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): 360 Minimal number of generators: 61 Number of equivalence classes of cusps: 30 Genus: 16 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -9/22 -7/22 -10/33 -5/22 -3/22 -1/8 0/1 1/7 1/6 2/11 1/5 2/9 1/4 3/11 2/7 3/10 1/3 4/11 2/5 5/11 1/2 6/11 5/9 7/11 2/3 23/33 8/11 9/11 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -6/7 0/1 1/2 -5/6 2/3 1/1 -9/11 1/1 -4/5 1/1 2/1 -7/9 2/1 1/0 -3/4 0/1 1/1 -8/11 1/1 -5/7 1/1 2/1 -7/10 1/1 2/1 -9/13 2/1 1/0 -2/3 1/1 1/0 -7/11 1/0 -5/8 0/1 1/0 -18/29 0/1 1/2 -13/21 0/1 1/1 -8/13 1/1 1/0 -3/5 0/1 1/1 -7/12 1/1 2/1 -4/7 2/1 1/0 -9/16 3/1 4/1 -14/25 3/1 1/0 -5/9 4/1 1/0 -6/11 1/0 -1/2 0/1 1/0 -5/11 1/0 -4/9 -3/1 1/0 -7/16 -3/1 -2/1 -3/7 -2/1 -1/1 -5/12 -2/1 -1/1 -7/17 -4/3 -1/1 -9/22 -1/1 -2/5 -1/1 0/1 -5/13 -1/2 0/1 -13/34 -1/3 0/1 -21/55 0/1 -8/21 -1/2 0/1 -3/8 -1/2 0/1 -4/11 0/1 -1/3 0/1 1/0 -7/22 1/0 -6/19 -1/1 1/0 -5/16 -1/1 0/1 -4/13 -1/1 1/0 -7/23 -2/1 -1/1 -10/33 -1/1 -3/10 -1/1 0/1 -2/7 -1/2 0/1 -3/11 0/1 -1/4 0/1 1/1 -3/13 2/1 1/0 -5/22 1/0 -2/9 -1/1 1/0 -1/5 -1/1 0/1 -2/11 0/1 -1/6 0/1 1/1 -1/7 -1/1 0/1 -3/22 0/1 -2/15 0/1 1/2 -1/8 0/1 1/0 0/1 0/1 1/0 1/7 0/1 1/1 1/6 -1/1 0/1 2/11 0/1 1/5 0/1 1/1 2/9 1/1 1/0 1/4 -1/1 0/1 3/11 0/1 2/7 0/1 1/2 3/10 0/1 1/1 4/13 1/1 1/0 1/3 0/1 1/0 4/11 0/1 3/8 0/1 1/2 11/29 0/1 1/3 8/21 0/1 1/2 5/13 0/1 1/2 2/5 0/1 1/1 5/12 1/1 2/1 3/7 1/1 2/1 7/16 2/1 3/1 11/25 2/1 1/0 4/9 3/1 1/0 5/11 1/0 1/2 0/1 1/0 6/11 1/0 5/9 -4/1 1/0 9/16 -4/1 -3/1 4/7 -2/1 1/0 7/12 -2/1 -1/1 10/17 -4/3 -1/1 13/22 -1/1 3/5 -1/1 0/1 8/13 -1/1 1/0 21/34 -2/1 -1/1 34/55 -1/1 13/21 -1/1 0/1 5/8 0/1 1/0 7/11 1/0 2/3 -1/1 1/0 15/22 1/0 13/19 -2/1 1/0 11/16 -2/1 -1/1 9/13 -2/1 1/0 16/23 -2/1 1/0 23/33 -2/1 7/10 -2/1 -1/1 5/7 -2/1 -1/1 8/11 -1/1 3/4 -1/1 0/1 10/13 1/1 1/0 17/22 1/0 7/9 -2/1 1/0 4/5 -2/1 -1/1 9/11 -1/1 5/6 -1/1 -2/3 6/7 -1/2 0/1 19/22 0/1 13/15 0/1 1/1 7/8 0/1 1/0 1/1 -1/1 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(109,96,-176,-155) (-1/1,-6/7) -> (-18/29,-13/21) Hyperbolic Matrix(87,74,154,131) (-6/7,-5/6) -> (9/16,4/7) Hyperbolic Matrix(109,90,132,109) (-5/6,-9/11) -> (9/11,5/6) Hyperbolic Matrix(89,72,110,89) (-9/11,-4/5) -> (4/5,9/11) Hyperbolic Matrix(43,34,110,87) (-4/5,-7/9) -> (5/13,2/5) Hyperbolic Matrix(21,16,-88,-67) (-7/9,-3/4) -> (-1/4,-3/13) Hyperbolic Matrix(65,48,88,65) (-3/4,-8/11) -> (8/11,3/4) Hyperbolic Matrix(111,80,154,111) (-8/11,-5/7) -> (5/7,8/11) Hyperbolic Matrix(65,46,154,109) (-5/7,-7/10) -> (5/12,3/7) Hyperbolic Matrix(219,152,-572,-397) (-7/10,-9/13) -> (-5/13,-13/34) Hyperbolic Matrix(197,136,-352,-243) (-9/13,-2/3) -> (-14/25,-5/9) Hyperbolic Matrix(43,28,66,43) (-2/3,-7/11) -> (7/11,2/3) Hyperbolic Matrix(111,70,176,111) (-7/11,-5/8) -> (5/8,7/11) Hyperbolic Matrix(45,28,-352,-219) (-5/8,-18/29) -> (-2/15,-1/8) Hyperbolic Matrix(175,108,-572,-353) (-13/21,-8/13) -> (-4/13,-7/23) Hyperbolic Matrix(23,14,110,67) (-8/13,-3/5) -> (1/5,2/9) Hyperbolic Matrix(109,64,-264,-155) (-3/5,-7/12) -> (-5/12,-7/17) Hyperbolic Matrix(45,26,154,89) (-7/12,-4/7) -> (2/7,3/10) Hyperbolic Matrix(131,74,154,87) (-4/7,-9/16) -> (5/6,6/7) Hyperbolic Matrix(221,124,-704,-395) (-9/16,-14/25) -> (-6/19,-5/16) Hyperbolic Matrix(109,60,198,109) (-5/9,-6/11) -> (6/11,5/9) Hyperbolic Matrix(23,12,44,23) (-6/11,-1/2) -> (1/2,6/11) Hyperbolic Matrix(21,10,44,21) (-1/2,-5/11) -> (5/11,1/2) Hyperbolic Matrix(89,40,198,89) (-5/11,-4/9) -> (4/9,5/11) Hyperbolic Matrix(109,48,-352,-155) (-4/9,-7/16) -> (-5/16,-4/13) Hyperbolic Matrix(23,10,154,67) (-7/16,-3/7) -> (1/7,1/6) Hyperbolic Matrix(109,46,154,65) (-3/7,-5/12) -> (7/10,5/7) Hyperbolic Matrix(287,118,484,199) (-7/17,-9/22) -> (13/22,3/5) Hyperbolic Matrix(285,116,484,197) (-9/22,-2/5) -> (10/17,13/22) Hyperbolic Matrix(87,34,110,43) (-2/5,-5/13) -> (7/9,4/5) Hyperbolic Matrix(1167,446,1672,639) (-13/34,-21/55) -> (23/33,7/10) Hyperbolic Matrix(1363,520,1958,747) (-21/55,-8/21) -> (16/23,23/33) Hyperbolic Matrix(21,8,-176,-67) (-8/21,-3/8) -> (-1/8,0/1) Hyperbolic Matrix(65,24,176,65) (-3/8,-4/11) -> (4/11,3/8) Hyperbolic Matrix(23,8,66,23) (-4/11,-1/3) -> (1/3,4/11) Hyperbolic Matrix(331,106,484,155) (-1/3,-7/22) -> (15/22,13/19) Hyperbolic Matrix(329,104,484,153) (-7/22,-6/19) -> (2/3,15/22) Hyperbolic Matrix(1211,368,1958,595) (-7/23,-10/33) -> (34/55,13/21) Hyperbolic Matrix(1033,312,1672,505) (-10/33,-3/10) -> (21/34,34/55) Hyperbolic Matrix(89,26,154,45) (-3/10,-2/7) -> (4/7,7/12) Hyperbolic Matrix(43,12,154,43) (-2/7,-3/11) -> (3/11,2/7) Hyperbolic Matrix(23,6,88,23) (-3/11,-1/4) -> (1/4,3/11) Hyperbolic Matrix(375,86,484,111) (-3/13,-5/22) -> (17/22,7/9) Hyperbolic Matrix(373,84,484,109) (-5/22,-2/9) -> (10/13,17/22) Hyperbolic Matrix(67,14,110,23) (-2/9,-1/5) -> (3/5,8/13) Hyperbolic Matrix(21,4,110,21) (-1/5,-2/11) -> (2/11,1/5) Hyperbolic Matrix(23,4,132,23) (-2/11,-1/6) -> (1/6,2/11) Hyperbolic Matrix(67,10,154,23) (-1/6,-1/7) -> (3/7,7/16) Hyperbolic Matrix(419,58,484,67) (-1/7,-3/22) -> (19/22,13/15) Hyperbolic Matrix(417,56,484,65) (-3/22,-2/15) -> (6/7,19/22) Hyperbolic Matrix(67,-8,176,-21) (0/1,1/7) -> (11/29,8/21) Hyperbolic Matrix(67,-16,88,-21) (2/9,1/4) -> (3/4,10/13) Hyperbolic Matrix(353,-108,572,-175) (3/10,4/13) -> (8/13,21/34) Hyperbolic Matrix(155,-48,352,-109) (4/13,1/3) -> (11/25,4/9) Hyperbolic Matrix(307,-116,352,-133) (3/8,11/29) -> (13/15,7/8) Hyperbolic Matrix(397,-152,572,-219) (8/21,5/13) -> (9/13,16/23) Hyperbolic Matrix(155,-64,264,-109) (2/5,5/12) -> (7/12,10/17) Hyperbolic Matrix(483,-212,704,-309) (7/16,11/25) -> (13/19,11/16) Hyperbolic Matrix(243,-136,352,-197) (5/9,9/16) -> (11/16,9/13) Hyperbolic Matrix(155,-96,176,-109) (13/21,5/8) -> (7/8,1/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(109,96,-176,-155) -> Matrix(1,0,0,1) Matrix(87,74,154,131) -> Matrix(5,-2,-2,1) Matrix(109,90,132,109) -> Matrix(5,-4,-6,5) Matrix(89,72,110,89) -> Matrix(3,-4,-2,3) Matrix(43,34,110,87) -> Matrix(1,-2,2,-3) Matrix(21,16,-88,-67) -> Matrix(1,0,0,1) Matrix(65,48,88,65) -> Matrix(1,0,-2,1) Matrix(111,80,154,111) -> Matrix(3,-4,-2,3) Matrix(65,46,154,109) -> Matrix(1,0,0,1) Matrix(219,152,-572,-397) -> Matrix(1,-2,-2,5) Matrix(197,136,-352,-243) -> Matrix(1,2,0,1) Matrix(43,28,66,43) -> Matrix(1,-2,0,1) Matrix(111,70,176,111) -> Matrix(1,0,0,1) Matrix(45,28,-352,-219) -> Matrix(1,0,0,1) Matrix(175,108,-572,-353) -> Matrix(1,-2,0,1) Matrix(23,14,110,67) -> Matrix(1,0,0,1) Matrix(109,64,-264,-155) -> Matrix(3,-4,-2,3) Matrix(45,26,154,89) -> Matrix(1,-2,2,-3) Matrix(131,74,154,87) -> Matrix(1,-2,-2,5) Matrix(221,124,-704,-395) -> Matrix(1,-4,0,1) Matrix(109,60,198,109) -> Matrix(1,-8,0,1) Matrix(23,12,44,23) -> Matrix(1,0,0,1) Matrix(21,10,44,21) -> Matrix(1,0,0,1) Matrix(89,40,198,89) -> Matrix(1,6,0,1) Matrix(109,48,-352,-155) -> Matrix(1,2,0,1) Matrix(23,10,154,67) -> Matrix(1,2,0,1) Matrix(109,46,154,65) -> Matrix(1,0,0,1) Matrix(287,118,484,199) -> Matrix(3,4,-4,-5) Matrix(285,116,484,197) -> Matrix(5,4,-4,-3) Matrix(87,34,110,43) -> Matrix(3,2,-2,-1) Matrix(1167,446,1672,639) -> Matrix(7,2,-4,-1) Matrix(1363,520,1958,747) -> Matrix(3,2,-2,-1) Matrix(21,8,-176,-67) -> Matrix(1,0,2,1) Matrix(65,24,176,65) -> Matrix(1,0,4,1) Matrix(23,8,66,23) -> Matrix(1,0,0,1) Matrix(331,106,484,155) -> Matrix(1,-2,0,1) Matrix(329,104,484,153) -> Matrix(1,0,0,1) Matrix(1211,368,1958,595) -> Matrix(1,2,-2,-3) Matrix(1033,312,1672,505) -> Matrix(3,2,-2,-1) Matrix(89,26,154,45) -> Matrix(3,2,-2,-1) Matrix(43,12,154,43) -> Matrix(1,0,4,1) Matrix(23,6,88,23) -> Matrix(1,0,-2,1) Matrix(375,86,484,111) -> Matrix(1,-4,0,1) Matrix(373,84,484,109) -> Matrix(1,2,0,1) Matrix(67,14,110,23) -> Matrix(1,0,0,1) Matrix(21,4,110,21) -> Matrix(1,0,2,1) Matrix(23,4,132,23) -> Matrix(1,0,-2,1) Matrix(67,10,154,23) -> Matrix(1,2,0,1) Matrix(419,58,484,67) -> Matrix(1,0,2,1) Matrix(417,56,484,65) -> Matrix(1,0,-4,1) Matrix(67,-8,176,-21) -> Matrix(1,0,2,1) Matrix(67,-16,88,-21) -> Matrix(1,0,0,1) Matrix(353,-108,572,-175) -> Matrix(1,-2,0,1) Matrix(155,-48,352,-109) -> Matrix(1,2,0,1) Matrix(307,-116,352,-133) -> Matrix(1,0,-2,1) Matrix(397,-152,572,-219) -> Matrix(5,-2,-2,1) Matrix(155,-64,264,-109) -> Matrix(3,-4,-2,3) Matrix(483,-212,704,-309) -> Matrix(1,-4,0,1) Matrix(243,-136,352,-197) -> Matrix(1,2,0,1) Matrix(155,-96,176,-109) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 12 Degree of the the map X: 12 Degree of the the map Y: 60 Permutation triple for Y: ((1,4,16,46,43,56,51,47,17,5,2)(3,10,36,57,60,53,29,8,7,28,11)(6,22,9,34,32,31,14,13,38,52,23)(12,20,19,15,26,25,33,55,59,54,39)(18,37,44,42,41,24,50,49,27,35,48); (1,2,8,32,54,49,58,37,33,9,3)(4,14,21,6,5,20,50,60,57,44,15)(7,26,52,38,12,11,18,17,45,16,27)(10,30,29,13,42,47,59,55,46,24,23)(22,51,53,25,40,39,36,43,31,48,35); (2,6,24,41,13,4,3,12,40,25,7)(5,18,31,8,30,10,9,35,16,15,19)(11,28,27,54,47,22,21,14,43,55,37)(17,42,57,39,32,34,33,53,50,46,45)(20,38,29,51,56,36,23,26,44,58,49)) ----------------------------------------------------------------------- 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): 180 Minimal number of generators: 31 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: 20 Genus: 6 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/11 1/5 2/9 1/4 3/11 2/7 1/3 4/11 2/5 5/11 1/2 6/11 34/55 7/11 2/3 8/11 9/11 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -5/6 2/3 1/1 -9/11 1/1 -4/5 1/1 2/1 -3/4 0/1 1/1 -2/3 1/1 1/0 -7/11 1/0 -5/8 0/1 1/0 -8/13 1/1 1/0 -3/5 0/1 1/1 -7/12 1/1 2/1 -4/7 2/1 1/0 -1/2 0/1 1/0 -5/11 1/0 -4/9 -3/1 1/0 -3/7 -2/1 -1/1 -2/5 -1/1 0/1 -1/3 0/1 1/0 -3/10 -1/1 0/1 -2/7 -1/2 0/1 -3/11 0/1 -1/4 0/1 1/1 -2/9 -1/1 1/0 -1/5 -1/1 0/1 0/1 0/1 1/0 1/6 -1/1 0/1 2/11 0/1 1/5 0/1 1/1 2/9 1/1 1/0 1/4 -1/1 0/1 3/11 0/1 2/7 0/1 1/2 3/10 0/1 1/1 4/13 1/1 1/0 1/3 0/1 1/0 4/11 0/1 3/8 0/1 1/2 2/5 0/1 1/1 3/7 1/1 2/1 4/9 3/1 1/0 5/11 1/0 1/2 0/1 1/0 6/11 1/0 5/9 -4/1 1/0 4/7 -2/1 1/0 7/12 -2/1 -1/1 3/5 -1/1 0/1 8/13 -1/1 1/0 21/34 -2/1 -1/1 34/55 -1/1 13/21 -1/1 0/1 5/8 0/1 1/0 7/11 1/0 2/3 -1/1 1/0 5/7 -2/1 -1/1 8/11 -1/1 3/4 -1/1 0/1 4/5 -2/1 -1/1 9/11 -1/1 5/6 -1/1 -2/3 1/1 -1/1 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(45,38,-77,-65) (-1/1,-5/6) -> (-3/5,-7/12) Hyperbolic Matrix(109,90,132,109) (-5/6,-9/11) -> (9/11,5/6) Hyperbolic Matrix(89,72,110,89) (-9/11,-4/5) -> (4/5,9/11) Hyperbolic Matrix(23,18,-55,-43) (-4/5,-3/4) -> (-3/7,-2/5) Hyperbolic Matrix(23,16,33,23) (-3/4,-2/3) -> (2/3,5/7) Hyperbolic Matrix(43,28,66,43) (-2/3,-7/11) -> (7/11,2/3) Hyperbolic Matrix(111,70,176,111) (-7/11,-5/8) -> (5/8,7/11) Hyperbolic Matrix(45,28,143,89) (-5/8,-8/13) -> (4/13,1/3) Hyperbolic Matrix(23,14,110,67) (-8/13,-3/5) -> (1/5,2/9) Hyperbolic Matrix(45,26,154,89) (-7/12,-4/7) -> (2/7,3/10) Hyperbolic Matrix(43,24,77,43) (-4/7,-1/2) -> (5/9,4/7) Hyperbolic Matrix(21,10,44,21) (-1/2,-5/11) -> (5/11,1/2) Hyperbolic Matrix(89,40,198,89) (-5/11,-4/9) -> (4/9,5/11) Hyperbolic Matrix(23,10,-99,-43) (-4/9,-3/7) -> (-1/4,-2/9) Hyperbolic Matrix(21,8,55,21) (-2/5,-1/3) -> (3/8,2/5) Hyperbolic Matrix(89,28,143,45) (-1/3,-3/10) -> (13/21,5/8) Hyperbolic Matrix(89,26,154,45) (-3/10,-2/7) -> (4/7,7/12) Hyperbolic Matrix(43,12,154,43) (-2/7,-3/11) -> (3/11,2/7) Hyperbolic Matrix(23,6,88,23) (-3/11,-1/4) -> (1/4,3/11) Hyperbolic Matrix(67,14,110,23) (-2/9,-1/5) -> (3/5,8/13) Hyperbolic Matrix(1,0,11,1) (-1/5,0/1) -> (0/1,1/6) Parabolic Matrix(23,-4,121,-21) (1/6,2/11) -> (2/11,1/5) Parabolic Matrix(43,-10,99,-23) (2/9,1/4) -> (3/7,4/9) Hyperbolic Matrix(353,-108,572,-175) (3/10,4/13) -> (8/13,21/34) Hyperbolic Matrix(45,-16,121,-43) (1/3,4/11) -> (4/11,3/8) Parabolic Matrix(43,-18,55,-23) (2/5,3/7) -> (3/4,4/5) Hyperbolic Matrix(67,-36,121,-65) (1/2,6/11) -> (6/11,5/9) Parabolic Matrix(65,-38,77,-45) (7/12,3/5) -> (5/6,1/1) Hyperbolic Matrix(1871,-1156,3025,-1869) (21/34,34/55) -> (34/55,13/21) Parabolic Matrix(89,-64,121,-87) (5/7,8/11) -> (8/11,3/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(45,38,-77,-65) -> Matrix(3,-2,2,-1) Matrix(109,90,132,109) -> Matrix(5,-4,-6,5) Matrix(89,72,110,89) -> Matrix(3,-4,-2,3) Matrix(23,18,-55,-43) -> Matrix(1,-2,0,1) Matrix(23,16,33,23) -> Matrix(1,-2,0,1) Matrix(43,28,66,43) -> Matrix(1,-2,0,1) Matrix(111,70,176,111) -> Matrix(1,0,0,1) Matrix(45,28,143,89) -> Matrix(1,0,0,1) Matrix(23,14,110,67) -> Matrix(1,0,0,1) Matrix(45,26,154,89) -> Matrix(1,-2,2,-3) Matrix(43,24,77,43) -> Matrix(1,-4,0,1) Matrix(21,10,44,21) -> Matrix(1,0,0,1) Matrix(89,40,198,89) -> Matrix(1,6,0,1) Matrix(23,10,-99,-43) -> Matrix(1,2,0,1) Matrix(21,8,55,21) -> Matrix(1,0,2,1) Matrix(89,28,143,45) -> Matrix(1,0,0,1) Matrix(89,26,154,45) -> Matrix(3,2,-2,-1) Matrix(43,12,154,43) -> Matrix(1,0,4,1) Matrix(23,6,88,23) -> Matrix(1,0,-2,1) Matrix(67,14,110,23) -> Matrix(1,0,0,1) Matrix(1,0,11,1) -> Matrix(1,0,0,1) Matrix(23,-4,121,-21) -> Matrix(1,0,2,1) Matrix(43,-10,99,-23) -> Matrix(1,2,0,1) Matrix(353,-108,572,-175) -> Matrix(1,-2,0,1) Matrix(45,-16,121,-43) -> Matrix(1,0,2,1) Matrix(43,-18,55,-23) -> Matrix(1,-2,0,1) Matrix(67,-36,121,-65) -> Matrix(1,-4,0,1) Matrix(65,-38,77,-45) -> Matrix(1,2,-2,-3) Matrix(1871,-1156,3025,-1869) -> Matrix(1,2,-2,-3) Matrix(89,-64,121,-87) -> Matrix(1,2,-2,-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): 6 ----------------------------------------------------------------------- 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 (0/1,1/0) 0 11 2/11 0/1 2 1 1/5 (0/1,1/1) 0 11 2/9 (1/1,1/0) 0 11 1/4 (-1/1,0/1) 0 11 3/11 0/1 3 1 2/7 (0/1,1/2) 0 11 3/10 (0/1,1/1) 0 11 1/3 (0/1,1/0) 0 11 4/11 0/1 2 1 2/5 (0/1,1/1) 0 11 3/7 (1/1,2/1) 0 11 4/9 (3/1,1/0) 0 11 5/11 1/0 3 1 1/2 (0/1,1/0) 0 11 6/11 1/0 4 1 4/7 (-2/1,1/0) 0 11 7/12 (-2/1,-1/1) 0 11 3/5 (-1/1,0/1) 0 11 8/13 (-1/1,1/0) 0 11 34/55 -1/1 2 1 13/21 (-1/1,0/1) 0 11 5/8 (0/1,1/0) 0 11 7/11 1/0 1 1 2/3 (-1/1,1/0) 0 11 8/11 -1/1 2 1 3/4 (-1/1,0/1) 0 11 4/5 (-2/1,-1/1) 0 11 9/11 -1/1 4 1 5/6 (-1/1,-2/3) 0 11 1/1 (-1/1,0/1) 0 11 1/0 0/1 1 1 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(1,0,11,-1) (0/1,2/11) -> (0/1,2/11) Reflection Matrix(21,-4,110,-21) (2/11,1/5) -> (2/11,1/5) Reflection Matrix(67,-14,110,-23) (1/5,2/9) -> (3/5,8/13) Glide Reflection Matrix(43,-10,99,-23) (2/9,1/4) -> (3/7,4/9) Hyperbolic Matrix(23,-6,88,-23) (1/4,3/11) -> (1/4,3/11) Reflection Matrix(43,-12,154,-43) (3/11,2/7) -> (3/11,2/7) Reflection Matrix(89,-26,154,-45) (2/7,3/10) -> (4/7,7/12) Glide Reflection Matrix(89,-28,143,-45) (3/10,1/3) -> (13/21,5/8) Glide Reflection Matrix(23,-8,66,-23) (1/3,4/11) -> (1/3,4/11) Reflection Matrix(21,-8,55,-21) (4/11,2/5) -> (4/11,2/5) Reflection Matrix(43,-18,55,-23) (2/5,3/7) -> (3/4,4/5) Hyperbolic Matrix(89,-40,198,-89) (4/9,5/11) -> (4/9,5/11) Reflection Matrix(21,-10,44,-21) (5/11,1/2) -> (5/11,1/2) Reflection Matrix(23,-12,44,-23) (1/2,6/11) -> (1/2,6/11) Reflection Matrix(43,-24,77,-43) (6/11,4/7) -> (6/11,4/7) Reflection Matrix(65,-38,77,-45) (7/12,3/5) -> (5/6,1/1) Hyperbolic Matrix(441,-272,715,-441) (8/13,34/55) -> (8/13,34/55) Reflection Matrix(1429,-884,2310,-1429) (34/55,13/21) -> (34/55,13/21) Reflection Matrix(111,-70,176,-111) (5/8,7/11) -> (5/8,7/11) Reflection Matrix(43,-28,66,-43) (7/11,2/3) -> (7/11,2/3) Reflection Matrix(23,-16,33,-23) (2/3,8/11) -> (2/3,8/11) Reflection Matrix(65,-48,88,-65) (8/11,3/4) -> (8/11,3/4) Reflection Matrix(89,-72,110,-89) (4/5,9/11) -> (4/5,9/11) Reflection Matrix(109,-90,132,-109) (9/11,5/6) -> (9/11,5/6) 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(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,11,-1) -> Matrix(1,0,0,-1) (0/1,2/11) -> (0/1,1/0) Matrix(21,-4,110,-21) -> Matrix(1,0,2,-1) (2/11,1/5) -> (0/1,1/1) Matrix(67,-14,110,-23) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(43,-10,99,-23) -> Matrix(1,2,0,1) 1/0 Matrix(23,-6,88,-23) -> Matrix(-1,0,2,1) (1/4,3/11) -> (-1/1,0/1) Matrix(43,-12,154,-43) -> Matrix(1,0,4,-1) (3/11,2/7) -> (0/1,1/2) Matrix(89,-26,154,-45) -> Matrix(3,-2,-2,1) Matrix(89,-28,143,-45) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(23,-8,66,-23) -> Matrix(1,0,0,-1) (1/3,4/11) -> (0/1,1/0) Matrix(21,-8,55,-21) -> Matrix(1,0,2,-1) (4/11,2/5) -> (0/1,1/1) Matrix(43,-18,55,-23) -> Matrix(1,-2,0,1) 1/0 Matrix(89,-40,198,-89) -> Matrix(-1,6,0,1) (4/9,5/11) -> (3/1,1/0) Matrix(21,-10,44,-21) -> Matrix(1,0,0,-1) (5/11,1/2) -> (0/1,1/0) Matrix(23,-12,44,-23) -> Matrix(1,0,0,-1) (1/2,6/11) -> (0/1,1/0) Matrix(43,-24,77,-43) -> Matrix(1,4,0,-1) (6/11,4/7) -> (-2/1,1/0) Matrix(65,-38,77,-45) -> Matrix(1,2,-2,-3) -1/1 Matrix(441,-272,715,-441) -> Matrix(1,2,0,-1) (8/13,34/55) -> (-1/1,1/0) Matrix(1429,-884,2310,-1429) -> Matrix(-1,0,2,1) (34/55,13/21) -> (-1/1,0/1) Matrix(111,-70,176,-111) -> Matrix(1,0,0,-1) (5/8,7/11) -> (0/1,1/0) Matrix(43,-28,66,-43) -> Matrix(1,2,0,-1) (7/11,2/3) -> (-1/1,1/0) Matrix(23,-16,33,-23) -> Matrix(1,2,0,-1) (2/3,8/11) -> (-1/1,1/0) Matrix(65,-48,88,-65) -> Matrix(-1,0,2,1) (8/11,3/4) -> (-1/1,0/1) Matrix(89,-72,110,-89) -> Matrix(3,4,-2,-3) (4/5,9/11) -> (-2/1,-1/1) Matrix(109,-90,132,-109) -> Matrix(5,4,-6,-5) (9/11,5/6) -> (-1/1,-2/3) 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.