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: 24 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -2/1 -30/19 -10/7 0/1 1/1 5/4 3/2 5/3 2/1 20/9 12/5 5/2 8/3 30/11 3/1 10/3 4/1 5/1 16/3 40/7 6/1 20/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 0/1 -11/2 -1/2 -5/1 -1/2 -4/1 -1/2 0/1 -3/1 -1/2 -14/5 -1/1 0/1 -11/4 1/0 -8/3 -1/2 -5/2 -1/2 -12/5 -1/2 -1/3 -19/8 -1/2 -26/11 -2/5 -1/3 -7/3 -1/4 -16/7 -1/2 -25/11 -1/2 -9/4 -1/2 -2/1 -1/3 0/1 -11/6 -1/2 -20/11 -1/3 -9/5 -1/4 -16/9 -1/4 -7/4 -1/2 -19/11 -1/4 -31/18 -1/4 -12/7 -1/3 -1/4 -5/3 -1/4 -8/5 -1/4 -19/12 -3/14 -30/19 -1/5 -41/26 -1/6 -11/7 -1/6 -3/2 -1/4 -10/7 0/1 -17/12 1/0 -24/17 -1/2 -7/5 -1/4 -4/3 -1/4 0/1 -5/4 -1/4 -16/13 -1/4 -11/9 -1/4 -17/14 -3/14 -40/33 -1/5 -23/19 -3/16 -6/5 -1/5 0/1 -13/11 -1/6 -20/17 0/1 -7/6 -1/4 -1/1 -1/6 0/1 0/1 1/1 1/6 6/5 0/1 1/5 11/9 1/4 5/4 1/4 4/3 0/1 1/4 3/2 1/4 14/9 0/1 1/5 11/7 1/6 8/5 1/4 5/3 1/4 12/7 1/4 1/3 19/11 1/4 26/15 2/7 1/3 7/4 1/2 16/9 1/4 25/14 1/4 9/5 1/4 2/1 0/1 1/3 11/5 1/4 20/9 1/3 9/4 1/2 16/7 1/2 7/3 1/4 19/8 1/2 31/13 1/2 12/5 1/3 1/2 5/2 1/2 8/3 1/2 19/7 3/4 30/11 1/1 41/15 1/0 11/4 1/0 3/1 1/2 10/3 0/1 17/5 1/6 24/7 1/4 7/2 1/2 4/1 0/1 1/2 5/1 1/2 16/3 1/2 11/2 1/2 17/3 3/4 40/7 1/1 23/4 3/2 6/1 0/1 1/1 13/2 1/0 20/3 0/1 7/1 1/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(11,80,-4,-29) (-6/1,1/0) -> (-14/5,-11/4) Hyperbolic Matrix(71,400,-30,-169) (-6/1,-11/2) -> (-19/8,-26/11) Hyperbolic Matrix(59,320,-26,-141) (-11/2,-5/1) -> (-25/11,-9/4) Hyperbolic 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(71,200,-60,-169) (-3/1,-14/5) -> (-6/5,-13/11) Hyperbolic Matrix(89,240,-56,-151) (-11/4,-8/3) -> (-8/5,-19/12) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(251,600,-146,-349) (-12/5,-19/8) -> (-31/18,-12/7) Hyperbolic Matrix(271,640,-224,-529) (-26/11,-7/3) -> (-23/19,-6/5) Hyperbolic Matrix(121,280,-86,-199) (-7/3,-16/7) -> (-24/17,-7/5) Hyperbolic Matrix(211,480,40,91) (-16/7,-25/11) -> (5/1,16/3) Hyperbolic Matrix(19,40,-10,-21) (-9/4,-2/1) -> (-2/1,-11/6) Parabolic Matrix(219,400,98,179) (-11/6,-20/11) -> (20/9,9/4) Hyperbolic Matrix(221,400,100,181) (-20/11,-9/5) -> (11/5,20/9) Hyperbolic Matrix(179,320,-146,-261) (-9/5,-16/9) -> (-16/13,-11/9) Hyperbolic Matrix(249,440,-176,-311) (-16/9,-7/4) -> (-17/12,-24/17) Hyperbolic Matrix(231,400,-190,-329) (-7/4,-19/11) -> (-11/9,-17/14) Hyperbolic Matrix(649,1120,-412,-711) (-19/11,-31/18) -> (-41/26,-11/7) Hyperbolic Matrix(71,120,42,71) (-12/7,-5/3) -> (5/3,12/7) Hyperbolic Matrix(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(1139,1800,-722,-1141) (-19/12,-30/19) -> (-30/19,-41/26) Parabolic Matrix(51,80,-44,-69) (-11/7,-3/2) -> (-7/6,-1/1) Hyperbolic Matrix(139,200,-98,-141) (-3/2,-10/7) -> (-10/7,-17/12) Parabolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(389,480,218,269) (-5/4,-16/13) -> (16/9,25/14) Hyperbolic Matrix(1319,1600,230,279) (-17/14,-40/33) -> (40/7,23/4) Hyperbolic Matrix(1321,1600,232,281) (-40/33,-23/19) -> (17/3,40/7) Hyperbolic 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(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(69,-80,44,-51) (1/1,6/5) -> (14/9,11/7) Hyperbolic Matrix(329,-400,190,-231) (6/5,11/9) -> (19/11,26/15) Hyperbolic Matrix(261,-320,146,-179) (11/9,5/4) -> (25/14,9/5) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(129,-200,20,-31) (3/2,14/9) -> (6/1,13/2) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(349,-600,146,-251) (12/7,19/11) -> (31/13,12/5) Hyperbolic Matrix(369,-640,64,-111) (26/15,7/4) -> (23/4,6/1) Hyperbolic Matrix(159,-280,46,-81) (7/4,16/9) -> (24/7,7/2) Hyperbolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(141,-320,26,-59) (9/4,16/7) -> (16/3,11/2) Hyperbolic Matrix(191,-440,56,-129) (16/7,7/3) -> (17/5,24/7) Hyperbolic Matrix(169,-400,30,-71) (7/3,19/8) -> (11/2,17/3) Hyperbolic Matrix(471,-1120,172,-409) (19/8,31/13) -> (41/15,11/4) Hyperbolic Matrix(661,-1800,242,-659) (19/7,30/11) -> (30/11,41/15) Parabolic Matrix(29,-80,4,-11) (11/4,3/1) -> (7/1,1/0) Hyperbolic Matrix(61,-200,18,-59) (3/1,10/3) -> (10/3,17/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,80,-4,-29) -> Matrix(1,0,0,1) Matrix(71,400,-30,-169) -> Matrix(3,2,-8,-5) Matrix(59,320,-26,-141) -> Matrix(3,2,-8,-5) Matrix(9,40,2,9) -> Matrix(1,0,4,1) Matrix(11,40,-8,-29) -> Matrix(1,0,-2,1) Matrix(71,200,-60,-169) -> Matrix(1,0,-4,1) Matrix(89,240,-56,-151) -> Matrix(3,2,-14,-9) Matrix(31,80,12,31) -> Matrix(1,0,4,1) Matrix(49,120,20,49) -> Matrix(5,2,12,5) Matrix(251,600,-146,-349) -> Matrix(5,2,-18,-7) Matrix(271,640,-224,-529) -> Matrix(5,2,-28,-11) Matrix(121,280,-86,-199) -> Matrix(1,0,0,1) Matrix(211,480,40,91) -> Matrix(1,0,4,1) Matrix(19,40,-10,-21) -> Matrix(1,0,0,1) Matrix(219,400,98,179) -> Matrix(5,2,12,5) Matrix(221,400,100,181) -> Matrix(7,2,24,7) Matrix(179,320,-146,-261) -> Matrix(7,2,-32,-9) Matrix(249,440,-176,-311) -> Matrix(1,0,2,1) Matrix(231,400,-190,-329) -> Matrix(7,2,-32,-9) Matrix(649,1120,-412,-711) -> Matrix(1,0,-2,1) Matrix(71,120,42,71) -> Matrix(7,2,24,7) Matrix(49,80,30,49) -> Matrix(1,0,8,1) Matrix(1139,1800,-722,-1141) -> Matrix(19,4,-100,-21) Matrix(51,80,-44,-69) -> Matrix(1,0,0,1) Matrix(139,200,-98,-141) -> Matrix(1,0,4,1) Matrix(31,40,24,31) -> Matrix(1,0,8,1) Matrix(389,480,218,269) -> Matrix(1,0,8,1) Matrix(1319,1600,230,279) -> Matrix(29,6,24,5) Matrix(1321,1600,232,281) -> Matrix(31,6,36,7) Matrix(339,400,50,59) -> Matrix(1,0,8,1) Matrix(341,400,52,61) -> Matrix(1,0,4,1) Matrix(1,0,2,1) -> Matrix(1,0,12,1) Matrix(69,-80,44,-51) -> Matrix(1,0,0,1) Matrix(329,-400,190,-231) -> Matrix(9,-2,32,-7) Matrix(261,-320,146,-179) -> Matrix(9,-2,32,-7) Matrix(29,-40,8,-11) -> Matrix(1,0,-2,1) Matrix(129,-200,20,-31) -> Matrix(1,0,-4,1) Matrix(151,-240,56,-89) -> Matrix(9,-2,14,-3) Matrix(349,-600,146,-251) -> Matrix(7,-2,18,-5) Matrix(369,-640,64,-111) -> Matrix(7,-2,4,-1) Matrix(159,-280,46,-81) -> Matrix(1,0,0,1) Matrix(21,-40,10,-19) -> Matrix(1,0,0,1) Matrix(141,-320,26,-59) -> Matrix(5,-2,8,-3) Matrix(191,-440,56,-129) -> Matrix(1,0,2,1) Matrix(169,-400,30,-71) -> Matrix(5,-2,8,-3) Matrix(471,-1120,172,-409) -> Matrix(1,0,-2,1) Matrix(661,-1800,242,-659) -> Matrix(5,-4,4,-3) Matrix(29,-80,4,-11) -> Matrix(1,0,0,1) Matrix(61,-200,18,-59) -> Matrix(1,0,4,1) 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): 12 Degree of the the map X: 12 Degree of the the map Y: 48 Permutation triple for Y: ((2,6,20,21,7)(3,12,31,13,4)(5,15,10,9,8)(11,28,14,18,23)(16,36,24,44,37)(19,33,29,22,41)(25,34,35,46,26)(30,32,38,39,43); (1,4,15,34,28,47,33,35,32,31,42,20,38,37,41,40,18,16,5,2)(3,10,27,30,12,29,46,48,44,22,21,39,17,8,7,23,36,45,25,11)(6,14,13,19)(9,26,43,24); (1,2,8,24,23,40,41,44,43,21,42,31,30,26,29,47,28,25,9,3)(4,14,34,45,36,18,6,5,17,39,20,19,37,48,46,33,13,32,27,10)(7,22,12,11)(15,16,38,35)) ----------------------------------------------------------------------- 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: 8 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 20/9 8/3 10/3 4/1 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 1/6 5/4 1/4 4/3 0/1 1/4 3/2 1/4 11/7 1/6 8/5 1/4 5/3 1/4 7/4 1/2 9/5 1/4 2/1 0/1 1/3 11/5 1/4 20/9 1/3 9/4 1/2 7/3 1/4 5/2 1/2 8/3 1/2 19/7 3/4 11/4 1/0 3/1 1/2 10/3 0/1 17/5 1/6 7/2 1/2 4/1 0/1 1/2 5/1 1/2 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(33,-40,19,-23) (1/1,5/4) -> (5/3,7/4) Hyperbolic Matrix(31,-40,7,-9) (5/4,4/3) -> (4/1,5/1) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(77,-120,43,-67) (3/2,11/7) -> (7/4,9/5) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(49,-80,19,-31) (8/5,5/3) -> (5/2,8/3) Hyperbolic Matrix(21,-40,10,-19) (9/5,2/1) -> (2/1,11/5) Parabolic Matrix(181,-400,81,-179) (11/5,20/9) -> (20/9,9/4) Parabolic Matrix(53,-120,19,-43) (9/4,7/3) -> (11/4,3/1) Hyperbolic Matrix(17,-40,3,-7) (7/3,5/2) -> (5/1,1/0) Hyperbolic Matrix(147,-400,43,-117) (19/7,11/4) -> (17/5,7/2) Hyperbolic Matrix(61,-200,18,-59) (3/1,10/3) -> (10/3,17/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,6,1) Matrix(33,-40,19,-23) -> Matrix(5,-1,16,-3) Matrix(31,-40,7,-9) -> Matrix(1,0,-2,1) Matrix(29,-40,8,-11) -> Matrix(1,0,-2,1) Matrix(77,-120,43,-67) -> Matrix(5,-1,16,-3) Matrix(151,-240,56,-89) -> Matrix(9,-2,14,-3) Matrix(49,-80,19,-31) -> Matrix(1,0,-2,1) Matrix(21,-40,10,-19) -> Matrix(1,0,0,1) Matrix(181,-400,81,-179) -> Matrix(7,-2,18,-5) Matrix(53,-120,19,-43) -> Matrix(3,-1,4,-1) Matrix(17,-40,3,-7) -> Matrix(3,-1,4,-1) Matrix(147,-400,43,-117) -> Matrix(1,-1,6,-5) Matrix(61,-200,18,-59) -> Matrix(1,0,4,1) 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): 6 ----------------------------------------------------------------------- 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 1 2/1 (0/1,1/3) 0 10 20/9 1/3 2 1 9/4 1/2 1 20 7/3 1/4 1 20 5/2 1/2 1 4 8/3 1/2 2 5 11/4 1/0 1 20 3/1 1/2 1 20 10/3 0/1 4 2 4/1 (0/1,1/2) 0 5 5/1 1/2 1 4 1/0 1/0 1 20 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(19,-40,9,-19) (2/1,20/9) -> (2/1,20/9) Reflection Matrix(161,-360,72,-161) (20/9,9/4) -> (20/9,9/4) Reflection Matrix(53,-120,19,-43) (9/4,7/3) -> (11/4,3/1) Hyperbolic Matrix(17,-40,3,-7) (7/3,5/2) -> (5/1,1/0) Hyperbolic Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(89,-240,33,-89) (8/3,30/11) -> (8/3,30/11) Reflection Matrix(73,-200,23,-63) (19/7,11/4) -> (3/1,13/4) Hyperbolic Matrix(49,-160,15,-49) (16/5,10/3) -> (16/5,10/3) Reflection Matrix(11,-40,3,-11) (10/3,4/1) -> (10/3,4/1) Reflection Matrix(9,-40,2,-9) (4/1,5/1) -> (4/1,5/1) 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(1,0,6,-1) (0/1,2/1) -> (0/1,1/3) Matrix(19,-40,9,-19) -> Matrix(1,0,6,-1) (2/1,20/9) -> (0/1,1/3) Matrix(161,-360,72,-161) -> Matrix(5,-2,12,-5) (20/9,9/4) -> (1/3,1/2) Matrix(53,-120,19,-43) -> Matrix(3,-1,4,-1) 1/2 Matrix(17,-40,3,-7) -> Matrix(3,-1,4,-1) 1/2 Matrix(31,-80,12,-31) -> Matrix(1,0,4,-1) (5/2,8/3) -> (0/1,1/2) Matrix(89,-240,33,-89) -> Matrix(3,-2,4,-3) (8/3,30/11) -> (1/2,1/1) Matrix(73,-200,23,-63) -> Matrix(1,-1,2,-1) (0/1,1/1).(1/2,1/0) Matrix(49,-160,15,-49) -> Matrix(1,0,0,-1) (16/5,10/3) -> (0/1,1/0) Matrix(11,-40,3,-11) -> Matrix(1,0,4,-1) (10/3,4/1) -> (0/1,1/2) Matrix(9,-40,2,-9) -> Matrix(1,0,4,-1) (4/1,5/1) -> (0/1,1/2) ----------------------------------------------------------------------- The pullback map has no extra symmetries.