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): 216 Minimal number of generators: 37 Number of equivalence classes of cusps: 18 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -9/2 -3/1 -9/5 -3/2 0/1 1/1 3/2 27/17 9/5 2/1 18/7 3/1 27/8 18/5 9/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -2/3 -5/1 -1/1 -9/2 -1/2 -4/1 0/1 -7/2 -3/8 -3/1 -1/3 -11/4 -1/2 -19/7 -1/3 -27/10 -1/3 -8/3 -4/13 -5/2 -1/4 -17/7 -1/5 -12/5 0/1 -7/3 -3/11 -9/4 -1/4 -2/1 -2/9 -9/5 -1/5 -7/4 -3/16 -12/7 0/1 -5/3 -1/5 -18/11 -2/11 -13/8 -5/28 -8/5 -4/23 -3/2 -1/6 -10/7 -2/11 -27/19 -1/6 -17/12 -1/6 -7/5 -3/19 -18/13 -2/13 -11/8 -3/20 -4/3 0/1 -9/7 -1/7 -5/4 -1/8 -11/9 -1/7 -6/5 -2/15 -1/1 -1/9 0/1 0/1 1/1 1/9 6/5 2/15 5/4 1/8 9/7 1/7 4/3 0/1 7/5 3/19 3/2 1/6 11/7 1/7 19/12 1/6 27/17 1/6 8/5 4/23 5/3 1/5 17/10 1/4 12/7 0/1 7/4 3/16 9/5 1/5 2/1 2/9 9/4 1/4 7/3 3/11 12/5 0/1 5/2 1/4 18/7 2/7 13/5 5/17 8/3 4/13 3/1 1/3 10/3 2/7 27/8 1/3 17/5 1/3 7/2 3/8 18/5 2/5 11/3 3/7 4/1 0/1 9/2 1/2 5/1 1/1 11/2 1/2 6/1 2/3 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,54,4,31) (-6/1,1/0) -> (12/7,7/4) Hyperbolic Matrix(29,162,-12,-67) (-6/1,-5/1) -> (-17/7,-12/5) Hyperbolic Matrix(23,108,10,47) (-5/1,-9/2) -> (9/4,7/3) Hyperbolic Matrix(13,54,6,25) (-9/2,-4/1) -> (2/1,9/4) Hyperbolic Matrix(29,108,-18,-67) (-4/1,-7/2) -> (-13/8,-8/5) Hyperbolic Matrix(17,54,-6,-19) (-7/2,-3/1) -> (-3/1,-11/4) Parabolic Matrix(139,378,82,223) (-11/4,-19/7) -> (5/3,17/10) Hyperbolic Matrix(359,972,106,287) (-19/7,-27/10) -> (27/8,17/5) Hyperbolic Matrix(181,486,54,145) (-27/10,-8/3) -> (10/3,27/8) Hyperbolic Matrix(41,108,-30,-79) (-8/3,-5/2) -> (-11/8,-4/3) Hyperbolic Matrix(155,378,98,239) (-5/2,-17/7) -> (11/7,19/12) Hyperbolic Matrix(23,54,20,47) (-12/5,-7/3) -> (1/1,6/5) Hyperbolic Matrix(47,108,10,23) (-7/3,-9/4) -> (9/2,5/1) Hyperbolic Matrix(25,54,6,13) (-9/4,-2/1) -> (4/1,9/2) Hyperbolic Matrix(29,54,22,41) (-2/1,-9/5) -> (9/7,4/3) Hyperbolic Matrix(61,108,48,85) (-9/5,-7/4) -> (5/4,9/7) Hyperbolic Matrix(31,54,4,7) (-7/4,-12/7) -> (6/1,1/0) Hyperbolic Matrix(95,162,-78,-133) (-12/7,-5/3) -> (-11/9,-6/5) Hyperbolic Matrix(197,324,76,125) (-5/3,-18/11) -> (18/7,13/5) Hyperbolic Matrix(199,324,78,127) (-18/11,-13/8) -> (5/2,18/7) Hyperbolic Matrix(35,54,-24,-37) (-8/5,-3/2) -> (-3/2,-10/7) Parabolic Matrix(341,486,214,305) (-10/7,-27/19) -> (27/17,8/5) Hyperbolic Matrix(685,972,432,613) (-27/19,-17/12) -> (19/12,27/17) Hyperbolic Matrix(115,162,22,31) (-17/12,-7/5) -> (5/1,11/2) Hyperbolic Matrix(233,324,64,89) (-7/5,-18/13) -> (18/5,11/3) Hyperbolic Matrix(235,324,66,91) (-18/13,-11/8) -> (7/2,18/5) Hyperbolic Matrix(41,54,22,29) (-4/3,-9/7) -> (9/5,2/1) Hyperbolic Matrix(85,108,48,61) (-9/7,-5/4) -> (7/4,9/5) Hyperbolic Matrix(131,162,38,47) (-5/4,-11/9) -> (17/5,7/2) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(133,-162,78,-95) (6/5,5/4) -> (17/10,12/7) Hyperbolic Matrix(79,-108,30,-41) (4/3,7/5) -> (13/5,8/3) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) Hyperbolic Matrix(67,-162,12,-29) (12/5,5/2) -> (11/2,6/1) Hyperbolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,54,4,31) -> Matrix(3,2,16,11) Matrix(29,162,-12,-67) -> Matrix(3,2,-14,-9) Matrix(23,108,10,47) -> Matrix(7,4,26,15) Matrix(13,54,6,25) -> Matrix(5,2,22,9) Matrix(29,108,-18,-67) -> Matrix(9,4,-52,-23) Matrix(17,54,-6,-19) -> Matrix(5,2,-18,-7) Matrix(139,378,82,223) -> Matrix(5,2,22,9) Matrix(359,972,106,287) -> Matrix(35,12,102,35) Matrix(181,486,54,145) -> Matrix(19,6,60,19) Matrix(41,108,-30,-79) -> Matrix(13,4,-88,-27) Matrix(155,378,98,239) -> Matrix(9,2,58,13) Matrix(23,54,20,47) -> Matrix(7,2,52,15) Matrix(47,108,10,23) -> Matrix(15,4,26,7) Matrix(25,54,6,13) -> Matrix(9,2,22,5) Matrix(29,54,22,41) -> Matrix(9,2,58,13) Matrix(61,108,48,85) -> Matrix(21,4,152,29) Matrix(31,54,4,7) -> Matrix(11,2,16,3) Matrix(95,162,-78,-133) -> Matrix(9,2,-68,-15) Matrix(197,324,76,125) -> Matrix(65,12,222,41) Matrix(199,324,78,127) -> Matrix(67,12,240,43) Matrix(35,54,-24,-37) -> Matrix(11,2,-72,-13) Matrix(341,486,214,305) -> Matrix(35,6,204,35) Matrix(685,972,432,613) -> Matrix(73,12,444,73) Matrix(115,162,22,31) -> Matrix(13,2,32,5) Matrix(233,324,64,89) -> Matrix(77,12,186,29) Matrix(235,324,66,91) -> Matrix(79,12,204,31) Matrix(41,54,22,29) -> Matrix(13,2,58,9) Matrix(85,108,48,61) -> Matrix(29,4,152,21) Matrix(131,162,38,47) -> Matrix(13,2,32,5) Matrix(47,54,20,23) -> Matrix(15,2,52,7) Matrix(1,0,2,1) -> Matrix(1,0,18,1) Matrix(133,-162,78,-95) -> Matrix(15,-2,68,-9) Matrix(79,-108,30,-41) -> Matrix(27,-4,88,-13) Matrix(37,-54,24,-35) -> Matrix(13,-2,72,-11) Matrix(67,-108,18,-29) -> Matrix(23,-4,52,-9) Matrix(67,-162,12,-29) -> Matrix(9,-2,14,-3) Matrix(19,-54,6,-17) -> Matrix(7,-2,18,-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): 18 Degree of the the map X: 18 Degree of the the map Y: 36 Permutation triple for Y: ((2,6,23,16,15,11,31,36,26,19,5,18,13,4,3,12,27,8,21,20,35,34,14,10,9,24,7)(17,29,22)(25,33,28); (1,4,16,25,24,9,29,20,19,30,14,13,33,36,31,22,6,21,32,15,34,28,27,12,17,5,2)(3,10,11)(7,26,8); (1,2,8,28,13,18,17,31,10,30,19,7,25,34,35,29,12,11,32,21,26,33,16,23,22,9,3)(4,14,15)(5,20,6)) ----------------------------------------------------------------------- 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, lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z. ----------------------------------------------------------------------- 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): 108 Minimal number of generators: 19 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: 12 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 6/5 3/2 2/1 9/4 18/7 3/1 27/8 18/5 9/2 6/1 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 6/5 2/15 5/4 1/8 9/7 1/7 4/3 0/1 7/5 3/19 3/2 1/6 11/7 1/7 19/12 1/6 27/17 1/6 8/5 4/23 5/3 1/5 17/10 1/4 12/7 0/1 7/4 3/16 9/5 1/5 2/1 2/9 9/4 1/4 7/3 3/11 12/5 0/1 5/2 1/4 18/7 2/7 13/5 5/17 8/3 4/13 3/1 1/3 10/3 2/7 27/8 1/3 17/5 1/3 7/2 3/8 18/5 2/5 11/3 3/7 4/1 0/1 9/2 1/2 5/1 1/1 11/2 1/2 6/1 2/3 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(47,-54,27,-31) (1/1,6/5) -> (12/7,7/4) Hyperbolic Matrix(133,-162,78,-95) (6/5,5/4) -> (17/10,12/7) Hyperbolic Matrix(85,-108,37,-47) (5/4,9/7) -> (9/4,7/3) Hyperbolic Matrix(41,-54,19,-25) (9/7,4/3) -> (2/1,9/4) Hyperbolic Matrix(79,-108,30,-41) (4/3,7/5) -> (13/5,8/3) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(239,-378,141,-223) (11/7,19/12) -> (5/3,17/10) Hyperbolic Matrix(613,-972,181,-287) (19/12,27/17) -> (27/8,17/5) Hyperbolic Matrix(305,-486,91,-145) (27/17,8/5) -> (10/3,27/8) Hyperbolic Matrix(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) Hyperbolic Matrix(61,-108,13,-23) (7/4,9/5) -> (9/2,5/1) Hyperbolic Matrix(29,-54,7,-13) (9/5,2/1) -> (4/1,9/2) Hyperbolic Matrix(23,-54,3,-7) (7/3,12/5) -> (6/1,1/0) Hyperbolic Matrix(67,-162,12,-29) (12/5,5/2) -> (11/2,6/1) Hyperbolic Matrix(127,-324,49,-125) (5/2,18/7) -> (18/7,13/5) Parabolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(47,-162,9,-31) (17/5,7/2) -> (5/1,11/2) Hyperbolic Matrix(91,-324,25,-89) (7/2,18/5) -> (18/5,11/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,9,1) Matrix(47,-54,27,-31) -> Matrix(15,-2,83,-11) Matrix(133,-162,78,-95) -> Matrix(15,-2,68,-9) Matrix(85,-108,37,-47) -> Matrix(29,-4,109,-15) Matrix(41,-54,19,-25) -> Matrix(13,-2,59,-9) Matrix(79,-108,30,-41) -> Matrix(27,-4,88,-13) Matrix(37,-54,24,-35) -> Matrix(13,-2,72,-11) Matrix(239,-378,141,-223) -> Matrix(13,-2,59,-9) Matrix(613,-972,181,-287) -> Matrix(73,-12,213,-35) Matrix(305,-486,91,-145) -> Matrix(35,-6,111,-19) Matrix(67,-108,18,-29) -> Matrix(23,-4,52,-9) Matrix(61,-108,13,-23) -> Matrix(21,-4,37,-7) Matrix(29,-54,7,-13) -> Matrix(9,-2,23,-5) Matrix(23,-54,3,-7) -> Matrix(7,-2,11,-3) Matrix(67,-162,12,-29) -> Matrix(9,-2,14,-3) Matrix(127,-324,49,-125) -> Matrix(43,-12,147,-41) Matrix(19,-54,6,-17) -> Matrix(7,-2,18,-5) Matrix(47,-162,9,-31) -> Matrix(5,-2,13,-5) Matrix(91,-324,25,-89) -> Matrix(31,-12,75,-29) 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): 18 ----------------------------------------------------------------------- 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 2/9 1 27 9/4 1/4 9 3 7/3 3/11 1 27 12/5 0/1 3 9 5/2 1/4 1 27 8/3 4/13 1 27 3/1 1/3 3 9 10/3 2/7 1 27 27/8 1/3 9 1 17/5 1/3 1 27 7/2 3/8 1 27 18/5 2/5 9 3 11/3 3/7 1 27 4/1 0/1 1 27 9/2 1/2 9 3 5/1 1/1 1 27 11/2 1/2 1 27 6/1 2/3 3 9 1/0 1/0 1 27 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(25,-54,6,-13) (2/1,9/4) -> (4/1,9/2) Glide Reflection Matrix(47,-108,10,-23) (9/4,7/3) -> (9/2,5/1) Glide Reflection Matrix(23,-54,3,-7) (7/3,12/5) -> (6/1,1/0) Hyperbolic Matrix(67,-162,12,-29) (12/5,5/2) -> (11/2,6/1) Hyperbolic Matrix(41,-108,11,-29) (5/2,8/3) -> (11/3,4/1) Glide Reflection Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(161,-540,48,-161) (10/3,27/8) -> (10/3,27/8) Reflection Matrix(271,-918,80,-271) (27/8,17/5) -> (27/8,17/5) Reflection Matrix(47,-162,9,-31) (17/5,7/2) -> (5/1,11/2) Hyperbolic Matrix(91,-324,25,-89) (7/2,18/5) -> (18/5,11/3) Parabolic 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(25,-54,6,-13) -> Matrix(9,-2,22,-5) Matrix(47,-108,10,-23) -> Matrix(15,-4,26,-7) Matrix(23,-54,3,-7) -> Matrix(7,-2,11,-3) Matrix(67,-162,12,-29) -> Matrix(9,-2,14,-3) Matrix(41,-108,11,-29) -> Matrix(13,-4,29,-9) Matrix(19,-54,6,-17) -> Matrix(7,-2,18,-5) 1/3 Matrix(161,-540,48,-161) -> Matrix(13,-4,42,-13) (10/3,27/8) -> (2/7,1/3) Matrix(271,-918,80,-271) -> Matrix(41,-14,120,-41) (27/8,17/5) -> (1/3,7/20) Matrix(47,-162,9,-31) -> Matrix(5,-2,13,-5) (0/1,2/5).(1/3,1/2) Matrix(91,-324,25,-89) -> Matrix(31,-12,75,-29) 2/5 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.