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): 432 Minimal number of generators: 73 Number of equivalence classes of cusps: 32 Genus: 21 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -3/1 -12/5 -3/2 0/1 1/1 9/7 3/2 27/17 9/5 2/1 9/4 12/5 5/2 18/7 8/3 36/13 3/1 54/17 36/11 10/3 27/8 7/2 18/5 4/1 9/2 14/3 5/1 16/3 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 0/1 1/1 -6/1 -1/1 -11/2 -2/3 -3/5 -5/1 -1/1 0/1 -14/3 -3/5 -9/2 -1/2 -4/1 -1/2 -1/3 -7/2 -1/3 -4/13 -17/5 -1/3 -2/7 -27/8 -1/4 -10/3 -1/3 -13/4 -2/7 -3/11 -16/5 -1/3 -1/4 -3/1 -1/4 -20/7 -1/4 -3/13 -17/6 -3/13 -2/9 -14/5 -3/13 -11/4 -2/9 -5/23 -19/7 -2/9 -5/23 -27/10 -3/14 -8/3 -3/14 -1/5 -5/2 -1/5 -4/21 -17/7 -5/27 -2/11 -12/5 -2/11 -19/8 -2/11 -3/17 -7/3 -3/17 -4/23 -16/7 -7/41 -1/6 -9/4 -1/6 -2/1 -1/7 -9/5 -1/8 -16/9 -1/8 -7/57 -23/13 -6/49 -5/41 -7/4 -4/33 -3/25 -19/11 -3/25 -2/17 -12/7 -2/17 -17/10 -2/17 -5/43 -22/13 -5/43 -5/3 -4/35 -1/9 -18/11 -1/9 -13/8 -1/9 -10/91 -8/5 -1/9 -3/28 -27/17 -3/28 -19/12 -5/47 -2/19 -11/7 -5/47 -2/19 -36/23 -2/19 -25/16 -2/19 -11/105 -14/9 -3/29 -17/11 -2/19 -3/29 -3/2 -1/10 -19/13 -2/21 -1/11 -54/37 -1/11 -35/24 -1/11 0/1 -16/11 -1/10 -1/11 -13/9 -3/31 -2/21 -36/25 -2/21 -23/16 -2/21 -5/53 -10/7 -1/11 -27/19 -1/10 -17/12 -2/21 -1/11 -7/5 -4/43 -1/11 -18/13 -1/11 -11/8 -1/11 -6/67 -4/3 -1/11 -1/12 -9/7 -1/12 -14/11 -3/37 -19/15 -3/37 -2/25 -5/4 -1/13 0/1 -16/13 -1/11 -1/12 -11/9 -3/37 -2/25 -6/5 -1/13 -7/6 -1/15 0/1 -1/1 -1/15 0/1 0/1 0/1 1/1 0/1 1/17 7/6 0/1 1/17 6/5 1/15 11/9 2/29 3/43 5/4 0/1 1/15 14/11 3/43 9/7 1/14 4/3 1/14 1/13 7/5 1/13 4/51 17/12 1/13 2/25 27/19 1/12 10/7 1/13 13/9 2/25 3/37 16/11 1/13 1/12 3/2 1/12 20/13 1/12 3/35 17/11 3/35 2/23 14/9 3/35 11/7 2/23 5/57 19/12 2/23 5/57 27/17 3/34 8/5 3/34 1/11 5/3 1/11 4/43 17/10 5/53 2/21 12/7 2/21 19/11 2/21 3/31 7/4 3/31 4/41 16/9 7/71 1/10 9/5 1/10 2/1 1/9 9/4 1/8 16/7 1/8 7/55 23/10 6/47 5/39 7/3 4/31 3/23 19/8 3/23 2/15 12/5 2/15 17/7 2/15 5/37 22/9 5/37 5/2 4/29 1/7 18/7 1/7 13/5 1/7 10/69 8/3 1/7 3/20 27/10 3/20 19/7 5/33 2/13 11/4 5/33 2/13 36/13 2/13 25/9 2/13 11/71 14/5 3/19 17/6 2/13 3/19 3/1 1/6 19/6 2/11 1/5 54/17 1/5 35/11 0/1 1/5 16/5 1/6 1/5 13/4 3/17 2/11 36/11 2/11 23/7 2/11 5/27 10/3 1/5 27/8 1/6 17/5 2/11 1/5 7/2 4/21 1/5 18/5 1/5 11/3 1/5 6/29 4/1 1/5 1/4 9/2 1/4 14/3 3/11 19/4 3/11 2/7 5/1 0/1 1/3 16/3 1/5 1/4 11/2 3/11 2/7 6/1 1/3 7/1 0/1 1/1 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(19,144,12,91) (-7/1,1/0) -> (11/7,19/12) Hyperbolic Matrix(17,108,14,89) (-7/1,-6/1) -> (6/5,11/9) Hyperbolic Matrix(19,108,16,91) (-6/1,-11/2) -> (7/6,6/5) Hyperbolic Matrix(53,288,-30,-163) (-11/2,-5/1) -> (-23/13,-7/4) Hyperbolic Matrix(37,180,-22,-107) (-5/1,-14/3) -> (-22/13,-5/3) Hyperbolic Matrix(55,252,12,55) (-14/3,-9/2) -> (9/2,14/3) Hyperbolic Matrix(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(19,72,-14,-53) (-4/1,-7/2) -> (-11/8,-4/3) Hyperbolic Matrix(73,252,42,145) (-7/2,-17/5) -> (19/11,7/4) Hyperbolic Matrix(287,972,106,359) (-17/5,-27/8) -> (27/10,19/7) Hyperbolic Matrix(161,540,48,161) (-27/8,-10/3) -> (10/3,27/8) Hyperbolic Matrix(109,360,-76,-251) (-10/3,-13/4) -> (-23/16,-10/7) Hyperbolic Matrix(179,576,78,251) (-13/4,-16/5) -> (16/7,23/10) Hyperbolic Matrix(35,108,-12,-37) (-16/5,-3/1) -> (-3/1,-20/7) Parabolic Matrix(341,972,-234,-667) (-20/7,-17/6) -> (-35/24,-16/11) Hyperbolic Matrix(179,504,38,107) (-17/6,-14/5) -> (14/3,19/4) Hyperbolic Matrix(181,504,-116,-323) (-14/5,-11/4) -> (-25/16,-14/9) Hyperbolic Matrix(53,144,46,125) (-11/4,-19/7) -> (1/1,7/6) Hyperbolic Matrix(359,972,106,287) (-19/7,-27/10) -> (27/8,17/5) Hyperbolic Matrix(161,432,60,161) (-27/10,-8/3) -> (8/3,27/10) Hyperbolic Matrix(55,144,-34,-89) (-8/3,-5/2) -> (-13/8,-8/5) Hyperbolic Matrix(73,180,-58,-143) (-5/2,-17/7) -> (-19/15,-5/4) Hyperbolic Matrix(179,432,104,251) (-17/7,-12/5) -> (12/7,19/11) Hyperbolic Matrix(181,432,106,253) (-12/5,-19/8) -> (17/10,12/7) Hyperbolic Matrix(107,252,76,179) (-19/8,-7/3) -> (7/5,17/12) Hyperbolic Matrix(125,288,-102,-235) (-7/3,-16/7) -> (-16/13,-11/9) Hyperbolic Matrix(127,288,56,127) (-16/7,-9/4) -> (9/4,16/7) Hyperbolic Matrix(17,36,8,17) (-9/4,-2/1) -> (2/1,9/4) Hyperbolic Matrix(19,36,10,19) (-2/1,-9/5) -> (9/5,2/1) Hyperbolic Matrix(161,288,90,161) (-9/5,-16/9) -> (16/9,9/5) Hyperbolic Matrix(325,576,224,397) (-16/9,-23/13) -> (13/9,16/11) Hyperbolic Matrix(145,252,42,73) (-7/4,-19/11) -> (17/5,7/2) Hyperbolic Matrix(251,432,104,179) (-19/11,-12/7) -> (12/5,17/7) Hyperbolic Matrix(253,432,106,181) (-12/7,-17/10) -> (19/8,12/5) Hyperbolic Matrix(361,612,128,217) (-17/10,-22/13) -> (14/5,17/6) Hyperbolic Matrix(197,324,76,125) (-5/3,-18/11) -> (18/7,13/5) Hyperbolic Matrix(199,324,78,127) (-18/11,-13/8) -> (5/2,18/7) Hyperbolic Matrix(271,432,170,271) (-8/5,-27/17) -> (27/17,8/5) Hyperbolic Matrix(613,972,432,685) (-27/17,-19/12) -> (17/12,27/19) Hyperbolic Matrix(91,144,12,19) (-19/12,-11/7) -> (7/1,1/0) Hyperbolic Matrix(827,1296,298,467) (-11/7,-36/23) -> (36/13,25/9) Hyperbolic Matrix(829,1296,300,469) (-36/23,-25/16) -> (11/4,36/13) Hyperbolic Matrix(395,612,162,251) (-14/9,-17/11) -> (17/7,22/9) Hyperbolic Matrix(71,108,-48,-73) (-17/11,-3/2) -> (-3/2,-19/13) Parabolic Matrix(1997,2916,628,917) (-19/13,-54/37) -> (54/17,35/11) Hyperbolic Matrix(1999,2916,630,919) (-54/37,-35/24) -> (19/6,54/17) Hyperbolic Matrix(199,288,38,55) (-16/11,-13/9) -> (5/1,16/3) Hyperbolic Matrix(899,1296,274,395) (-13/9,-36/25) -> (36/11,23/7) Hyperbolic Matrix(901,1296,276,397) (-36/25,-23/16) -> (13/4,36/11) Hyperbolic Matrix(379,540,266,379) (-10/7,-27/19) -> (27/19,10/7) Hyperbolic Matrix(685,972,432,613) (-27/19,-17/12) -> (19/12,27/17) Hyperbolic Matrix(179,252,76,107) (-17/12,-7/5) -> (7/3,19/8) Hyperbolic Matrix(233,324,64,89) (-7/5,-18/13) -> (18/5,11/3) Hyperbolic Matrix(235,324,66,91) (-18/13,-11/8) -> (7/2,18/5) Hyperbolic Matrix(55,72,42,55) (-4/3,-9/7) -> (9/7,4/3) Hyperbolic Matrix(197,252,154,197) (-9/7,-14/11) -> (14/11,9/7) Hyperbolic Matrix(397,504,256,325) (-14/11,-19/15) -> (17/11,14/9) Hyperbolic Matrix(233,288,72,89) (-5/4,-16/13) -> (16/5,13/4) Hyperbolic Matrix(89,108,14,17) (-11/9,-6/5) -> (6/1,7/1) Hyperbolic Matrix(91,108,16,19) (-6/5,-7/6) -> (11/2,6/1) Hyperbolic Matrix(125,144,46,53) (-7/6,-1/1) -> (19/7,11/4) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(235,-288,102,-125) (11/9,5/4) -> (23/10,7/3) Hyperbolic Matrix(143,-180,58,-73) (5/4,14/11) -> (22/9,5/2) Hyperbolic Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(251,-360,76,-109) (10/7,13/9) -> (23/7,10/3) Hyperbolic Matrix(73,-108,48,-71) (16/11,3/2) -> (3/2,20/13) Parabolic Matrix(631,-972,198,-305) (20/13,17/11) -> (35/11,16/5) Hyperbolic Matrix(323,-504,116,-181) (14/9,11/7) -> (25/9,14/5) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(107,-180,22,-37) (5/3,17/10) -> (19/4,5/1) Hyperbolic Matrix(163,-288,30,-53) (7/4,16/9) -> (16/3,11/2) Hyperbolic Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(19,144,12,91) -> Matrix(3,2,34,23) Matrix(17,108,14,89) -> Matrix(1,2,14,29) Matrix(19,108,16,91) -> Matrix(3,2,46,31) Matrix(53,288,-30,-163) -> Matrix(11,6,-90,-49) Matrix(37,180,-22,-107) -> Matrix(5,4,-44,-35) Matrix(55,252,12,55) -> Matrix(11,6,42,23) Matrix(17,72,4,17) -> Matrix(5,2,22,9) Matrix(19,72,-14,-53) -> Matrix(5,2,-58,-23) Matrix(73,252,42,145) -> Matrix(27,8,280,83) Matrix(287,972,106,359) -> Matrix(29,8,192,53) Matrix(161,540,48,161) -> Matrix(7,2,38,11) Matrix(109,360,-76,-251) -> Matrix(13,4,-140,-43) Matrix(179,576,78,251) -> Matrix(31,8,244,63) Matrix(35,108,-12,-37) -> Matrix(15,4,-64,-17) Matrix(341,972,-234,-667) -> Matrix(9,2,-86,-19) Matrix(179,504,38,107) -> Matrix(1,0,8,1) Matrix(181,504,-116,-323) -> Matrix(53,12,-508,-115) Matrix(53,144,46,125) -> Matrix(9,2,130,29) Matrix(359,972,106,287) -> Matrix(37,8,208,45) Matrix(161,432,60,161) -> Matrix(29,6,198,41) Matrix(55,144,-34,-89) -> Matrix(29,6,-266,-55) Matrix(73,180,-58,-143) -> Matrix(21,4,-268,-51) Matrix(179,432,104,251) -> Matrix(87,16,908,167) Matrix(181,432,106,253) -> Matrix(89,16,940,169) Matrix(107,252,76,179) -> Matrix(45,8,568,101) Matrix(125,288,-102,-235) -> Matrix(35,6,-426,-73) Matrix(127,288,56,127) -> Matrix(83,14,658,111) Matrix(17,36,8,17) -> Matrix(13,2,110,17) Matrix(19,36,10,19) -> Matrix(15,2,142,19) Matrix(161,288,90,161) -> Matrix(113,14,1138,141) Matrix(325,576,224,397) -> Matrix(65,8,788,97) Matrix(145,252,42,73) -> Matrix(67,8,360,43) Matrix(251,432,104,179) -> Matrix(135,16,1004,119) Matrix(253,432,106,181) -> Matrix(137,16,1036,121) Matrix(361,612,128,217) -> Matrix(35,4,236,27) Matrix(197,324,76,125) -> Matrix(125,14,866,97) Matrix(199,324,78,127) -> Matrix(127,14,898,99) Matrix(271,432,170,271) -> Matrix(55,6,614,67) Matrix(613,972,432,685) -> Matrix(75,8,928,99) Matrix(91,144,12,19) -> Matrix(19,2,66,7) Matrix(827,1296,298,467) -> Matrix(303,32,1960,207) Matrix(829,1296,300,469) -> Matrix(305,32,1992,209) Matrix(395,612,162,251) -> Matrix(37,4,268,29) Matrix(71,108,-48,-73) -> Matrix(39,4,-400,-41) Matrix(1997,2916,628,917) -> Matrix(21,2,94,9) Matrix(1999,2916,630,919) -> Matrix(23,2,126,11) Matrix(199,288,38,55) -> Matrix(21,2,94,9) Matrix(899,1296,274,395) -> Matrix(167,16,908,87) Matrix(901,1296,276,397) -> Matrix(169,16,940,89) Matrix(379,540,266,379) -> Matrix(21,2,262,25) Matrix(685,972,432,613) -> Matrix(83,8,944,91) Matrix(179,252,76,107) -> Matrix(85,8,648,61) Matrix(233,324,64,89) -> Matrix(109,10,534,49) Matrix(235,324,66,91) -> Matrix(111,10,566,51) Matrix(55,72,42,55) -> Matrix(23,2,310,27) Matrix(197,252,154,197) -> Matrix(73,6,1034,85) Matrix(397,504,256,325) -> Matrix(1,0,24,1) Matrix(233,288,72,89) -> Matrix(23,2,126,11) Matrix(89,108,14,17) -> Matrix(25,2,62,5) Matrix(91,108,16,19) -> Matrix(27,2,94,7) Matrix(125,144,46,53) -> Matrix(25,2,162,13) Matrix(1,0,2,1) -> Matrix(1,0,32,1) Matrix(235,-288,102,-125) -> Matrix(85,-6,666,-47) Matrix(143,-180,58,-73) -> Matrix(59,-4,428,-29) Matrix(53,-72,14,-19) -> Matrix(27,-2,122,-9) Matrix(251,-360,76,-109) -> Matrix(51,-4,268,-21) Matrix(73,-108,48,-71) -> Matrix(49,-4,576,-47) Matrix(631,-972,198,-305) -> Matrix(23,-2,150,-13) Matrix(323,-504,116,-181) -> Matrix(139,-12,892,-77) Matrix(89,-144,34,-55) -> Matrix(67,-6,458,-41) Matrix(107,-180,22,-37) -> Matrix(43,-4,140,-13) Matrix(163,-288,30,-53) -> Matrix(61,-6,234,-23) Matrix(37,-108,12,-35) -> Matrix(25,-4,144,-23) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 18 Degree of the the map X: 36 Degree of the the map Y: 72 ----------------------------------------------------------------------- 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): 216 Minimal number of generators: 37 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: 24 Genus: 7 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 6/5 3/2 12/7 2/1 9/4 12/5 18/7 8/3 36/13 3/1 54/17 36/11 10/3 27/8 18/5 4/1 9/2 14/3 5/1 16/3 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 0/1 1/17 7/6 0/1 1/17 6/5 1/15 11/9 2/29 3/43 5/4 0/1 1/15 14/11 3/43 9/7 1/14 4/3 1/14 1/13 7/5 1/13 4/51 17/12 1/13 2/25 27/19 1/12 10/7 1/13 13/9 2/25 3/37 16/11 1/13 1/12 3/2 1/12 20/13 1/12 3/35 17/11 3/35 2/23 14/9 3/35 11/7 2/23 5/57 19/12 2/23 5/57 27/17 3/34 8/5 3/34 1/11 5/3 1/11 4/43 17/10 5/53 2/21 12/7 2/21 19/11 2/21 3/31 7/4 3/31 4/41 16/9 7/71 1/10 9/5 1/10 2/1 1/9 9/4 1/8 16/7 1/8 7/55 23/10 6/47 5/39 7/3 4/31 3/23 19/8 3/23 2/15 12/5 2/15 17/7 2/15 5/37 22/9 5/37 5/2 4/29 1/7 18/7 1/7 13/5 1/7 10/69 8/3 1/7 3/20 27/10 3/20 19/7 5/33 2/13 11/4 5/33 2/13 36/13 2/13 25/9 2/13 11/71 14/5 3/19 17/6 2/13 3/19 3/1 1/6 19/6 2/11 1/5 54/17 1/5 35/11 0/1 1/5 16/5 1/6 1/5 13/4 3/17 2/11 36/11 2/11 23/7 2/11 5/27 10/3 1/5 27/8 1/6 17/5 2/11 1/5 7/2 4/21 1/5 18/5 1/5 11/3 1/5 6/29 4/1 1/5 1/4 9/2 1/4 14/3 3/11 19/4 3/11 2/7 5/1 0/1 1/3 16/3 1/5 1/4 11/2 3/11 2/7 6/1 1/3 7/1 0/1 1/1 1/0 0/1 1/1 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(125,-144,79,-91) (1/1,7/6) -> (11/7,19/12) Hyperbolic Matrix(91,-108,75,-89) (7/6,6/5) -> (6/5,11/9) Parabolic Matrix(235,-288,102,-125) (11/9,5/4) -> (23/10,7/3) Hyperbolic Matrix(143,-180,58,-73) (5/4,14/11) -> (22/9,5/2) Hyperbolic Matrix(197,-252,43,-55) (14/11,9/7) -> (9/2,14/3) Hyperbolic Matrix(55,-72,13,-17) (9/7,4/3) -> (4/1,9/2) Hyperbolic Matrix(53,-72,14,-19) (4/3,7/5) -> (11/3,4/1) Hyperbolic Matrix(179,-252,103,-145) (7/5,17/12) -> (19/11,7/4) Hyperbolic Matrix(685,-972,253,-359) (17/12,27/19) -> (27/10,19/7) Hyperbolic Matrix(379,-540,113,-161) (27/19,10/7) -> (10/3,27/8) Hyperbolic Matrix(251,-360,76,-109) (10/7,13/9) -> (23/7,10/3) Hyperbolic Matrix(397,-576,173,-251) (13/9,16/11) -> (16/7,23/10) Hyperbolic Matrix(73,-108,48,-71) (16/11,3/2) -> (3/2,20/13) Parabolic Matrix(631,-972,198,-305) (20/13,17/11) -> (35/11,16/5) Hyperbolic Matrix(325,-504,69,-107) (17/11,14/9) -> (14/3,19/4) Hyperbolic Matrix(323,-504,116,-181) (14/9,11/7) -> (25/9,14/5) Hyperbolic Matrix(613,-972,181,-287) (19/12,27/17) -> (27/8,17/5) Hyperbolic Matrix(271,-432,101,-161) (27/17,8/5) -> (8/3,27/10) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(107,-180,22,-37) (5/3,17/10) -> (19/4,5/1) Hyperbolic Matrix(253,-432,147,-251) (17/10,12/7) -> (12/7,19/11) Parabolic Matrix(163,-288,30,-53) (7/4,16/9) -> (16/3,11/2) Hyperbolic Matrix(161,-288,71,-127) (16/9,9/5) -> (9/4,16/7) Hyperbolic Matrix(19,-36,9,-17) (9/5,2/1) -> (2/1,9/4) Parabolic Matrix(107,-252,31,-73) (7/3,19/8) -> (17/5,7/2) Hyperbolic Matrix(181,-432,75,-179) (19/8,12/5) -> (12/5,17/7) Parabolic Matrix(251,-612,89,-217) (17/7,22/9) -> (14/5,17/6) Hyperbolic Matrix(127,-324,49,-125) (5/2,18/7) -> (18/7,13/5) Parabolic Matrix(53,-144,7,-19) (19/7,11/4) -> (7/1,1/0) Hyperbolic Matrix(469,-1296,169,-467) (11/4,36/13) -> (36/13,25/9) Parabolic Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic Matrix(919,-2916,289,-917) (19/6,54/17) -> (54/17,35/11) Parabolic Matrix(89,-288,17,-55) (16/5,13/4) -> (5/1,16/3) Hyperbolic Matrix(397,-1296,121,-395) (13/4,36/11) -> (36/11,23/7) Parabolic Matrix(91,-324,25,-89) (7/2,18/5) -> (18/5,11/3) Parabolic Matrix(19,-108,3,-17) (11/2,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,16,1) Matrix(125,-144,79,-91) -> Matrix(29,-2,334,-23) Matrix(91,-108,75,-89) -> Matrix(31,-2,450,-29) Matrix(235,-288,102,-125) -> Matrix(85,-6,666,-47) Matrix(143,-180,58,-73) -> Matrix(59,-4,428,-29) Matrix(197,-252,43,-55) -> Matrix(85,-6,326,-23) Matrix(55,-72,13,-17) -> Matrix(27,-2,122,-9) Matrix(53,-72,14,-19) -> Matrix(27,-2,122,-9) Matrix(179,-252,103,-145) -> Matrix(101,-8,1048,-83) Matrix(685,-972,253,-359) -> Matrix(99,-8,656,-53) Matrix(379,-540,113,-161) -> Matrix(25,-2,138,-11) Matrix(251,-360,76,-109) -> Matrix(51,-4,268,-21) Matrix(397,-576,173,-251) -> Matrix(97,-8,764,-63) Matrix(73,-108,48,-71) -> Matrix(49,-4,576,-47) Matrix(631,-972,198,-305) -> Matrix(23,-2,150,-13) Matrix(325,-504,69,-107) -> Matrix(1,0,-8,1) Matrix(323,-504,116,-181) -> Matrix(139,-12,892,-77) Matrix(613,-972,181,-287) -> Matrix(91,-8,512,-45) Matrix(271,-432,101,-161) -> Matrix(67,-6,458,-41) Matrix(89,-144,34,-55) -> Matrix(67,-6,458,-41) Matrix(107,-180,22,-37) -> Matrix(43,-4,140,-13) Matrix(253,-432,147,-251) -> Matrix(169,-16,1764,-167) Matrix(163,-288,30,-53) -> Matrix(61,-6,234,-23) Matrix(161,-288,71,-127) -> Matrix(141,-14,1118,-111) Matrix(19,-36,9,-17) -> Matrix(19,-2,162,-17) Matrix(107,-252,31,-73) -> Matrix(61,-8,328,-43) Matrix(181,-432,75,-179) -> Matrix(121,-16,900,-119) Matrix(251,-612,89,-217) -> Matrix(29,-4,196,-27) Matrix(127,-324,49,-125) -> Matrix(99,-14,686,-97) Matrix(53,-144,7,-19) -> Matrix(13,-2,46,-7) Matrix(469,-1296,169,-467) -> Matrix(209,-32,1352,-207) Matrix(37,-108,12,-35) -> Matrix(25,-4,144,-23) Matrix(919,-2916,289,-917) -> Matrix(11,-2,50,-9) Matrix(89,-288,17,-55) -> Matrix(11,-2,50,-9) Matrix(397,-1296,121,-395) -> Matrix(89,-16,484,-87) Matrix(91,-324,25,-89) -> Matrix(51,-10,250,-49) Matrix(19,-108,3,-17) -> Matrix(7,-2,18,-5) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 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: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 9 ----------------------------------------------------------------------- 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 16 1 2/1 1/9 2 9 9/4 1/8 8 2 16/7 (1/8,7/55) 0 9 7/3 (4/31,3/23) 0 18 19/8 (3/23,2/15) 0 18 12/5 2/15 4 3 17/7 (2/15,5/37) 0 18 5/2 (4/29,1/7) 0 18 18/7 1/7 14 1 8/3 (1/7,3/20) 0 9 27/10 3/20 2 2 19/7 (5/33,2/13) 0 18 11/4 (5/33,2/13) 0 18 36/13 2/13 8 1 14/5 3/19 2 9 17/6 (2/13,3/19) 0 18 3/1 1/6 2 6 19/6 (2/11,1/5) 0 18 54/17 1/5 2 1 16/5 (1/6,1/5) 0 9 13/4 (3/17,2/11) 0 18 36/11 2/11 4 1 10/3 1/5 2 9 27/8 1/6 2 2 17/5 (2/11,1/5) 0 18 7/2 (4/21,1/5) 0 18 18/5 1/5 10 1 4/1 (1/5,1/4) 0 9 9/2 1/4 4 2 14/3 3/11 2 9 19/4 (3/11,2/7) 0 18 5/1 (0/1,1/3) 0 18 16/3 (1/5,1/4) 0 9 11/2 (3/11,2/7) 0 18 6/1 1/3 2 3 7/1 (0/1,1/1) 0 18 1/0 (0/1,1/1) 0 18 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(17,-36,8,-17) (2/1,9/4) -> (2/1,9/4) Reflection Matrix(127,-288,56,-127) (9/4,16/7) -> (9/4,16/7) Reflection Matrix(125,-288,23,-53) (16/7,7/3) -> (16/3,11/2) Glide Reflection Matrix(107,-252,31,-73) (7/3,19/8) -> (17/5,7/2) Hyperbolic Matrix(181,-432,75,-179) (19/8,12/5) -> (12/5,17/7) Parabolic Matrix(73,-180,15,-37) (17/7,5/2) -> (19/4,5/1) Glide Reflection Matrix(71,-180,28,-71) (5/2,18/7) -> (5/2,18/7) Reflection Matrix(55,-144,21,-55) (18/7,8/3) -> (18/7,8/3) Reflection Matrix(161,-432,60,-161) (8/3,27/10) -> (8/3,27/10) Reflection Matrix(359,-972,106,-287) (27/10,19/7) -> (27/8,17/5) Glide Reflection Matrix(53,-144,7,-19) (19/7,11/4) -> (7/1,1/0) Hyperbolic Matrix(287,-792,104,-287) (11/4,36/13) -> (11/4,36/13) Reflection Matrix(181,-504,65,-181) (36/13,14/5) -> (36/13,14/5) Reflection Matrix(179,-504,38,-107) (14/5,17/6) -> (14/3,19/4) Glide Reflection Matrix(37,-108,12,-35) (17/6,3/1) -> (3/1,19/6) Parabolic Matrix(647,-2052,204,-647) (19/6,54/17) -> (19/6,54/17) Reflection Matrix(271,-864,85,-271) (54/17,16/5) -> (54/17,16/5) Reflection Matrix(89,-288,17,-55) (16/5,13/4) -> (5/1,16/3) Hyperbolic Matrix(287,-936,88,-287) (13/4,36/11) -> (13/4,36/11) Reflection Matrix(109,-360,33,-109) (36/11,10/3) -> (36/11,10/3) Reflection Matrix(161,-540,48,-161) (10/3,27/8) -> (10/3,27/8) Reflection Matrix(71,-252,20,-71) (7/2,18/5) -> (7/2,18/5) Reflection Matrix(19,-72,5,-19) (18/5,4/1) -> (18/5,4/1) Reflection Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(55,-252,12,-55) (9/2,14/3) -> (9/2,14/3) Reflection Matrix(19,-108,3,-17) (11/2,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,2,-1) (0/1,1/0) -> (0/1,1/1) Matrix(1,0,1,-1) -> Matrix(1,0,18,-1) (0/1,2/1) -> (0/1,1/9) Matrix(17,-36,8,-17) -> Matrix(17,-2,144,-17) (2/1,9/4) -> (1/9,1/8) Matrix(127,-288,56,-127) -> Matrix(111,-14,880,-111) (9/4,16/7) -> (1/8,7/55) Matrix(125,-288,23,-53) -> Matrix(47,-6,180,-23) Matrix(107,-252,31,-73) -> Matrix(61,-8,328,-43) Matrix(181,-432,75,-179) -> Matrix(121,-16,900,-119) 2/15 Matrix(73,-180,15,-37) -> Matrix(29,-4,94,-13) Matrix(71,-180,28,-71) -> Matrix(57,-8,406,-57) (5/2,18/7) -> (4/29,1/7) Matrix(55,-144,21,-55) -> Matrix(41,-6,280,-41) (18/7,8/3) -> (1/7,3/20) Matrix(161,-432,60,-161) -> Matrix(41,-6,280,-41) (8/3,27/10) -> (1/7,3/20) Matrix(359,-972,106,-287) -> Matrix(53,-8,298,-45) Matrix(53,-144,7,-19) -> Matrix(13,-2,46,-7) Matrix(287,-792,104,-287) -> Matrix(131,-20,858,-131) (11/4,36/13) -> (5/33,2/13) Matrix(181,-504,65,-181) -> Matrix(77,-12,494,-77) (36/13,14/5) -> (2/13,3/19) Matrix(179,-504,38,-107) -> Matrix(1,0,10,-1) *** -> (0/1,1/5) Matrix(37,-108,12,-35) -> Matrix(25,-4,144,-23) 1/6 Matrix(647,-2052,204,-647) -> Matrix(21,-4,110,-21) (19/6,54/17) -> (2/11,1/5) Matrix(271,-864,85,-271) -> Matrix(11,-2,60,-11) (54/17,16/5) -> (1/6,1/5) Matrix(89,-288,17,-55) -> Matrix(11,-2,50,-9) 1/5 Matrix(287,-936,88,-287) -> Matrix(67,-12,374,-67) (13/4,36/11) -> (3/17,2/11) Matrix(109,-360,33,-109) -> Matrix(21,-4,110,-21) (36/11,10/3) -> (2/11,1/5) Matrix(161,-540,48,-161) -> Matrix(11,-2,60,-11) (10/3,27/8) -> (1/6,1/5) Matrix(71,-252,20,-71) -> Matrix(41,-8,210,-41) (7/2,18/5) -> (4/21,1/5) Matrix(19,-72,5,-19) -> Matrix(9,-2,40,-9) (18/5,4/1) -> (1/5,1/4) Matrix(17,-72,4,-17) -> Matrix(9,-2,40,-9) (4/1,9/2) -> (1/5,1/4) Matrix(55,-252,12,-55) -> Matrix(23,-6,88,-23) (9/2,14/3) -> (1/4,3/11) Matrix(19,-108,3,-17) -> Matrix(7,-2,18,-5) 1/3 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.