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 0/1 1/1 11/9 11/8 3/2 11/7 11/6 2/1 11/5 22/9 5/2 55/21 11/4 3/1 22/7 33/10 10/3 7/2 11/3 4/1 13/3 22/5 9/2 5/1 11/2 6/1 7/1 22/3 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 0/1 1/2 -6/1 0/1 1/0 -11/2 0/1 -5/1 0/1 1/4 -9/2 3/10 1/3 -4/1 2/5 1/2 -11/3 1/2 -7/2 1/2 4/7 -10/3 1/2 2/3 -13/4 1/2 1/1 -3/1 1/2 1/1 -11/4 1/1 -8/3 1/1 1/0 -29/11 0/1 1/2 -21/8 0/1 1/2 -13/5 1/2 1/1 -5/2 0/1 1/0 -12/5 0/1 1/2 -7/3 0/1 1/2 -16/7 0/1 1/4 -25/11 1/3 1/2 -9/4 1/3 1/2 -11/5 1/2 -2/1 1/2 1/1 -11/6 1/1 -9/5 1/1 7/6 -16/9 6/5 5/4 -7/4 4/3 3/2 -12/7 3/2 2/1 -17/10 7/4 2/1 -22/13 2/1 -5/3 2/1 1/0 -13/8 3/1 7/2 -34/21 7/2 4/1 -55/34 4/1 -21/13 4/1 1/0 -8/5 5/1 1/0 -11/7 1/0 -3/2 -1/1 1/0 -22/15 -1/1 -19/13 -1/1 -1/2 -16/11 0/1 1/0 -13/9 -1/1 -1/2 -23/16 -2/3 -1/2 -33/23 -1/2 -10/7 -1/2 0/1 -7/5 -1/2 0/1 -11/8 0/1 -4/3 0/1 1/2 -13/10 5/6 1/1 -22/17 1/1 -9/7 1/1 3/2 -5/4 2/1 1/0 -11/9 1/0 -6/5 -2/1 1/0 -7/6 -1/2 0/1 -22/19 0/1 -15/13 0/1 1/2 -8/7 1/1 1/0 -1/1 0/1 1/0 0/1 0/1 1/1 0/1 1/4 7/6 0/1 1/6 6/5 2/9 1/4 11/9 1/4 5/4 1/4 2/7 9/7 3/10 1/3 4/3 0/1 1/2 11/8 0/1 7/5 0/1 1/6 10/7 0/1 1/6 13/9 1/6 1/5 3/2 1/5 1/4 11/7 1/4 8/5 1/4 5/19 29/18 4/15 7/26 21/13 1/4 4/15 13/8 7/26 3/11 5/3 1/4 2/7 12/7 2/7 3/10 7/4 3/10 4/13 16/9 5/16 6/19 25/14 5/16 1/3 9/5 7/22 1/3 11/6 1/3 2/1 1/3 1/2 11/5 1/2 9/4 1/2 1/1 16/7 0/1 1/0 7/3 0/1 1/2 12/5 0/1 1/2 17/7 0/1 1/0 22/9 0/1 5/2 0/1 1/4 13/5 1/3 1/2 34/13 2/5 1/2 55/21 1/2 21/8 0/1 1/2 8/3 1/4 1/3 11/4 1/3 3/1 1/3 1/2 22/7 1/3 19/6 1/3 3/8 16/5 3/8 2/5 13/4 1/3 1/2 23/7 3/8 2/5 33/10 2/5 10/3 2/5 1/2 7/2 4/9 1/2 11/3 1/2 4/1 1/2 2/3 13/3 5/6 1/1 22/5 1/1 9/2 1/1 3/2 5/1 0/1 1/0 11/2 0/1 6/1 0/1 1/4 7/1 0/1 1/2 22/3 0/1 15/2 0/1 1/6 8/1 1/4 1/3 1/0 0/1 1/2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(21,176,-8,-67) (-7/1,1/0) -> (-29/11,-21/8) Hyperbolic Matrix(23,154,10,67) (-7/1,-6/1) -> (16/7,7/3) Hyperbolic Matrix(23,132,4,23) (-6/1,-11/2) -> (11/2,6/1) Hyperbolic Matrix(21,110,4,21) (-11/2,-5/1) -> (5/1,11/2) Hyperbolic Matrix(23,110,14,67) (-5/1,-9/2) -> (13/8,5/3) Hyperbolic Matrix(21,88,-16,-67) (-9/2,-4/1) -> (-4/3,-13/10) Hyperbolic Matrix(23,88,6,23) (-4/1,-11/3) -> (11/3,4/1) Hyperbolic Matrix(43,154,12,43) (-11/3,-7/2) -> (7/2,11/3) Hyperbolic Matrix(45,154,26,89) (-7/2,-10/3) -> (12/7,7/4) Hyperbolic Matrix(175,572,-108,-353) (-10/3,-13/4) -> (-13/8,-34/21) Hyperbolic Matrix(109,352,-48,-155) (-13/4,-3/1) -> (-25/11,-9/4) Hyperbolic Matrix(23,66,8,23) (-3/1,-11/4) -> (11/4,3/1) Hyperbolic Matrix(65,176,24,65) (-11/4,-8/3) -> (8/3,11/4) Hyperbolic Matrix(133,352,-116,-307) (-8/3,-29/11) -> (-15/13,-8/7) Hyperbolic Matrix(219,572,-152,-397) (-21/8,-13/5) -> (-13/9,-23/16) Hyperbolic Matrix(43,110,34,87) (-13/5,-5/2) -> (5/4,9/7) Hyperbolic Matrix(109,264,-64,-155) (-5/2,-12/5) -> (-12/7,-17/10) Hyperbolic Matrix(65,154,46,109) (-12/5,-7/3) -> (7/5,10/7) Hyperbolic Matrix(67,154,10,23) (-7/3,-16/7) -> (6/1,7/1) Hyperbolic Matrix(309,704,-212,-483) (-16/7,-25/11) -> (-19/13,-16/11) Hyperbolic Matrix(89,198,40,89) (-9/4,-11/5) -> (11/5,9/4) Hyperbolic Matrix(21,44,10,21) (-11/5,-2/1) -> (2/1,11/5) Hyperbolic Matrix(23,44,12,23) (-2/1,-11/6) -> (11/6,2/1) Hyperbolic Matrix(109,198,60,109) (-11/6,-9/5) -> (9/5,11/6) Hyperbolic Matrix(197,352,-136,-243) (-9/5,-16/9) -> (-16/11,-13/9) Hyperbolic Matrix(87,154,74,131) (-16/9,-7/4) -> (7/6,6/5) Hyperbolic Matrix(89,154,26,45) (-7/4,-12/7) -> (10/3,7/2) Hyperbolic Matrix(285,484,116,197) (-17/10,-22/13) -> (22/9,5/2) Hyperbolic Matrix(287,484,118,199) (-22/13,-5/3) -> (17/7,22/9) Hyperbolic Matrix(67,110,14,23) (-5/3,-13/8) -> (9/2,5/1) Hyperbolic Matrix(1033,1672,312,505) (-34/21,-55/34) -> (33/10,10/3) Hyperbolic Matrix(1211,1958,368,595) (-55/34,-21/13) -> (23/7,33/10) Hyperbolic Matrix(109,176,-96,-155) (-21/13,-8/5) -> (-8/7,-1/1) Hyperbolic Matrix(111,176,70,111) (-8/5,-11/7) -> (11/7,8/5) Hyperbolic Matrix(43,66,28,43) (-11/7,-3/2) -> (3/2,11/7) Hyperbolic Matrix(329,484,104,153) (-3/2,-22/15) -> (22/7,19/6) Hyperbolic Matrix(331,484,106,155) (-22/15,-19/13) -> (3/1,22/7) Hyperbolic Matrix(1363,1958,520,747) (-23/16,-33/23) -> (55/21,21/8) Hyperbolic Matrix(1167,1672,446,639) (-33/23,-10/7) -> (34/13,55/21) Hyperbolic Matrix(109,154,46,65) (-10/7,-7/5) -> (7/3,12/5) Hyperbolic Matrix(111,154,80,111) (-7/5,-11/8) -> (11/8,7/5) Hyperbolic Matrix(65,88,48,65) (-11/8,-4/3) -> (4/3,11/8) Hyperbolic Matrix(373,484,84,109) (-13/10,-22/17) -> (22/5,9/2) Hyperbolic Matrix(375,484,86,111) (-22/17,-9/7) -> (13/3,22/5) Hyperbolic Matrix(87,110,34,43) (-9/7,-5/4) -> (5/2,13/5) Hyperbolic Matrix(89,110,72,89) (-5/4,-11/9) -> (11/9,5/4) Hyperbolic Matrix(109,132,90,109) (-11/9,-6/5) -> (6/5,11/9) Hyperbolic Matrix(131,154,74,87) (-6/5,-7/6) -> (7/4,16/9) Hyperbolic Matrix(417,484,56,65) (-7/6,-22/19) -> (22/3,15/2) Hyperbolic Matrix(419,484,58,67) (-22/19,-15/13) -> (7/1,22/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(155,-176,96,-109) (1/1,7/6) -> (29/18,21/13) Hyperbolic Matrix(67,-88,16,-21) (9/7,4/3) -> (4/1,13/3) Hyperbolic Matrix(397,-572,152,-219) (10/7,13/9) -> (13/5,34/13) Hyperbolic Matrix(243,-352,136,-197) (13/9,3/2) -> (25/14,9/5) Hyperbolic Matrix(219,-352,28,-45) (8/5,29/18) -> (15/2,8/1) Hyperbolic Matrix(353,-572,108,-175) (21/13,13/8) -> (13/4,23/7) Hyperbolic Matrix(155,-264,64,-109) (5/3,12/7) -> (12/5,17/7) Hyperbolic Matrix(395,-704,124,-221) (16/9,25/14) -> (19/6,16/5) Hyperbolic Matrix(155,-352,48,-109) (9/4,16/7) -> (16/5,13/4) Hyperbolic Matrix(67,-176,8,-21) (21/8,8/3) -> (8/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(21,176,-8,-67) -> Matrix(1,0,0,1) Matrix(23,154,10,67) -> Matrix(1,0,0,1) Matrix(23,132,4,23) -> Matrix(1,0,4,1) Matrix(21,110,4,21) -> Matrix(1,0,-4,1) Matrix(23,110,14,67) -> Matrix(9,-2,32,-7) Matrix(21,88,-16,-67) -> Matrix(5,-2,8,-3) Matrix(23,88,6,23) -> Matrix(9,-4,16,-7) Matrix(43,154,12,43) -> Matrix(15,-8,32,-17) Matrix(45,154,26,89) -> Matrix(13,-8,44,-27) Matrix(175,572,-108,-353) -> Matrix(13,-10,4,-3) Matrix(109,352,-48,-155) -> Matrix(3,-2,8,-5) Matrix(23,66,8,23) -> Matrix(3,-2,8,-5) Matrix(65,176,24,65) -> Matrix(1,-2,4,-7) Matrix(133,352,-116,-307) -> Matrix(1,0,0,1) Matrix(219,572,-152,-397) -> Matrix(3,-2,-4,3) Matrix(43,110,34,87) -> Matrix(1,-2,4,-7) Matrix(109,264,-64,-155) -> Matrix(7,-2,4,-1) Matrix(65,154,46,109) -> Matrix(1,0,4,1) Matrix(67,154,10,23) -> Matrix(1,0,0,1) Matrix(309,704,-212,-483) -> Matrix(1,0,-4,1) Matrix(89,198,40,89) -> Matrix(5,-2,8,-3) Matrix(21,44,10,21) -> Matrix(3,-2,8,-5) Matrix(23,44,12,23) -> Matrix(3,-2,8,-5) Matrix(109,198,60,109) -> Matrix(13,-14,40,-43) Matrix(197,352,-136,-243) -> Matrix(5,-6,-4,5) Matrix(87,154,74,131) -> Matrix(3,-4,16,-21) Matrix(89,154,26,45) -> Matrix(5,-8,12,-19) Matrix(285,484,116,197) -> Matrix(1,-2,8,-15) Matrix(287,484,118,199) -> Matrix(1,-2,0,1) Matrix(67,110,14,23) -> Matrix(1,-2,0,1) Matrix(1033,1672,312,505) -> Matrix(5,-18,12,-43) Matrix(1211,1958,368,595) -> Matrix(3,-14,8,-37) Matrix(109,176,-96,-155) -> Matrix(1,-4,0,1) Matrix(111,176,70,111) -> Matrix(1,-10,4,-39) Matrix(43,66,28,43) -> Matrix(1,2,4,9) Matrix(329,484,104,153) -> Matrix(3,4,8,11) Matrix(331,484,106,155) -> Matrix(1,0,4,1) Matrix(1363,1958,520,747) -> Matrix(3,2,4,3) Matrix(1167,1672,446,639) -> Matrix(5,2,12,5) Matrix(109,154,46,65) -> Matrix(1,0,4,1) Matrix(111,154,80,111) -> Matrix(1,0,8,1) Matrix(65,88,48,65) -> Matrix(1,0,0,1) Matrix(373,484,84,109) -> Matrix(9,-8,8,-7) Matrix(375,484,86,111) -> Matrix(7,-8,8,-9) Matrix(87,110,34,43) -> Matrix(1,-2,4,-7) Matrix(89,110,72,89) -> Matrix(1,-4,4,-15) Matrix(109,132,90,109) -> Matrix(1,4,4,17) Matrix(131,154,74,87) -> Matrix(5,4,16,13) Matrix(417,484,56,65) -> Matrix(1,0,8,1) Matrix(419,484,58,67) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(155,-176,96,-109) -> Matrix(17,-4,64,-15) Matrix(67,-88,16,-21) -> Matrix(5,-2,8,-3) Matrix(397,-572,152,-219) -> Matrix(11,-2,28,-5) Matrix(243,-352,136,-197) -> Matrix(29,-6,92,-19) Matrix(219,-352,28,-45) -> Matrix(15,-4,64,-17) Matrix(353,-572,108,-175) -> Matrix(37,-10,100,-27) Matrix(155,-264,64,-109) -> Matrix(7,-2,4,-1) Matrix(395,-704,124,-221) -> Matrix(25,-8,72,-23) Matrix(155,-352,48,-109) -> Matrix(3,-2,8,-5) Matrix(67,-176,8,-21) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 10 Degree of the the map X: 20 Degree of the the map Y: 60 Permutation triple for Y: ((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); (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)) ----------------------------------------------------------------------- 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 1/1 11/9 11/8 3/2 11/7 9/5 11/6 2/1 11/5 7/3 55/21 11/4 3/1 11/3 4/1 5/1 11/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 0/1 1/0 -11/2 0/1 -5/1 0/1 1/4 -4/1 2/5 1/2 -3/1 1/2 1/1 -11/4 1/1 -8/3 1/1 1/0 -13/5 1/2 1/1 -5/2 0/1 1/0 -12/5 0/1 1/2 -7/3 0/1 1/2 -2/1 1/2 1/1 -11/6 1/1 -9/5 1/1 7/6 -7/4 4/3 3/2 -5/3 2/1 1/0 -3/2 -1/1 1/0 -10/7 -1/2 0/1 -7/5 -1/2 0/1 -11/8 0/1 -4/3 0/1 1/2 -9/7 1/1 3/2 -5/4 2/1 1/0 -1/1 0/1 1/0 0/1 0/1 1/1 0/1 1/4 6/5 2/9 1/4 11/9 1/4 5/4 1/4 2/7 9/7 3/10 1/3 4/3 0/1 1/2 11/8 0/1 7/5 0/1 1/6 10/7 0/1 1/6 13/9 1/6 1/5 3/2 1/5 1/4 11/7 1/4 8/5 1/4 5/19 5/3 1/4 2/7 7/4 3/10 4/13 9/5 7/22 1/3 11/6 1/3 2/1 1/3 1/2 11/5 1/2 9/4 1/2 1/1 7/3 0/1 1/2 12/5 0/1 1/2 5/2 0/1 1/4 13/5 1/3 1/2 34/13 2/5 1/2 55/21 1/2 21/8 0/1 1/2 8/3 1/4 1/3 11/4 1/3 3/1 1/3 1/2 7/2 4/9 1/2 11/3 1/2 4/1 1/2 2/3 5/1 0/1 1/0 11/2 0/1 6/1 0/1 1/4 1/0 0/1 1/2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(12,77,-5,-32) (-6/1,1/0) -> (-5/2,-12/5) Hyperbolic Matrix(23,132,4,23) (-6/1,-11/2) -> (11/2,6/1) Hyperbolic Matrix(21,110,4,21) (-11/2,-5/1) -> (5/1,11/2) Hyperbolic Matrix(12,55,-7,-32) (-5/1,-4/1) -> (-7/4,-5/3) Hyperbolic Matrix(10,33,3,10) (-4/1,-3/1) -> (3/1,7/2) Hyperbolic Matrix(23,66,8,23) (-3/1,-11/4) -> (11/4,3/1) Hyperbolic Matrix(65,176,24,65) (-11/4,-8/3) -> (8/3,11/4) Hyperbolic Matrix(54,143,37,98) (-8/3,-13/5) -> (13/9,3/2) Hyperbolic Matrix(43,110,34,87) (-13/5,-5/2) -> (5/4,9/7) Hyperbolic Matrix(65,154,46,109) (-12/5,-7/3) -> (7/5,10/7) Hyperbolic Matrix(34,77,15,34) (-7/3,-2/1) -> (9/4,7/3) Hyperbolic Matrix(23,44,12,23) (-2/1,-11/6) -> (11/6,2/1) Hyperbolic Matrix(109,198,60,109) (-11/6,-9/5) -> (9/5,11/6) Hyperbolic Matrix(56,99,-43,-76) (-9/5,-7/4) -> (-4/3,-9/7) Hyperbolic Matrix(34,55,21,34) (-5/3,-3/2) -> (8/5,5/3) Hyperbolic Matrix(98,143,37,54) (-3/2,-10/7) -> (21/8,8/3) Hyperbolic Matrix(109,154,46,65) (-10/7,-7/5) -> (7/3,12/5) Hyperbolic Matrix(111,154,80,111) (-7/5,-11/8) -> (11/8,7/5) Hyperbolic Matrix(65,88,48,65) (-11/8,-4/3) -> (4/3,11/8) Hyperbolic Matrix(87,110,34,43) (-9/7,-5/4) -> (5/2,13/5) Hyperbolic Matrix(10,11,9,10) (-5/4,-1/1) -> (1/1,6/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(100,-121,81,-98) (6/5,11/9) -> (11/9,5/4) Parabolic Matrix(76,-99,43,-56) (9/7,4/3) -> (7/4,9/5) Hyperbolic Matrix(397,-572,152,-219) (10/7,13/9) -> (13/5,34/13) Hyperbolic Matrix(78,-121,49,-76) (3/2,11/7) -> (11/7,8/5) Parabolic Matrix(32,-55,7,-12) (5/3,7/4) -> (4/1,5/1) Hyperbolic Matrix(56,-121,25,-54) (2/1,11/5) -> (11/5,9/4) Parabolic Matrix(32,-77,5,-12) (12/5,5/2) -> (6/1,1/0) Hyperbolic Matrix(1156,-3025,441,-1154) (34/13,55/21) -> (55/21,21/8) Parabolic Matrix(34,-121,9,-32) (7/2,11/3) -> (11/3,4/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(12,77,-5,-32) -> Matrix(1,0,0,1) Matrix(23,132,4,23) -> Matrix(1,0,4,1) Matrix(21,110,4,21) -> Matrix(1,0,-4,1) Matrix(12,55,-7,-32) -> Matrix(7,-2,4,-1) Matrix(10,33,3,10) -> Matrix(3,-2,8,-5) Matrix(23,66,8,23) -> Matrix(3,-2,8,-5) Matrix(65,176,24,65) -> Matrix(1,-2,4,-7) Matrix(54,143,37,98) -> Matrix(1,0,4,1) Matrix(43,110,34,87) -> Matrix(1,-2,4,-7) Matrix(65,154,46,109) -> Matrix(1,0,4,1) Matrix(34,77,15,34) -> Matrix(1,0,0,1) Matrix(23,44,12,23) -> Matrix(3,-2,8,-5) Matrix(109,198,60,109) -> Matrix(13,-14,40,-43) Matrix(56,99,-43,-76) -> Matrix(3,-4,4,-5) Matrix(34,55,21,34) -> Matrix(1,-4,4,-15) Matrix(98,143,37,54) -> Matrix(1,0,4,1) Matrix(109,154,46,65) -> Matrix(1,0,4,1) Matrix(111,154,80,111) -> Matrix(1,0,8,1) Matrix(65,88,48,65) -> Matrix(1,0,0,1) Matrix(87,110,34,43) -> Matrix(1,-2,4,-7) Matrix(10,11,9,10) -> Matrix(1,0,4,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(100,-121,81,-98) -> Matrix(17,-4,64,-15) Matrix(76,-99,43,-56) -> Matrix(11,-4,36,-13) Matrix(397,-572,152,-219) -> Matrix(11,-2,28,-5) Matrix(78,-121,49,-76) -> Matrix(25,-6,96,-23) Matrix(32,-55,7,-12) -> Matrix(7,-2,4,-1) Matrix(56,-121,25,-54) -> Matrix(5,-2,8,-3) Matrix(32,-77,5,-12) -> Matrix(1,0,0,1) Matrix(1156,-3025,441,-1154) -> Matrix(5,-2,8,-3) Matrix(34,-121,9,-32) -> Matrix(13,-6,24,-11) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 5 ----------------------------------------------------------------------- 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 2 1 1/1 (0/1,1/4) 0 11 11/9 1/4 4 1 5/4 (1/4,2/7) 0 11 9/7 (3/10,1/3) 0 11 4/3 (0/1,1/2) 0 11 11/8 0/1 4 1 7/5 (0/1,1/6) 0 11 10/7 (0/1,1/6) 0 11 3/2 (1/5,1/4) 0 11 11/7 1/4 6 1 5/3 (1/4,2/7) 0 11 7/4 (3/10,4/13) 0 11 9/5 (7/22,1/3) 0 11 11/6 1/3 8 1 2/1 (1/3,1/2) 0 11 11/5 1/2 2 1 7/3 (0/1,1/2) 0 11 12/5 (0/1,1/2) 0 11 5/2 (0/1,1/4) 0 11 13/5 (1/3,1/2) 0 11 55/21 1/2 2 1 21/8 (0/1,1/2) 0 11 8/3 (1/4,1/3) 0 11 11/4 1/3 2 1 3/1 (1/3,1/2) 0 11 11/3 1/2 6 1 4/1 (1/2,2/3) 0 11 5/1 (0/1,1/0) 0 11 11/2 0/1 4 1 6/1 (0/1,1/4) 0 11 1/0 (0/1,1/2) 0 11 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(10,-11,9,-10) (1/1,11/9) -> (1/1,11/9) Reflection Matrix(89,-110,72,-89) (11/9,5/4) -> (11/9,5/4) Reflection Matrix(87,-110,34,-43) (5/4,9/7) -> (5/2,13/5) Glide Reflection Matrix(76,-99,43,-56) (9/7,4/3) -> (7/4,9/5) Hyperbolic Matrix(65,-88,48,-65) (4/3,11/8) -> (4/3,11/8) Reflection Matrix(111,-154,80,-111) (11/8,7/5) -> (11/8,7/5) Reflection Matrix(109,-154,46,-65) (7/5,10/7) -> (7/3,12/5) Glide Reflection Matrix(98,-143,37,-54) (10/7,3/2) -> (21/8,8/3) Glide Reflection Matrix(43,-66,28,-43) (3/2,11/7) -> (3/2,11/7) Reflection Matrix(34,-55,21,-34) (11/7,5/3) -> (11/7,5/3) Reflection Matrix(32,-55,7,-12) (5/3,7/4) -> (4/1,5/1) Hyperbolic Matrix(109,-198,60,-109) (9/5,11/6) -> (9/5,11/6) Reflection Matrix(23,-44,12,-23) (11/6,2/1) -> (11/6,2/1) Reflection Matrix(21,-44,10,-21) (2/1,11/5) -> (2/1,11/5) Reflection Matrix(34,-77,15,-34) (11/5,7/3) -> (11/5,7/3) Reflection Matrix(32,-77,5,-12) (12/5,5/2) -> (6/1,1/0) Hyperbolic Matrix(274,-715,105,-274) (13/5,55/21) -> (13/5,55/21) Reflection Matrix(881,-2310,336,-881) (55/21,21/8) -> (55/21,21/8) Reflection Matrix(65,-176,24,-65) (8/3,11/4) -> (8/3,11/4) Reflection Matrix(23,-66,8,-23) (11/4,3/1) -> (11/4,3/1) Reflection Matrix(10,-33,3,-10) (3/1,11/3) -> (3/1,11/3) Reflection Matrix(23,-88,6,-23) (11/3,4/1) -> (11/3,4/1) Reflection Matrix(21,-110,4,-21) (5/1,11/2) -> (5/1,11/2) Reflection Matrix(23,-132,4,-23) (11/2,6/1) -> (11/2,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,4,-1) (0/1,1/0) -> (0/1,1/2) Matrix(1,0,2,-1) -> Matrix(1,0,8,-1) (0/1,1/1) -> (0/1,1/4) Matrix(10,-11,9,-10) -> Matrix(1,0,8,-1) (1/1,11/9) -> (0/1,1/4) Matrix(89,-110,72,-89) -> Matrix(15,-4,56,-15) (11/9,5/4) -> (1/4,2/7) Matrix(87,-110,34,-43) -> Matrix(7,-2,24,-7) *** -> (1/4,1/3) Matrix(76,-99,43,-56) -> Matrix(11,-4,36,-13) 1/3 Matrix(65,-88,48,-65) -> Matrix(1,0,4,-1) (4/3,11/8) -> (0/1,1/2) Matrix(111,-154,80,-111) -> Matrix(1,0,12,-1) (11/8,7/5) -> (0/1,1/6) Matrix(109,-154,46,-65) -> Matrix(1,0,8,-1) *** -> (0/1,1/4) Matrix(98,-143,37,-54) -> Matrix(1,0,8,-1) *** -> (0/1,1/4) Matrix(43,-66,28,-43) -> Matrix(9,-2,40,-9) (3/2,11/7) -> (1/5,1/4) Matrix(34,-55,21,-34) -> Matrix(15,-4,56,-15) (11/7,5/3) -> (1/4,2/7) Matrix(32,-55,7,-12) -> Matrix(7,-2,4,-1) Matrix(109,-198,60,-109) -> Matrix(43,-14,132,-43) (9/5,11/6) -> (7/22,1/3) Matrix(23,-44,12,-23) -> Matrix(5,-2,12,-5) (11/6,2/1) -> (1/3,1/2) Matrix(21,-44,10,-21) -> Matrix(5,-2,12,-5) (2/1,11/5) -> (1/3,1/2) Matrix(34,-77,15,-34) -> Matrix(1,0,4,-1) (11/5,7/3) -> (0/1,1/2) Matrix(32,-77,5,-12) -> Matrix(1,0,0,1) Matrix(274,-715,105,-274) -> Matrix(5,-2,12,-5) (13/5,55/21) -> (1/3,1/2) Matrix(881,-2310,336,-881) -> Matrix(1,0,4,-1) (55/21,21/8) -> (0/1,1/2) Matrix(65,-176,24,-65) -> Matrix(7,-2,24,-7) (8/3,11/4) -> (1/4,1/3) Matrix(23,-66,8,-23) -> Matrix(5,-2,12,-5) (11/4,3/1) -> (1/3,1/2) Matrix(10,-33,3,-10) -> Matrix(5,-2,12,-5) (3/1,11/3) -> (1/3,1/2) Matrix(23,-88,6,-23) -> Matrix(7,-4,12,-7) (11/3,4/1) -> (1/2,2/3) Matrix(21,-110,4,-21) -> Matrix(1,0,0,-1) (5/1,11/2) -> (0/1,1/0) Matrix(23,-132,4,-23) -> Matrix(1,0,8,-1) (11/2,6/1) -> (0/1,1/4) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.