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 -1/4 -4/1 -1/5 -3/1 -3/20 -11/4 -11/80 -8/3 -2/15 -5/2 -1/8 -7/3 -7/60 -9/4 -9/80 -2/1 -1/10 -11/6 -11/120 -20/11 -1/11 -9/5 -9/100 -7/4 -7/80 -5/3 -1/12 -8/5 -2/25 -19/12 -19/240 -11/7 -11/140 -3/2 -3/40 -10/7 -1/14 -17/12 -17/240 -7/5 -7/100 -4/3 -1/15 -5/4 -1/16 -1/1 -1/20 0/1 0/1 1/1 1/20 5/4 1/16 4/3 1/15 3/2 3/40 11/7 11/140 8/5 2/25 5/3 1/12 7/4 7/80 9/5 9/100 2/1 1/10 11/5 11/100 20/9 1/9 9/4 9/80 7/3 7/60 5/2 1/8 8/3 2/15 19/7 19/140 11/4 11/80 3/1 3/20 10/3 1/6 17/5 17/100 7/2 7/40 4/1 1/5 5/1 1/4 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(7,2,80,23) Matrix(9,40,2,9) -> Matrix(9,2,40,9) Matrix(11,40,-8,-29) -> Matrix(11,2,-160,-29) Matrix(43,120,24,67) -> Matrix(43,6,480,67) Matrix(89,240,-56,-151) -> Matrix(89,12,-1120,-151) Matrix(31,80,12,31) -> Matrix(31,4,240,31) Matrix(17,40,14,33) -> Matrix(17,2,280,33) Matrix(53,120,34,77) -> Matrix(53,6,680,77) Matrix(19,40,-10,-21) -> Matrix(19,2,-200,-21) Matrix(219,400,98,179) -> Matrix(219,20,1960,179) Matrix(221,400,100,181) -> Matrix(221,20,2000,181) Matrix(67,120,24,43) -> Matrix(67,6,480,43) Matrix(23,40,4,7) -> Matrix(23,2,80,7) Matrix(49,80,30,49) -> Matrix(49,4,600,49) Matrix(253,400,74,117) -> Matrix(253,20,1480,117) Matrix(77,120,34,53) -> Matrix(77,6,680,53) Matrix(139,200,-98,-141) -> Matrix(139,10,-1960,-141) Matrix(283,400,104,147) -> Matrix(283,20,2080,147) Matrix(31,40,24,31) -> Matrix(31,2,480,31) Matrix(33,40,14,17) -> Matrix(33,2,280,17) Matrix(1,0,2,1) -> Matrix(1,0,40,1) Matrix(29,-40,8,-11) -> Matrix(29,-2,160,-11) Matrix(151,-240,56,-89) -> Matrix(151,-12,1120,-89) Matrix(21,-40,10,-19) -> Matrix(21,-2,200,-19) Matrix(61,-200,18,-59) -> Matrix(61,-10,360,-59) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 12 Genus: 7 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 24 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, lambda1 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1, lambda2, lambda1+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): 36 Minimal number of generators: 7 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: 6 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 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 1/20 4/3 1/15 3/2 3/40 5/3 1/12 2/1 1/10 5/2 1/8 8/3 2/15 3/1 3/20 10/3 1/6 7/2 7/40 4/1 1/5 5/1 1/4 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(17,-20,6,-7) (1/1,4/3) -> (8/3,3/1) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(13,-20,2,-3) (3/2,5/3) -> (5/1,1/0) Hyperbolic Matrix(11,-20,5,-9) (5/3,2/1) -> (2/1,5/2) Parabolic Matrix(23,-60,5,-13) (5/2,8/3) -> (4/1,5/1) Hyperbolic Matrix(31,-100,9,-29) (3/1,10/3) -> (10/3,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,20,1) Matrix(17,-20,6,-7) -> Matrix(17,-1,120,-7) Matrix(29,-40,8,-11) -> Matrix(29,-2,160,-11) Matrix(13,-20,2,-3) -> Matrix(13,-1,40,-3) Matrix(11,-20,5,-9) -> Matrix(11,-1,100,-9) Matrix(23,-60,5,-13) -> Matrix(23,-3,100,-13) Matrix(31,-100,9,-29) -> Matrix(31,-5,180,-29) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 36 Minimal number of generators: 7 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: 6 Genus: 1 Degree of H/liftables -> H/(image of liftables): 1 ----------------------------------------------------------------------- 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 20 1 2/1 1/10 2 10 5/2 1/8 5 4 3/1 3/20 1 20 10/3 1/6 10 2 4/1 1/5 4 5 5/1 1/4 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(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(7,-20,1,-3) (5/2,3/1) -> (5/1,1/0) Glide Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) 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(1,0,20,-1) (0/1,2/1) -> (0/1,1/10) Matrix(9,-20,4,-9) -> Matrix(9,-1,80,-9) (2/1,5/2) -> (1/10,1/8) Matrix(7,-20,1,-3) -> Matrix(7,-1,20,-3) Matrix(19,-60,6,-19) -> Matrix(19,-3,120,-19) (3/1,10/3) -> (3/20,1/6) Matrix(11,-40,3,-11) -> Matrix(11,-2,60,-11) (10/3,4/1) -> (1/6,1/5) Matrix(9,-40,2,-9) -> Matrix(9,-2,40,-9) (4/1,5/1) -> (1/5,1/4) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.