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 -3/5 -6/1 -4/7 -11/2 -1/2 -5/1 -3/7 -9/2 -1/2 -4/1 -2/5 -11/3 -1/3 -7/2 -3/10 -10/3 -4/15 -13/4 -1/4 -3/1 -1/3 -11/4 -1/4 -8/3 -2/9 -29/11 -5/23 -21/8 -5/24 -13/5 -1/5 -5/2 -1/6 -12/5 -2/5 -7/3 -1/5 -16/7 -2/9 -25/11 -1/3 -9/4 -1/4 -11/5 -1/5 -2/1 0/1 -11/6 -1/4 -9/5 -1/5 -16/9 -2/9 -7/4 -1/4 -12/7 -2/13 -17/10 -1/10 -22/13 0/1 -5/3 -1/3 -13/8 -1/4 -34/21 -4/15 -55/34 -1/4 -21/13 -5/21 -8/5 -2/9 -11/7 -1/5 -3/2 -1/6 -22/15 0/1 -19/13 -1/3 -16/11 -2/9 -13/9 -1/5 -23/16 -5/24 -33/23 -1/5 -10/7 -4/21 -7/5 -3/17 -11/8 -1/6 -4/3 -2/13 -13/10 -1/6 -22/17 -2/13 -9/7 -1/7 -5/4 -3/20 -11/9 -1/7 -6/5 -4/29 -7/6 -3/22 -22/19 -2/15 -15/13 -7/53 -8/7 -2/15 -1/1 -1/9 0/1 0/1 1/1 1/9 7/6 3/22 6/5 4/29 11/9 1/7 5/4 3/20 9/7 1/7 4/3 2/13 11/8 1/6 7/5 3/17 10/7 4/21 13/9 1/5 3/2 1/6 11/7 1/5 8/5 2/9 29/18 5/22 21/13 5/21 13/8 1/4 5/3 1/3 12/7 2/13 7/4 1/4 16/9 2/9 25/14 1/6 9/5 1/5 11/6 1/4 2/1 0/1 11/5 1/5 9/4 1/4 16/7 2/9 7/3 1/5 12/5 2/5 17/7 -1/1 22/9 0/1 5/2 1/6 13/5 1/5 34/13 4/21 55/21 1/5 21/8 5/24 8/3 2/9 11/4 1/4 3/1 1/3 22/7 0/1 19/6 1/6 16/5 2/9 13/4 1/4 23/7 5/21 33/10 1/4 10/3 4/15 7/2 3/10 11/3 1/3 4/1 2/5 13/3 1/3 22/5 2/5 9/2 1/2 5/1 3/7 11/2 1/2 6/1 4/7 7/1 3/5 22/3 2/3 15/2 7/10 8/1 2/3 1/0 1/0 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(5,4,-24,-19) Matrix(23,154,10,67) -> Matrix(3,2,10,7) Matrix(23,132,4,23) -> Matrix(15,8,28,15) Matrix(21,110,4,21) -> Matrix(13,6,28,13) Matrix(23,110,14,67) -> Matrix(5,2,22,9) Matrix(21,88,-16,-67) -> Matrix(1,0,-4,1) Matrix(23,88,6,23) -> Matrix(11,4,30,11) Matrix(43,154,12,43) -> Matrix(19,6,60,19) Matrix(45,154,26,89) -> Matrix(7,2,38,11) Matrix(175,572,-108,-353) -> Matrix(1,0,0,1) Matrix(109,352,-48,-155) -> Matrix(1,0,0,1) Matrix(23,66,8,23) -> Matrix(7,2,24,7) Matrix(65,176,24,65) -> Matrix(17,4,72,17) Matrix(133,352,-116,-307) -> Matrix(17,4,-132,-31) Matrix(219,572,-152,-397) -> Matrix(1,0,0,1) Matrix(43,110,34,87) -> Matrix(9,2,58,13) Matrix(109,264,-64,-155) -> Matrix(1,0,-4,1) Matrix(65,154,46,109) -> Matrix(7,2,38,11) Matrix(67,154,10,23) -> Matrix(7,2,10,3) Matrix(309,704,-212,-483) -> Matrix(1,0,0,1) Matrix(89,198,40,89) -> Matrix(9,2,40,9) Matrix(21,44,10,21) -> Matrix(1,0,10,1) Matrix(23,44,12,23) -> Matrix(1,0,8,1) Matrix(109,198,60,109) -> Matrix(9,2,40,9) Matrix(197,352,-136,-243) -> Matrix(1,0,0,1) Matrix(87,154,74,131) -> Matrix(11,2,82,15) Matrix(89,154,26,45) -> Matrix(11,2,38,7) Matrix(285,484,116,197) -> Matrix(1,0,16,1) Matrix(287,484,118,199) -> Matrix(1,0,2,1) Matrix(67,110,14,23) -> Matrix(9,2,22,5) Matrix(1033,1672,312,505) -> Matrix(31,8,120,31) Matrix(1211,1958,368,595) -> Matrix(41,10,168,41) Matrix(109,176,-96,-155) -> Matrix(17,4,-132,-31) Matrix(111,176,70,111) -> Matrix(19,4,90,19) Matrix(43,66,28,43) -> Matrix(11,2,60,11) Matrix(329,484,104,153) -> Matrix(1,0,12,1) Matrix(331,484,106,155) -> Matrix(1,0,6,1) Matrix(1363,1958,520,747) -> Matrix(49,10,240,49) Matrix(1167,1672,446,639) -> Matrix(41,8,210,41) Matrix(109,154,46,65) -> Matrix(11,2,38,7) Matrix(111,154,80,111) -> Matrix(35,6,204,35) Matrix(65,88,48,65) -> Matrix(25,4,156,25) Matrix(373,484,84,109) -> Matrix(25,4,56,9) Matrix(375,484,86,111) -> Matrix(27,4,74,11) Matrix(87,110,34,43) -> Matrix(13,2,58,9) Matrix(89,110,72,89) -> Matrix(41,6,280,41) Matrix(109,132,90,109) -> Matrix(57,8,406,57) Matrix(131,154,74,87) -> Matrix(15,2,82,11) Matrix(417,484,56,65) -> Matrix(149,20,216,29) Matrix(419,484,58,67) -> Matrix(151,20,234,31) Matrix(1,0,2,1) -> Matrix(1,0,18,1) Matrix(155,-176,96,-109) -> Matrix(31,-4,132,-17) Matrix(67,-88,16,-21) -> Matrix(1,0,-4,1) Matrix(397,-572,152,-219) -> Matrix(1,0,0,1) Matrix(243,-352,136,-197) -> Matrix(1,0,0,1) Matrix(219,-352,28,-45) -> Matrix(19,-4,24,-5) Matrix(353,-572,108,-175) -> Matrix(1,0,0,1) Matrix(155,-264,64,-109) -> Matrix(1,0,-4,1) Matrix(395,-704,124,-221) -> Matrix(1,0,0,1) Matrix(155,-352,48,-109) -> Matrix(1,0,0,1) Matrix(67,-176,8,-21) -> Matrix(19,-4,24,-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 cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 30 Degree of the the map X: 30 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): 60 Minimal number of generators: 11 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: 10 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 11/5 11/4 3/1 11/3 4/1 5/1 11/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/9 5/4 3/20 4/3 2/13 3/2 1/6 5/3 1/3 7/4 1/4 2/1 0/1 11/5 1/5 9/4 1/4 7/3 1/5 5/2 1/6 8/3 2/9 11/4 1/4 3/1 1/3 7/2 3/10 11/3 1/3 4/1 2/5 5/1 3/7 11/2 1/2 6/1 4/7 1/0 1/0 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(10,-11,1,-1) (1/1,5/4) -> (6/1,1/0) Hyperbolic Matrix(43,-55,18,-23) (5/4,4/3) -> (7/3,5/2) Hyperbolic Matrix(23,-33,7,-10) (4/3,3/2) -> (3/1,7/2) Hyperbolic Matrix(34,-55,13,-21) (3/2,5/3) -> (5/2,8/3) Hyperbolic Matrix(32,-55,7,-12) (5/3,7/4) -> (4/1,5/1) Hyperbolic Matrix(43,-77,19,-34) (7/4,2/1) -> (9/4,7/3) Hyperbolic Matrix(56,-121,25,-54) (2/1,11/5) -> (11/5,9/4) Parabolic Matrix(45,-121,16,-43) (8/3,11/4) -> (11/4,3/1) Parabolic Matrix(34,-121,9,-32) (7/2,11/3) -> (11/3,4/1) Parabolic Matrix(23,-121,4,-21) (5/1,11/2) -> (11/2,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,9,1) Matrix(10,-11,1,-1) -> Matrix(8,-1,9,-1) Matrix(43,-55,18,-23) -> Matrix(7,-1,22,-3) Matrix(23,-33,7,-10) -> Matrix(5,-1,21,-4) Matrix(34,-55,13,-21) -> Matrix(4,-1,21,-5) Matrix(32,-55,7,-12) -> Matrix(6,-1,13,-2) Matrix(43,-77,19,-34) -> Matrix(5,-1,21,-4) Matrix(56,-121,25,-54) -> Matrix(6,-1,25,-4) Matrix(45,-121,16,-43) -> Matrix(13,-3,48,-11) Matrix(34,-121,9,-32) -> Matrix(16,-5,45,-14) Matrix(23,-121,4,-21) -> Matrix(15,-7,28,-13) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 1 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: 1 Number of equivalence classes of cusps: 1 Genus: 0 Degree of H/liftables -> H/(image of liftables): 30 ----------------------------------------------------------------------- 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 9 1 2/1 0/1 1 11 11/5 1/5 1 1 7/3 1/5 1 11 5/2 1/6 1 11 11/4 1/4 3 1 3/1 1/3 1 11 11/3 1/3 5 1 4/1 2/5 1 11 5/1 3/7 1 11 11/2 1/2 7 1 1/0 1/0 1 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) 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(23,-55,5,-12) (7/3,5/2) -> (4/1,5/1) Glide Reflection Matrix(21,-55,8,-21) (5/2,11/4) -> (5/2,11/4) 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(-1,11,0,1) (11/2,1/0) -> (11/2,1/0) 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(1,0,1,-1) -> Matrix(1,0,9,-1) (0/1,2/1) -> (0/1,2/9) Matrix(21,-44,10,-21) -> Matrix(1,0,10,-1) (2/1,11/5) -> (0/1,1/5) Matrix(34,-77,15,-34) -> Matrix(4,-1,15,-4) (11/5,7/3) -> (1/5,1/3) Matrix(23,-55,5,-12) -> Matrix(3,-1,5,-2) Matrix(21,-55,8,-21) -> Matrix(5,-1,24,-5) (5/2,11/4) -> (1/6,1/4) Matrix(23,-66,8,-23) -> Matrix(7,-2,24,-7) (11/4,3/1) -> (1/4,1/3) Matrix(10,-33,3,-10) -> Matrix(4,-1,15,-4) (3/1,11/3) -> (1/5,1/3) Matrix(23,-88,6,-23) -> Matrix(11,-4,30,-11) (11/3,4/1) -> (1/3,2/5) Matrix(21,-110,4,-21) -> Matrix(13,-6,28,-13) (5/1,11/2) -> (3/7,1/2) Matrix(-1,11,0,1) -> Matrix(-1,1,0,1) (11/2,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.