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): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -4/1 -3/1 -2/1 -3/2 -1/1 -3/4 -3/5 0/1 1/2 3/5 3/4 1/1 6/5 5/4 3/2 2/1 12/5 5/2 3/1 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -2/1 0/1 -5/1 -1/1 1/1 -4/1 -2/1 0/1 -3/1 -1/1 1/1 -8/3 -2/1 0/1 -5/2 -1/2 1/0 -12/5 0/1 -7/3 -1/1 1/1 -2/1 0/1 -7/4 1/0 -12/7 0/1 -5/3 1/1 -3/2 1/0 -7/5 -1/1 -4/3 -2/1 0/1 -5/4 1/0 -6/5 -2/1 0/1 -1/1 -1/1 1/1 -6/7 -2/1 0/1 -5/6 -1/2 1/0 -4/5 -2/1 0/1 -3/4 -1/2 1/0 -8/11 -2/1 0/1 -5/7 -1/1 -2/3 0/1 -5/8 1/0 -3/5 -1/1 1/1 -7/12 1/0 -4/7 -2/1 0/1 -5/9 -1/1 1/1 -6/11 -2/1 0/1 -1/2 -1/2 1/0 0/1 0/1 1/2 1/2 1/0 5/9 1/1 4/7 0/1 2/3 3/5 1/1 5/8 1/0 2/3 0/1 5/7 1/3 1/1 3/4 1/2 1/0 7/9 1/3 1/1 4/5 0/1 2/3 1/1 1/1 6/5 0/1 2/1 5/4 1/0 9/7 1/1 4/3 0/1 2/1 3/2 1/0 8/5 -2/1 0/1 5/3 -1/1 1/1 12/7 0/1 7/4 1/0 2/1 0/1 7/3 1/1 12/5 0/1 5/2 1/2 1/0 3/1 1/1 7/2 3/2 1/0 18/5 0/1 2/1 11/3 1/1 4/1 0/1 2/1 5/1 1/1 11/2 3/2 1/0 6/1 0/1 2/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(11,60,-20,-109) (-6/1,-5/1) -> (-5/9,-6/11) Hyperbolic Matrix(13,60,8,37) (-5/1,-4/1) -> (8/5,5/3) Hyperbolic Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(23,60,-28,-73) (-8/3,-5/2) -> (-5/6,-4/5) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(61,144,36,85) (-12/5,-7/3) -> (5/3,12/7) Hyperbolic Matrix(11,24,16,35) (-7/3,-2/1) -> (2/3,5/7) Hyperbolic Matrix(13,24,20,37) (-2/1,-7/4) -> (5/8,2/3) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(35,48,8,11) (-7/5,-4/3) -> (4/1,5/1) Hyperbolic Matrix(37,48,-64,-83) (-4/3,-5/4) -> (-7/12,-4/7) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(11,12,-12,-13) (-6/5,-1/1) -> (-1/1,-6/7) Parabolic Matrix(157,132,44,37) (-6/7,-5/6) -> (7/2,18/5) Hyperbolic Matrix(47,36,-64,-49) (-4/5,-3/4) -> (-3/4,-8/11) Parabolic Matrix(83,60,148,107) (-8/11,-5/7) -> (5/9,4/7) Hyperbolic Matrix(35,24,16,11) (-5/7,-2/3) -> (2/1,7/3) Hyperbolic Matrix(37,24,20,13) (-2/3,-5/8) -> (7/4,2/1) Hyperbolic Matrix(59,36,-100,-61) (-5/8,-3/5) -> (-3/5,-7/12) Parabolic Matrix(85,48,108,61) (-4/7,-5/9) -> (7/9,4/5) Hyperbolic Matrix(133,72,24,13) (-6/11,-1/2) -> (11/2,6/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(109,-60,20,-11) (1/2,5/9) -> (5/1,11/2) Hyperbolic Matrix(83,-48,64,-37) (4/7,3/5) -> (9/7,4/3) Hyperbolic Matrix(97,-60,76,-47) (3/5,5/8) -> (5/4,9/7) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(59,-48,16,-13) (4/5,1/1) -> (11/3,4/1) Hyperbolic Matrix(73,-84,20,-23) (1/1,6/5) -> (18/5,11/3) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,2,0,1) Matrix(11,60,-20,-109) -> Matrix(1,0,0,1) Matrix(13,60,8,37) -> Matrix(1,0,0,1) Matrix(11,36,-4,-13) -> Matrix(1,0,0,1) Matrix(23,60,-28,-73) -> Matrix(1,0,0,1) Matrix(49,120,20,49) -> Matrix(1,0,2,1) Matrix(61,144,36,85) -> Matrix(1,0,0,1) Matrix(11,24,16,35) -> Matrix(1,0,2,1) Matrix(13,24,20,37) -> Matrix(1,0,0,1) Matrix(97,168,56,97) -> Matrix(1,0,0,1) Matrix(85,144,36,61) -> Matrix(1,0,0,1) Matrix(23,36,-16,-25) -> Matrix(1,-2,0,1) Matrix(35,48,8,11) -> Matrix(1,2,0,1) Matrix(37,48,-64,-83) -> Matrix(1,0,0,1) Matrix(49,60,40,49) -> Matrix(1,2,0,1) Matrix(11,12,-12,-13) -> Matrix(1,0,0,1) Matrix(157,132,44,37) -> Matrix(1,2,0,1) Matrix(47,36,-64,-49) -> Matrix(1,0,0,1) Matrix(83,60,148,107) -> Matrix(1,0,2,1) Matrix(35,24,16,11) -> Matrix(1,0,2,1) Matrix(37,24,20,13) -> Matrix(1,0,0,1) Matrix(59,36,-100,-61) -> Matrix(1,0,0,1) Matrix(85,48,108,61) -> Matrix(1,0,2,1) Matrix(133,72,24,13) -> Matrix(1,2,0,1) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(109,-60,20,-11) -> Matrix(3,-2,2,-1) Matrix(83,-48,64,-37) -> Matrix(3,-2,2,-1) Matrix(97,-60,76,-47) -> Matrix(1,0,0,1) Matrix(49,-36,64,-47) -> Matrix(1,0,0,1) Matrix(59,-48,16,-13) -> Matrix(3,-2,2,-1) Matrix(73,-84,20,-23) -> Matrix(1,0,0,1) Matrix(25,-36,16,-23) -> Matrix(1,-2,0,1) Matrix(13,-36,4,-11) -> Matrix(3,-2,2,-1) 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): 4 Degree of the the map X: 4 Degree of the the map Y: 32 Permutation triple for Y: ((1,2)(3,10,21,6,20,11)(4,15,8,7,16,5)(9,24,23,19,14,13)(12,26,25,22,18,17)(27,32)(28,29)(30,31); (1,5,18,32,19,6)(2,8,26,27,9,3)(4,14)(7,24)(10,13,31,25,15,28)(11,12)(16,29,20,23,30,17)(21,22); (1,3,12,30,13,4)(2,6,22,31,23,7)(5,17)(8,25)(9,10)(11,29,15,14,32,26)(16,24,27,18,21,28)(19,20)) ----------------------------------------------------------------------- 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): 64 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: 4 Number of equivalence classes of cusps: 8 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/1 -2/1 -3/2 -1/1 0/1 1/2 3/4 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -1/1 1/1 -5/2 -1/2 1/0 -2/1 0/1 -5/3 1/1 -3/2 1/0 -4/3 -2/1 0/1 -5/4 1/0 -1/1 -1/1 1/1 -4/5 -2/1 0/1 -3/4 -1/2 1/0 -2/3 0/1 -3/5 -1/1 1/1 -4/7 -2/1 0/1 -1/2 -1/2 1/0 0/1 0/1 1/2 1/2 1/0 3/5 1/1 2/3 0/1 5/7 1/3 1/1 3/4 1/2 1/0 1/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(2,9,-3,-13) (-3/1,1/0) -> (-3/4,-2/3) Hyperbolic Matrix(10,27,-13,-35) (-3/1,-5/2) -> (-4/5,-3/4) Hyperbolic Matrix(11,27,-9,-22) (-5/2,-2/1) -> (-5/4,-1/1) Hyperbolic Matrix(22,39,-13,-23) (-2/1,-12/7) -> (-9/5,-5/3) Elliptic Matrix(13,21,21,34) (-5/3,-3/2) -> (3/5,2/3) Hyperbolic Matrix(11,15,19,26) (-3/2,-4/3) -> (1/2,3/5) Hyperbolic Matrix(26,33,37,47) (-4/3,-5/4) -> (2/3,5/7) Hyperbolic Matrix(25,21,-31,-26) (-1/1,-9/11) -> (-6/7,-4/5) Elliptic Matrix(14,9,17,11) (-2/3,-3/5) -> (3/4,1/1) Hyperbolic Matrix(46,27,63,37) (-3/5,-4/7) -> (5/7,3/4) Hyperbolic Matrix(37,21,-67,-38) (-4/7,-6/11) -> (-9/16,-1/2) Elliptic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(2,-3,1,-1) (1/1,3/1) -> (3/2,1/0) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(2,9,-3,-13) -> Matrix(0,-1,1,1) Matrix(10,27,-13,-35) -> Matrix(0,-1,1,1) Matrix(11,27,-9,-22) -> Matrix(1,1,-1,0) Matrix(22,39,-13,-23) -> Matrix(0,1,-1,1) Matrix(13,21,21,34) -> Matrix(1,-1,1,0) Matrix(11,15,19,26) -> Matrix(1,1,1,2) Matrix(26,33,37,47) -> Matrix(0,1,-1,1) Matrix(25,21,-31,-26) -> Matrix(1,1,-1,0) Matrix(14,9,17,11) -> Matrix(0,1,-1,1) Matrix(46,27,63,37) -> Matrix(0,1,-1,1) Matrix(37,21,-67,-38) -> Matrix(1,1,-1,0) Matrix(1,0,4,1) -> Matrix(1,0,2,1) Matrix(2,-3,1,-1) -> Matrix(0,1,-1,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 2 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: 2 Number of equivalence classes of cusps: 1 Genus: 0 Degree of H/liftables -> H/(image of liftables): 4 ----------------------------------------------------------------------- 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 -3/1 0 2 -5/2 0 6 -2/1 0/1 1 6 -3/2 1/0 1 2 -5/4 1/0 1 6 -1/1 0 6 -4/5 0 6 -3/4 0 2 -2/3 0/1 1 6 0/1 0/1 1 2 1/0 1/0 1 6 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(2,9,-3,-13) (-3/1,1/0) -> (-3/4,-2/3) Hyperbolic Matrix(10,27,-13,-35) (-3/1,-5/2) -> (-4/5,-3/4) Hyperbolic Matrix(11,27,-9,-22) (-5/2,-2/1) -> (-5/4,-1/1) Hyperbolic Matrix(13,24,-7,-13) (-2/1,-12/7) -> (-2/1,-12/7) Reflection Matrix(13,21,-8,-13) (-7/4,-3/2) -> (-7/4,-3/2) Reflection Matrix(11,15,-8,-11) (-3/2,-5/4) -> (-3/2,-5/4) Reflection Matrix(25,21,-31,-26) (-1/1,-9/11) -> (-6/7,-4/5) Elliptic Matrix(-1,0,3,1) (-2/3,0/1) -> (-2/3,0/1) Reflection Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(-1,3,0,1) (3/2,1/0) -> (3/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(2,9,-3,-13) -> Matrix(0,-1,1,1) (-1/1,1/1).(-1/2,1/0).(-2/1,0/1) Matrix(10,27,-13,-35) -> Matrix(0,-1,1,1) (-1/1,1/1).(-1/2,1/0).(-2/1,0/1) Matrix(11,27,-9,-22) -> Matrix(1,1,-1,0) (-1/1,1/1).(-1/2,1/0).(-2/1,0/1) Matrix(13,24,-7,-13) -> Matrix(1,0,1,-1) (-2/1,-12/7) -> (0/1,2/1) Matrix(13,21,-8,-13) -> Matrix(-1,1,0,1) (-7/4,-3/2) -> (1/2,1/0) Matrix(11,15,-8,-11) -> Matrix(1,1,0,-1) (-3/2,-5/4) -> (-1/2,1/0) Matrix(25,21,-31,-26) -> Matrix(1,1,-1,0) (-1/1,1/1).(-1/2,1/0).(-2/1,0/1) Matrix(-1,0,3,1) -> Matrix(-1,0,1,1) (-2/3,0/1) -> (-2/1,0/1) Matrix(1,0,1,-1) -> Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Matrix(-1,3,0,1) -> Matrix(-1,1,0,1) (3/2,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.