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): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 16 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 15/11 3/2 2/1 12/5 5/2 30/11 3/1 10/3 15/4 4/1 5/1 6/1 20/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/5 -5/1 -1/6 -4/1 -2/15 -15/4 -1/8 -26/7 -13/105 -11/3 -11/90 -18/5 -3/25 -7/2 -7/60 -10/3 -1/9 -3/1 -1/10 -11/4 -11/120 -8/3 -4/45 -5/2 -1/12 -12/5 -2/25 -19/8 -19/240 -7/3 -7/90 -9/4 -3/40 -20/9 -2/27 -11/5 -11/150 -13/6 -13/180 -2/1 -1/15 -11/6 -11/180 -20/11 -2/33 -9/5 -3/50 -7/4 -7/120 -26/15 -13/225 -19/11 -19/330 -12/7 -2/35 -5/3 -1/18 -8/5 -4/75 -19/12 -19/360 -30/19 -1/19 -11/7 -11/210 -3/2 -1/20 -10/7 -1/21 -17/12 -17/360 -24/17 -4/85 -7/5 -7/150 -11/8 -11/240 -26/19 -13/285 -15/11 -1/22 -4/3 -2/45 -5/4 -1/24 -6/5 -1/25 -13/11 -13/330 -20/17 -2/51 -7/6 -7/180 -1/1 -1/30 0/1 0/1 1/1 1/30 6/5 1/25 5/4 1/24 4/3 2/45 15/11 1/22 26/19 13/285 11/8 11/240 18/13 3/65 7/5 7/150 10/7 1/21 3/2 1/20 11/7 11/210 8/5 4/75 5/3 1/18 12/7 2/35 19/11 19/330 7/4 7/120 9/5 3/50 20/11 2/33 11/6 11/180 13/7 13/210 2/1 1/15 11/5 11/150 20/9 2/27 9/4 3/40 7/3 7/90 26/11 13/165 19/8 19/240 12/5 2/25 5/2 1/12 8/3 4/45 19/7 19/210 30/11 1/11 11/4 11/120 3/1 1/10 10/3 1/9 17/5 17/150 24/7 4/35 7/2 7/60 11/3 11/90 26/7 13/105 15/4 1/8 4/1 2/15 5/1 1/6 6/1 1/5 13/2 13/60 20/3 2/9 7/1 7/30 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,60,-2,-17) (-6/1,1/0) -> (-18/5,-7/2) Hyperbolic Matrix(11,60,2,11) (-6/1,-5/1) -> (5/1,6/1) Hyperbolic Matrix(13,60,8,37) (-5/1,-4/1) -> (8/5,5/3) Hyperbolic Matrix(31,120,8,31) (-4/1,-15/4) -> (15/4,4/1) Hyperbolic Matrix(209,780,56,209) (-15/4,-26/7) -> (26/7,15/4) Hyperbolic Matrix(97,360,52,193) (-26/7,-11/3) -> (13/7,2/1) Hyperbolic Matrix(83,300,-70,-253) (-11/3,-18/5) -> (-6/5,-13/11) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(19,60,6,19) (-10/3,-3/1) -> (3/1,10/3) 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(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(277,660,-196,-467) (-12/5,-19/8) -> (-17/12,-24/17) Hyperbolic Matrix(127,300,-58,-137) (-19/8,-7/3) -> (-11/5,-13/6) Hyperbolic Matrix(53,120,34,77) (-7/3,-9/4) -> (3/2,11/7) Hyperbolic Matrix(161,360,72,161) (-9/4,-20/9) -> (20/9,9/4) Hyperbolic Matrix(217,480,-184,-407) (-20/9,-11/5) -> (-13/11,-20/17) Hyperbolic Matrix(167,360,122,263) (-13/6,-2/1) -> (26/19,11/8) Hyperbolic Matrix(163,300,-94,-173) (-2/1,-11/6) -> (-7/4,-26/15) Hyperbolic Matrix(197,360,-168,-307) (-11/6,-20/11) -> (-20/17,-7/6) Hyperbolic Matrix(199,360,110,199) (-20/11,-9/5) -> (9/5,20/11) Hyperbolic Matrix(67,120,24,43) (-9/5,-7/4) -> (11/4,3/1) Hyperbolic Matrix(451,780,122,211) (-26/15,-19/11) -> (11/3,26/7) Hyperbolic Matrix(313,540,-222,-383) (-19/11,-12/7) -> (-24/17,-7/5) Hyperbolic Matrix(71,120,42,71) (-12/7,-5/3) -> (5/3,12/7) Hyperbolic Matrix(37,60,8,13) (-5/3,-8/5) -> (4/1,5/1) Hyperbolic Matrix(569,900,208,329) (-19/12,-30/19) -> (30/11,11/4) Hyperbolic Matrix(571,900,210,331) (-30/19,-11/7) -> (19/7,30/11) Hyperbolic Matrix(77,120,34,53) (-11/7,-3/2) -> (9/4,7/3) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(43,60,-38,-53) (-7/5,-11/8) -> (-7/6,-1/1) Hyperbolic Matrix(569,780,240,329) (-11/8,-26/19) -> (26/11,19/8) Hyperbolic Matrix(571,780,418,571) (-26/19,-15/11) -> (15/11,26/19) Hyperbolic Matrix(89,120,66,89) (-15/11,-4/3) -> (4/3,15/11) Hyperbolic Matrix(47,60,18,23) (-4/3,-5/4) -> (5/2,8/3) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(53,-60,38,-43) (1/1,6/5) -> (18/13,7/5) Hyperbolic Matrix(217,-300,34,-47) (11/8,18/13) -> (6/1,13/2) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(383,-660,112,-193) (12/7,19/11) -> (17/5,24/7) Hyperbolic Matrix(173,-300,94,-163) (19/11,7/4) -> (11/6,13/7) Hyperbolic Matrix(263,-480,40,-73) (20/11,11/6) -> (13/2,20/3) Hyperbolic Matrix(137,-300,58,-127) (2/1,11/5) -> (7/3,26/11) Hyperbolic Matrix(163,-360,24,-53) (11/5,20/9) -> (20/3,7/1) Hyperbolic Matrix(227,-540,66,-157) (19/8,12/5) -> (24/7,7/2) Hyperbolic Matrix(17,-60,2,-7) (7/2,11/3) -> (7/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,60,-2,-17) -> Matrix(7,2,-60,-17) Matrix(11,60,2,11) -> Matrix(11,2,60,11) Matrix(13,60,8,37) -> Matrix(13,2,240,37) Matrix(31,120,8,31) -> Matrix(31,4,240,31) Matrix(209,780,56,209) -> Matrix(209,26,1680,209) Matrix(97,360,52,193) -> Matrix(97,12,1560,193) Matrix(83,300,-70,-253) -> Matrix(83,10,-2100,-253) Matrix(71,240,-50,-169) -> Matrix(71,8,-1500,-169) Matrix(19,60,6,19) -> Matrix(19,2,180,19) Matrix(43,120,24,67) -> Matrix(43,4,720,67) Matrix(89,240,-56,-151) -> Matrix(89,8,-1680,-151) Matrix(23,60,18,47) -> Matrix(23,2,540,47) Matrix(49,120,20,49) -> Matrix(49,4,600,49) Matrix(277,660,-196,-467) -> Matrix(277,22,-5880,-467) Matrix(127,300,-58,-137) -> Matrix(127,10,-1740,-137) Matrix(53,120,34,77) -> Matrix(53,4,1020,77) Matrix(161,360,72,161) -> Matrix(161,12,2160,161) Matrix(217,480,-184,-407) -> Matrix(217,16,-5520,-407) Matrix(167,360,122,263) -> Matrix(167,12,3660,263) Matrix(163,300,-94,-173) -> Matrix(163,10,-2820,-173) Matrix(197,360,-168,-307) -> Matrix(197,12,-5040,-307) Matrix(199,360,110,199) -> Matrix(199,12,3300,199) Matrix(67,120,24,43) -> Matrix(67,4,720,43) Matrix(451,780,122,211) -> Matrix(451,26,3660,211) Matrix(313,540,-222,-383) -> Matrix(313,18,-6660,-383) Matrix(71,120,42,71) -> Matrix(71,4,1260,71) Matrix(37,60,8,13) -> Matrix(37,2,240,13) Matrix(569,900,208,329) -> Matrix(569,30,6240,329) Matrix(571,900,210,331) -> Matrix(571,30,6300,331) Matrix(77,120,34,53) -> Matrix(77,4,1020,53) Matrix(41,60,28,41) -> Matrix(41,2,840,41) Matrix(43,60,-38,-53) -> Matrix(43,2,-1140,-53) Matrix(569,780,240,329) -> Matrix(569,26,7200,329) Matrix(571,780,418,571) -> Matrix(571,26,12540,571) Matrix(89,120,66,89) -> Matrix(89,4,1980,89) Matrix(47,60,18,23) -> Matrix(47,2,540,23) Matrix(49,60,40,49) -> Matrix(49,2,1200,49) Matrix(1,0,2,1) -> Matrix(1,0,60,1) Matrix(53,-60,38,-43) -> Matrix(53,-2,1140,-43) Matrix(217,-300,34,-47) -> Matrix(217,-10,1020,-47) Matrix(169,-240,50,-71) -> Matrix(169,-8,1500,-71) Matrix(151,-240,56,-89) -> Matrix(151,-8,1680,-89) Matrix(383,-660,112,-193) -> Matrix(383,-22,3360,-193) Matrix(173,-300,94,-163) -> Matrix(173,-10,2820,-163) Matrix(263,-480,40,-73) -> Matrix(263,-16,1200,-73) Matrix(137,-300,58,-127) -> Matrix(137,-10,1740,-127) Matrix(163,-360,24,-53) -> Matrix(163,-12,720,-53) Matrix(227,-540,66,-157) -> Matrix(227,-18,1980,-157) Matrix(17,-60,2,-7) -> Matrix(17,-2,60,-7) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 16 Genus: 17 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 48 Degree of the the map Y: 48 Permutation triple for Y: ((2,6,22,43,47,35,13,4,3,12,34,40,45,23,7)(5,11,32,14,8,27,30,10,9,20,25,16,15,38,18)(17,39,28,29,37)(19,26,31)(21,36,33,24,41)(44,48,46); (1,4,16,37,15,31,36,47,27,39,46,45,24,23,32,42,20,12,33,34,48,29,30,43,41,19,18,17,5,2)(3,10,26,8,7,25,35,44,22,11)(6,9,28,14,13,21)(38,40); (1,2,8,28,27,26,41,45,38,37,48,35,36,13,25,42,32,22,21,43,44,39,18,40,33,31,10,29,9,3)(4,14,23,46,34,20,6,5,19,15)(7,24,12,11,17,16)(30,47)) ----------------------------------------------------------------------- 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): 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 3/1 10/3 15/4 5/1 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 1/30 4/3 2/45 15/11 1/22 11/8 11/240 7/5 7/150 10/7 1/21 3/2 1/20 5/3 1/18 2/1 1/15 7/3 7/90 19/8 19/240 12/5 2/25 5/2 1/12 3/1 1/10 10/3 1/9 17/5 17/150 7/2 7/60 11/3 11/90 15/4 1/8 4/1 2/15 5/1 1/6 6/1 1/5 13/2 13/60 7/1 7/30 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(23,-30,10,-13) (1/1,4/3) -> (2/1,7/3) Hyperbolic Matrix(89,-120,23,-31) (4/3,15/11) -> (15/4,4/1) Hyperbolic Matrix(241,-330,65,-89) (15/11,11/8) -> (11/3,15/4) Hyperbolic Matrix(151,-210,64,-89) (11/8,7/5) -> (7/3,19/8) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(41,-60,13,-19) (10/7,3/2) -> (3/1,10/3) Hyperbolic Matrix(19,-30,7,-11) (3/2,5/3) -> (5/2,3/1) Hyperbolic Matrix(17,-30,4,-7) (5/3,2/1) -> (4/1,5/1) Hyperbolic Matrix(113,-270,18,-43) (19/8,12/5) -> (6/1,13/2) Hyperbolic Matrix(37,-90,7,-17) (12/5,5/2) -> (5/1,6/1) Hyperbolic Matrix(79,-270,12,-41) (17/5,7/2) -> (13/2,7/1) Hyperbolic Matrix(17,-60,2,-7) (7/2,11/3) -> (7/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,30,1) Matrix(23,-30,10,-13) -> Matrix(23,-1,300,-13) Matrix(89,-120,23,-31) -> Matrix(89,-4,690,-31) Matrix(241,-330,65,-89) -> Matrix(241,-11,1950,-89) Matrix(151,-210,64,-89) -> Matrix(151,-7,1920,-89) Matrix(169,-240,50,-71) -> Matrix(169,-8,1500,-71) Matrix(41,-60,13,-19) -> Matrix(41,-2,390,-19) Matrix(19,-30,7,-11) -> Matrix(19,-1,210,-11) Matrix(17,-30,4,-7) -> Matrix(17,-1,120,-7) Matrix(113,-270,18,-43) -> Matrix(113,-9,540,-43) Matrix(37,-90,7,-17) -> Matrix(37,-3,210,-17) Matrix(79,-270,12,-41) -> Matrix(79,-9,360,-41) Matrix(17,-60,2,-7) -> Matrix(17,-2,60,-7) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM 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 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 30 1 2/1 1/15 2 15 5/2 1/12 5 6 3/1 1/10 3 10 10/3 1/9 10 3 7/2 7/60 1 30 11/3 11/90 1 30 15/4 1/8 15 2 4/1 2/15 2 15 5/1 1/6 5 6 6/1 1/5 6 5 7/1 7/30 1 30 1/0 1/0 1 30 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(13,-30,3,-7) (2/1,5/2) -> (4/1,5/1) Glide Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) Reflection Matrix(71,-240,21,-71) (10/3,24/7) -> (10/3,24/7) Reflection Matrix(61,-210,9,-31) (17/5,7/2) -> (13/2,7/1) Glide Reflection Matrix(17,-60,2,-7) (7/2,11/3) -> (7/1,1/0) Hyperbolic Matrix(89,-330,24,-89) (11/3,15/4) -> (11/3,15/4) Reflection Matrix(31,-120,8,-31) (15/4,4/1) -> (15/4,4/1) Reflection Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection Matrix(19,-120,3,-19) (6/1,20/3) -> (6/1,20/3) 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(1,0,30,-1) (0/1,2/1) -> (0/1,1/15) Matrix(13,-30,3,-7) -> Matrix(13,-1,90,-7) Matrix(11,-30,4,-11) -> Matrix(11,-1,120,-11) (5/2,3/1) -> (1/12,1/10) Matrix(19,-60,6,-19) -> Matrix(19,-2,180,-19) (3/1,10/3) -> (1/10,1/9) Matrix(71,-240,21,-71) -> Matrix(71,-8,630,-71) (10/3,24/7) -> (1/9,4/35) Matrix(61,-210,9,-31) -> Matrix(61,-7,270,-31) Matrix(17,-60,2,-7) -> Matrix(17,-2,60,-7) Matrix(89,-330,24,-89) -> Matrix(89,-11,720,-89) (11/3,15/4) -> (11/90,1/8) Matrix(31,-120,8,-31) -> Matrix(31,-4,240,-31) (15/4,4/1) -> (1/8,2/15) Matrix(11,-60,2,-11) -> Matrix(11,-2,60,-11) (5/1,6/1) -> (1/6,1/5) Matrix(19,-120,3,-19) -> Matrix(19,-4,90,-19) (6/1,20/3) -> (1/5,2/9) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.