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: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -4/1 -11/3 -3/1 -5/2 -2/1 -5/3 -5/4 -1/1 -5/7 -5/9 0/1 1/2 5/9 5/7 1/1 5/4 10/7 3/2 5/3 2/1 20/9 7/3 5/2 3/1 10/3 7/2 11/3 4/1 5/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -1/2 1/0 -9/2 1/0 -4/1 -1/2 1/0 -11/3 -1/1 0/1 -7/2 -1/2 -10/3 0/1 -3/1 -1/1 0/1 -5/2 -1/2 1/0 -7/3 -1/1 0/1 -9/4 1/0 -20/9 -1/1 -11/5 -1/1 -2/3 -2/1 -1/2 1/0 -5/3 -1/2 -8/5 -1/2 -3/8 -19/12 -3/8 -30/19 -1/3 -11/7 -1/3 0/1 -3/2 -1/2 -10/7 -1/3 -7/5 -1/3 0/1 -11/8 -1/4 -4/3 -1/2 -1/4 -9/7 -1/3 0/1 -5/4 -1/2 -1/4 -1/1 -1/3 0/1 -5/6 -1/2 -1/4 -9/11 -1/3 0/1 -4/5 -1/2 -1/4 -11/14 -1/2 -7/9 -1/3 0/1 -10/13 -1/3 -3/4 -1/4 -5/7 -1/4 -7/10 -1/4 -9/13 -2/9 -1/5 -2/3 -1/4 -1/6 -9/14 -1/6 -7/11 -1/5 0/1 -5/8 -1/4 -1/6 -3/5 -1/5 0/1 -10/17 0/1 -7/12 -1/4 -11/19 -1/5 0/1 -4/7 -1/4 -1/6 -5/9 -1/4 -1/6 -6/11 -1/4 -1/6 -1/2 -1/6 0/1 0/1 1/2 1/6 5/9 1/6 1/4 4/7 1/6 1/4 7/12 1/4 3/5 0/1 1/5 2/3 1/6 1/4 5/7 1/4 8/11 1/4 3/10 3/4 1/4 7/9 0/1 1/3 4/5 1/4 1/2 1/1 0/1 1/3 6/5 1/4 1/2 5/4 1/4 1/2 14/11 1/4 1/2 9/7 0/1 1/3 4/3 1/4 1/2 15/11 1/4 1/2 11/8 1/4 7/5 0/1 1/3 17/12 1/4 10/7 1/3 3/2 1/2 5/3 1/2 7/4 1/2 16/9 1/2 3/4 9/5 2/3 1/1 20/11 1/1 11/6 1/0 2/1 1/2 1/0 11/5 2/3 1/1 20/9 1/1 9/4 1/0 7/3 0/1 1/1 5/2 1/2 1/0 13/5 0/1 1/1 21/8 1/2 8/3 1/2 1/0 3/1 0/1 1/1 13/4 1/0 10/3 0/1 7/2 1/2 18/5 1/2 3/4 11/3 0/1 1/1 4/1 1/2 1/0 5/1 1/2 1/0 6/1 1/2 1/0 7/1 0/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(9,50,-2,-11) (-5/1,1/0) -> (-5/1,-9/2) Parabolic Matrix(19,80,-24,-101) (-9/2,-4/1) -> (-4/5,-11/14) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(39,140,-56,-201) (-11/3,-7/2) -> (-7/10,-9/13) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(19,60,-32,-101) (-10/3,-3/1) -> (-3/5,-10/17) Hyperbolic Matrix(11,30,-18,-49) (-3/1,-5/2) -> (-5/8,-3/5) Hyperbolic Matrix(29,70,-46,-111) (-5/2,-7/3) -> (-7/11,-5/8) Hyperbolic Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(161,360,72,161) (-9/4,-20/9) -> (20/9,9/4) Hyperbolic Matrix(181,400,100,221) (-20/9,-11/5) -> (9/5,20/11) Hyperbolic Matrix(101,220,28,61) (-11/5,-2/1) -> (18/5,11/3) Hyperbolic Matrix(29,50,-18,-31) (-2/1,-5/3) -> (-5/3,-8/5) Parabolic Matrix(201,320,76,121) (-8/5,-19/12) -> (21/8,8/3) Hyperbolic Matrix(449,710,246,389) (-19/12,-30/19) -> (20/11,11/6) Hyperbolic Matrix(349,550,158,249) (-30/19,-11/7) -> (11/5,20/9) Hyperbolic Matrix(71,110,-122,-189) (-11/7,-3/2) -> (-7/12,-11/19) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(99,140,-128,-181) (-10/7,-7/5) -> (-7/9,-10/13) Hyperbolic Matrix(101,140,44,61) (-7/5,-11/8) -> (9/4,7/3) Hyperbolic Matrix(51,70,-94,-129) (-11/8,-4/3) -> (-6/11,-1/2) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(71,90,-86,-109) (-9/7,-5/4) -> (-5/6,-9/11) Hyperbolic Matrix(9,10,-10,-11) (-5/4,-1/1) -> (-1/1,-5/6) Parabolic Matrix(221,180,124,101) (-9/11,-4/5) -> (16/9,9/5) Hyperbolic Matrix(371,290,142,111) (-11/14,-7/9) -> (13/5,21/8) Hyperbolic Matrix(261,200,184,141) (-10/13,-3/4) -> (17/12,10/7) Hyperbolic Matrix(69,50,-98,-71) (-3/4,-5/7) -> (-5/7,-7/10) Parabolic Matrix(59,40,28,19) (-9/13,-2/3) -> (2/1,11/5) Hyperbolic Matrix(61,40,32,21) (-2/3,-9/14) -> (11/6,2/1) Hyperbolic Matrix(109,70,14,9) (-9/14,-7/11) -> (7/1,1/0) Hyperbolic Matrix(341,200,104,61) (-10/17,-7/12) -> (13/4,10/3) Hyperbolic Matrix(329,190,258,149) (-11/19,-4/7) -> (14/11,9/7) Hyperbolic Matrix(89,50,-162,-91) (-4/7,-5/9) -> (-5/9,-6/11) Parabolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(129,-70,94,-51) (1/2,5/9) -> (15/11,11/8) Hyperbolic Matrix(141,-80,104,-59) (5/9,4/7) -> (4/3,15/11) Hyperbolic Matrix(241,-140,136,-79) (4/7,7/12) -> (7/4,16/9) Hyperbolic Matrix(101,-60,32,-19) (7/12,3/5) -> (3/1,13/4) Hyperbolic Matrix(49,-30,18,-11) (3/5,2/3) -> (8/3,3/1) Hyperbolic Matrix(71,-50,98,-69) (2/3,5/7) -> (5/7,8/11) Parabolic Matrix(149,-110,42,-31) (8/11,3/4) -> (7/2,18/5) Hyperbolic Matrix(181,-140,128,-99) (3/4,7/9) -> (7/5,17/12) Hyperbolic Matrix(89,-70,14,-11) (7/9,4/5) -> (6/1,7/1) Hyperbolic Matrix(11,-10,10,-9) (4/5,1/1) -> (1/1,6/5) Parabolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(9,50,-2,-11) -> Matrix(1,0,0,1) Matrix(19,80,-24,-101) -> Matrix(1,0,-2,1) Matrix(21,80,16,61) -> Matrix(1,0,4,1) Matrix(39,140,-56,-201) -> Matrix(3,2,-14,-9) Matrix(41,140,12,41) -> Matrix(1,0,4,1) Matrix(19,60,-32,-101) -> Matrix(1,0,-4,1) Matrix(11,30,-18,-49) -> Matrix(1,0,-4,1) Matrix(29,70,-46,-111) -> Matrix(1,0,-4,1) Matrix(61,140,44,101) -> Matrix(1,0,4,1) Matrix(161,360,72,161) -> Matrix(1,2,0,1) Matrix(181,400,100,221) -> Matrix(5,4,6,5) Matrix(101,220,28,61) -> Matrix(3,2,4,3) Matrix(29,50,-18,-31) -> Matrix(3,2,-8,-5) Matrix(201,320,76,121) -> Matrix(5,2,2,1) Matrix(449,710,246,389) -> Matrix(11,4,8,3) Matrix(349,550,158,249) -> Matrix(7,2,10,3) Matrix(71,110,-122,-189) -> Matrix(1,0,-2,1) Matrix(41,60,28,41) -> Matrix(5,2,12,5) Matrix(99,140,-128,-181) -> Matrix(1,0,0,1) Matrix(101,140,44,61) -> Matrix(1,0,4,1) Matrix(51,70,-94,-129) -> Matrix(1,0,-2,1) Matrix(61,80,16,21) -> Matrix(1,0,4,1) Matrix(71,90,-86,-109) -> Matrix(1,0,0,1) Matrix(9,10,-10,-11) -> Matrix(1,0,0,1) Matrix(221,180,124,101) -> Matrix(7,2,10,3) Matrix(371,290,142,111) -> Matrix(1,0,4,1) Matrix(261,200,184,141) -> Matrix(7,2,24,7) Matrix(69,50,-98,-71) -> Matrix(7,2,-32,-9) Matrix(59,40,28,19) -> Matrix(1,0,6,1) Matrix(61,40,32,21) -> Matrix(1,0,6,1) Matrix(109,70,14,9) -> Matrix(1,0,6,1) Matrix(341,200,104,61) -> Matrix(1,0,4,1) Matrix(329,190,258,149) -> Matrix(1,0,8,1) Matrix(89,50,-162,-91) -> Matrix(1,0,0,1) Matrix(1,0,4,1) -> Matrix(1,0,12,1) Matrix(129,-70,94,-51) -> Matrix(1,0,-2,1) Matrix(141,-80,104,-59) -> Matrix(1,0,-2,1) Matrix(241,-140,136,-79) -> Matrix(9,-2,14,-3) Matrix(101,-60,32,-19) -> Matrix(1,0,-4,1) Matrix(49,-30,18,-11) -> Matrix(1,0,-4,1) Matrix(71,-50,98,-69) -> Matrix(9,-2,32,-7) Matrix(149,-110,42,-31) -> Matrix(1,0,-2,1) Matrix(181,-140,128,-99) -> Matrix(1,0,0,1) Matrix(89,-70,14,-11) -> Matrix(1,0,-2,1) Matrix(11,-10,10,-9) -> Matrix(1,0,0,1) Matrix(81,-100,64,-79) -> Matrix(1,0,0,1) Matrix(31,-50,18,-29) -> Matrix(5,-2,8,-3) Matrix(41,-100,16,-39) -> Matrix(1,0,0,1) Matrix(11,-50,2,-9) -> Matrix(1,0,0,1) 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): 8 Degree of the the map X: 8 Degree of the the map Y: 48 Permutation triple for Y: ((1,2)(3,10,32,40,15,6,21,38,33,11)(4,16,41,26,8,7,25,42,17,5)(9,29,14,13,37,20,46,24,23,30)(12,35,19,18,45,22,43,28,27,36)(31,44)(34,39)(47,48); (1,5,19,20,6)(2,8,28,9,3)(4,14,39,22,15)(7,24,34,12,11)(16,40,27,31,30)(18,44,37,25,33)(29,43,42,48,38)(32,46,35,41,47); (1,3,12,36,40,47,42,37,13,4)(2,6,22,45,33,48,41,30,23,7)(5,17,43,39,24,32,10,9,31,18)(8,26,35,34,14,38,21,20,44,27)(11,25)(15,16)(19,46)(28,29)) ----------------------------------------------------------------------- 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): 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: 12 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 5/4 3/2 5/3 2/1 5/2 3/1 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 -1/2 1/0 -4/1 -1/2 1/0 -3/1 -1/1 0/1 -5/2 -1/2 1/0 -2/1 -1/2 1/0 -5/3 -1/2 -3/2 -1/2 -10/7 -1/3 -7/5 -1/3 0/1 -4/3 -1/2 -1/4 -5/4 -1/2 -1/4 -1/1 -1/3 0/1 0/1 0/1 1/1 0/1 1/3 5/4 1/4 1/2 4/3 1/4 1/2 3/2 1/2 5/3 1/2 2/1 1/2 1/0 5/2 1/2 1/0 3/1 0/1 1/1 10/3 0/1 7/2 1/2 4/1 1/2 1/0 5/1 1/2 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,10,0,1) (-5/1,1/0) -> (5/1,1/0) Parabolic 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(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(11,20,6,11) (-2/1,-5/3) -> (5/3,2/1) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(69,100,20,29) (-3/2,-10/7) -> (10/3,7/2) Hyperbolic Matrix(71,100,22,31) (-10/7,-7/5) -> (3/1,10/3) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(9,10,8,9) (-5/4,-1/1) -> (1/1,5/4) 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 IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,10,0,1) -> Matrix(1,1,0,1) Matrix(9,40,2,9) -> Matrix(1,1,0,1) Matrix(11,40,-8,-29) -> Matrix(1,1,-4,-3) Matrix(11,30,4,11) -> Matrix(1,0,2,1) Matrix(9,20,4,9) -> Matrix(1,1,0,1) Matrix(11,20,6,11) -> Matrix(1,1,0,1) Matrix(19,30,12,19) -> Matrix(3,1,8,3) Matrix(69,100,20,29) -> Matrix(3,1,8,3) Matrix(71,100,22,31) -> Matrix(3,1,2,1) Matrix(31,40,24,31) -> Matrix(3,1,8,3) Matrix(9,10,8,9) -> Matrix(1,0,6,1) Matrix(1,0,2,1) -> Matrix(1,0,6,1) Matrix(29,-40,8,-11) -> Matrix(3,-1,4,-1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 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 0/1 0/1 6 2 1/1 (0/1,1/3) 0 10 5/4 (0/1,1/3).(1/4,1/2) 0 2 4/3 (1/4,1/2) 0 10 3/2 1/2 1 10 5/3 1/2 2 2 2/1 (1/2,1/0) 0 10 5/2 (0/1,1/1).(1/2,1/0) 0 2 3/1 (0/1,1/1) 0 10 10/3 0/1 2 2 7/2 1/2 1 10 4/1 (1/2,1/0) 0 10 5/1 (1/2,1/0) 0 2 1/0 1/0 1 10 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(9,-10,8,-9) (1/1,5/4) -> (1/1,5/4) Reflection Matrix(31,-40,24,-31) (5/4,4/3) -> (5/4,4/3) Reflection Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) Reflection Matrix(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) 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(41,-140,12,-41) (10/3,7/2) -> (10/3,7/2) Reflection Matrix(9,-40,2,-9) (4/1,5/1) -> (4/1,5/1) Reflection Matrix(-1,10,0,1) (5/1,1/0) -> (5/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,6,-1) (0/1,1/1) -> (0/1,1/3) Matrix(9,-10,8,-9) -> Matrix(1,0,6,-1) (1/1,5/4) -> (0/1,1/3) Matrix(31,-40,24,-31) -> Matrix(3,-1,8,-3) (5/4,4/3) -> (1/4,1/2) Matrix(29,-40,8,-11) -> Matrix(3,-1,4,-1) 1/2 Matrix(19,-30,12,-19) -> Matrix(3,-1,8,-3) (3/2,5/3) -> (1/4,1/2) Matrix(11,-20,6,-11) -> Matrix(-1,1,0,1) (5/3,2/1) -> (1/2,1/0) Matrix(9,-20,4,-9) -> Matrix(-1,1,0,1) (2/1,5/2) -> (1/2,1/0) Matrix(11,-30,4,-11) -> Matrix(1,0,2,-1) (5/2,3/1) -> (0/1,1/1) Matrix(19,-60,6,-19) -> Matrix(1,0,2,-1) (3/1,10/3) -> (0/1,1/1) Matrix(41,-140,12,-41) -> Matrix(1,0,4,-1) (10/3,7/2) -> (0/1,1/2) Matrix(9,-40,2,-9) -> Matrix(-1,1,0,1) (4/1,5/1) -> (1/2,1/0) Matrix(-1,10,0,1) -> Matrix(-1,1,0,1) (5/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.