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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 12 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/1 -10/7 0/1 1/1 2/1 20/9 5/2 8/3 10/3 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -1/2 -4/1 -2/5 0/1 -3/1 -1/4 -11/4 -1/2 -8/3 -2/7 -5/2 -1/4 -7/3 -3/14 -9/4 -1/4 -2/1 -1/5 -11/6 -1/6 -20/11 -2/11 0/1 -9/5 -1/6 -7/4 -3/16 -5/3 -1/6 -8/5 -2/13 -19/12 -5/34 -11/7 -1/8 -3/2 -1/6 -10/7 -1/7 -17/12 -9/64 -7/5 -3/22 -4/3 -2/15 0/1 -5/4 -1/8 -1/1 -1/10 0/1 0/1 1/1 1/10 5/4 1/8 4/3 0/1 2/15 3/2 1/6 11/7 1/8 8/5 2/13 5/3 1/6 7/4 3/16 9/5 1/6 2/1 1/5 11/5 1/4 20/9 0/1 2/9 9/4 1/4 7/3 3/14 5/2 1/4 8/3 2/7 19/7 5/16 11/4 1/2 3/1 1/4 10/3 1/3 17/5 9/26 7/2 3/8 4/1 0/1 2/5 5/1 1/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,40,4,23) (-5/1,1/0) -> (5/3,7/4) Hyperbolic Matrix(9,40,2,9) (-5/1,-4/1) -> (4/1,5/1) Hyperbolic Matrix(11,40,-8,-29) (-4/1,-3/1) -> (-7/5,-4/3) Hyperbolic Matrix(43,120,24,67) (-3/1,-11/4) -> (7/4,9/5) Hyperbolic Matrix(89,240,-56,-151) (-11/4,-8/3) -> (-8/5,-19/12) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(17,40,14,33) (-5/2,-7/3) -> (1/1,5/4) Hyperbolic Matrix(53,120,34,77) (-7/3,-9/4) -> (3/2,11/7) Hyperbolic Matrix(19,40,-10,-21) (-9/4,-2/1) -> (-2/1,-11/6) Parabolic Matrix(219,400,98,179) (-11/6,-20/11) -> (20/9,9/4) Hyperbolic Matrix(221,400,100,181) (-20/11,-9/5) -> (11/5,20/9) Hyperbolic Matrix(67,120,24,43) (-9/5,-7/4) -> (11/4,3/1) Hyperbolic Matrix(23,40,4,7) (-7/4,-5/3) -> (5/1,1/0) Hyperbolic Matrix(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(253,400,74,117) (-19/12,-11/7) -> (17/5,7/2) Hyperbolic Matrix(77,120,34,53) (-11/7,-3/2) -> (9/4,7/3) Hyperbolic Matrix(139,200,-98,-141) (-3/2,-10/7) -> (-10/7,-17/12) Parabolic Matrix(283,400,104,147) (-17/12,-7/5) -> (19/7,11/4) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(33,40,14,17) (-5/4,-1/1) -> (7/3,5/2) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(61,-200,18,-59) (3/1,10/3) -> (10/3,17/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,40,4,23) -> Matrix(3,2,16,11) Matrix(9,40,2,9) -> Matrix(5,2,12,5) Matrix(11,40,-8,-29) -> Matrix(5,2,-38,-15) Matrix(43,120,24,67) -> Matrix(7,2,38,11) Matrix(89,240,-56,-151) -> Matrix(13,4,-88,-27) Matrix(31,80,12,31) -> Matrix(15,4,56,15) Matrix(17,40,14,33) -> Matrix(9,2,76,17) Matrix(53,120,34,77) -> Matrix(9,2,58,13) Matrix(19,40,-10,-21) -> Matrix(9,2,-50,-11) Matrix(219,400,98,179) -> Matrix(1,0,10,1) Matrix(221,400,100,181) -> Matrix(1,0,10,1) Matrix(67,120,24,43) -> Matrix(11,2,38,7) Matrix(23,40,4,7) -> Matrix(11,2,16,3) Matrix(49,80,30,49) -> Matrix(25,4,156,25) Matrix(253,400,74,117) -> Matrix(55,8,158,23) Matrix(77,120,34,53) -> Matrix(13,2,58,9) Matrix(139,200,-98,-141) -> Matrix(69,10,-490,-71) Matrix(283,400,104,147) -> Matrix(57,8,178,25) Matrix(31,40,24,31) -> Matrix(15,2,112,15) Matrix(33,40,14,17) -> Matrix(17,2,76,9) Matrix(1,0,2,1) -> Matrix(1,0,20,1) Matrix(29,-40,8,-11) -> Matrix(15,-2,38,-5) Matrix(151,-240,56,-89) -> Matrix(27,-4,88,-13) Matrix(21,-40,10,-19) -> Matrix(11,-2,50,-9) Matrix(61,-200,18,-59) -> Matrix(31,-10,90,-29) 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: 24 Permutation triple for Y: ((2,6,18,19,7)(3,12,23,13,4)(5,15,10,9,8)(11,22,14,17,20); (1,4,15,19,22,16,8,7,20,23,24,18,11,3,10,21,17,12,5,2)(6,14,13,9); (1,2,8,13,20,21,10,4,14,19,24,23,17,6,5,16,22,18,9,3)(7,15,12,11)) ----------------------------------------------------------------------- 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): 72 Minimal number of generators: 13 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: 8 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 20/9 8/3 10/3 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/10 5/4 1/8 4/3 0/1 2/15 3/2 1/6 11/7 1/8 8/5 2/13 5/3 1/6 7/4 3/16 9/5 1/6 2/1 1/5 11/5 1/4 20/9 0/1 2/9 9/4 1/4 7/3 3/14 5/2 1/4 8/3 2/7 19/7 5/16 11/4 1/2 3/1 1/4 10/3 1/3 17/5 9/26 7/2 3/8 4/1 0/1 2/5 5/1 1/2 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(33,-40,19,-23) (1/1,5/4) -> (5/3,7/4) Hyperbolic Matrix(31,-40,7,-9) (5/4,4/3) -> (4/1,5/1) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(77,-120,43,-67) (3/2,11/7) -> (7/4,9/5) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(49,-80,19,-31) (8/5,5/3) -> (5/2,8/3) Hyperbolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(181,-400,81,-179) (11/5,20/9) -> (20/9,9/4) Parabolic Matrix(53,-120,19,-43) (9/4,7/3) -> (11/4,3/1) Hyperbolic Matrix(17,-40,3,-7) (7/3,5/2) -> (5/1,1/0) Hyperbolic Matrix(147,-400,43,-117) (19/7,11/4) -> (17/5,7/2) Hyperbolic Matrix(61,-200,18,-59) (3/1,10/3) -> (10/3,17/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,10,1) Matrix(33,-40,19,-23) -> Matrix(17,-2,94,-11) Matrix(31,-40,7,-9) -> Matrix(15,-2,38,-5) Matrix(29,-40,8,-11) -> Matrix(15,-2,38,-5) Matrix(77,-120,43,-67) -> Matrix(13,-2,72,-11) Matrix(151,-240,56,-89) -> Matrix(27,-4,88,-13) Matrix(49,-80,19,-31) -> Matrix(25,-4,94,-15) Matrix(21,-40,10,-19) -> Matrix(11,-2,50,-9) Matrix(181,-400,81,-179) -> Matrix(1,0,0,1) Matrix(53,-120,19,-43) -> Matrix(9,-2,32,-7) Matrix(17,-40,3,-7) -> Matrix(9,-2,14,-3) Matrix(147,-400,43,-117) -> Matrix(25,-8,72,-23) Matrix(61,-200,18,-59) -> Matrix(31,-10,90,-29) 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 10 1 2/1 1/5 2 10 20/9 (1/5,1/4) 0 1 9/4 1/4 1 20 7/3 3/14 1 20 5/2 1/4 5 4 8/3 2/7 2 5 11/4 1/2 1 20 3/1 1/4 1 20 10/3 1/3 10 2 4/1 (1/3,1/2) 0 5 5/1 1/2 5 4 1/0 1/0 1 20 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(19,-40,9,-19) (2/1,20/9) -> (2/1,20/9) Reflection Matrix(161,-360,72,-161) (20/9,9/4) -> (20/9,9/4) Reflection Matrix(53,-120,19,-43) (9/4,7/3) -> (11/4,3/1) Hyperbolic Matrix(17,-40,3,-7) (7/3,5/2) -> (5/1,1/0) Hyperbolic Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(89,-240,33,-89) (8/3,30/11) -> (8/3,30/11) Reflection Matrix(73,-200,23,-63) (19/7,11/4) -> (3/1,13/4) Hyperbolic Matrix(49,-160,15,-49) (16/5,10/3) -> (16/5,10/3) Reflection Matrix(11,-40,3,-11) (10/3,4/1) -> (10/3,4/1) Reflection Matrix(9,-40,2,-9) (4/1,5/1) -> (4/1,5/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,10,-1) (0/1,2/1) -> (0/1,1/5) Matrix(19,-40,9,-19) -> Matrix(9,-2,40,-9) (2/1,20/9) -> (1/5,1/4) Matrix(161,-360,72,-161) -> Matrix(9,-2,40,-9) (20/9,9/4) -> (1/5,1/4) Matrix(53,-120,19,-43) -> Matrix(9,-2,32,-7) 1/4 Matrix(17,-40,3,-7) -> Matrix(9,-2,14,-3) Matrix(31,-80,12,-31) -> Matrix(15,-4,56,-15) (5/2,8/3) -> (1/4,2/7) Matrix(89,-240,33,-89) -> Matrix(13,-4,42,-13) (8/3,30/11) -> (2/7,1/3) Matrix(73,-200,23,-63) -> Matrix(5,-2,18,-7) 1/3 Matrix(49,-160,15,-49) -> Matrix(25,-8,78,-25) (16/5,10/3) -> (4/13,1/3) Matrix(11,-40,3,-11) -> Matrix(5,-2,12,-5) (10/3,4/1) -> (1/3,1/2) Matrix(9,-40,2,-9) -> Matrix(5,-2,12,-5) (4/1,5/1) -> (1/3,1/2)