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 -4/1 -10/3 -30/11 -8/3 0/1 1/1 5/4 3/2 5/3 2/1 5/2 8/3 30/11 20/7 3/1 10/3 18/5 40/11 4/1 14/3 5/1 6/1 20/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -3/2 -6/1 -1/1 0/1 -5/1 -1/2 1/0 -14/3 -1/1 0/1 -9/2 1/0 -4/1 -1/2 1/0 -11/3 1/0 -18/5 -1/1 0/1 -7/2 -1/2 -10/3 -1/2 1/0 -3/1 -1/2 -14/5 0/1 1/1 -11/4 1/0 -30/11 1/0 -19/7 1/0 -8/3 -1/2 1/0 -21/8 1/0 -13/5 -1/2 -18/7 -1/1 0/1 -5/2 -1/2 1/0 -2/1 -1/1 0/1 -5/3 -1/2 1/0 -18/11 -1/1 0/1 -13/8 1/0 -21/13 -1/2 -8/5 -1/2 1/0 -19/12 -1/2 -30/19 -1/2 -11/7 -1/2 -14/9 -1/3 0/1 -17/11 -1/6 -20/13 0/1 -3/2 1/0 -10/7 -1/2 1/0 -7/5 1/0 -18/13 -1/1 0/1 -29/21 1/0 -40/29 -1/1 -11/8 -1/2 -4/3 -1/2 1/0 -9/7 -1/2 -14/11 -1/1 0/1 -5/4 -1/2 1/0 -6/5 -1/1 0/1 -13/11 -3/2 -20/17 -1/1 -7/6 -3/4 -1/1 -1/2 0/1 0/1 1/1 1/2 7/6 3/4 6/5 0/1 1/1 5/4 1/2 1/0 14/11 0/1 1/1 9/7 1/2 4/3 1/2 1/0 11/8 1/2 18/13 0/1 1/1 7/5 1/0 10/7 1/2 1/0 3/2 1/0 14/9 0/1 1/3 11/7 1/2 30/19 1/2 19/12 1/2 8/5 1/2 1/0 21/13 1/2 13/8 1/0 18/11 0/1 1/1 5/3 1/2 1/0 2/1 0/1 1/1 5/2 1/2 1/0 18/7 0/1 1/1 13/5 1/2 21/8 1/0 8/3 1/2 1/0 19/7 1/0 30/11 1/0 11/4 1/0 14/5 -1/1 0/1 17/6 -1/4 20/7 0/1 3/1 1/2 10/3 1/2 1/0 7/2 1/2 18/5 0/1 1/1 29/8 1/2 40/11 1/1 11/3 1/0 4/1 1/2 1/0 9/2 1/0 14/3 0/1 1/1 5/1 1/2 1/0 6/1 0/1 1/1 13/2 3/4 20/3 1/1 7/1 3/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(21,160,8,61) (-7/1,1/0) -> (13/5,21/8) Hyperbolic Matrix(19,120,-16,-101) (-7/1,-6/1) -> (-6/5,-13/11) Hyperbolic Matrix(11,60,2,11) (-6/1,-5/1) -> (5/1,6/1) Hyperbolic Matrix(29,140,6,29) (-5/1,-14/3) -> (14/3,5/1) Hyperbolic Matrix(61,280,22,101) (-14/3,-9/2) -> (11/4,14/5) Hyperbolic Matrix(19,80,14,59) (-9/2,-4/1) -> (4/3,11/8) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(199,720,-144,-521) (-11/3,-18/5) -> (-18/13,-29/21) Hyperbolic Matrix(101,360,62,221) (-18/5,-7/2) -> (13/8,18/11) 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(99,280,-64,-181) (-3/1,-14/5) -> (-14/9,-17/11) Hyperbolic Matrix(101,280,22,61) (-14/5,-11/4) -> (9/2,14/3) Hyperbolic Matrix(329,900,208,569) (-11/4,-30/11) -> (30/19,19/12) Hyperbolic Matrix(331,900,210,571) (-30/11,-19/7) -> (11/7,30/19) Hyperbolic Matrix(119,320,74,199) (-19/7,-8/3) -> (8/5,21/13) Hyperbolic Matrix(121,320,76,201) (-8/3,-21/8) -> (19/12,8/5) Hyperbolic Matrix(61,160,8,21) (-21/8,-13/5) -> (7/1,1/0) Hyperbolic Matrix(139,360,100,259) (-13/5,-18/7) -> (18/13,7/5) Hyperbolic Matrix(71,180,28,71) (-18/7,-5/2) -> (5/2,18/7) 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(109,180,66,109) (-5/3,-18/11) -> (18/11,5/3) Hyperbolic Matrix(221,360,62,101) (-18/11,-13/8) -> (7/2,18/5) Hyperbolic Matrix(99,160,86,139) (-13/8,-21/13) -> (1/1,7/6) Hyperbolic Matrix(199,320,74,119) (-21/13,-8/5) -> (8/3,19/7) Hyperbolic Matrix(201,320,76,121) (-8/5,-19/12) -> (21/8,8/3) Hyperbolic Matrix(569,900,208,329) (-19/12,-30/19) -> (30/11,11/4) Hyperbolic Matrix(571,900,210,331) (-30/19,-11/7) -> (19/7,30/11) Hyperbolic Matrix(179,280,140,219) (-11/7,-14/9) -> (14/11,9/7) Hyperbolic Matrix(259,400,90,139) (-17/11,-20/13) -> (20/7,3/1) Hyperbolic Matrix(261,400,92,141) (-20/13,-3/2) -> (17/6,20/7) 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(259,360,100,139) (-7/5,-18/13) -> (18/7,13/5) 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(59,80,14,19) (-11/8,-4/3) -> (4/1,9/2) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(219,280,140,179) (-9/7,-14/11) -> (14/9,11/7) Hyperbolic Matrix(111,140,88,111) (-14/11,-5/4) -> (5/4,14/11) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) 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(139,160,86,99) (-7/6,-1/1) -> (21/13,13/8) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(101,-120,16,-19) (7/6,6/5) -> (6/1,13/2) Hyperbolic Matrix(521,-720,144,-199) (11/8,18/13) -> (18/5,29/8) Hyperbolic Matrix(181,-280,64,-99) (3/2,14/9) -> (14/5,17/6) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(21,160,8,61) -> Matrix(1,2,0,1) Matrix(19,120,-16,-101) -> Matrix(1,0,0,1) Matrix(11,60,2,11) -> Matrix(1,0,2,1) Matrix(29,140,6,29) -> Matrix(1,0,2,1) Matrix(61,280,22,101) -> Matrix(1,0,0,1) Matrix(19,80,14,59) -> Matrix(1,0,2,1) Matrix(21,80,16,61) -> Matrix(1,0,2,1) Matrix(199,720,-144,-521) -> Matrix(1,0,0,1) Matrix(101,360,62,221) -> Matrix(1,0,2,1) Matrix(29,100,20,69) -> Matrix(1,0,2,1) Matrix(31,100,22,71) -> Matrix(1,0,2,1) Matrix(99,280,-64,-181) -> Matrix(1,0,-4,1) Matrix(101,280,22,61) -> Matrix(1,0,0,1) Matrix(329,900,208,569) -> Matrix(1,-2,2,-3) Matrix(331,900,210,571) -> Matrix(1,2,2,5) Matrix(119,320,74,199) -> Matrix(1,0,2,1) Matrix(121,320,76,201) -> Matrix(1,0,2,1) Matrix(61,160,8,21) -> Matrix(1,2,0,1) Matrix(139,360,100,259) -> Matrix(1,0,2,1) Matrix(71,180,28,71) -> Matrix(1,0,2,1) Matrix(9,20,4,9) -> Matrix(1,0,2,1) Matrix(11,20,6,11) -> Matrix(1,0,2,1) Matrix(109,180,66,109) -> Matrix(1,0,2,1) Matrix(221,360,62,101) -> Matrix(1,0,2,1) Matrix(99,160,86,139) -> Matrix(3,2,4,3) Matrix(199,320,74,119) -> Matrix(1,0,2,1) Matrix(201,320,76,121) -> Matrix(1,0,2,1) Matrix(569,900,208,329) -> Matrix(3,2,-2,-1) Matrix(571,900,210,331) -> Matrix(5,2,2,1) Matrix(179,280,140,219) -> Matrix(1,0,4,1) Matrix(259,400,90,139) -> Matrix(1,0,8,1) Matrix(261,400,92,141) -> Matrix(1,0,-4,1) Matrix(69,100,20,29) -> Matrix(1,0,2,1) Matrix(71,100,22,31) -> Matrix(1,0,2,1) Matrix(259,360,100,139) -> Matrix(1,0,2,1) Matrix(1159,1600,318,439) -> Matrix(1,2,0,1) Matrix(1161,1600,320,441) -> Matrix(3,2,4,3) Matrix(59,80,14,19) -> Matrix(1,0,2,1) Matrix(61,80,16,21) -> Matrix(1,0,2,1) Matrix(219,280,140,179) -> Matrix(1,0,4,1) Matrix(111,140,88,111) -> Matrix(1,0,2,1) Matrix(49,60,40,49) -> Matrix(1,0,2,1) Matrix(339,400,50,59) -> Matrix(5,6,4,5) Matrix(341,400,52,61) -> Matrix(7,6,8,7) Matrix(139,160,86,99) -> Matrix(3,2,4,3) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(101,-120,16,-19) -> Matrix(1,0,0,1) Matrix(521,-720,144,-199) -> Matrix(1,0,0,1) Matrix(181,-280,64,-99) -> 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): 8 Degree of the the map X: 8 Degree of the the map Y: 48 Permutation triple for Y: ((2,6,4,3,7)(5,16,10,9,17)(8,24,14,13,25)(11,31,20,19,32)(12,35,22,21,36)(15,41,26,27,40)(28,30,37,39,43)(29,38,33,42,34); (1,4,14,40,31,44,19,39,24,42,47,29,16,37,36,45,35,15,5,2)(3,10,18,17,33,32,41,46,27,21,38,13,23,8,7,22,43,48,30,11)(6,20,34,12)(9,26,25,28); (1,2,8,26,32,44,31,30,25,38,47,42,17,28,22,45,36,27,9,3)(4,12,37,48,43,19,6,5,18,10,29,20,40,46,41,35,34,24,23,13)(7,11,33,21)(14,39,16,15)) ----------------------------------------------------------------------- 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 2/1 5/2 8/3 30/11 3/1 10/3 4/1 5/1 6/1 20/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/2 6/5 0/1 1/1 5/4 1/2 1/0 4/3 1/2 1/0 3/2 1/0 14/9 0/1 1/3 11/7 1/2 8/5 1/2 1/0 5/3 1/2 1/0 2/1 0/1 1/1 5/2 1/2 1/0 8/3 1/2 1/0 19/7 1/0 30/11 1/0 11/4 1/0 3/1 1/2 10/3 1/2 1/0 7/2 1/2 4/1 1/2 1/0 5/1 1/2 1/0 6/1 0/1 1/1 13/2 3/4 20/3 1/1 7/1 3/2 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(69,-80,44,-51) (1/1,6/5) -> (14/9,11/7) Hyperbolic Matrix(49,-60,9,-11) (6/5,5/4) -> (5/1,6/1) 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(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(49,-80,19,-31) (8/5,5/3) -> (5/2,8/3) Hyperbolic Matrix(11,-20,5,-9) (5/3,2/1) -> (2/1,5/2) Parabolic Matrix(331,-900,121,-329) (19/7,30/11) -> (30/11,11/4) Parabolic Matrix(29,-80,4,-11) (11/4,3/1) -> (7/1,1/0) Hyperbolic Matrix(31,-100,9,-29) (3/1,10/3) -> (10/3,7/2) Parabolic Matrix(61,-400,9,-59) (13/2,20/3) -> (20/3,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,2,1) Matrix(69,-80,44,-51) -> Matrix(1,-1,4,-3) Matrix(49,-60,9,-11) -> Matrix(1,0,0,1) Matrix(31,-40,7,-9) -> Matrix(1,-1,2,-1) Matrix(29,-40,8,-11) -> Matrix(1,-1,2,-1) Matrix(129,-200,20,-31) -> Matrix(3,-1,4,-1) Matrix(151,-240,56,-89) -> Matrix(1,-1,2,-1) Matrix(49,-80,19,-31) -> Matrix(1,-1,2,-1) Matrix(11,-20,5,-9) -> Matrix(1,0,0,1) Matrix(331,-900,121,-329) -> Matrix(1,-2,0,1) Matrix(29,-80,4,-11) -> Matrix(1,1,0,1) Matrix(31,-100,9,-29) -> Matrix(1,0,0,1) Matrix(61,-400,9,-59) -> Matrix(7,-6,6,-5) 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 2 1 2/1 (0/1,1/1) 0 10 5/2 (0/1,1/1).(1/2,1/0) 0 4 8/3 (1/2,1/0) 0 5 30/11 1/0 4 2 11/4 1/0 1 20 3/1 1/2 1 20 10/3 (1/2,1/0) 0 2 4/1 (1/2,1/0) 0 5 5/1 (0/1,1/1).(1/2,1/0) 0 4 6/1 (0/1,1/1) 0 10 20/3 1/1 6 1 7/1 3/2 1 20 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(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection 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(241,-660,88,-241) (30/11,11/4) -> (30/11,11/4) Reflection Matrix(29,-80,4,-11) (11/4,3/1) -> (7/1,1/0) Hyperbolic Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) 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 Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection Matrix(19,-120,3,-19) (6/1,20/3) -> (6/1,20/3) Reflection Matrix(41,-280,6,-41) (20/3,7/1) -> (20/3,7/1) 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(1,0,2,-1) (0/1,2/1) -> (0/1,1/1) Matrix(9,-20,4,-9) -> Matrix(1,0,2,-1) (2/1,5/2) -> (0/1,1/1) Matrix(31,-80,12,-31) -> Matrix(-1,1,0,1) (5/2,8/3) -> (1/2,1/0) Matrix(89,-240,33,-89) -> Matrix(-1,1,0,1) (8/3,30/11) -> (1/2,1/0) Matrix(241,-660,88,-241) -> Matrix(1,1,0,-1) (30/11,11/4) -> (-1/2,1/0) Matrix(29,-80,4,-11) -> Matrix(1,1,0,1) 1/0 Matrix(19,-60,6,-19) -> Matrix(-1,1,0,1) (3/1,10/3) -> (1/2,1/0) Matrix(11,-40,3,-11) -> Matrix(-1,1,0,1) (10/3,4/1) -> (1/2,1/0) Matrix(9,-40,2,-9) -> Matrix(-1,1,0,1) (4/1,5/1) -> (1/2,1/0) Matrix(11,-60,2,-11) -> Matrix(1,0,2,-1) (5/1,6/1) -> (0/1,1/1) Matrix(19,-120,3,-19) -> Matrix(1,0,2,-1) (6/1,20/3) -> (0/1,1/1) Matrix(41,-280,6,-41) -> Matrix(5,-6,4,-5) (20/3,7/1) -> (1/1,3/2) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.