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 -4/1 -3/1 -5/2 -9/4 -20/9 -2/1 -5/4 -1/1 -5/6 -5/8 0/1 1/2 5/9 5/8 5/7 3/4 5/6 1/1 5/4 10/7 3/2 5/3 7/4 2/1 5/2 3/1 10/3 7/2 4/1 9/2 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 0/1 -9/2 1/5 -4/1 1/3 1/1 -11/3 1/3 1/1 -7/2 1/1 -10/3 1/1 -3/1 -1/1 1/1 -5/2 -1/1 1/1 -7/3 -1/1 1/1 -9/4 0/1 1/0 -20/9 1/0 -11/5 -3/1 -1/1 -2/1 -1/1 1/1 -11/6 1/1 -20/11 1/0 -9/5 -3/1 -1/1 -7/4 -1/1 1/0 -5/3 -1/1 -3/2 -1/1 -10/7 -1/1 -7/5 -1/1 -3/5 -11/8 -2/3 -1/2 -4/3 -1/1 -1/3 -5/4 -1/1 -1/2 0/1 -6/5 -1/1 -1/3 -1/1 -1/1 -1/3 -5/6 -1/3 -9/11 -1/3 -3/11 -4/5 -1/3 -1/5 -11/14 -1/1 -7/9 -1/1 -1/3 -10/13 -1/3 -3/4 -1/3 0/1 -5/7 -1/3 -7/10 -1/3 -9/13 -1/3 -3/11 -2/3 -1/3 -1/5 -5/8 -1/3 -1/4 0/1 -8/13 -1/3 -1/5 -11/18 -1/5 -3/5 -1/3 -1/5 -10/17 -1/3 -7/12 -1/3 -1/4 -11/19 -1/3 -1/5 -4/7 -1/3 -3/11 -9/16 -4/15 -1/4 -5/9 -1/4 -1/2 -1/5 0/1 0/1 1/2 1/5 5/9 1/4 4/7 3/11 1/3 7/12 1/4 1/3 3/5 1/5 1/3 5/8 0/1 1/4 1/3 7/11 1/5 1/3 9/14 1/5 2/3 1/5 1/3 7/10 1/3 5/7 1/3 3/4 0/1 1/3 7/9 1/3 1/1 4/5 1/5 1/3 5/6 1/3 6/7 1/3 3/7 1/1 1/3 1/1 5/4 0/1 1/2 1/1 9/7 1/3 1/1 4/3 1/3 1/1 15/11 1/2 11/8 1/2 2/3 7/5 3/5 1/1 17/12 2/3 1/1 10/7 1/1 3/2 1/1 5/3 1/1 7/4 1/1 1/0 16/9 -1/1 1/1 9/5 1/1 3/1 20/11 1/0 11/6 -1/1 2/1 -1/1 1/1 5/2 -1/1 1/1 8/3 -1/1 1/1 19/7 1/1 3/1 30/11 1/0 11/4 0/1 1/0 3/1 -1/1 1/1 13/4 -1/1 1/0 10/3 -1/1 7/2 -1/1 18/5 -1/1 -3/5 11/3 -1/1 -1/3 4/1 -1/1 -1/3 13/3 -1/3 -1/5 9/2 -1/5 5/1 0/1 1/0 0/1 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(19,90,4,19) (-5/1,-9/2) -> (9/2,5/1) Hyperbolic 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(19,50,-8,-21) (-3/1,-5/2) -> (-5/2,-7/3) Parabolic Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(219,490,80,179) (-9/4,-20/9) -> (30/11,11/4) 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(21,40,32,61) (-2/1,-11/6) -> (9/14,2/3) Hyperbolic Matrix(241,440,132,241) (-11/6,-20/11) -> (20/11,11/6) Hyperbolic Matrix(359,650,132,239) (-20/11,-9/5) -> (19/7,30/11) Hyperbolic Matrix(79,140,-136,-241) (-9/5,-7/4) -> (-7/12,-11/19) Hyperbolic Matrix(41,70,24,41) (-7/4,-5/3) -> (5/3,7/4) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) 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(79,110,28,39) (-7/5,-11/8) -> (11/4,3/1) Hyperbolic Matrix(59,80,-104,-141) (-11/8,-4/3) -> (-4/7,-9/16) Hyperbolic Matrix(39,50,-32,-41) (-4/3,-5/4) -> (-5/4,-6/5) Parabolic Matrix(59,70,16,19) (-6/5,-1/1) -> (11/3,4/1) Hyperbolic Matrix(59,50,-72,-61) (-1/1,-5/6) -> (-5/6,-9/11) Parabolic Matrix(221,180,124,101) (-9/11,-4/5) -> (16/9,9/5) Hyperbolic Matrix(179,140,280,219) (-11/14,-7/9) -> (7/11,9/14) Hyperbolic Matrix(261,200,184,141) (-10/13,-3/4) -> (17/12,10/7) Hyperbolic Matrix(41,30,56,41) (-3/4,-5/7) -> (5/7,3/4) Hyperbolic Matrix(99,70,140,99) (-5/7,-7/10) -> (7/10,5/7) Hyperbolic Matrix(161,110,60,41) (-9/13,-2/3) -> (8/3,19/7) Hyperbolic Matrix(79,50,-128,-81) (-2/3,-5/8) -> (-5/8,-8/13) Parabolic Matrix(179,110,96,59) (-8/13,-11/18) -> (11/6,2/1) Hyperbolic Matrix(279,170,64,39) (-11/18,-3/5) -> (13/3,9/2) Hyperbolic Matrix(341,200,104,61) (-10/17,-7/12) -> (13/4,10/3) Hyperbolic Matrix(121,70,140,81) (-11/19,-4/7) -> (6/7,1/1) Hyperbolic Matrix(339,190,248,139) (-9/16,-5/9) -> (15/11,11/8) Hyperbolic Matrix(19,10,36,19) (-5/9,-1/2) -> (1/2,5/9) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic 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(81,-50,128,-79) (3/5,5/8) -> (5/8,7/11) Parabolic Matrix(201,-140,56,-39) (2/3,7/10) -> (7/2,18/5) Hyperbolic Matrix(181,-140,128,-99) (3/4,7/9) -> (7/5,17/12) Hyperbolic Matrix(101,-80,24,-19) (7/9,4/5) -> (4/1,13/3) Hyperbolic Matrix(61,-50,72,-59) (4/5,5/6) -> (5/6,6/7) Parabolic Matrix(41,-50,32,-39) (1/1,5/4) -> (5/4,9/7) Parabolic Matrix(21,-50,8,-19) (2/1,5/2) -> (5/2,8/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,10,0,1) -> Matrix(1,0,0,1) Matrix(19,90,4,19) -> Matrix(1,0,-10,1) Matrix(19,80,-24,-101) -> Matrix(1,0,-6,1) Matrix(21,80,16,61) -> Matrix(1,0,0,1) Matrix(39,140,-56,-201) -> Matrix(3,-2,-10,7) Matrix(41,140,12,41) -> Matrix(1,0,-2,1) Matrix(19,60,-32,-101) -> Matrix(1,0,-4,1) Matrix(19,50,-8,-21) -> Matrix(1,0,0,1) Matrix(61,140,44,101) -> Matrix(1,-2,2,-3) Matrix(219,490,80,179) -> Matrix(1,0,0,1) Matrix(181,400,100,221) -> Matrix(1,4,0,1) Matrix(101,220,28,61) -> Matrix(1,2,-2,-3) Matrix(21,40,32,61) -> Matrix(1,0,4,1) Matrix(241,440,132,241) -> Matrix(1,-2,0,1) Matrix(359,650,132,239) -> Matrix(1,4,0,1) Matrix(79,140,-136,-241) -> Matrix(1,2,-4,-7) Matrix(41,70,24,41) -> Matrix(1,2,0,1) Matrix(19,30,12,19) -> Matrix(1,0,2,1) Matrix(41,60,28,41) -> Matrix(3,2,4,3) Matrix(99,140,-128,-181) -> Matrix(3,2,-8,-5) Matrix(79,110,28,39) -> Matrix(3,2,-2,-1) Matrix(59,80,-104,-141) -> Matrix(5,2,-18,-7) Matrix(39,50,-32,-41) -> Matrix(1,0,0,1) Matrix(59,70,16,19) -> Matrix(1,0,0,1) Matrix(59,50,-72,-61) -> Matrix(5,2,-18,-7) Matrix(221,180,124,101) -> Matrix(1,0,4,1) Matrix(179,140,280,219) -> Matrix(1,0,6,1) Matrix(261,200,184,141) -> Matrix(7,2,10,3) Matrix(41,30,56,41) -> Matrix(1,0,6,1) Matrix(99,70,140,99) -> Matrix(7,2,24,7) Matrix(161,110,60,41) -> Matrix(1,0,4,1) Matrix(79,50,-128,-81) -> Matrix(1,0,0,1) Matrix(179,110,96,59) -> Matrix(1,0,4,1) Matrix(279,170,64,39) -> Matrix(1,0,0,1) Matrix(341,200,104,61) -> Matrix(7,2,-4,-1) Matrix(121,70,140,81) -> Matrix(1,0,6,1) Matrix(339,190,248,139) -> Matrix(23,6,42,11) Matrix(19,10,36,19) -> Matrix(9,2,40,9) Matrix(1,0,4,1) -> Matrix(1,0,10,1) Matrix(141,-80,104,-59) -> Matrix(7,-2,18,-5) Matrix(241,-140,136,-79) -> Matrix(7,-2,4,-1) Matrix(101,-60,32,-19) -> Matrix(1,0,-4,1) Matrix(81,-50,128,-79) -> Matrix(1,0,0,1) Matrix(201,-140,56,-39) -> Matrix(7,-2,-10,3) Matrix(181,-140,128,-99) -> Matrix(5,-2,8,-3) Matrix(101,-80,24,-19) -> Matrix(1,0,-6,1) Matrix(61,-50,72,-59) -> Matrix(7,-2,18,-5) Matrix(41,-50,32,-39) -> Matrix(1,0,0,1) Matrix(21,-50,8,-19) -> 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): 10 Degree of the the map X: 10 Degree of the the map Y: 48 Permutation triple for Y: ((1,2)(3,10,32,46,21,6,20,38,33,11)(4,15,40,26,8,7,25,41,16,5)(9,29,14,13,37,19,45,24,23,30)(12,35,18,17,44,22,42,28,27,36)(31,43)(34,39)(47,48); (1,5,18,35,40,47,32,45,19,6)(2,8,28,42,41,48,38,29,9,3)(4,14,39,22,21,46,27,31,30,15)(7,24,34,12,11,33,17,43,37,25)(10,16)(13,36)(20,26)(23,44); (1,3,12,13,4)(2,6,22,23,7)(5,10,9,31,17)(8,20,19,43,27)(14,38,26,35,34)(16,42,39,24,32)(30,44,33,48,40)(36,46,47,41,37)) ----------------------------------------------------------------------- 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 1/2 5/8 3/4 5/6 1/1 5/4 3/2 5/3 2/1 5/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 1/1 -5/3 -1/1 -3/2 -1/1 -1/1 -1/1 -1/3 -3/4 -1/3 0/1 -5/7 -1/3 -2/3 -1/3 -1/5 -1/2 -1/5 0/1 0/1 1/2 1/5 3/5 1/5 1/3 5/8 0/1 1/4 1/3 2/3 1/5 1/3 3/4 0/1 1/3 4/5 1/5 1/3 5/6 1/3 1/1 1/3 1/1 5/4 0/1 1/2 1/1 4/3 1/3 1/1 3/2 1/1 5/3 1/1 7/4 1/1 1/0 2/1 -1/1 1/1 5/2 -1/1 1/1 3/1 -1/1 1/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,5,0,1) (-2/1,1/0) -> (3/1,1/0) Parabolic Matrix(11,20,-16,-29) (-2/1,-5/3) -> (-5/7,-2/3) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(11,15,8,11) (-3/2,-1/1) -> (4/3,3/2) Hyperbolic Matrix(19,15,24,19) (-1/1,-3/4) -> (3/4,4/5) Hyperbolic Matrix(69,50,40,29) (-3/4,-5/7) -> (5/3,7/4) Hyperbolic Matrix(9,5,16,9) (-2/3,-1/2) -> (1/2,3/5) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(41,-25,64,-39) (3/5,5/8) -> (5/8,2/3) Parabolic Matrix(29,-20,16,-11) (2/3,3/4) -> (7/4,2/1) Hyperbolic Matrix(31,-25,36,-29) (4/5,5/6) -> (5/6,1/1) Parabolic Matrix(21,-25,16,-19) (1/1,5/4) -> (5/4,4/3) Parabolic Matrix(11,-25,4,-9) (2/1,5/2) -> (5/2,3/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,5,0,1) -> Matrix(1,0,0,1) Matrix(11,20,-16,-29) -> Matrix(0,-1,1,4) Matrix(19,30,12,19) -> Matrix(1,0,2,1) Matrix(11,15,8,11) -> Matrix(2,1,3,2) Matrix(19,15,24,19) -> Matrix(1,0,6,1) Matrix(69,50,40,29) -> Matrix(2,1,-1,0) Matrix(9,5,16,9) -> Matrix(4,1,15,4) Matrix(1,0,4,1) -> Matrix(1,0,10,1) Matrix(41,-25,64,-39) -> Matrix(1,0,0,1) Matrix(29,-20,16,-11) -> Matrix(4,-1,1,0) Matrix(31,-25,36,-29) -> Matrix(4,-1,9,-2) Matrix(21,-25,16,-19) -> Matrix(1,0,0,1) Matrix(11,-25,4,-9) -> Matrix(0,-1,1,0) 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): 5 ----------------------------------------------------------------------- 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 5 2 1/2 1/5 2 10 5/8 (1/5,1/3) 0 2 2/3 (1/5,1/3) 0 10 3/4 (0/1,1/3) 0 10 5/6 1/3 2 2 1/1 (1/3,1/1) 0 10 5/4 (1/3,1/1) 0 2 3/2 1/1 2 10 5/3 1/1 1 2 7/4 (1/1,1/0) 0 10 2/1 (-1/1,1/1) 0 10 5/2 (-1/1,1/1).(0/1,1/0) 0 2 1/0 (0/1,1/0) 0 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(9,-5,16,-9) (1/2,5/8) -> (1/2,5/8) Reflection Matrix(31,-20,48,-31) (5/8,2/3) -> (5/8,2/3) Reflection Matrix(29,-20,16,-11) (2/3,3/4) -> (7/4,2/1) Hyperbolic Matrix(19,-15,24,-19) (3/4,5/6) -> (3/4,5/6) Reflection Matrix(11,-10,12,-11) (5/6,1/1) -> (5/6,1/1) Reflection Matrix(9,-10,8,-9) (1/1,5/4) -> (1/1,5/4) Reflection Matrix(11,-15,8,-11) (5/4,3/2) -> (5/4,3/2) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(41,-70,24,-41) (5/3,7/4) -> (5/3,7/4) Reflection Matrix(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(-1,5,0,1) (5/2,1/0) -> (5/2,1/0) 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(1,0,4,-1) -> Matrix(1,0,10,-1) (0/1,1/2) -> (0/1,1/5) Matrix(9,-5,16,-9) -> Matrix(4,-1,15,-4) (1/2,5/8) -> (1/5,1/3) Matrix(31,-20,48,-31) -> Matrix(4,-1,15,-4) (5/8,2/3) -> (1/5,1/3) Matrix(29,-20,16,-11) -> Matrix(4,-1,1,0) Matrix(19,-15,24,-19) -> Matrix(1,0,6,-1) (3/4,5/6) -> (0/1,1/3) Matrix(11,-10,12,-11) -> Matrix(2,-1,3,-2) (5/6,1/1) -> (1/3,1/1) Matrix(9,-10,8,-9) -> Matrix(2,-1,3,-2) (1/1,5/4) -> (1/3,1/1) Matrix(11,-15,8,-11) -> Matrix(2,-1,3,-2) (5/4,3/2) -> (1/3,1/1) Matrix(19,-30,12,-19) -> Matrix(1,0,2,-1) (3/2,5/3) -> (0/1,1/1) Matrix(41,-70,24,-41) -> Matrix(-1,2,0,1) (5/3,7/4) -> (1/1,1/0) Matrix(9,-20,4,-9) -> Matrix(0,1,1,0) (2/1,5/2) -> (-1/1,1/1) Matrix(-1,5,0,1) -> Matrix(1,0,0,-1) (5/2,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.