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): 180 Minimal number of generators: 31 Number of equivalence classes of cusps: 18 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -10/3 -5/2 -5/3 -5/4 0/1 1/1 5/4 5/3 25/14 2/1 5/2 10/3 25/7 40/11 5/1 20/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -1/5 -9/2 -9/50 -13/3 -13/75 -4/1 -4/25 -7/2 -7/50 -10/3 -2/15 -3/1 -3/25 -5/2 -1/10 -7/3 -7/75 -16/7 -16/175 -25/11 -1/11 -9/4 -9/100 -2/1 -2/25 -5/3 -1/15 -8/5 -8/125 -3/2 -3/50 -10/7 -2/35 -7/5 -7/125 -25/18 -1/18 -18/13 -18/325 -29/21 -29/525 -40/29 -8/145 -11/8 -11/200 -4/3 -4/75 -5/4 -1/20 -6/5 -6/125 -13/11 -13/275 -20/17 -4/85 -7/6 -7/150 -1/1 -1/25 0/1 0/1 1/1 1/25 5/4 1/20 9/7 9/175 13/10 13/250 4/3 4/75 7/5 7/125 10/7 2/35 3/2 3/50 5/3 1/15 7/4 7/100 16/9 16/225 25/14 1/14 9/5 9/125 2/1 2/25 5/2 1/10 8/3 8/75 3/1 3/25 10/3 2/15 7/2 7/50 25/7 1/7 18/5 18/125 29/8 29/200 40/11 8/55 11/3 11/75 4/1 4/25 5/1 1/5 6/1 6/25 13/2 13/50 20/3 4/15 7/1 7/25 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(57,250,44,193) (-9/2,-13/3) -> (9/7,13/10) Hyperbolic Matrix(83,350,-60,-253) (-13/3,-4/1) -> (-18/13,-29/21) Hyperbolic Matrix(13,50,-6,-23) (-4/1,-7/2) -> (-9/4,-2/1) Hyperbolic Matrix(29,100,20,69) (-7/2,-10/3) -> (10/7,3/2) Hyperbolic Matrix(31,100,22,71) (-10/3,-3/1) -> (7/5,10/7) Hyperbolic Matrix(19,50,-8,-21) (-3/1,-5/2) -> (-5/2,-7/3) Parabolic Matrix(109,250,-92,-211) (-7/3,-16/7) -> (-6/5,-13/11) Hyperbolic Matrix(373,850,104,237) (-16/7,-25/11) -> (25/7,18/5) Hyperbolic Matrix(177,400,50,113) (-25/11,-9/4) -> (7/2,25/7) Hyperbolic Matrix(29,50,-18,-31) (-2/1,-5/3) -> (-5/3,-8/5) Parabolic Matrix(63,100,-46,-73) (-8/5,-3/2) -> (-11/8,-4/3) 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(287,400,160,223) (-7/5,-25/18) -> (25/14,9/5) Hyperbolic Matrix(613,850,344,477) (-25/18,-18/13) -> (16/9,25/14) Hyperbolic Matrix(1159,1600,318,439) (-29/21,-40/29) -> (40/11,11/3) Hyperbolic Matrix(1161,1600,320,441) (-40/29,-11/8) -> (29/8,40/11) Hyperbolic Matrix(39,50,-32,-41) (-4/3,-5/4) -> (-5/4,-6/5) Parabolic Matrix(339,400,50,59) (-13/11,-20/17) -> (20/3,7/1) Hyperbolic Matrix(341,400,52,61) (-20/17,-7/6) -> (13/2,20/3) Hyperbolic Matrix(43,50,6,7) (-7/6,-1/1) -> (7/1,1/0) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(41,-50,32,-39) (1/1,5/4) -> (5/4,9/7) Parabolic Matrix(267,-350,74,-97) (13/10,4/3) -> (18/5,29/8) Hyperbolic Matrix(37,-50,20,-27) (4/3,7/5) -> (9/5,2/1) Hyperbolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(141,-250,22,-39) (7/4,16/9) -> (6/1,13/2) Hyperbolic Matrix(21,-50,8,-19) (2/1,5/2) -> (5/2,8/3) Parabolic Matrix(37,-100,10,-27) (8/3,3/1) -> (11/3,4/1) Hyperbolic 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(9,2,-50,-11) Matrix(57,250,44,193) -> Matrix(57,10,1100,193) Matrix(83,350,-60,-253) -> Matrix(83,14,-1500,-253) Matrix(13,50,-6,-23) -> Matrix(13,2,-150,-23) Matrix(29,100,20,69) -> Matrix(29,4,500,69) Matrix(31,100,22,71) -> Matrix(31,4,550,71) Matrix(19,50,-8,-21) -> Matrix(19,2,-200,-21) Matrix(109,250,-92,-211) -> Matrix(109,10,-2300,-211) Matrix(373,850,104,237) -> Matrix(373,34,2600,237) Matrix(177,400,50,113) -> Matrix(177,16,1250,113) Matrix(29,50,-18,-31) -> Matrix(29,2,-450,-31) Matrix(63,100,-46,-73) -> Matrix(63,4,-1150,-73) Matrix(69,100,20,29) -> Matrix(69,4,500,29) Matrix(71,100,22,31) -> Matrix(71,4,550,31) Matrix(287,400,160,223) -> Matrix(287,16,4000,223) Matrix(613,850,344,477) -> Matrix(613,34,8600,477) Matrix(1159,1600,318,439) -> Matrix(1159,64,7950,439) Matrix(1161,1600,320,441) -> Matrix(1161,64,8000,441) Matrix(39,50,-32,-41) -> Matrix(39,2,-800,-41) Matrix(339,400,50,59) -> Matrix(339,16,1250,59) Matrix(341,400,52,61) -> Matrix(341,16,1300,61) Matrix(43,50,6,7) -> Matrix(43,2,150,7) Matrix(1,0,2,1) -> Matrix(1,0,50,1) Matrix(41,-50,32,-39) -> Matrix(41,-2,800,-39) Matrix(267,-350,74,-97) -> Matrix(267,-14,1850,-97) Matrix(37,-50,20,-27) -> Matrix(37,-2,500,-27) Matrix(31,-50,18,-29) -> Matrix(31,-2,450,-29) Matrix(141,-250,22,-39) -> Matrix(141,-10,550,-39) Matrix(21,-50,8,-19) -> Matrix(21,-2,200,-19) Matrix(37,-100,10,-27) -> Matrix(37,-4,250,-27) Matrix(11,-50,2,-9) -> Matrix(11,-2,50,-9) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 180 Minimal number of generators: 31 Number of equivalence classes of cusps: 18 Genus: 7 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 30 Degree of the the map Y: 30 Permutation triple for Y: ((2,6,16,5,15,25,28,22,24,14,13,12,4,3,11,10,9,21,23,27,26,20,8,18,7); (1,4,14,24,26,30,22,11,3,10,17,16,27,23,13,19,8,7,18,25,29,21,15,5,2); (1,2,8,20,24,30,26,16,6,5,17,10,22,28,18,19,13,4,12,23,29,25,21,9,3)) ----------------------------------------------------------------------- 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, lambda1, lambda2, lambda1+lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z+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): 30 Minimal number of generators: 7 Number of equivalence classes of elliptic points of order 2: 2 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 6 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 5/4 5/3 5/2 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 1/25 5/4 1/20 4/3 4/75 3/2 3/50 5/3 1/15 2/1 2/25 5/2 1/10 3/1 3/25 4/1 4/25 5/1 1/5 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(21,-25,16,-19) (1/1,5/4) -> (5/4,4/3) Parabolic Matrix(18,-25,13,-18) (4/3,3/2) -> (4/3,3/2) Elliptic Matrix(16,-25,9,-14) (3/2,5/3) -> (5/3,2/1) Parabolic Matrix(11,-25,4,-9) (2/1,5/2) -> (5/2,3/1) Parabolic Matrix(7,-25,2,-7) (3/1,4/1) -> (3/1,4/1) Elliptic Matrix(6,-25,1,-4) (4/1,5/1) -> (5/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,25,1) Matrix(21,-25,16,-19) -> Matrix(21,-1,400,-19) Matrix(18,-25,13,-18) -> Matrix(18,-1,325,-18) Matrix(16,-25,9,-14) -> Matrix(16,-1,225,-14) Matrix(11,-25,4,-9) -> Matrix(11,-1,100,-9) Matrix(7,-25,2,-7) -> Matrix(7,-1,50,-7) Matrix(6,-25,1,-4) -> Matrix(6,-1,25,-4) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 30 Minimal number of generators: 7 Number of equivalence classes of elliptic points of order 2: 2 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 6 Genus: 0 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 25 1 2/1 2/25 1 25 5/2 1/10 5 5 3/1 3/25 1 25 4/1 4/25 1 25 5/1 1/5 5 5 1/0 1/0 1 25 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(11,-25,4,-9) (2/1,5/2) -> (5/2,3/1) Parabolic Matrix(7,-25,2,-7) (3/1,4/1) -> (3/1,4/1) Elliptic Matrix(6,-25,1,-4) (4/1,5/1) -> (5/1,1/0) Parabolic 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(1,0,25,-1) (0/1,2/1) -> (0/1,2/25) Matrix(11,-25,4,-9) -> Matrix(11,-1,100,-9) 1/10 Matrix(7,-25,2,-7) -> Matrix(7,-1,50,-7) (0/1,1/7).(1/8,1/6) Matrix(6,-25,1,-4) -> Matrix(6,-1,25,-4) 1/5 ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.