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: 24 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/3 -17/27 -7/18 -1/3 -8/27 -5/18 -1/6 -1/7 0/1 1/6 1/5 2/9 1/4 2/7 1/3 3/8 2/5 4/9 1/2 5/9 2/3 7/9 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -5/6 0/1 -4/5 1/1 1/0 -7/9 1/0 -3/4 -2/1 1/0 -5/7 1/0 -2/3 -1/1 -7/11 -3/4 -12/19 -2/3 -1/2 -17/27 -1/2 -5/8 -1/1 -1/2 -3/5 -3/4 -7/12 -2/3 -11/19 -5/8 -4/7 -3/5 -1/2 -5/9 -1/2 -1/2 -1/2 0/1 -4/9 0/1 -3/7 1/0 -5/12 0/1 -2/5 -1/1 1/0 -7/18 -1/1 -5/13 -3/4 -3/8 -1/1 -1/2 -1/3 -1/2 -3/10 -1/2 -1/3 -8/27 -1/3 -5/17 -1/4 -2/7 -1/2 -1/3 -5/18 -1/3 -3/11 -1/4 -1/4 -1/4 0/1 -2/9 0/1 -1/5 1/0 -1/6 0/1 -1/7 1/0 0/1 -1/2 0/1 1/6 0/1 1/5 -1/4 2/9 0/1 1/4 0/1 1/0 2/7 -1/1 -1/2 1/3 -1/2 4/11 -1/2 -1/3 7/19 -3/8 10/27 -1/3 3/8 -1/2 -1/3 2/5 -1/3 -1/4 5/12 0/1 8/19 -1/2 0/1 3/7 -1/4 4/9 0/1 1/2 -1/2 0/1 5/9 -1/2 4/7 -1/2 -3/7 7/12 -2/5 3/5 -3/8 11/18 -1/3 8/13 -1/3 -1/4 5/8 -1/2 -1/3 2/3 -1/3 7/10 -1/3 -3/10 19/27 -3/10 12/17 -3/10 -2/7 5/7 -1/4 13/18 -1/3 8/11 -1/3 -3/10 3/4 -2/7 -1/4 7/9 -1/4 4/5 -1/4 -1/5 5/6 0/1 6/7 -1/2 -1/3 1/1 -1/4 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(73,62,-126,-107) (-1/1,-5/6) -> (-7/12,-11/19) Hyperbolic Matrix(37,30,90,73) (-5/6,-4/5) -> (2/5,5/12) Hyperbolic Matrix(71,56,90,71) (-4/5,-7/9) -> (7/9,4/5) Hyperbolic Matrix(55,42,72,55) (-7/9,-3/4) -> (3/4,7/9) Hyperbolic Matrix(19,14,-72,-53) (-3/4,-5/7) -> (-3/11,-1/4) Hyperbolic Matrix(35,24,-54,-37) (-5/7,-2/3) -> (-2/3,-7/11) Parabolic Matrix(145,92,342,217) (-7/11,-12/19) -> (8/19,3/7) Hyperbolic Matrix(685,432,972,613) (-12/19,-17/27) -> (19/27,12/17) Hyperbolic Matrix(341,214,486,305) (-17/27,-5/8) -> (7/10,19/27) Hyperbolic Matrix(55,34,-144,-89) (-5/8,-3/5) -> (-5/13,-3/8) Hyperbolic Matrix(17,10,90,53) (-3/5,-7/12) -> (1/6,1/5) Hyperbolic Matrix(125,72,342,197) (-11/19,-4/7) -> (4/11,7/19) Hyperbolic Matrix(71,40,126,71) (-4/7,-5/9) -> (5/9,4/7) Hyperbolic Matrix(19,10,36,19) (-5/9,-1/2) -> (1/2,5/9) Hyperbolic Matrix(17,8,36,17) (-1/2,-4/9) -> (4/9,1/2) Hyperbolic Matrix(55,24,126,55) (-4/9,-3/7) -> (3/7,4/9) Hyperbolic Matrix(19,8,-126,-53) (-3/7,-5/12) -> (-1/6,-1/7) Hyperbolic Matrix(73,30,90,37) (-5/12,-2/5) -> (4/5,5/6) Hyperbolic Matrix(199,78,324,127) (-2/5,-7/18) -> (11/18,8/13) Hyperbolic Matrix(197,76,324,125) (-7/18,-5/13) -> (3/5,11/18) Hyperbolic Matrix(17,6,-54,-19) (-3/8,-1/3) -> (-1/3,-3/10) Parabolic Matrix(181,54,486,145) (-3/10,-8/27) -> (10/27,3/8) Hyperbolic Matrix(359,106,972,287) (-8/27,-5/17) -> (7/19,10/27) Hyperbolic Matrix(109,32,126,37) (-5/17,-2/7) -> (6/7,1/1) Hyperbolic Matrix(235,66,324,91) (-2/7,-5/18) -> (13/18,8/11) Hyperbolic Matrix(233,64,324,89) (-5/18,-3/11) -> (5/7,13/18) Hyperbolic Matrix(17,4,72,17) (-1/4,-2/9) -> (2/9,1/4) Hyperbolic Matrix(19,4,90,19) (-2/9,-1/5) -> (1/5,2/9) Hyperbolic Matrix(53,10,90,17) (-1/5,-1/6) -> (7/12,3/5) Hyperbolic Matrix(89,12,126,17) (-1/7,0/1) -> (12/17,5/7) Hyperbolic Matrix(53,-8,126,-19) (0/1,1/6) -> (5/12,8/19) Hyperbolic Matrix(53,-14,72,-19) (1/4,2/7) -> (8/11,3/4) Hyperbolic Matrix(19,-6,54,-17) (2/7,1/3) -> (1/3,4/11) Parabolic Matrix(89,-34,144,-55) (3/8,2/5) -> (8/13,5/8) Hyperbolic Matrix(107,-62,126,-73) (4/7,7/12) -> (5/6,6/7) Hyperbolic Matrix(37,-24,54,-35) (5/8,2/3) -> (2/3,7/10) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(73,62,-126,-107) -> Matrix(5,2,-8,-3) Matrix(37,30,90,73) -> Matrix(1,0,-4,1) Matrix(71,56,90,71) -> Matrix(1,-2,-4,9) Matrix(55,42,72,55) -> Matrix(1,4,-4,-15) Matrix(19,14,-72,-53) -> Matrix(1,2,-4,-7) Matrix(35,24,-54,-37) -> Matrix(3,4,-4,-5) Matrix(145,92,342,217) -> Matrix(3,2,-8,-5) Matrix(685,432,972,613) -> Matrix(13,8,-44,-27) Matrix(341,214,486,305) -> Matrix(1,2,-4,-7) Matrix(55,34,-144,-89) -> Matrix(1,0,0,1) Matrix(17,10,90,53) -> Matrix(3,2,-8,-5) Matrix(125,72,342,197) -> Matrix(7,4,-16,-9) Matrix(71,40,126,71) -> Matrix(11,6,-24,-13) Matrix(19,10,36,19) -> Matrix(1,0,0,1) Matrix(17,8,36,17) -> Matrix(1,0,0,1) Matrix(55,24,126,55) -> Matrix(1,0,-4,1) Matrix(19,8,-126,-53) -> Matrix(1,0,0,1) Matrix(73,30,90,37) -> Matrix(1,0,-4,1) Matrix(199,78,324,127) -> Matrix(1,2,-4,-7) Matrix(197,76,324,125) -> Matrix(7,6,-20,-17) Matrix(17,6,-54,-19) -> Matrix(3,2,-8,-5) Matrix(181,54,486,145) -> Matrix(1,0,0,1) Matrix(359,106,972,287) -> Matrix(13,4,-36,-11) Matrix(109,32,126,37) -> Matrix(1,0,0,1) Matrix(235,66,324,91) -> Matrix(11,4,-36,-13) Matrix(233,64,324,89) -> Matrix(1,0,0,1) Matrix(17,4,72,17) -> Matrix(1,0,4,1) Matrix(19,4,90,19) -> Matrix(1,0,-4,1) Matrix(53,10,90,17) -> Matrix(3,2,-8,-5) Matrix(89,12,126,17) -> Matrix(1,2,-4,-7) Matrix(53,-8,126,-19) -> Matrix(1,0,0,1) Matrix(53,-14,72,-19) -> Matrix(1,2,-4,-7) Matrix(19,-6,54,-17) -> Matrix(3,2,-8,-5) Matrix(89,-34,144,-55) -> Matrix(1,0,0,1) Matrix(107,-62,126,-73) -> Matrix(5,2,-8,-3) Matrix(37,-24,54,-35) -> Matrix(11,4,-36,-13) 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): 6 Degree of the the map X: 12 Degree of the the map Y: 36 Permutation triple for Y: ((1,4,16,33,36,31,17,5,2)(3,10,11)(6,21,32,14,13,28,27,9,22)(7,26,8)(12,29,20,19,30,15,34,25,24); (1,2,8,28,34,35,29,9,3)(4,14,15)(5,20,6)(7,25,16,18,17,12,11,32,21)(10,30,19,26,33,13,23,22,31); (2,6,23,13,4,3,12,24,7)(5,18,16,15,10,9,27,8,19)(11,31,36,26,21,20,35,34,14)(17,22,29)(25,28,33)) ----------------------------------------------------------------------- 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): 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: 16 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/3 -1/3 0/1 1/5 2/9 1/4 1/3 3/7 4/9 1/2 5/9 2/3 19/27 7/9 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -3/4 -2/1 1/0 -5/7 1/0 -2/3 -1/1 -7/11 -3/4 -5/8 -1/1 -1/2 -3/5 -3/4 -1/2 -1/2 0/1 -4/9 0/1 -3/7 1/0 -2/5 -1/1 1/0 -1/3 -1/2 -1/4 -1/4 0/1 -2/9 0/1 -1/5 1/0 0/1 -1/2 0/1 1/5 -1/4 2/9 0/1 1/4 0/1 1/0 1/3 -1/2 2/5 -1/3 -1/4 3/7 -1/4 4/9 0/1 1/2 -1/2 0/1 5/9 -1/2 4/7 -1/2 -3/7 3/5 -3/8 5/8 -1/2 -1/3 2/3 -1/3 7/10 -1/3 -3/10 19/27 -3/10 12/17 -3/10 -2/7 5/7 -1/4 3/4 -2/7 -1/4 7/9 -1/4 4/5 -1/4 -1/5 1/1 -1/4 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(8,7,9,8) (-1/1,-3/4) -> (4/5,1/1) Hyperbolic Matrix(26,19,-63,-46) (-3/4,-5/7) -> (-3/7,-2/5) Hyperbolic Matrix(35,24,-54,-37) (-5/7,-2/3) -> (-2/3,-7/11) Parabolic Matrix(172,109,243,154) (-7/11,-5/8) -> (12/17,5/7) Hyperbolic Matrix(8,5,-45,-28) (-5/8,-3/5) -> (-1/5,0/1) Hyperbolic Matrix(26,15,45,26) (-3/5,-1/2) -> (4/7,3/5) Hyperbolic Matrix(17,8,36,17) (-1/2,-4/9) -> (4/9,1/2) Hyperbolic Matrix(55,24,126,55) (-4/9,-3/7) -> (3/7,4/9) Hyperbolic Matrix(8,3,-27,-10) (-2/5,-1/3) -> (-1/3,-1/4) Parabolic Matrix(17,4,72,17) (-1/4,-2/9) -> (2/9,1/4) Hyperbolic Matrix(19,4,90,19) (-2/9,-1/5) -> (1/5,2/9) Hyperbolic Matrix(28,-5,45,-8) (0/1,1/5) -> (3/5,5/8) Hyperbolic Matrix(10,-3,27,-8) (1/4,1/3) -> (1/3,2/5) Parabolic Matrix(46,-19,63,-26) (2/5,3/7) -> (5/7,3/4) Hyperbolic Matrix(46,-25,81,-44) (1/2,5/9) -> (5/9,4/7) Parabolic Matrix(37,-24,54,-35) (5/8,2/3) -> (2/3,7/10) Parabolic Matrix(514,-361,729,-512) (7/10,19/27) -> (19/27,12/17) Parabolic Matrix(64,-49,81,-62) (3/4,7/9) -> (7/9,4/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-4,1) Matrix(8,7,9,8) -> Matrix(1,1,-4,-3) Matrix(26,19,-63,-46) -> Matrix(1,1,0,1) Matrix(35,24,-54,-37) -> Matrix(3,4,-4,-5) Matrix(172,109,243,154) -> Matrix(7,5,-24,-17) Matrix(8,5,-45,-28) -> Matrix(1,1,-4,-3) Matrix(26,15,45,26) -> Matrix(5,3,-12,-7) Matrix(17,8,36,17) -> Matrix(1,0,0,1) Matrix(55,24,126,55) -> Matrix(1,0,-4,1) Matrix(8,3,-27,-10) -> Matrix(1,1,-4,-3) Matrix(17,4,72,17) -> Matrix(1,0,4,1) Matrix(19,4,90,19) -> Matrix(1,0,-4,1) Matrix(28,-5,45,-8) -> Matrix(1,1,-4,-3) Matrix(10,-3,27,-8) -> Matrix(1,1,-4,-3) Matrix(46,-19,63,-26) -> Matrix(5,1,-16,-3) Matrix(46,-25,81,-44) -> Matrix(5,3,-12,-7) Matrix(37,-24,54,-35) -> Matrix(11,4,-36,-13) Matrix(514,-361,729,-512) -> Matrix(29,9,-100,-31) Matrix(64,-49,81,-62) -> Matrix(11,3,-48,-13) 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 (-1/2,0/1) 0 9 1/5 -1/4 1 9 2/9 0/1 4 1 1/4 (0/1,1/0) 0 9 1/3 -1/2 1 3 2/5 (-1/3,-1/4) 0 9 3/7 -1/4 1 9 4/9 0/1 2 1 1/2 (-1/2,0/1) 0 9 5/9 -1/2 3 1 3/5 -3/8 1 9 5/8 (-1/2,-1/3) 0 9 2/3 -1/3 2 3 7/10 (-1/3,-3/10) 0 9 19/27 -3/10 1 1 5/7 -1/4 1 9 3/4 (-2/7,-1/4) 0 9 7/9 -1/4 3 1 1/1 -1/4 1 9 1/0 0/1 2 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(28,-5,45,-8) (0/1,1/5) -> (3/5,5/8) Hyperbolic Matrix(19,-4,90,-19) (1/5,2/9) -> (1/5,2/9) Reflection Matrix(17,-4,72,-17) (2/9,1/4) -> (2/9,1/4) Reflection Matrix(10,-3,27,-8) (1/4,1/3) -> (1/3,2/5) Parabolic Matrix(46,-19,63,-26) (2/5,3/7) -> (5/7,3/4) Hyperbolic Matrix(55,-24,126,-55) (3/7,4/9) -> (3/7,4/9) Reflection Matrix(17,-8,36,-17) (4/9,1/2) -> (4/9,1/2) Reflection Matrix(19,-10,36,-19) (1/2,5/9) -> (1/2,5/9) Reflection Matrix(26,-15,45,-26) (5/9,3/5) -> (5/9,3/5) Reflection Matrix(37,-24,54,-35) (5/8,2/3) -> (2/3,7/10) Parabolic Matrix(379,-266,540,-379) (7/10,19/27) -> (7/10,19/27) Reflection Matrix(134,-95,189,-134) (19/27,5/7) -> (19/27,5/7) Reflection Matrix(55,-42,72,-55) (3/4,7/9) -> (3/4,7/9) Reflection Matrix(8,-7,9,-8) (7/9,1/1) -> (7/9,1/1) 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(-1,0,4,1) (0/1,1/0) -> (-1/2,0/1) Matrix(28,-5,45,-8) -> Matrix(1,1,-4,-3) -1/2 Matrix(19,-4,90,-19) -> Matrix(-1,0,8,1) (1/5,2/9) -> (-1/4,0/1) Matrix(17,-4,72,-17) -> Matrix(1,0,0,-1) (2/9,1/4) -> (0/1,1/0) Matrix(10,-3,27,-8) -> Matrix(1,1,-4,-3) -1/2 Matrix(46,-19,63,-26) -> Matrix(5,1,-16,-3) -1/4 Matrix(55,-24,126,-55) -> Matrix(-1,0,8,1) (3/7,4/9) -> (-1/4,0/1) Matrix(17,-8,36,-17) -> Matrix(-1,0,4,1) (4/9,1/2) -> (-1/2,0/1) Matrix(19,-10,36,-19) -> Matrix(-1,0,4,1) (1/2,5/9) -> (-1/2,0/1) Matrix(26,-15,45,-26) -> Matrix(7,3,-16,-7) (5/9,3/5) -> (-1/2,-3/8) Matrix(37,-24,54,-35) -> Matrix(11,4,-36,-13) -1/3 Matrix(379,-266,540,-379) -> Matrix(19,6,-60,-19) (7/10,19/27) -> (-1/3,-3/10) Matrix(134,-95,189,-134) -> Matrix(11,3,-40,-11) (19/27,5/7) -> (-3/10,-1/4) Matrix(55,-42,72,-55) -> Matrix(15,4,-56,-15) (3/4,7/9) -> (-2/7,-1/4) Matrix(8,-7,9,-8) -> Matrix(3,1,-8,-3) (7/9,1/1) -> (-1/2,-1/4) Matrix(-1,2,0,1) -> Matrix(-1,0,8,1) (1/1,1/0) -> (-1/4,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.