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 -1/2 -9/20 -3/10 -11/40 -1/4 -3/16 0/1 1/5 1/4 3/10 3/8 2/5 9/20 1/2 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/40 -5/6 3/100 -9/11 11/360 -4/5 1/32 -3/4 1/30 -8/11 11/320 -13/18 9/260 -5/7 7/200 -7/10 1/28 -2/3 3/80 -9/14 7/180 -7/11 11/280 -5/8 1/25 -3/5 1/24 -4/7 7/160 -9/16 2/45 -14/25 5/112 -5/9 9/200 -11/20 1/22 -6/11 11/240 -1/2 1/20 -5/11 11/200 -9/20 1/18 -4/9 9/160 -7/16 2/35 -3/7 7/120 -5/12 3/50 -2/5 1/16 -3/8 1/15 -7/19 19/280 -4/11 11/160 -1/3 3/40 -5/16 2/25 -4/13 13/160 -3/10 1/12 -5/17 17/200 -2/7 7/80 -5/18 9/100 -8/29 29/320 -11/40 1/11 -3/11 11/120 -1/4 1/10 -2/9 9/80 -1/5 1/8 -3/16 2/15 -2/11 11/80 -1/6 3/20 -1/7 7/40 0/1 1/0 1/6 -3/20 2/11 -11/80 1/5 -1/8 1/4 -1/10 3/11 -11/120 5/18 -9/100 2/7 -7/80 3/10 -1/12 1/3 -3/40 5/14 -7/100 4/11 -11/160 3/8 -1/15 2/5 -1/16 3/7 -7/120 7/16 -2/35 11/25 -5/88 4/9 -9/160 9/20 -1/18 5/11 -11/200 1/2 -1/20 6/11 -11/240 11/20 -1/22 5/9 -9/200 9/16 -2/45 4/7 -7/160 7/12 -3/70 3/5 -1/24 5/8 -1/25 12/19 -19/480 7/11 -11/280 2/3 -3/80 11/16 -2/55 9/13 -13/360 7/10 -1/28 12/17 -17/480 5/7 -7/200 13/18 -9/260 21/29 -29/840 29/40 -1/29 8/11 -11/320 3/4 -1/30 7/9 -9/280 4/5 -1/32 13/16 -2/65 9/11 -11/360 5/6 -3/100 6/7 -7/240 1/1 -1/40 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(51,44,-80,-69) (-1/1,-5/6) -> (-9/14,-7/11) Hyperbolic Matrix(173,142,240,197) (-5/6,-9/11) -> (5/7,13/18) Hyperbolic Matrix(179,146,-320,-261) (-9/11,-4/5) -> (-14/25,-5/9) Hyperbolic Matrix(33,26,-80,-63) (-4/5,-3/4) -> (-5/12,-2/5) Hyperbolic Matrix(93,68,160,117) (-3/4,-8/11) -> (4/7,7/12) Hyperbolic Matrix(199,144,-720,-521) (-8/11,-13/18) -> (-5/18,-8/29) Hyperbolic Matrix(197,142,240,173) (-13/18,-5/7) -> (9/11,5/6) Hyperbolic Matrix(71,50,-240,-169) (-5/7,-7/10) -> (-3/10,-5/17) Hyperbolic Matrix(49,34,-160,-111) (-7/10,-2/3) -> (-4/13,-3/10) Hyperbolic Matrix(87,56,160,103) (-2/3,-9/14) -> (1/2,6/11) Hyperbolic Matrix(89,56,-240,-151) (-7/11,-5/8) -> (-3/8,-7/19) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(17,10,-80,-47) (-3/5,-4/7) -> (-2/9,-1/5) Hyperbolic Matrix(99,56,-320,-181) (-4/7,-9/16) -> (-5/16,-4/13) Hyperbolic Matrix(389,218,480,269) (-9/16,-14/25) -> (4/5,13/16) Hyperbolic Matrix(181,100,400,221) (-5/9,-11/20) -> (9/20,5/11) Hyperbolic Matrix(179,98,400,219) (-11/20,-6/11) -> (4/9,9/20) Hyperbolic Matrix(67,36,80,43) (-6/11,-1/2) -> (5/6,6/7) Hyperbolic Matrix(57,26,160,73) (-1/2,-5/11) -> (1/3,5/14) Hyperbolic Matrix(221,100,400,181) (-5/11,-9/20) -> (11/20,5/9) Hyperbolic Matrix(219,98,400,179) (-9/20,-4/9) -> (6/11,11/20) Hyperbolic Matrix(59,26,-320,-141) (-4/9,-7/16) -> (-3/16,-2/11) Hyperbolic Matrix(51,22,-160,-69) (-7/16,-3/7) -> (-1/3,-5/16) Hyperbolic Matrix(43,18,160,67) (-3/7,-5/12) -> (1/4,3/11) Hyperbolic Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(283,104,400,147) (-7/19,-4/11) -> (12/17,5/7) Hyperbolic Matrix(11,4,-80,-29) (-4/11,-1/3) -> (-1/7,0/1) Hyperbolic Matrix(253,74,400,117) (-5/17,-2/7) -> (12/19,7/11) Hyperbolic Matrix(43,12,240,67) (-2/7,-5/18) -> (1/6,2/11) Hyperbolic Matrix(1161,320,1600,441) (-8/29,-11/40) -> (29/40,8/11) Hyperbolic Matrix(1159,318,1600,439) (-11/40,-3/11) -> (21/29,29/40) Hyperbolic Matrix(61,16,80,21) (-3/11,-1/4) -> (3/4,7/9) Hyperbolic Matrix(59,14,80,19) (-1/4,-2/9) -> (8/11,3/4) Hyperbolic Matrix(211,40,480,91) (-1/5,-3/16) -> (7/16,11/25) Hyperbolic Matrix(67,12,240,43) (-2/11,-1/6) -> (5/18,2/7) Hyperbolic Matrix(37,6,80,13) (-1/6,-1/7) -> (5/11,1/2) Hyperbolic Matrix(29,-4,80,-11) (0/1,1/6) -> (5/14,4/11) Hyperbolic Matrix(141,-26,320,-59) (2/11,1/5) -> (11/25,4/9) Hyperbolic Matrix(47,-10,80,-17) (1/5,1/4) -> (7/12,3/5) Hyperbolic Matrix(521,-144,720,-199) (3/11,5/18) -> (13/18,21/29) Hyperbolic Matrix(169,-50,240,-71) (2/7,3/10) -> (7/10,12/17) Hyperbolic Matrix(111,-34,160,-49) (3/10,1/3) -> (9/13,7/10) Hyperbolic Matrix(151,-56,240,-89) (4/11,3/8) -> (5/8,12/19) Hyperbolic Matrix(63,-26,80,-33) (2/5,3/7) -> (7/9,4/5) Hyperbolic Matrix(221,-96,320,-139) (3/7,7/16) -> (11/16,9/13) Hyperbolic Matrix(261,-146,320,-179) (5/9,9/16) -> (13/16,9/11) Hyperbolic Matrix(109,-62,160,-91) (9/16,4/7) -> (2/3,11/16) Hyperbolic Matrix(69,-44,80,-51) (7/11,2/3) -> (6/7,1/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-80,1) Matrix(51,44,-80,-69) -> Matrix(69,-2,1760,-51) Matrix(173,142,240,197) -> Matrix(197,-6,-5680,173) Matrix(179,146,-320,-261) -> Matrix(261,-8,5840,-179) Matrix(33,26,-80,-63) -> Matrix(63,-2,1040,-33) Matrix(93,68,160,117) -> Matrix(117,-4,-2720,93) Matrix(199,144,-720,-521) -> Matrix(521,-18,5760,-199) Matrix(197,142,240,173) -> Matrix(173,-6,-5680,197) Matrix(71,50,-240,-169) -> Matrix(169,-6,2000,-71) Matrix(49,34,-160,-111) -> Matrix(111,-4,1360,-49) Matrix(87,56,160,103) -> Matrix(103,-4,-2240,87) Matrix(89,56,-240,-151) -> Matrix(151,-6,2240,-89) Matrix(49,30,80,49) -> Matrix(49,-2,-1200,49) Matrix(17,10,-80,-47) -> Matrix(47,-2,400,-17) Matrix(99,56,-320,-181) -> Matrix(181,-8,2240,-99) Matrix(389,218,480,269) -> Matrix(269,-12,-8720,389) Matrix(181,100,400,221) -> Matrix(221,-10,-4000,181) Matrix(179,98,400,219) -> Matrix(219,-10,-3920,179) Matrix(67,36,80,43) -> Matrix(43,-2,-1440,67) Matrix(57,26,160,73) -> Matrix(73,-4,-1040,57) Matrix(221,100,400,181) -> Matrix(181,-10,-4000,221) Matrix(219,98,400,179) -> Matrix(179,-10,-3920,219) Matrix(59,26,-320,-141) -> Matrix(141,-8,1040,-59) Matrix(51,22,-160,-69) -> Matrix(69,-4,880,-51) Matrix(43,18,160,67) -> Matrix(67,-4,-720,43) Matrix(31,12,80,31) -> Matrix(31,-2,-480,31) Matrix(283,104,400,147) -> Matrix(147,-10,-4160,283) Matrix(11,4,-80,-29) -> Matrix(29,-2,160,-11) Matrix(253,74,400,117) -> Matrix(117,-10,-2960,253) Matrix(43,12,240,67) -> Matrix(67,-6,-480,43) Matrix(1161,320,1600,441) -> Matrix(441,-40,-12800,1161) Matrix(1159,318,1600,439) -> Matrix(439,-40,-12720,1159) Matrix(61,16,80,21) -> Matrix(21,-2,-640,61) Matrix(59,14,80,19) -> Matrix(19,-2,-560,59) Matrix(211,40,480,91) -> Matrix(91,-12,-1600,211) Matrix(67,12,240,43) -> Matrix(43,-6,-480,67) Matrix(37,6,80,13) -> Matrix(13,-2,-240,37) Matrix(29,-4,80,-11) -> Matrix(11,2,-160,-29) Matrix(141,-26,320,-59) -> Matrix(59,8,-1040,-141) Matrix(47,-10,80,-17) -> Matrix(17,2,-400,-47) Matrix(521,-144,720,-199) -> Matrix(199,18,-5760,-521) Matrix(169,-50,240,-71) -> Matrix(71,6,-2000,-169) Matrix(111,-34,160,-49) -> Matrix(49,4,-1360,-111) Matrix(151,-56,240,-89) -> Matrix(89,6,-2240,-151) Matrix(63,-26,80,-33) -> Matrix(33,2,-1040,-63) Matrix(221,-96,320,-139) -> Matrix(139,8,-3840,-221) Matrix(261,-146,320,-179) -> Matrix(179,8,-5840,-261) Matrix(109,-62,160,-91) -> Matrix(91,4,-2480,-109) Matrix(69,-44,80,-51) -> Matrix(51,2,-1760,-69) 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: ((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); (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)) ----------------------------------------------------------------------- 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 1/5 1/4 3/8 1/2 11/20 7/10 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 1/0 1/5 -1/8 1/4 -1/10 1/3 -3/40 4/11 -11/160 3/8 -1/15 2/5 -1/16 3/7 -7/120 4/9 -9/160 1/2 -1/20 6/11 -11/240 11/20 -1/22 5/9 -9/200 4/7 -7/160 3/5 -1/24 5/8 -1/25 12/19 -19/480 7/11 -11/280 2/3 -3/80 7/10 -1/28 12/17 -17/480 5/7 -7/200 3/4 -1/30 4/5 -1/32 1/1 -1/40 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(17,-3,40,-7) (0/1,1/5) -> (2/5,3/7) Hyperbolic Matrix(31,-7,40,-9) (1/5,1/4) -> (3/4,4/5) Hyperbolic Matrix(29,-8,40,-11) (1/4,1/3) -> (5/7,3/4) Hyperbolic Matrix(53,-19,120,-43) (1/3,4/11) -> (3/7,4/9) Hyperbolic Matrix(151,-56,240,-89) (4/11,3/8) -> (5/8,12/19) Hyperbolic Matrix(49,-19,80,-31) (3/8,2/5) -> (3/5,5/8) Hyperbolic Matrix(21,-10,40,-19) (4/9,1/2) -> (1/2,6/11) Parabolic Matrix(221,-121,400,-219) (6/11,11/20) -> (11/20,5/9) Parabolic Matrix(77,-43,120,-67) (5/9,4/7) -> (7/11,2/3) Hyperbolic Matrix(33,-19,40,-23) (4/7,3/5) -> (4/5,1/1) Hyperbolic Matrix(283,-179,400,-253) (12/19,7/11) -> (12/17,5/7) Hyperbolic Matrix(141,-98,200,-139) (2/3,7/10) -> (7/10,12/17) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-40,1) Matrix(17,-3,40,-7) -> Matrix(7,1,-120,-17) Matrix(31,-7,40,-9) -> Matrix(9,1,-280,-31) Matrix(29,-8,40,-11) -> Matrix(11,1,-320,-29) Matrix(53,-19,120,-43) -> Matrix(43,3,-760,-53) Matrix(151,-56,240,-89) -> Matrix(89,6,-2240,-151) Matrix(49,-19,80,-31) -> Matrix(31,2,-760,-49) Matrix(21,-10,40,-19) -> Matrix(19,1,-400,-21) Matrix(221,-121,400,-219) -> Matrix(219,10,-4840,-221) Matrix(77,-43,120,-67) -> Matrix(67,3,-1720,-77) Matrix(33,-19,40,-23) -> Matrix(23,1,-760,-33) Matrix(283,-179,400,-253) -> Matrix(253,10,-7160,-283) Matrix(141,-98,200,-139) -> Matrix(139,5,-3920,-141) 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 1/0 1 40 1/5 -1/8 5 8 1/4 -1/10 4 10 3/10 -1/12 10 4 1/3 -3/40 1 40 4/11 -11/160 1 40 3/8 -1/15 8 5 2/5 -1/16 5 8 3/7 -7/120 1 40 4/9 -9/160 1 40 9/20 -1/18 20 2 1/2 -1/20 2 20 1/0 0/1 40 1 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(17,-3,40,-7) (0/1,1/5) -> (2/5,3/7) Hyperbolic Matrix(9,-2,40,-9) (1/5,1/4) -> (1/5,1/4) Reflection Matrix(11,-3,40,-11) (1/4,3/10) -> (1/4,3/10) Reflection Matrix(49,-15,160,-49) (3/10,5/16) -> (3/10,5/16) Reflection Matrix(73,-23,200,-63) (4/13,1/3) -> (4/11,7/19) Hyperbolic Matrix(53,-19,120,-43) (1/3,4/11) -> (3/7,4/9) Hyperbolic Matrix(89,-33,240,-89) (11/30,3/8) -> (11/30,3/8) Reflection Matrix(31,-12,80,-31) (3/8,2/5) -> (3/8,2/5) Reflection Matrix(161,-72,360,-161) (4/9,9/20) -> (4/9,9/20) Reflection Matrix(19,-9,40,-19) (9/20,1/2) -> (9/20,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) 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(17,-3,40,-7) -> Matrix(7,1,-120,-17) Matrix(9,-2,40,-9) -> Matrix(9,1,-80,-9) (1/5,1/4) -> (-1/8,-1/10) Matrix(11,-3,40,-11) -> Matrix(11,1,-120,-11) (1/4,3/10) -> (-1/10,-1/12) Matrix(49,-15,160,-49) -> Matrix(49,4,-600,-49) (3/10,5/16) -> (-1/12,-2/25) Matrix(73,-23,200,-63) -> Matrix(63,5,-920,-73) Matrix(53,-19,120,-43) -> Matrix(43,3,-760,-53) Matrix(89,-33,240,-89) -> Matrix(89,6,-1320,-89) (11/30,3/8) -> (-3/44,-1/15) Matrix(31,-12,80,-31) -> Matrix(31,2,-480,-31) (3/8,2/5) -> (-1/15,-1/16) Matrix(161,-72,360,-161) -> Matrix(161,9,-2880,-161) (4/9,9/20) -> (-9/160,-1/18) Matrix(19,-9,40,-19) -> Matrix(19,1,-360,-19) (9/20,1/2) -> (-1/18,-1/20) Matrix(-1,1,0,1) -> Matrix(-1,0,40,1) (1/2,1/0) -> (-1/20,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.