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 -3/8 -6/1 -1/3 -11/2 -1/4 -5/1 -1/4 -14/3 -4/15 -1/4 -9/2 -1/4 -4/1 -1/4 -1/5 -7/2 -3/16 -17/5 -7/40 -27/8 -1/6 -10/3 -1/6 0/1 -13/4 -1/4 -16/5 -1/4 -1/5 -3/1 -1/6 -20/7 -3/20 -1/7 -17/6 -5/36 -14/5 -1/8 0/1 -11/4 -1/4 -19/7 -5/28 -27/10 -1/6 -8/3 -1/6 -1/7 -5/2 -1/8 -17/7 -3/28 -12/5 0/1 -19/8 -3/16 -7/3 -1/8 -16/7 -1/7 -1/8 -9/4 -1/8 -2/1 -1/8 0/1 -9/5 -1/8 -16/9 -1/8 -1/9 -23/13 -1/8 -7/4 -1/8 -19/11 -3/32 -12/7 0/1 -17/10 -3/20 -22/13 -2/15 -1/8 -5/3 -1/8 -18/11 -1/9 -13/8 -3/28 -8/5 -1/9 -1/10 -27/17 -1/10 -19/12 -5/52 -11/7 -1/12 -36/23 0/1 -25/16 -1/4 -14/9 -1/8 0/1 -17/11 -5/44 -3/2 -1/10 -19/13 -7/76 -54/37 -1/11 -35/24 -9/100 -16/11 -1/11 -1/12 -13/9 -1/12 -36/25 0/1 -23/16 -1/8 -10/7 -1/10 0/1 -27/19 -1/10 -17/12 -7/72 -7/5 -3/32 -18/13 -1/11 -11/8 -5/56 -4/3 -1/11 -1/12 -9/7 -1/12 -14/11 -1/12 -4/49 -19/15 -11/136 -5/4 -1/12 -16/13 -1/12 -1/13 -11/9 -1/12 -6/5 -1/13 -7/6 -3/40 -1/1 -1/16 0/1 0/1 1/1 1/16 7/6 3/40 6/5 1/13 11/9 1/12 5/4 1/12 14/11 4/49 1/12 9/7 1/12 4/3 1/12 1/11 7/5 3/32 17/12 7/72 27/19 1/10 10/7 0/1 1/10 13/9 1/12 16/11 1/12 1/11 3/2 1/10 20/13 3/28 1/9 17/11 5/44 14/9 0/1 1/8 11/7 1/12 19/12 5/52 27/17 1/10 8/5 1/10 1/9 5/3 1/8 17/10 3/20 12/7 0/1 19/11 3/32 7/4 1/8 16/9 1/9 1/8 9/5 1/8 2/1 0/1 1/8 9/4 1/8 16/7 1/8 1/7 23/10 1/8 7/3 1/8 19/8 3/16 12/5 0/1 17/7 3/28 22/9 2/17 1/8 5/2 1/8 18/7 1/7 13/5 3/20 8/3 1/7 1/6 27/10 1/6 19/7 5/28 11/4 1/4 36/13 0/1 25/9 1/12 14/5 0/1 1/8 17/6 5/36 3/1 1/6 19/6 7/36 54/17 1/5 35/11 9/44 16/5 1/5 1/4 13/4 1/4 36/11 0/1 23/7 1/8 10/3 0/1 1/6 27/8 1/6 17/5 7/40 7/2 3/16 18/5 1/5 11/3 5/24 4/1 1/5 1/4 9/2 1/4 14/3 1/4 4/15 19/4 11/40 5/1 1/4 16/3 1/4 1/3 11/2 1/4 6/1 1/3 7/1 3/8 1/0 1/0 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(5,2,52,21) Matrix(17,108,14,89) -> Matrix(11,4,140,51) Matrix(19,108,16,91) -> Matrix(13,4,172,53) Matrix(53,288,-30,-163) -> Matrix(7,2,-60,-17) Matrix(37,180,-22,-107) -> Matrix(7,2,-60,-17) Matrix(55,252,12,55) -> Matrix(31,8,120,31) Matrix(17,72,4,17) -> Matrix(9,2,40,9) Matrix(19,72,-14,-53) -> Matrix(9,2,-104,-23) Matrix(73,252,42,145) -> Matrix(11,2,104,19) Matrix(287,972,106,359) -> Matrix(35,6,204,35) Matrix(161,540,48,161) -> Matrix(1,0,12,1) Matrix(109,360,-76,-251) -> Matrix(1,0,-4,1) Matrix(179,576,78,251) -> Matrix(1,0,12,1) Matrix(35,108,-12,-37) -> Matrix(11,2,-72,-13) Matrix(341,972,-234,-667) -> Matrix(27,4,-304,-45) Matrix(179,504,38,107) -> Matrix(31,4,116,15) Matrix(181,504,-116,-323) -> Matrix(1,0,0,1) Matrix(53,144,46,125) -> Matrix(11,2,148,27) Matrix(359,972,106,287) -> Matrix(35,6,204,35) Matrix(161,432,60,161) -> Matrix(13,2,84,13) Matrix(55,144,-34,-89) -> Matrix(13,2,-124,-19) Matrix(73,180,-58,-143) -> Matrix(15,2,-188,-25) Matrix(179,432,104,251) -> Matrix(1,0,20,1) Matrix(181,432,106,253) -> Matrix(1,0,12,1) Matrix(107,252,76,179) -> Matrix(13,2,136,21) Matrix(125,288,-102,-235) -> Matrix(15,2,-188,-25) Matrix(127,288,56,127) -> Matrix(15,2,112,15) Matrix(17,36,8,17) -> Matrix(1,0,16,1) Matrix(19,36,10,19) -> Matrix(1,0,16,1) Matrix(161,288,90,161) -> Matrix(17,2,144,17) Matrix(325,576,224,397) -> Matrix(1,0,20,1) Matrix(145,252,42,73) -> Matrix(19,2,104,11) Matrix(251,432,104,179) -> Matrix(1,0,20,1) Matrix(253,432,106,181) -> Matrix(1,0,12,1) Matrix(361,612,128,217) -> Matrix(15,2,112,15) Matrix(197,324,76,125) -> Matrix(35,4,236,27) Matrix(199,324,78,127) -> Matrix(37,4,268,29) Matrix(271,432,170,271) -> Matrix(19,2,180,19) Matrix(613,972,432,685) -> Matrix(61,6,620,61) Matrix(91,144,12,19) -> Matrix(21,2,52,5) Matrix(827,1296,298,467) -> Matrix(1,0,24,1) Matrix(829,1296,300,469) -> Matrix(1,0,8,1) Matrix(395,612,162,251) -> Matrix(17,2,144,17) Matrix(71,108,-48,-73) -> Matrix(19,2,-200,-21) Matrix(1997,2916,628,917) -> Matrix(175,16,864,79) Matrix(1999,2916,630,919) -> Matrix(177,16,896,81) Matrix(199,288,38,55) -> Matrix(23,2,80,7) Matrix(899,1296,274,395) -> Matrix(1,0,20,1) Matrix(901,1296,276,397) -> Matrix(1,0,12,1) Matrix(379,540,266,379) -> Matrix(1,0,20,1) Matrix(685,972,432,613) -> Matrix(61,6,620,61) Matrix(179,252,76,107) -> Matrix(21,2,136,13) Matrix(233,324,64,89) -> Matrix(87,8,424,39) Matrix(235,324,66,91) -> Matrix(89,8,456,41) Matrix(55,72,42,55) -> Matrix(23,2,264,23) Matrix(197,252,154,197) -> Matrix(97,8,1176,97) Matrix(397,504,256,325) -> Matrix(49,4,404,33) Matrix(233,288,72,89) -> Matrix(25,2,112,9) Matrix(89,108,14,17) -> Matrix(51,4,140,11) Matrix(91,108,16,19) -> Matrix(53,4,172,13) Matrix(125,144,46,53) -> Matrix(27,2,148,11) Matrix(1,0,2,1) -> Matrix(1,0,32,1) Matrix(235,-288,102,-125) -> Matrix(25,-2,188,-15) Matrix(143,-180,58,-73) -> Matrix(25,-2,188,-15) Matrix(53,-72,14,-19) -> Matrix(23,-2,104,-9) Matrix(251,-360,76,-109) -> Matrix(1,0,-4,1) Matrix(73,-108,48,-71) -> Matrix(21,-2,200,-19) Matrix(631,-972,198,-305) -> Matrix(37,-4,176,-19) Matrix(323,-504,116,-181) -> Matrix(1,0,0,1) Matrix(89,-144,34,-55) -> Matrix(19,-2,124,-13) Matrix(107,-180,22,-37) -> Matrix(17,-2,60,-7) Matrix(163,-288,30,-53) -> Matrix(17,-2,60,-7) Matrix(37,-108,12,-35) -> Matrix(13,-2,72,-11) 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): 108 Minimal number of generators: 19 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: 16 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 6/5 3/2 2/1 9/4 18/7 8/3 3/1 27/8 18/5 4/1 9/2 5/1 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 1/16 6/5 1/13 5/4 1/12 9/7 1/12 4/3 1/12 1/11 7/5 3/32 3/2 1/10 11/7 1/12 19/12 5/52 27/17 1/10 8/5 1/10 1/9 5/3 1/8 12/7 0/1 19/11 3/32 7/4 1/8 9/5 1/8 2/1 0/1 1/8 9/4 1/8 7/3 1/8 12/5 0/1 5/2 1/8 18/7 1/7 13/5 3/20 8/3 1/7 1/6 3/1 1/6 10/3 0/1 1/6 27/8 1/6 17/5 7/40 7/2 3/16 18/5 1/5 11/3 5/24 4/1 1/5 1/4 9/2 1/4 5/1 1/4 6/1 1/3 7/1 3/8 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(107,-126,62,-73) (1/1,6/5) -> (12/7,19/11) Hyperbolic Matrix(73,-90,43,-53) (6/5,5/4) -> (5/3,12/7) Hyperbolic Matrix(71,-90,15,-19) (5/4,9/7) -> (9/2,5/1) 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(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(217,-342,125,-197) (11/7,19/12) -> (19/11,7/4) Hyperbolic Matrix(613,-972,181,-287) (19/12,27/17) -> (27/8,17/5) Hyperbolic Matrix(305,-486,91,-145) (27/17,8/5) -> (10/3,27/8) Hyperbolic Matrix(89,-144,34,-55) (8/5,5/3) -> (13/5,8/3) Hyperbolic Matrix(71,-126,31,-55) (7/4,9/5) -> (9/4,7/3) Hyperbolic Matrix(19,-36,9,-17) (9/5,2/1) -> (2/1,9/4) Parabolic Matrix(53,-126,8,-19) (7/3,12/5) -> (6/1,7/1) Hyperbolic Matrix(37,-90,7,-17) (12/5,5/2) -> (5/1,6/1) Hyperbolic Matrix(127,-324,49,-125) (5/2,18/7) -> (18/7,13/5) Parabolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(37,-126,5,-17) (17/5,7/2) -> (7/1,1/0) Hyperbolic Matrix(91,-324,25,-89) (7/2,18/5) -> (18/5,11/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,16,1) Matrix(107,-126,62,-73) -> Matrix(13,-1,144,-11) Matrix(73,-90,43,-53) -> Matrix(13,-1,92,-7) Matrix(71,-90,15,-19) -> Matrix(37,-3,136,-11) Matrix(55,-72,13,-17) -> Matrix(23,-2,104,-9) Matrix(53,-72,14,-19) -> Matrix(23,-2,104,-9) Matrix(37,-54,24,-35) -> Matrix(11,-1,100,-9) Matrix(217,-342,125,-197) -> Matrix(11,-1,100,-9) Matrix(613,-972,181,-287) -> Matrix(61,-6,356,-35) Matrix(305,-486,91,-145) -> Matrix(9,-1,64,-7) Matrix(89,-144,34,-55) -> Matrix(19,-2,124,-13) Matrix(71,-126,31,-55) -> Matrix(9,-1,64,-7) Matrix(19,-36,9,-17) -> Matrix(1,0,0,1) Matrix(53,-126,8,-19) -> Matrix(5,-1,16,-3) Matrix(37,-90,7,-17) -> Matrix(9,-1,28,-3) Matrix(127,-324,49,-125) -> Matrix(29,-4,196,-27) Matrix(19,-54,6,-17) -> Matrix(7,-1,36,-5) Matrix(37,-126,5,-17) -> Matrix(17,-3,40,-7) Matrix(91,-324,25,-89) -> Matrix(41,-8,200,-39) 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 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 (0/1,1/8) 0 9 9/4 1/8 1 2 7/3 1/8 1 18 12/5 0/1 4 3 5/2 1/8 1 18 18/7 1/7 4 1 8/3 (1/7,1/6) 0 9 3/1 1/6 1 6 10/3 (0/1,1/6) 0 9 27/8 1/6 7 2 17/5 7/40 1 18 7/2 3/16 1 18 18/5 1/5 8 1 4/1 (1/5,1/4) 0 9 9/2 1/4 5 2 5/1 1/4 1 18 6/1 1/3 4 3 7/1 3/8 1 18 1/0 1/0 1 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(55,-126,24,-55) (9/4,7/3) -> (9/4,7/3) Reflection Matrix(53,-126,8,-19) (7/3,12/5) -> (6/1,7/1) Hyperbolic Matrix(37,-90,7,-17) (12/5,5/2) -> (5/1,6/1) Hyperbolic 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(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic Matrix(161,-540,48,-161) (10/3,27/8) -> (10/3,27/8) Reflection Matrix(271,-918,80,-271) (27/8,17/5) -> (27/8,17/5) Reflection Matrix(37,-126,5,-17) (17/5,7/2) -> (7/1,1/0) Hyperbolic 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(19,-90,4,-19) (9/2,5/1) -> (9/2,5/1) 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(1,0,16,-1) (0/1,2/1) -> (0/1,1/8) Matrix(17,-36,8,-17) -> Matrix(1,0,16,-1) (2/1,9/4) -> (0/1,1/8) Matrix(55,-126,24,-55) -> Matrix(7,-1,48,-7) (9/4,7/3) -> (1/8,1/6) Matrix(53,-126,8,-19) -> Matrix(5,-1,16,-3) 1/4 Matrix(37,-90,7,-17) -> Matrix(9,-1,28,-3) Matrix(71,-180,28,-71) -> Matrix(15,-2,112,-15) (5/2,18/7) -> (1/8,1/7) Matrix(55,-144,21,-55) -> Matrix(13,-2,84,-13) (18/7,8/3) -> (1/7,1/6) Matrix(19,-54,6,-17) -> Matrix(7,-1,36,-5) 1/6 Matrix(161,-540,48,-161) -> Matrix(1,0,12,-1) (10/3,27/8) -> (0/1,1/6) Matrix(271,-918,80,-271) -> Matrix(41,-7,240,-41) (27/8,17/5) -> (1/6,7/40) Matrix(37,-126,5,-17) -> Matrix(17,-3,40,-7) Matrix(71,-252,20,-71) -> Matrix(31,-6,160,-31) (7/2,18/5) -> (3/16,1/5) 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(19,-90,4,-19) -> Matrix(11,-3,40,-11) (9/2,5/1) -> (1/4,3/10) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.