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: 16 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -10/3 -20/9 -2/1 0/1 1/1 2/1 20/9 5/2 8/3 10/3 40/11 4/1 5/1 16/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -3/20 -11/2 -11/80 -5/1 -1/8 -4/1 -1/10 -11/3 -11/120 -18/5 -9/100 -7/2 -7/80 -10/3 -1/12 -3/1 -3/40 -14/5 -7/100 -11/4 -11/160 -8/3 -1/15 -5/2 -1/16 -7/3 -7/120 -16/7 -2/35 -25/11 -5/88 -9/4 -9/160 -20/9 -1/18 -11/5 -11/200 -2/1 -1/20 -11/6 -11/240 -20/11 -1/22 -9/5 -9/200 -16/9 -2/45 -7/4 -7/160 -12/7 -3/70 -5/3 -1/24 -8/5 -1/25 -19/12 -19/480 -11/7 -11/280 -3/2 -3/80 -16/11 -2/55 -13/9 -13/360 -10/7 -1/28 -17/12 -17/480 -7/5 -7/200 -18/13 -9/260 -29/21 -29/840 -40/29 -1/29 -11/8 -11/320 -4/3 -1/30 -9/7 -9/280 -5/4 -1/32 -16/13 -2/65 -11/9 -11/360 -6/5 -3/100 -7/6 -7/240 -1/1 -1/40 0/1 0/1 1/1 1/40 6/5 3/100 11/9 11/360 5/4 1/32 4/3 1/30 11/8 11/320 18/13 9/260 7/5 7/200 10/7 1/28 3/2 3/80 14/9 7/180 11/7 11/280 8/5 1/25 5/3 1/24 7/4 7/160 16/9 2/45 25/14 5/112 9/5 9/200 20/11 1/22 11/6 11/240 2/1 1/20 11/5 11/200 20/9 1/18 9/4 9/160 16/7 2/35 7/3 7/120 12/5 3/50 5/2 1/16 8/3 1/15 19/7 19/280 11/4 11/160 3/1 3/40 16/5 2/25 13/4 13/160 10/3 1/12 17/5 17/200 7/2 7/80 18/5 9/100 29/8 29/320 40/11 1/11 11/3 11/120 4/1 1/10 9/2 9/80 5/1 1/8 16/3 2/15 11/2 11/80 6/1 3/20 7/1 7/40 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(43,240,12,67) (-6/1,-11/2) -> (7/2,18/5) Hyperbolic Matrix(59,320,-26,-141) (-11/2,-5/1) -> (-25/11,-9/4) Hyperbolic Matrix(17,80,-10,-47) (-5/1,-4/1) -> (-12/7,-5/3) Hyperbolic Matrix(43,160,18,67) (-4/1,-11/3) -> (7/3,12/5) Hyperbolic Matrix(199,720,-144,-521) (-11/3,-18/5) -> (-18/13,-29/21) Hyperbolic Matrix(67,240,12,43) (-18/5,-7/2) -> (11/2,6/1) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(49,160,-34,-111) (-10/3,-3/1) -> (-13/9,-10/7) Hyperbolic Matrix(57,160,26,73) (-3/1,-14/5) -> (2/1,11/5) 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(33,80,-26,-63) (-5/2,-7/3) -> (-9/7,-5/4) Hyperbolic Matrix(139,320,-96,-221) (-7/3,-16/7) -> (-16/11,-13/9) Hyperbolic Matrix(211,480,40,91) (-16/7,-25/11) -> (5/1,16/3) Hyperbolic Matrix(179,400,98,219) (-9/4,-20/9) -> (20/11,11/6) Hyperbolic Matrix(181,400,100,221) (-20/9,-11/5) -> (9/5,20/11) Hyperbolic Matrix(37,80,6,13) (-11/5,-2/1) -> (6/1,7/1) Hyperbolic Matrix(87,160,56,103) (-2/1,-11/6) -> (3/2,14/9) Hyperbolic 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(91,160,-62,-109) (-16/9,-7/4) -> (-3/2,-16/11) Hyperbolic Matrix(93,160,68,117) (-7/4,-12/7) -> (4/3,11/8) Hyperbolic Matrix(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(253,400,74,117) (-19/12,-11/7) -> (17/5,7/2) Hyperbolic Matrix(51,80,-44,-69) (-11/7,-3/2) -> (-7/6,-1/1) Hyperbolic Matrix(283,400,104,147) (-17/12,-7/5) -> (19/7,11/4) Hyperbolic Matrix(173,240,142,197) (-7/5,-18/13) -> (6/5,11/9) 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(389,480,218,269) (-5/4,-16/13) -> (16/9,25/14) Hyperbolic Matrix(197,240,142,173) (-11/9,-6/5) -> (18/13,7/5) Hyperbolic Matrix(67,80,36,43) (-6/5,-7/6) -> (11/6,2/1) 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(261,-320,146,-179) (11/9,5/4) -> (25/14,9/5) Hyperbolic Matrix(63,-80,26,-33) (5/4,4/3) -> (12/5,5/2) Hyperbolic Matrix(521,-720,144,-199) (11/8,18/13) -> (18/5,29/8) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(111,-160,34,-49) (10/7,3/2) -> (13/4,10/3) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(47,-80,10,-17) (5/3,7/4) -> (9/2,5/1) Hyperbolic Matrix(181,-320,56,-99) (7/4,16/9) -> (16/5,13/4) Hyperbolic Matrix(141,-320,26,-59) (9/4,16/7) -> (16/3,11/2) Hyperbolic Matrix(69,-160,22,-51) (16/7,7/3) -> (3/1,16/5) Hyperbolic Matrix(29,-80,4,-11) (11/4,3/1) -> (7/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,80,-4,-29) -> Matrix(11,2,-160,-29) Matrix(43,240,12,67) -> Matrix(43,6,480,67) Matrix(59,320,-26,-141) -> Matrix(59,8,-1040,-141) Matrix(17,80,-10,-47) -> Matrix(17,2,-400,-47) Matrix(43,160,18,67) -> Matrix(43,4,720,67) Matrix(199,720,-144,-521) -> Matrix(199,18,-5760,-521) Matrix(67,240,12,43) -> Matrix(67,6,480,43) Matrix(71,240,-50,-169) -> Matrix(71,6,-2000,-169) Matrix(49,160,-34,-111) -> Matrix(49,4,-1360,-111) Matrix(57,160,26,73) -> Matrix(57,4,1040,73) Matrix(89,240,-56,-151) -> Matrix(89,6,-2240,-151) Matrix(31,80,12,31) -> Matrix(31,2,480,31) Matrix(33,80,-26,-63) -> Matrix(33,2,-1040,-63) Matrix(139,320,-96,-221) -> Matrix(139,8,-3840,-221) Matrix(211,480,40,91) -> Matrix(211,12,1600,91) Matrix(179,400,98,219) -> Matrix(179,10,3920,219) Matrix(181,400,100,221) -> Matrix(181,10,4000,221) Matrix(37,80,6,13) -> Matrix(37,2,240,13) Matrix(87,160,56,103) -> Matrix(87,4,2240,103) Matrix(219,400,98,179) -> Matrix(219,10,3920,179) Matrix(221,400,100,181) -> Matrix(221,10,4000,181) Matrix(179,320,-146,-261) -> Matrix(179,8,-5840,-261) Matrix(91,160,-62,-109) -> Matrix(91,4,-2480,-109) Matrix(93,160,68,117) -> Matrix(93,4,2720,117) Matrix(49,80,30,49) -> Matrix(49,2,1200,49) Matrix(253,400,74,117) -> Matrix(253,10,2960,117) Matrix(51,80,-44,-69) -> Matrix(51,2,-1760,-69) Matrix(283,400,104,147) -> Matrix(283,10,4160,147) Matrix(173,240,142,197) -> Matrix(173,6,5680,197) Matrix(1159,1600,318,439) -> Matrix(1159,40,12720,439) Matrix(1161,1600,320,441) -> Matrix(1161,40,12800,441) Matrix(59,80,14,19) -> Matrix(59,2,560,19) Matrix(61,80,16,21) -> Matrix(61,2,640,21) Matrix(389,480,218,269) -> Matrix(389,12,8720,269) Matrix(197,240,142,173) -> Matrix(197,6,5680,173) Matrix(67,80,36,43) -> Matrix(67,2,1440,43) Matrix(1,0,2,1) -> Matrix(1,0,80,1) Matrix(69,-80,44,-51) -> Matrix(69,-2,1760,-51) Matrix(261,-320,146,-179) -> Matrix(261,-8,5840,-179) Matrix(63,-80,26,-33) -> Matrix(63,-2,1040,-33) Matrix(521,-720,144,-199) -> Matrix(521,-18,5760,-199) Matrix(169,-240,50,-71) -> Matrix(169,-6,2000,-71) Matrix(111,-160,34,-49) -> Matrix(111,-4,1360,-49) Matrix(151,-240,56,-89) -> Matrix(151,-6,2240,-89) Matrix(47,-80,10,-17) -> Matrix(47,-2,400,-17) Matrix(181,-320,56,-99) -> Matrix(181,-8,2240,-99) Matrix(141,-320,26,-59) -> Matrix(141,-8,1040,-59) Matrix(69,-160,22,-51) -> Matrix(69,-4,880,-51) Matrix(29,-80,4,-11) -> Matrix(29,-2,160,-11) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 16 Genus: 17 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 48 Degree of the the map Y: 48 Permutation triple for Y: ((2,6,23,39,41,43,30,29,24,7)(3,12,36,28,31,40,17,37,13,4)(5,18,10,9,19)(8,27,16,15,22)(11,34,14,21,25)(20,26)(32,33)(35,38,42,44,47); (1,4,16,39,34,20,19,41,38,37,46,29,11,3,10,33,35,12,27,43,48,31,18,24,44,26,8,7,25,36,45,23,42,40,15,32,21,17,5,2)(6,14,13,9,30,47,28,22); (1,2,8,28,25,32,10,31,47,24,46,37,21,6,5,20,44,23,22,40,48,43,19,13,38,33,15,4,14,39,45,36,35,30,27,26,34,29,9,3)(7,18,17,42,41,16,12,11)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. 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. ----------------------------------------------------------------------- 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 1/40 5/4 1/32 4/3 1/30 3/2 3/80 11/7 11/280 8/5 1/25 5/3 1/24 7/4 7/160 9/5 9/200 2/1 1/20 11/5 11/200 20/9 1/18 9/4 9/160 7/3 7/120 5/2 1/16 8/3 1/15 19/7 19/280 11/4 11/160 3/1 3/40 10/3 1/12 17/5 17/200 7/2 7/80 4/1 1/10 5/1 1/8 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(1,0,40,1) Matrix(33,-40,19,-23) -> Matrix(33,-1,760,-23) Matrix(31,-40,7,-9) -> Matrix(31,-1,280,-9) Matrix(29,-40,8,-11) -> Matrix(29,-1,320,-11) Matrix(77,-120,43,-67) -> Matrix(77,-3,1720,-67) Matrix(151,-240,56,-89) -> Matrix(151,-6,2240,-89) Matrix(49,-80,19,-31) -> Matrix(49,-2,760,-31) Matrix(21,-40,10,-19) -> Matrix(21,-1,400,-19) Matrix(181,-400,81,-179) -> Matrix(181,-10,3240,-179) Matrix(53,-120,19,-43) -> Matrix(53,-3,760,-43) Matrix(17,-40,3,-7) -> Matrix(17,-1,120,-7) Matrix(147,-400,43,-117) -> Matrix(147,-10,1720,-117) Matrix(61,-200,18,-59) -> Matrix(61,-5,720,-59) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM 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 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 40 1 2/1 1/20 2 20 20/9 1/18 20 2 9/4 9/160 1 40 7/3 7/120 1 40 5/2 1/16 5 8 8/3 1/15 8 5 11/4 11/160 1 40 3/1 3/40 1 40 10/3 1/12 10 4 4/1 1/10 4 10 5/1 1/8 5 8 1/0 1/0 1 40 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(1,0,40,-1) (0/1,2/1) -> (0/1,1/20) Matrix(19,-40,9,-19) -> Matrix(19,-1,360,-19) (2/1,20/9) -> (1/20,1/18) Matrix(161,-360,72,-161) -> Matrix(161,-9,2880,-161) (20/9,9/4) -> (1/18,9/160) Matrix(53,-120,19,-43) -> Matrix(53,-3,760,-43) Matrix(17,-40,3,-7) -> Matrix(17,-1,120,-7) Matrix(31,-80,12,-31) -> Matrix(31,-2,480,-31) (5/2,8/3) -> (1/16,1/15) Matrix(89,-240,33,-89) -> Matrix(89,-6,1320,-89) (8/3,30/11) -> (1/15,3/44) Matrix(73,-200,23,-63) -> Matrix(73,-5,920,-63) Matrix(49,-160,15,-49) -> Matrix(49,-4,600,-49) (16/5,10/3) -> (2/25,1/12) Matrix(11,-40,3,-11) -> Matrix(11,-1,120,-11) (10/3,4/1) -> (1/12,1/10) Matrix(9,-40,2,-9) -> Matrix(9,-1,80,-9) (4/1,5/1) -> (1/10,1/8) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.