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 -11/30 -3/20 0/1 1/6 1/5 1/4 4/15 3/10 1/3 2/5 5/12 1/2 2/3 11/15 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/30 -5/6 1/25 -4/5 1/24 -3/4 2/45 -11/15 1/22 -19/26 13/285 -8/11 11/240 -13/18 3/65 -5/7 7/150 -7/10 1/21 -2/3 1/20 -7/11 11/210 -5/8 4/75 -3/5 1/18 -7/12 2/35 -11/19 19/330 -4/7 7/120 -5/9 3/50 -11/20 2/33 -6/11 11/180 -7/13 13/210 -1/2 1/15 -5/11 11/150 -9/20 2/27 -4/9 3/40 -3/7 7/90 -11/26 13/165 -8/19 19/240 -5/12 2/25 -2/5 1/12 -3/8 4/45 -7/19 19/210 -11/30 1/11 -4/11 11/120 -1/3 1/10 -3/10 1/9 -5/17 17/150 -7/24 4/35 -2/7 7/60 -3/11 11/90 -7/26 13/105 -4/15 1/8 -1/4 2/15 -1/5 1/6 -1/6 1/5 -2/13 13/60 -3/20 2/9 -1/7 7/30 0/1 1/0 1/6 -1/5 1/5 -1/6 1/4 -2/15 4/15 -1/8 7/26 -13/105 3/11 -11/90 5/18 -3/25 2/7 -7/60 3/10 -1/9 1/3 -1/10 4/11 -11/120 3/8 -4/45 2/5 -1/12 5/12 -2/25 8/19 -19/240 3/7 -7/90 4/9 -3/40 9/20 -2/27 5/11 -11/150 6/13 -13/180 1/2 -1/15 6/11 -11/180 11/20 -2/33 5/9 -3/50 4/7 -7/120 15/26 -13/225 11/19 -19/330 7/12 -2/35 3/5 -1/18 5/8 -4/75 12/19 -19/360 19/30 -1/19 7/11 -11/210 2/3 -1/20 7/10 -1/21 12/17 -17/360 17/24 -4/85 5/7 -7/150 8/11 -11/240 19/26 -13/285 11/15 -1/22 3/4 -2/45 4/5 -1/24 5/6 -1/25 11/13 -13/330 17/20 -2/51 6/7 -7/180 1/1 -1/30 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(43,38,-60,-53) (-1/1,-5/6) -> (-13/18,-5/7) Hyperbolic Matrix(49,40,60,49) (-5/6,-4/5) -> (4/5,5/6) Hyperbolic Matrix(23,18,60,47) (-4/5,-3/4) -> (3/8,2/5) Hyperbolic Matrix(89,66,120,89) (-3/4,-11/15) -> (11/15,3/4) Hyperbolic Matrix(571,418,780,571) (-11/15,-19/26) -> (19/26,11/15) Hyperbolic Matrix(167,122,360,263) (-19/26,-8/11) -> (6/13,1/2) Hyperbolic Matrix(47,34,-300,-217) (-8/11,-13/18) -> (-1/6,-2/13) Hyperbolic Matrix(71,50,-240,-169) (-5/7,-7/10) -> (-3/10,-5/17) Hyperbolic Matrix(41,28,60,41) (-7/10,-2/3) -> (2/3,7/10) Hyperbolic Matrix(53,34,120,77) (-2/3,-7/11) -> (3/7,4/9) Hyperbolic Matrix(89,56,-240,-151) (-7/11,-5/8) -> (-3/8,-7/19) Hyperbolic Matrix(13,8,60,37) (-5/8,-3/5) -> (1/5,1/4) Hyperbolic Matrix(71,42,120,71) (-3/5,-7/12) -> (7/12,3/5) Hyperbolic Matrix(193,112,-660,-383) (-7/12,-11/19) -> (-5/17,-7/24) Hyperbolic Matrix(163,94,-300,-173) (-11/19,-4/7) -> (-6/11,-7/13) Hyperbolic Matrix(43,24,120,67) (-4/7,-5/9) -> (1/3,4/11) Hyperbolic Matrix(199,110,360,199) (-5/9,-11/20) -> (11/20,5/9) Hyperbolic Matrix(73,40,-480,-263) (-11/20,-6/11) -> (-2/13,-3/20) Hyperbolic Matrix(97,52,360,193) (-7/13,-1/2) -> (7/26,3/11) Hyperbolic Matrix(127,58,-300,-137) (-1/2,-5/11) -> (-3/7,-11/26) Hyperbolic Matrix(53,24,-360,-163) (-5/11,-9/20) -> (-3/20,-1/7) Hyperbolic Matrix(161,72,360,161) (-9/20,-4/9) -> (4/9,9/20) Hyperbolic Matrix(77,34,120,53) (-4/9,-3/7) -> (7/11,2/3) Hyperbolic Matrix(569,240,780,329) (-11/26,-8/19) -> (8/11,19/26) Hyperbolic Matrix(157,66,-540,-227) (-8/19,-5/12) -> (-7/24,-2/7) Hyperbolic Matrix(49,20,120,49) (-5/12,-2/5) -> (2/5,5/12) Hyperbolic Matrix(47,18,60,23) (-2/5,-3/8) -> (3/4,4/5) Hyperbolic Matrix(571,210,900,331) (-7/19,-11/30) -> (19/30,7/11) Hyperbolic Matrix(569,208,900,329) (-11/30,-4/11) -> (12/19,19/30) Hyperbolic Matrix(67,24,120,43) (-4/11,-1/3) -> (5/9,4/7) Hyperbolic Matrix(19,6,60,19) (-1/3,-3/10) -> (3/10,1/3) Hyperbolic Matrix(7,2,-60,-17) (-2/7,-3/11) -> (-1/7,0/1) Hyperbolic Matrix(451,122,780,211) (-3/11,-7/26) -> (15/26,11/19) Hyperbolic Matrix(209,56,780,209) (-7/26,-4/15) -> (4/15,7/26) Hyperbolic Matrix(31,8,120,31) (-4/15,-1/4) -> (1/4,4/15) Hyperbolic Matrix(37,8,60,13) (-1/4,-1/5) -> (3/5,5/8) Hyperbolic Matrix(11,2,60,11) (-1/5,-1/6) -> (1/6,1/5) Hyperbolic Matrix(17,-2,60,-7) (0/1,1/6) -> (5/18,2/7) Hyperbolic Matrix(253,-70,300,-83) (3/11,5/18) -> (5/6,11/13) Hyperbolic Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(151,-56,240,-89) (4/11,3/8) -> (5/8,12/19) Hyperbolic Matrix(467,-196,660,-277) (5/12,8/19) -> (12/17,17/24) Hyperbolic Matrix(137,-58,300,-127) (8/19,3/7) -> (5/11,6/13) Hyperbolic Matrix(407,-184,480,-217) (9/20,5/11) -> (11/13,17/20) Hyperbolic Matrix(173,-94,300,-163) (1/2,6/11) -> (4/7,15/26) Hyperbolic Matrix(307,-168,360,-197) (6/11,11/20) -> (17/20,6/7) Hyperbolic Matrix(383,-222,540,-313) (11/19,7/12) -> (17/24,5/7) Hyperbolic Matrix(53,-38,60,-43) (5/7,8/11) -> (6/7,1/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-60,1) Matrix(43,38,-60,-53) -> Matrix(53,-2,1140,-43) Matrix(49,40,60,49) -> Matrix(49,-2,-1200,49) Matrix(23,18,60,47) -> Matrix(47,-2,-540,23) Matrix(89,66,120,89) -> Matrix(89,-4,-1980,89) Matrix(571,418,780,571) -> Matrix(571,-26,-12540,571) Matrix(167,122,360,263) -> Matrix(263,-12,-3660,167) Matrix(47,34,-300,-217) -> Matrix(217,-10,1020,-47) Matrix(71,50,-240,-169) -> Matrix(169,-8,1500,-71) Matrix(41,28,60,41) -> Matrix(41,-2,-840,41) Matrix(53,34,120,77) -> Matrix(77,-4,-1020,53) Matrix(89,56,-240,-151) -> Matrix(151,-8,1680,-89) Matrix(13,8,60,37) -> Matrix(37,-2,-240,13) Matrix(71,42,120,71) -> Matrix(71,-4,-1260,71) Matrix(193,112,-660,-383) -> Matrix(383,-22,3360,-193) Matrix(163,94,-300,-173) -> Matrix(173,-10,2820,-163) Matrix(43,24,120,67) -> Matrix(67,-4,-720,43) Matrix(199,110,360,199) -> Matrix(199,-12,-3300,199) Matrix(73,40,-480,-263) -> Matrix(263,-16,1200,-73) Matrix(97,52,360,193) -> Matrix(193,-12,-1560,97) Matrix(127,58,-300,-137) -> Matrix(137,-10,1740,-127) Matrix(53,24,-360,-163) -> Matrix(163,-12,720,-53) Matrix(161,72,360,161) -> Matrix(161,-12,-2160,161) Matrix(77,34,120,53) -> Matrix(53,-4,-1020,77) Matrix(569,240,780,329) -> Matrix(329,-26,-7200,569) Matrix(157,66,-540,-227) -> Matrix(227,-18,1980,-157) Matrix(49,20,120,49) -> Matrix(49,-4,-600,49) Matrix(47,18,60,23) -> Matrix(23,-2,-540,47) Matrix(571,210,900,331) -> Matrix(331,-30,-6300,571) Matrix(569,208,900,329) -> Matrix(329,-30,-6240,569) Matrix(67,24,120,43) -> Matrix(43,-4,-720,67) Matrix(19,6,60,19) -> Matrix(19,-2,-180,19) Matrix(7,2,-60,-17) -> Matrix(17,-2,60,-7) Matrix(451,122,780,211) -> Matrix(211,-26,-3660,451) Matrix(209,56,780,209) -> Matrix(209,-26,-1680,209) Matrix(31,8,120,31) -> Matrix(31,-4,-240,31) Matrix(37,8,60,13) -> Matrix(13,-2,-240,37) Matrix(11,2,60,11) -> Matrix(11,-2,-60,11) Matrix(17,-2,60,-7) -> Matrix(7,2,-60,-17) Matrix(253,-70,300,-83) -> Matrix(83,10,-2100,-253) Matrix(169,-50,240,-71) -> Matrix(71,8,-1500,-169) Matrix(151,-56,240,-89) -> Matrix(89,8,-1680,-151) Matrix(467,-196,660,-277) -> Matrix(277,22,-5880,-467) Matrix(137,-58,300,-127) -> Matrix(127,10,-1740,-137) Matrix(407,-184,480,-217) -> Matrix(217,16,-5520,-407) Matrix(173,-94,300,-163) -> Matrix(163,10,-2820,-173) Matrix(307,-168,360,-197) -> Matrix(197,12,-5040,-307) Matrix(383,-222,540,-313) -> Matrix(313,18,-6660,-383) Matrix(53,-38,60,-43) -> Matrix(43,2,-1140,-53) 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: ((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); (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)) ----------------------------------------------------------------------- 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 4/15 3/10 1/3 2/5 1/2 5/6 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/4 -2/15 4/15 -1/8 3/11 -11/90 2/7 -7/60 3/10 -1/9 1/3 -1/10 2/5 -1/12 1/2 -1/15 4/7 -7/120 11/19 -19/330 7/12 -2/35 3/5 -1/18 2/3 -1/20 7/10 -1/21 12/17 -17/360 5/7 -7/150 8/11 -11/240 11/15 -1/22 3/4 -2/45 4/5 -1/24 5/6 -1/25 11/13 -13/330 6/7 -7/180 1/1 -1/30 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(17,-4,30,-7) (0/1,1/4) -> (1/2,4/7) Hyperbolic Matrix(89,-23,120,-31) (1/4,4/15) -> (11/15,3/4) Hyperbolic Matrix(241,-65,330,-89) (4/15,3/11) -> (8/11,11/15) Hyperbolic Matrix(121,-34,210,-59) (3/11,2/7) -> (4/7,11/19) Hyperbolic Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(41,-13,60,-19) (3/10,1/3) -> (2/3,7/10) Hyperbolic Matrix(19,-7,30,-11) (1/3,2/5) -> (3/5,2/3) Hyperbolic Matrix(23,-10,30,-13) (2/5,1/2) -> (3/4,4/5) Hyperbolic Matrix(227,-132,270,-157) (11/19,7/12) -> (5/6,11/13) Hyperbolic Matrix(73,-43,90,-53) (7/12,3/5) -> (4/5,5/6) Hyperbolic Matrix(229,-162,270,-191) (12/17,5/7) -> (11/13,6/7) Hyperbolic Matrix(53,-38,60,-43) (5/7,8/11) -> (6/7,1/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-30,1) Matrix(17,-4,30,-7) -> Matrix(7,1,-120,-17) Matrix(89,-23,120,-31) -> Matrix(31,4,-690,-89) Matrix(241,-65,330,-89) -> Matrix(89,11,-1950,-241) Matrix(121,-34,210,-59) -> Matrix(59,7,-1020,-121) Matrix(169,-50,240,-71) -> Matrix(71,8,-1500,-169) Matrix(41,-13,60,-19) -> Matrix(19,2,-390,-41) Matrix(19,-7,30,-11) -> Matrix(11,1,-210,-19) Matrix(23,-10,30,-13) -> Matrix(13,1,-300,-23) Matrix(227,-132,270,-157) -> Matrix(157,9,-3960,-227) Matrix(73,-43,90,-53) -> Matrix(53,3,-1290,-73) Matrix(229,-162,270,-191) -> Matrix(191,9,-4860,-229) Matrix(53,-38,60,-43) -> Matrix(43,2,-1140,-53) 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 1/0 1 30 1/4 -2/15 2 15 4/15 -1/8 15 2 3/11 -11/90 1 30 2/7 -7/60 1 30 3/10 -1/9 10 3 1/3 -1/10 3 10 2/5 -1/12 5 6 5/12 -2/25 6 5 8/19 -19/240 1 30 3/7 -7/90 1 30 1/2 -1/15 2 15 1/0 0/1 30 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(13,-3,30,-7) (0/1,1/4) -> (3/7,1/2) Glide Reflection Matrix(31,-8,120,-31) (1/4,4/15) -> (1/4,4/15) Reflection Matrix(89,-24,330,-89) (4/15,3/11) -> (4/15,3/11) Reflection Matrix(89,-25,210,-59) (3/11,2/7) -> (8/19,3/7) Glide Reflection Matrix(227,-66,540,-157) (2/7,5/17) -> (13/31,8/19) Hyperbolic Matrix(71,-21,240,-71) (7/24,3/10) -> (7/24,3/10) Reflection Matrix(19,-6,60,-19) (3/10,1/3) -> (3/10,1/3) Reflection Matrix(11,-4,30,-11) (1/3,2/5) -> (1/3,2/5) Reflection Matrix(49,-20,120,-49) (2/5,5/12) -> (2/5,5/12) Reflection Matrix(251,-105,600,-251) (5/12,21/50) -> (5/12,21/50) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(13,-3,30,-7) -> Matrix(7,1,-90,-13) Matrix(31,-8,120,-31) -> Matrix(31,4,-240,-31) (1/4,4/15) -> (-2/15,-1/8) Matrix(89,-24,330,-89) -> Matrix(89,11,-720,-89) (4/15,3/11) -> (-1/8,-11/90) Matrix(89,-25,210,-59) -> Matrix(59,7,-750,-89) Matrix(227,-66,540,-157) -> Matrix(157,18,-1980,-227) Matrix(71,-21,240,-71) -> Matrix(71,8,-630,-71) (7/24,3/10) -> (-4/35,-1/9) Matrix(19,-6,60,-19) -> Matrix(19,2,-180,-19) (3/10,1/3) -> (-1/9,-1/10) Matrix(11,-4,30,-11) -> Matrix(11,1,-120,-11) (1/3,2/5) -> (-1/10,-1/12) Matrix(49,-20,120,-49) -> Matrix(49,4,-600,-49) (2/5,5/12) -> (-1/12,-2/25) Matrix(251,-105,600,-251) -> Matrix(251,20,-3150,-251) (5/12,21/50) -> (-2/25,-5/63) Matrix(-1,1,0,1) -> Matrix(-1,0,30,1) (1/2,1/0) -> (-1/15,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.