matrix(1,2,-2,-3) Matrix(2633,4494,706,1205) -> Matrix(1,0,-2,1) Matrix(419,714,98,167) -> Matrix(1,0,-2,1) Matrix(223,378,82,139) -> Matrix(1,2,0,1) Matrix(127,210,26,43) -> Matrix(1,2,-4,-7) Matrix(335,546,154,251) -> Matrix(1,2,-2,-3) Matrix(337,546,208,337) -> Matrix(3,4,-4,-5) Matrix(209,336,130,209) -> Matrix(1,2,-2,-3) Matrix(211,336,-184,-293) -> Matrix(1,-2,0,1) Matrix(503,798,370,587) -> Matrix(1,2,-2,-3) Matrix(239,378,98,155) -> Matrix(1,2,0,1) Matrix(349,546,62,97) -> Matrix(3,4,-4,-5) Matrix(55,84,36,55) -> Matrix(1,2,-2,-3) Matrix(29,42,20,29) -> Matrix(1,0,0,1) Matrix(181,252,130,181) -> Matrix(1,0,-4,1) Matrix(335,462,182,251) -> Matrix(1,0,-2,1) Matrix(337,462,62,85) -> Matrix(1,0,0,1) Matrix(3289,4494,1362,1861) -> Matrix(1,0,-2,1) Matrix(923,1260,430,587) -> Matrix(1,0,-2,1) Matrix(587,798,370,503) -> Matrix(1,2,-2,-3) Matrix(1301,1764,340,461) -> Matrix(1,0,-2,1) Matrix(1303,1764,342,463) -> Matrix(1,0,-2,1) Matrix(127,168,96,127) -> Matrix(1,0,0,1) Matrix(545,714,416,545) -> Matrix(1,0,-10,1) Matrix(547,714,226,295) -> Matrix(1,0,-2,1) Matrix(97,126,10,13) -> Matrix(1,0,-2,1) Matrix(293,378,162,209) -> Matrix(1,0,-4,1) Matrix(197,252,154,197) -> Matrix(1,0,-4,1) Matrix(167,210,66,83) -> Matrix(1,-2,0,1) Matrix(169,210,136,169) -> Matrix(1,-4,-2,9) Matrix(545,672,442,545) -> Matrix(1,10,-2,-19) Matrix(377,462,102,125) -> Matrix(1,2,-2,-3) Matrix(449,546,162,197) -> Matrix(3,2,-2,-1) Matrix(71,84,60,71) -> Matrix(1,-2,-2,5) Matrix(617,714,388,449) -> Matrix(1,8,-2,-15) Matrix(113,126,26,29) -> Matrix(1,0,-2,1) Matrix(1,0,2,1) -> Matrix(1,2,-2,-3) Matrix(293,-336,184,-211) -> Matrix(3,2,-8,-5) Matrix(139,-168,24,-29) -> Matrix(1,0,0,1) Matrix(265,-336,56,-71) -> Matrix(5,2,-18,-7) Matrix(125,-168,32,-43) -> Matrix(1,0,0,1) Matrix(307,-420,144,-197) -> Matrix(1,0,2,1) Matrix(547,-756,212,-293) -> Matrix(1,0,2,1) Matrix(323,-504,116,-181) -> Matrix(7,6,-6,-5) Matrix(463,-756,128,-209) -> Matrix(3,2,-8,-5) Matrix(307,-504,120,-197) -> Matrix(3,2,-2,-1) Matrix(491,-840,204,-349) -> Matrix(3,2,-2,-1) Matrix(461,-840,208,-379) -> Matrix(1,0,0,1) Matrix(223,-420,60,-113) -> Matrix(3,2,-8,-5) Matrix(125,-336,16,-43) -> 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): 24 Degree of the the map X: 24 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: 24 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 28/13 42/19 7/3 12/5 21/8 14/5 3/1 7/2 18/5 42/11 4/1 21/5 9/2 14/3 5/1 21/4 11/2 6/1 7/1 15/2 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/1 1/1 -1/1 -1/2 8/7 -3/5 7/6 -1/2 6/5 -1/3 11/9 -2/3 -1/2 16/13 -5/9 21/17 -1/2 5/4 -1/2 -2/5 14/11 -1/3 9/7 -1/3 22/17 -1/5 13/10 -1/3 0/1 17/13 -1/6 0/1 21/16 0/1 4/3 -1/1 19/14 -1/2 0/1 15/11 -1/3 26/19 -1/5 11/8 -1/2 0/1 18/13 -1/3 7/5 0/1 3/2 -1/1 14/9 -1/1 11/7 -3/4 -2/3 19/12 -2/3 -1/2 27/17 -3/5 35/22 -1/2 8/5 -1/1 21/13 -1/1 13/8 -1/1 -2/3 18/11 -1/1 5/3 -2/3 -1/2 17/10 -4/7 -1/2 29/17 -1/1 -1/2 12/7 -3/5 7/4 -1/2 9/5 -1/3 20/11 -1/1 11/6 -1/2 0/1 13/7 -1/1 -1/2 2/1 -1/3 15/7 -1/1 28/13 -1/1 13/6 -1/1 0/1 11/5 -1/2 0/1 42/19 -1/1 -1/3 31/14 -1/2 0/1 20/9 -1/1 9/4 -1/3 7/3 0/1 12/5 1/1 41/17 -1/1 1/0 70/29 -1/1 29/12 -1/1 0/1 17/7 0/1 1/2 5/2 0/1 1/0 23/9 0/1 1/0 18/7 -1/1 31/12 0/1 1/0 13/5 -1/1 1/0 21/8 -1/1 8/3 -1/1 27/10 1/1 19/7 0/1 1/0 11/4 -2/1 1/0 25/9 -3/2 -4/3 14/5 -1/1 3/1 -1/1 7/2 -1/2 18/5 -1/3 29/8 -1/3 0/1 11/3 -1/2 0/1 26/7 -3/7 41/11 -1/2 -1/3 56/15 -1/3 15/4 -1/3 19/5 -1/2 0/1 42/11 -1/1 -1/3 23/6 -1/2 0/1 4/1 -1/1 21/5 -1/2 17/4 -1/2 -4/9 13/3 -1/2 -1/3 22/5 -3/7 9/2 -1/3 14/3 -1/3 19/4 -1/4 0/1 5/1 -1/4 0/1 21/4 0/1 16/3 1/3 11/2 0/1 1/0 17/3 -2/3 -1/2 6/1 -1/3 7/1 0/1 15/2 1/3 8/1 1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(113,-126,87,-97) (1/1,8/7) -> (22/17,13/10) Hyperbolic Matrix(293,-336,184,-211) (8/7,7/6) -> (35/22,8/5) Hyperbolic Matrix(71,-84,11,-13) (7/6,6/5) -> (6/1,7/1) Hyperbolic Matrix(139,-168,24,-29) (6/5,11/9) -> (17/3,6/1) Hyperbolic Matrix(377,-462,275,-337) (11/9,16/13) -> (26/19,11/8) Hyperbolic Matrix(545,-672,103,-127) (16/13,21/17) -> (21/4,16/3) Hyperbolic Matrix(169,-210,33,-41) (21/17,5/4) -> (5/1,21/4) Hyperbolic Matrix(265,-336,56,-71) (5/4,14/11) -> (14/3,19/4) Hyperbolic Matrix(197,-252,43,-55) (14/11,9/7) -> (9/2,14/3) Hyperbolic Matrix(293,-378,131,-169) (9/7,22/17) -> (20/9,9/4) Hyperbolic Matrix(547,-714,321,-419) (13/10,17/13) -> (17/10,29/17) Hyperbolic Matrix(545,-714,129,-169) (17/13,21/16) -> (21/5,17/4) Hyperbolic Matrix(127,-168,31,-41) (21/16,4/3) -> (4/1,21/5) Hyperbolic Matrix(125,-168,32,-43) (4/3,19/14) -> (23/6,4/1) Hyperbolic Matrix(587,-798,217,-295) (19/14,15/11) -> (27/10,19/7) Hyperbolic Matrix(307,-420,144,-197) (15/11,26/19) -> (2/1,15/7) Hyperbolic Matrix(547,-756,212,-293) (11/8,18/13) -> (18/7,31/12) Hyperbolic Matrix(181,-252,51,-71) (18/13,7/5) -> (7/2,18/5) Hyperbolic Matrix(29,-42,9,-13) (7/5,3/2) -> (3/1,7/2) Hyperbolic Matrix(55,-84,19,-29) (3/2,14/9) -> (14/5,3/1) Hyperbolic Matrix(323,-504,116,-181) (14/9,11/7) -> (25/9,14/5) Hyperbolic Matrix(239,-378,141,-223) (11/7,19/12) -> (5/3,17/10) Hyperbolic Matrix(503,-798,133,-211) (19/12,27/17) -> (15/4,19/5) Hyperbolic Matrix(449,-714,61,-97) (27/17,35/22) -> (7/1,15/2) Hyperbolic Matrix(209,-336,79,-127) (8/5,21/13) -> (21/8,8/3) Hyperbolic Matrix(337,-546,129,-209) (21/13,13/8) -> (13/5,21/8) Hyperbolic Matrix(463,-756,128,-209) (13/8,18/11) -> (18/5,29/8) Hyperbolic Matrix(307,-504,120,-197) (18/11,5/3) -> (23/9,18/7) Hyperbolic Matrix(491,-840,204,-349) (29/17,12/7) -> (12/5,41/17) Hyperbolic Matrix(97,-168,41,-71) (12/7,7/4) -> (7/3,12/5) Hyperbolic Matrix(71,-126,31,-55) (7/4,9/5) -> (9/4,7/3) Hyperbolic Matrix(209,-378,47,-85) (9/5,20/11) -> (22/5,9/2) Hyperbolic Matrix(461,-840,208,-379) (20/11,11/6) -> (31/14,20/9) Hyperbolic Matrix(251,-462,69,-127) (11/6,13/7) -> (29/8,11/3) Hyperbolic Matrix(223,-420,60,-113) (13/7,2/1) -> (26/7,41/11) Hyperbolic Matrix(587,-1260,157,-337) (15/7,28/13) -> (56/15,15/4) Hyperbolic Matrix(953,-2058,395,-853) (28/13,13/6) -> (41/17,70/29) Hyperbolic Matrix(251,-546,97,-211) (13/6,11/5) -> (31/12,13/5) Hyperbolic Matrix(799,-1764,361,-797) (11/5,42/19) -> (42/19,31/14) Parabolic Matrix(1861,-4494,499,-1205) (70/29,29/12) -> (41/11,56/15) Hyperbolic Matrix(295,-714,69,-167) (29/12,17/7) -> (17/4,13/3) Hyperbolic Matrix(155,-378,57,-139) (17/7,5/2) -> (19/7,11/4) Hyperbolic Matrix(83,-210,17,-43) (5/2,23/9) -> (19/4,5/1) Hyperbolic Matrix(125,-336,16,-43) (8/3,27/10) -> (15/2,8/1) Hyperbolic Matrix(197,-546,35,-97) (11/4,25/9) -> (11/2,17/3) Hyperbolic Matrix(125,-462,23,-85) (11/3,26/7) -> (16/3,11/2) Hyperbolic Matrix(463,-1764,121,-461) (19/5,42/11) -> (42/11,23/6) Parabolic Matrix(29,-126,3,-13) (13/3,22/5) -> (8/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(0,-1,1,2) Matrix(113,-126,87,-97) -> Matrix(2,1,-5,-2) Matrix(293,-336,184,-211) -> Matrix(3,2,-8,-5) Matrix(71,-84,11,-13) -> Matrix(2,1,-9,-4) Matrix(139,-168,24,-29) -> Matrix(1,0,0,1) Matrix(377,-462,275,-337) -> Matrix(2,1,-1,0) Matrix(545,-672,103,-127) -> Matrix(2,1,15,8) Matrix(169,-210,33,-41) -> Matrix(2,1,-13,-6) Matrix(265,-336,56,-71) -> Matrix(5,2,-18,-7) Matrix(197,-252,43,-55) -> Matrix(4,1,-9,-2) Matrix(293,-378,131,-169) -> Matrix(4,1,-9,-2) Matrix(547,-714,321,-419) -> Matrix(2,1,-5,-2) Matrix(545,-714,129,-169) -> Matrix(10,1,-21,-2) Matrix(127,-168,31,-41) -> Matrix(0,-1,1,2) Matrix(125,-168,32,-43) -> Matrix(1,0,0,1) Matrix(587,-798,217,-295) -> Matrix(2,1,-1,0) Matrix(307,-420,144,-197) -> Matrix(1,0,2,1) Matrix(547,-756,212,-293) -> Matrix(1,0,2,1) Matrix(181,-252,51,-71) -> Matrix(4,1,-9,-2) Matrix(29,-42,9,-13) -> Matrix(0,-1,1,2) Matrix(55,-84,19,-29) -> Matrix(2,1,-1,0) Matrix(323,-504,116,-181) -> Matrix(7,6,-6,-5) Matrix(239,-378,141,-223) -> Matrix(8,5,-13,-8) Matrix(503,-798,133,-211) -> Matrix(2,1,-1,0) Matrix(449,-714,61,-97) -> Matrix(2,1,11,6) Matrix(209,-336,79,-127) -> Matrix(2,1,-1,0) Matrix(337,-546,129,-209) -> Matrix(4,3,-3,-2) Matrix(463,-756,128,-209) -> Matrix(3,2,-8,-5) Matrix(307,-504,120,-197) -> Matrix(3,2,-2,-1) Matrix(491,-840,204,-349) -> Matrix(3,2,-2,-1) Matrix(97,-168,41,-71) -> Matrix(2,1,7,4) Matrix(71,-126,31,-55) -> Matrix(2,1,-9,-4) Matrix(209,-378,47,-85) -> Matrix(4,1,-9,-2) Matrix(461,-840,208,-379) -> Matrix(1,0,0,1) Matrix(251,-462,69,-127) -> Matrix(2,1,-5,-2) Matrix(223,-420,60,-113) -> Matrix(3,2,-8,-5) Matrix(587,-1260,157,-337) -> Matrix(2,1,-5,-2) Matrix(953,-2058,395,-853) -> Matrix(2,1,-1,0) Matrix(251,-546,97,-211) -> Matrix(2,1,-1,0) Matrix(799,-1764,361,-797) -> Matrix(2,1,-5,-2) Matrix(1861,-4494,499,-1205) -> Matrix(2,1,-5,-2) Matrix(295,-714,69,-167) -> Matrix(2,1,-5,-2) Matrix(155,-378,57,-139) -> Matrix(0,-1,1,0) Matrix(83,-210,17,-43) -> Matrix(0,-1,1,4) Matrix(125,-336,16,-43) -> Matrix(1,0,2,1) Matrix(197,-546,35,-97) -> Matrix(2,3,-3,-4) Matrix(125,-462,23,-85) -> Matrix(2,1,-1,0) Matrix(463,-1764,121,-461) -> Matrix(2,1,-5,-2) Matrix(29,-126,3,-13) -> Matrix(2,1,-5,-2) 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): 24 ----------------------------------------------------------------------- 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 1 2/1 -1/3 1 21 13/6 (-1/1,0/1) 0 21 11/5 (-1/2,0/1) 0 21 42/19 (-1/1,-1/3).(-1/2,0/1) 0 1 20/9 -1/1 1 21 9/4 -1/3 1 7 7/3 0/1 4 3 12/5 1/1 1 7 70/29 -1/1 1 3 29/12 (-1/1,0/1) 0 21 17/7 (0/1,1/2) 0 21 5/2 (0/1,1/0) 0 21 18/7 -1/1 1 7 13/5 (-1/1,1/0) 0 21 21/8 -1/1 1 1 8/3 -1/1 1 21 27/10 1/1 1 7 19/7 (0/1,1/0) 0 21 11/4 (-2/1,1/0) 0 21 14/5 -1/1 5 3 3/1 -1/1 1 7 7/2 -1/2 2 3 18/5 -1/3 1 7 29/8 (-1/3,0/1) 0 21 11/3 (-1/2,0/1) 0 21 26/7 -3/7 1 21 41/11 (-1/2,-1/3) 0 21 56/15 -1/3 1 3 15/4 -1/3 1 7 19/5 (-1/2,0/1) 0 21 42/11 (-1/1,-1/3).(-1/2,0/1) 0 1 4/1 -1/1 1 21 21/5 -1/2 5 1 17/4 (-1/2,-4/9) 0 21 13/3 (-1/2,-1/3) 0 21 22/5 -3/7 1 21 9/2 -1/3 1 7 14/3 -1/3 3 3 5/1 (-1/4,0/1) 0 21 21/4 0/1 7 1 16/3 1/3 1 21 11/2 (0/1,1/0) 0 21 6/1 -1/3 1 7 7/1 0/1 6 3 15/2 1/3 1 7 8/1 1/1 1 21 1/0 (-1/1,0/1) 0 21 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(197,-420,53,-113) (2/1,13/6) -> (26/7,41/11) Glide Reflection Matrix(211,-462,58,-127) (13/6,11/5) -> (29/8,11/3) Glide Reflection Matrix(419,-924,190,-419) (11/5,42/19) -> (11/5,42/19) Reflection Matrix(379,-840,171,-379) (42/19,20/9) -> (42/19,20/9) Reflection Matrix(169,-378,38,-85) (20/9,9/4) -> (22/5,9/2) Glide Reflection Matrix(55,-126,24,-55) (9/4,7/3) -> (9/4,7/3) Reflection Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(349,-840,145,-349) (12/5,70/29) -> (12/5,70/29) Reflection Matrix(1861,-4494,499,-1205) (70/29,29/12) -> (41/11,56/15) Hyperbolic Matrix(295,-714,69,-167) (29/12,17/7) -> (17/4,13/3) Hyperbolic Matrix(155,-378,57,-139) (17/7,5/2) -> (19/7,11/4) Hyperbolic Matrix(83,-210,17,-43) (5/2,23/9) -> (19/4,5/1) Hyperbolic Matrix(197,-504,77,-197) (28/11,18/7) -> (28/11,18/7) Reflection Matrix(293,-756,81,-209) (18/7,13/5) -> (18/5,29/8) Glide Reflection Matrix(209,-546,80,-209) (13/5,21/8) -> (13/5,21/8) Reflection Matrix(127,-336,48,-127) (21/8,8/3) -> (21/8,8/3) Reflection Matrix(125,-336,16,-43) (8/3,27/10) -> (15/2,8/1) Hyperbolic Matrix(295,-798,78,-211) (27/10,19/7) -> (15/4,19/5) Glide Reflection Matrix(167,-462,30,-83) (11/4,25/9) -> (11/2,17/3) Glide Reflection Matrix(181,-504,65,-181) (36/13,14/5) -> (36/13,14/5) Reflection Matrix(29,-84,10,-29) (14/5,3/1) -> (14/5,3/1) Reflection Matrix(13,-42,4,-13) (3/1,7/2) -> (3/1,7/2) Reflection Matrix(71,-252,20,-71) (7/2,18/5) -> (7/2,18/5) Reflection Matrix(125,-462,23,-85) (11/3,26/7) -> (16/3,11/2) Hyperbolic Matrix(449,-1680,120,-449) (56/15,15/4) -> (56/15,15/4) Reflection Matrix(419,-1596,110,-419) (19/5,42/11) -> (19/5,42/11) Reflection Matrix(43,-168,11,-43) (42/11,4/1) -> (42/11,4/1) Reflection Matrix(41,-168,10,-41) (4/1,21/5) -> (4/1,21/5) Reflection Matrix(169,-714,40,-169) (21/5,17/4) -> (21/5,17/4) Reflection Matrix(29,-126,3,-13) (13/3,22/5) -> (8/1,1/0) Hyperbolic Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) Reflection Matrix(71,-336,15,-71) (14/3,24/5) -> (14/3,24/5) Reflection Matrix(41,-210,8,-41) (5/1,21/4) -> (5/1,21/4) Reflection Matrix(127,-672,24,-127) (21/4,16/3) -> (21/4,16/3) Reflection Matrix(29,-168,5,-29) (28/5,6/1) -> (28/5,6/1) Reflection Matrix(13,-84,2,-13) (6/1,7/1) -> (6/1,7/1) Reflection Matrix(29,-210,4,-29) (7/1,15/2) -> (7/1,15/2) 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(1,0,1,-1) -> Matrix(2,1,-3,-2) (0/1,2/1) -> (-1/1,-1/3) Matrix(197,-420,53,-113) -> Matrix(0,1,1,-2) Matrix(211,-462,58,-127) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(419,-924,190,-419) -> Matrix(-1,0,4,1) (11/5,42/19) -> (-1/2,0/1) Matrix(379,-840,171,-379) -> Matrix(2,1,-3,-2) (42/19,20/9) -> (-1/1,-1/3) Matrix(169,-378,38,-85) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(55,-126,24,-55) -> Matrix(-1,0,6,1) (9/4,7/3) -> (-1/3,0/1) Matrix(71,-168,30,-71) -> Matrix(1,0,2,-1) (7/3,12/5) -> (0/1,1/1) Matrix(349,-840,145,-349) -> Matrix(0,1,1,0) (12/5,70/29) -> (-1/1,1/1) Matrix(1861,-4494,499,-1205) -> Matrix(2,1,-5,-2) (-1/1,-1/3).(-1/2,0/1) Matrix(295,-714,69,-167) -> Matrix(2,1,-5,-2) (-1/1,-1/3).(-1/2,0/1) Matrix(155,-378,57,-139) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) Matrix(83,-210,17,-43) -> Matrix(0,-1,1,4) Matrix(197,-504,77,-197) -> Matrix(0,1,1,0) (28/11,18/7) -> (-1/1,1/1) Matrix(293,-756,81,-209) -> Matrix(0,1,1,-2) Matrix(209,-546,80,-209) -> Matrix(1,2,0,-1) (13/5,21/8) -> (-1/1,1/0) Matrix(127,-336,48,-127) -> Matrix(-1,0,2,1) (21/8,8/3) -> (-1/1,0/1) Matrix(125,-336,16,-43) -> Matrix(1,0,2,1) 0/1 Matrix(295,-798,78,-211) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(167,-462,30,-83) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(181,-504,65,-181) -> Matrix(4,5,-3,-4) (36/13,14/5) -> (-5/3,-1/1) Matrix(29,-84,10,-29) -> Matrix(-1,0,2,1) (14/5,3/1) -> (-1/1,0/1) Matrix(13,-42,4,-13) -> Matrix(3,2,-4,-3) (3/1,7/2) -> (-1/1,-1/2) Matrix(71,-252,20,-71) -> Matrix(5,2,-12,-5) (7/2,18/5) -> (-1/2,-1/3) Matrix(125,-462,23,-85) -> Matrix(2,1,-1,0) -1/1 Matrix(449,-1680,120,-449) -> Matrix(5,2,-12,-5) (56/15,15/4) -> (-1/2,-1/3) Matrix(419,-1596,110,-419) -> Matrix(-1,0,4,1) (19/5,42/11) -> (-1/2,0/1) Matrix(43,-168,11,-43) -> Matrix(2,1,-3,-2) (42/11,4/1) -> (-1/1,-1/3) Matrix(41,-168,10,-41) -> Matrix(3,2,-4,-3) (4/1,21/5) -> (-1/1,-1/2) Matrix(169,-714,40,-169) -> Matrix(17,8,-36,-17) (21/5,17/4) -> (-1/2,-4/9) Matrix(29,-126,3,-13) -> Matrix(2,1,-5,-2) (-1/1,-1/3).(-1/2,0/1) Matrix(55,-252,12,-55) -> Matrix(5,2,-12,-5) (9/2,14/3) -> (-1/2,-1/3) Matrix(71,-336,15,-71) -> Matrix(4,1,-15,-4) (14/3,24/5) -> (-1/3,-1/5) Matrix(41,-210,8,-41) -> Matrix(-1,0,8,1) (5/1,21/4) -> (-1/4,0/1) Matrix(127,-672,24,-127) -> Matrix(1,0,6,-1) (21/4,16/3) -> (0/1,1/3) Matrix(29,-168,5,-29) -> Matrix(2,1,-3,-2) (28/5,6/1) -> (-1/1,-1/3) Matrix(13,-84,2,-13) -> Matrix(-1,0,6,1) (6/1,7/1) -> (-1/3,0/1) Matrix(29,-210,4,-29) -> Matrix(1,0,6,-1) (7/1,15/2) -> (0/1,1/3) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.