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: 40 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -2/5 -1/3 -4/13 -1/4 -2/11 -1/7 0/1 1/5 2/7 1/3 4/11 5/13 8/19 1/2 7/11 5/7 4/5 1/1 5/4 3/2 11/7 5/3 2/1 19/8 5/2 13/5 8/3 3/1 16/5 55/17 10/3 7/2 4/1 14/3 5/1 6/1 7/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 1/1 -4/9 0/1 1/0 -3/7 1/1 -8/19 2/1 1/0 -5/12 -1/1 -2/5 0/1 -7/18 1/1 -5/13 1/1 -8/21 2/3 1/1 -3/8 1/1 -7/19 1/0 -4/11 0/1 1/0 -1/3 1/1 -4/13 1/0 -7/23 -1/1 -3/10 1/1 -2/7 0/1 1/0 -5/18 1/1 -8/29 0/1 -3/11 1/1 -7/26 1/1 -4/15 2/1 1/0 -5/19 1/1 -1/4 1/0 -4/17 -2/1 -1/1 -3/13 -1/1 -2/9 -1/2 0/1 -3/14 1/3 -4/19 0/1 -1/5 1/1 -2/11 1/0 -3/17 -1/1 -1/6 -1/1 -2/13 0/1 1/0 -3/20 -1/1 -1/7 0/1 -1/8 -1/1 0/1 0/1 1/0 1/7 -1/1 2/13 0/1 1/6 1/1 2/11 -1/1 0/1 1/5 0/1 2/9 0/1 1/0 1/4 1/1 2/7 1/0 3/10 -1/1 1/3 -1/1 5/14 -1/1 4/11 0/1 3/8 -1/3 5/13 0/1 7/18 1/7 2/5 0/1 1/2 5/12 1/1 8/19 1/1 11/26 1/1 3/7 1/1 1/2 1/0 4/7 -2/1 1/0 11/19 -2/1 7/12 -1/1 10/17 -1/1 3/5 -1/1 14/23 -1/2 0/1 11/18 -1/3 8/13 0/1 21/34 1/1 13/21 -1/1 5/8 -1/1 7/11 0/1 9/14 1/1 2/3 0/1 1/0 7/10 1/1 12/17 2/1 1/0 5/7 1/0 13/18 -1/1 8/11 -1/1 1/0 11/15 -1/1 3/4 -1/1 4/5 1/0 5/6 -3/1 11/13 -2/1 6/7 -2/1 -1/1 7/8 1/0 1/1 -1/1 8/7 0/1 7/6 -1/1 6/5 -1/2 0/1 5/4 0/1 14/11 0/1 1/2 23/18 1/1 9/7 1/1 4/3 0/1 1/0 7/5 1/0 10/7 -1/1 1/0 33/23 -1/1 23/16 1/0 13/9 -1/1 3/2 -1/1 11/7 -1/1 19/12 -1/1 8/5 -1/1 -2/3 29/18 -3/5 21/13 -1/1 13/8 -1/2 5/3 -1/1 2/1 0/1 7/3 1/1 26/11 2/1 1/0 19/8 1/0 50/21 0/1 1/0 31/13 1/1 12/5 1/1 1/0 5/2 1/1 13/5 1/0 21/8 -5/1 8/3 -2/1 1/0 11/4 1/0 25/9 -1/1 39/14 1/1 53/19 1/0 14/5 -1/1 1/0 17/6 -1/1 3/1 -1/1 16/5 0/1 29/9 1/1 42/13 1/1 1/0 55/17 1/0 13/4 -1/1 23/7 0/1 10/3 -1/1 0/1 7/2 0/1 18/5 0/1 1/1 47/13 0/1 29/8 1/1 11/3 1/1 4/1 0/1 1/0 9/2 1/1 14/3 1/1 1/0 5/1 1/0 6/1 -2/1 1/0 7/1 -1/1 8/1 -1/1 1/0 -1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(67,30,96,43) (-1/2,-4/9) -> (2/3,7/10) Hyperbolic Matrix(127,56,34,15) (-4/9,-3/7) -> (11/3,4/1) Hyperbolic Matrix(123,52,-466,-197) (-3/7,-8/19) -> (-4/15,-5/19) Hyperbolic Matrix(495,208,188,79) (-8/19,-5/12) -> (21/8,8/3) Hyperbolic Matrix(59,24,-150,-61) (-5/12,-2/5) -> (-2/5,-7/18) Parabolic Matrix(237,92,322,125) (-7/18,-5/13) -> (11/15,3/4) Hyperbolic Matrix(115,44,-494,-189) (-5/13,-8/21) -> (-4/17,-3/13) Hyperbolic Matrix(463,176,292,111) (-8/21,-3/8) -> (19/12,8/5) Hyperbolic Matrix(287,106,398,147) (-3/8,-7/19) -> (5/7,13/18) Hyperbolic Matrix(283,104,400,147) (-7/19,-4/11) -> (12/17,5/7) Hyperbolic Matrix(111,40,86,31) (-4/11,-1/3) -> (9/7,4/3) Hyperbolic Matrix(103,32,-338,-105) (-1/3,-4/13) -> (-4/13,-7/23) Parabolic Matrix(877,266,544,165) (-7/23,-3/10) -> (29/18,21/13) Hyperbolic Matrix(103,30,24,7) (-3/10,-2/7) -> (4/1,9/2) Hyperbolic Matrix(99,28,152,43) (-2/7,-5/18) -> (9/14,2/3) Hyperbolic Matrix(347,96,300,83) (-5/18,-8/29) -> (8/7,7/6) Hyperbolic Matrix(51,14,346,95) (-8/29,-3/11) -> (1/7,2/13) Hyperbolic Matrix(199,54,468,127) (-3/11,-7/26) -> (11/26,3/7) Hyperbolic Matrix(97,26,-638,-171) (-7/26,-4/15) -> (-2/13,-3/20) Hyperbolic Matrix(489,128,340,89) (-5/19,-1/4) -> (23/16,13/9) Hyperbolic Matrix(143,34,164,39) (-1/4,-4/17) -> (6/7,7/8) Hyperbolic Matrix(141,32,22,5) (-3/13,-2/9) -> (6/1,7/1) Hyperbolic Matrix(91,20,232,51) (-2/9,-3/14) -> (7/18,2/5) Hyperbolic Matrix(47,10,296,63) (-3/14,-4/19) -> (2/13,1/6) Hyperbolic Matrix(137,28,44,9) (-4/19,-1/5) -> (3/1,16/5) Hyperbolic Matrix(43,8,-242,-45) (-1/5,-2/11) -> (-2/11,-3/17) Parabolic Matrix(307,54,108,19) (-3/17,-1/6) -> (17/6,3/1) Hyperbolic Matrix(383,60,300,47) (-1/6,-2/13) -> (14/11,23/18) Hyperbolic Matrix(1143,170,316,47) (-3/20,-1/7) -> (47/13,29/8) Hyperbolic Matrix(123,16,146,19) (-1/7,-1/8) -> (5/6,11/13) Hyperbolic Matrix(123,14,202,23) (-1/8,0/1) -> (14/23,11/18) Hyperbolic Matrix(189,-26,80,-11) (0/1,1/7) -> (7/3,26/11) Hyperbolic Matrix(367,-66,228,-41) (1/6,2/11) -> (8/5,29/18) Hyperbolic Matrix(303,-56,92,-17) (2/11,1/5) -> (23/7,10/3) Hyperbolic Matrix(119,-26,206,-45) (1/5,2/9) -> (4/7,11/19) Hyperbolic Matrix(87,-20,74,-17) (2/9,1/4) -> (7/6,6/5) Hyperbolic Matrix(29,-8,98,-27) (1/4,2/7) -> (2/7,3/10) Parabolic Matrix(139,-42,96,-29) (3/10,1/3) -> (13/9,3/2) Hyperbolic Matrix(241,-84,66,-23) (1/3,5/14) -> (29/8,11/3) Hyperbolic Matrix(343,-124,556,-201) (5/14,4/11) -> (8/13,21/34) Hyperbolic Matrix(185,-68,302,-111) (4/11,3/8) -> (11/18,8/13) Hyperbolic Matrix(131,-50,338,-129) (3/8,5/13) -> (5/13,7/18) Parabolic Matrix(179,-74,254,-105) (2/5,5/12) -> (7/10,12/17) Hyperbolic Matrix(305,-128,722,-303) (5/12,8/19) -> (8/19,11/26) Parabolic Matrix(101,-44,62,-27) (3/7,1/2) -> (13/8,5/3) Hyperbolic Matrix(93,-52,34,-19) (1/2,4/7) -> (8/3,11/4) Hyperbolic Matrix(523,-304,160,-93) (11/19,7/12) -> (13/4,23/7) Hyperbolic Matrix(113,-66,12,-7) (7/12,10/17) -> (8/1,1/0) Hyperbolic Matrix(159,-94,22,-13) (10/17,3/5) -> (7/1,8/1) Hyperbolic Matrix(963,-584,404,-245) (3/5,14/23) -> (50/21,31/13) Hyperbolic Matrix(1669,-1032,600,-371) (21/34,13/21) -> (25/9,39/14) Hyperbolic Matrix(555,-344,434,-269) (13/21,5/8) -> (23/18,9/7) Hyperbolic Matrix(155,-98,242,-153) (5/8,7/11) -> (7/11,9/14) Parabolic Matrix(439,-318,156,-113) (13/18,8/11) -> (14/5,17/6) Hyperbolic Matrix(521,-380,218,-159) (8/11,11/15) -> (31/13,12/5) Hyperbolic Matrix(41,-32,50,-39) (3/4,4/5) -> (4/5,5/6) Parabolic Matrix(563,-480,156,-133) (11/13,6/7) -> (18/5,47/13) Hyperbolic Matrix(287,-254,200,-177) (7/8,1/1) -> (33/23,23/16) Hyperbolic Matrix(219,-248,68,-77) (1/1,8/7) -> (16/5,29/9) Hyperbolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(45,-62,8,-11) (4/3,7/5) -> (5/1,6/1) Hyperbolic Matrix(105,-148,22,-31) (7/5,10/7) -> (14/3,5/1) Hyperbolic Matrix(1169,-1676,362,-519) (10/7,33/23) -> (29/9,42/13) Hyperbolic Matrix(155,-242,98,-153) (3/2,11/7) -> (11/7,19/12) Parabolic Matrix(343,-556,124,-201) (21/13,13/8) -> (11/4,25/9) Hyperbolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(609,-1444,256,-607) (26/11,19/8) -> (19/8,50/21) Parabolic Matrix(73,-178,16,-39) (12/5,5/2) -> (9/2,14/3) Hyperbolic Matrix(131,-338,50,-129) (5/2,13/5) -> (13/5,21/8) Parabolic Matrix(1017,-2834,314,-875) (39/14,53/19) -> (55/17,13/4) Hyperbolic Matrix(1073,-2996,332,-927) (53/19,14/5) -> (42/13,55/17) Hyperbolic Matrix(57,-196,16,-55) (10/3,7/2) -> (7/2,18/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,2,1) Matrix(67,30,96,43) -> Matrix(1,0,0,1) Matrix(127,56,34,15) -> Matrix(1,0,0,1) Matrix(123,52,-466,-197) -> Matrix(1,0,0,1) Matrix(495,208,188,79) -> Matrix(1,-4,0,1) Matrix(59,24,-150,-61) -> Matrix(1,0,2,1) Matrix(237,92,322,125) -> Matrix(1,0,-2,1) Matrix(115,44,-494,-189) -> Matrix(1,0,-2,1) Matrix(463,176,292,111) -> Matrix(5,-4,-6,5) Matrix(287,106,398,147) -> Matrix(1,-2,0,1) Matrix(283,104,400,147) -> Matrix(1,2,0,1) Matrix(111,40,86,31) -> Matrix(1,0,0,1) Matrix(103,32,-338,-105) -> Matrix(1,-2,0,1) Matrix(877,266,544,165) -> Matrix(1,2,-2,-3) Matrix(103,30,24,7) -> Matrix(1,0,0,1) Matrix(99,28,152,43) -> Matrix(1,0,0,1) Matrix(347,96,300,83) -> Matrix(1,0,-2,1) Matrix(51,14,346,95) -> Matrix(1,0,-2,1) Matrix(199,54,468,127) -> Matrix(1,0,0,1) Matrix(97,26,-638,-171) -> Matrix(1,-2,0,1) Matrix(489,128,340,89) -> Matrix(1,-2,0,1) Matrix(143,34,164,39) -> Matrix(1,0,0,1) Matrix(141,32,22,5) -> Matrix(3,2,-2,-1) Matrix(91,20,232,51) -> Matrix(1,0,4,1) Matrix(47,10,296,63) -> Matrix(1,0,-2,1) Matrix(137,28,44,9) -> Matrix(1,0,-2,1) Matrix(43,8,-242,-45) -> Matrix(1,-2,0,1) Matrix(307,54,108,19) -> Matrix(1,0,0,1) Matrix(383,60,300,47) -> Matrix(1,0,2,1) Matrix(1143,170,316,47) -> Matrix(1,0,2,1) Matrix(123,16,146,19) -> Matrix(1,-2,0,1) Matrix(123,14,202,23) -> Matrix(1,0,-2,1) Matrix(189,-26,80,-11) -> Matrix(1,2,0,1) Matrix(367,-66,228,-41) -> Matrix(1,2,-2,-3) Matrix(303,-56,92,-17) -> Matrix(1,0,0,1) Matrix(119,-26,206,-45) -> Matrix(1,-2,0,1) Matrix(87,-20,74,-17) -> Matrix(1,0,-2,1) Matrix(29,-8,98,-27) -> Matrix(1,-2,0,1) Matrix(139,-42,96,-29) -> Matrix(1,0,0,1) Matrix(241,-84,66,-23) -> Matrix(1,0,2,1) Matrix(343,-124,556,-201) -> Matrix(1,0,2,1) Matrix(185,-68,302,-111) -> Matrix(1,0,0,1) Matrix(131,-50,338,-129) -> Matrix(1,0,10,1) Matrix(179,-74,254,-105) -> Matrix(3,-2,2,-1) Matrix(305,-128,722,-303) -> Matrix(5,-4,4,-3) Matrix(101,-44,62,-27) -> Matrix(1,0,-2,1) Matrix(93,-52,34,-19) -> Matrix(1,0,0,1) Matrix(523,-304,160,-93) -> Matrix(1,2,-2,-3) Matrix(113,-66,12,-7) -> Matrix(1,2,-2,-3) Matrix(159,-94,22,-13) -> Matrix(3,2,-2,-1) Matrix(963,-584,404,-245) -> Matrix(1,0,2,1) Matrix(1669,-1032,600,-371) -> Matrix(1,0,0,1) Matrix(555,-344,434,-269) -> Matrix(1,0,2,1) Matrix(155,-98,242,-153) -> Matrix(1,0,2,1) Matrix(439,-318,156,-113) -> Matrix(1,0,0,1) Matrix(521,-380,218,-159) -> Matrix(1,2,0,1) Matrix(41,-32,50,-39) -> Matrix(1,-2,0,1) Matrix(563,-480,156,-133) -> Matrix(1,2,0,1) Matrix(287,-254,200,-177) -> Matrix(1,0,0,1) Matrix(219,-248,68,-77) -> Matrix(1,0,2,1) Matrix(81,-100,64,-79) -> Matrix(1,0,4,1) Matrix(45,-62,8,-11) -> Matrix(1,-2,0,1) Matrix(105,-148,22,-31) -> Matrix(1,2,0,1) Matrix(1169,-1676,362,-519) -> Matrix(1,2,0,1) Matrix(155,-242,98,-153) -> Matrix(5,6,-6,-7) Matrix(343,-556,124,-201) -> Matrix(3,2,-2,-1) Matrix(13,-24,6,-11) -> Matrix(1,0,2,1) Matrix(609,-1444,256,-607) -> Matrix(1,-2,0,1) Matrix(73,-178,16,-39) -> Matrix(1,0,0,1) Matrix(131,-338,50,-129) -> Matrix(1,-6,0,1) Matrix(1017,-2834,314,-875) -> Matrix(1,-2,0,1) Matrix(1073,-2996,332,-927) -> Matrix(1,2,0,1) Matrix(57,-196,16,-55) -> Matrix(1,0,2,1) 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): 16 Degree of the the map X: 16 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 -1/1 0/1 1 1 0/1 (0/1,1/0) 0 18 1/7 -1/1 1 9 1/6 1/1 1 18 2/11 (-1/1,0/1) 0 18 1/5 0/1 1 3 2/9 (0/1,1/0) 0 18 1/4 1/1 1 18 2/7 1/0 2 2 3/10 -1/1 1 18 1/3 -1/1 1 9 5/14 -1/1 1 18 4/11 0/1 2 6 3/8 -1/3 1 18 5/13 0/1 5 1 2/5 (0/1,1/2) 0 18 5/12 1/1 1 18 8/19 1/1 4 2 3/7 1/1 1 9 1/2 1/0 1 6 4/7 (-2/1,1/0) 0 18 11/19 -2/1 1 3 7/12 -1/1 1 18 10/17 -1/1 4 2 3/5 -1/1 1 9 11/18 -1/3 1 18 8/13 0/1 2 6 21/34 1/1 1 18 13/21 -1/1 1 9 5/8 -1/1 1 18 7/11 0/1 1 1 2/3 (0/1,1/0) 0 18 5/7 1/0 2 3 8/11 (-1/1,1/0) 0 18 3/4 -1/1 1 18 4/5 1/0 2 2 5/6 -3/1 1 18 6/7 (-2/1,-1/1) 0 18 1/1 -1/1 1 9 7/6 -1/1 1 18 6/5 (-1/2,0/1) 0 18 5/4 0/1 2 2 4/3 (0/1,1/0) 0 18 7/5 1/0 2 3 10/7 (-1/1,1/0) 0 18 13/9 -1/1 1 9 3/2 -1/1 1 18 11/7 -1/1 3 1 8/5 (-1/1,-2/3) 0 18 21/13 -1/1 1 9 13/8 -1/2 1 6 5/3 -1/1 1 9 2/1 0/1 2 6 7/3 1/1 1 9 26/11 (2/1,1/0) 0 18 19/8 1/0 1 2 12/5 (1/1,1/0) 0 18 5/2 1/1 1 18 13/5 1/0 3 1 8/3 (-2/1,1/0) 0 18 11/4 1/0 1 6 25/9 -1/1 1 9 39/14 1/1 1 18 53/19 1/0 2 1 14/5 (-1/1,1/0) 0 18 3/1 -1/1 1 9 13/4 -1/1 1 18 23/7 0/1 1 3 10/3 (-1/1,0/1) 0 18 7/2 0/1 1 2 4/1 (0/1,1/0) 0 18 5/1 1/0 2 3 6/1 (-2/1,1/0) 0 18 1/0 -1/1 1 18 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(189,-26,80,-11) (0/1,1/7) -> (7/3,26/11) Hyperbolic Matrix(95,-14,156,-23) (1/7,1/6) -> (3/5,11/18) Glide Reflection Matrix(91,-16,108,-19) (1/6,2/11) -> (5/6,6/7) Glide Reflection Matrix(303,-56,92,-17) (2/11,1/5) -> (23/7,10/3) Hyperbolic Matrix(119,-26,206,-45) (1/5,2/9) -> (4/7,11/19) Hyperbolic Matrix(87,-20,74,-17) (2/9,1/4) -> (7/6,6/5) Hyperbolic Matrix(29,-8,98,-27) (1/4,2/7) -> (2/7,3/10) Parabolic Matrix(139,-42,96,-29) (3/10,1/3) -> (13/9,3/2) Hyperbolic Matrix(81,-28,26,-9) (1/3,5/14) -> (3/1,13/4) Glide Reflection Matrix(343,-124,556,-201) (5/14,4/11) -> (8/13,21/34) Hyperbolic Matrix(185,-68,302,-111) (4/11,3/8) -> (11/18,8/13) Hyperbolic Matrix(79,-30,208,-79) (3/8,5/13) -> (3/8,5/13) Reflection Matrix(51,-20,130,-51) (5/13,2/5) -> (5/13,2/5) Reflection Matrix(77,-32,12,-5) (2/5,5/12) -> (6/1,1/0) Glide Reflection Matrix(253,-106,432,-181) (5/12,8/19) -> (7/12,10/17) Glide Reflection Matrix(127,-54,214,-91) (8/19,3/7) -> (10/17,3/5) Glide Reflection Matrix(101,-44,62,-27) (3/7,1/2) -> (13/8,5/3) Hyperbolic Matrix(93,-52,34,-19) (1/2,4/7) -> (8/3,11/4) Hyperbolic Matrix(523,-304,160,-93) (11/19,7/12) -> (13/4,23/7) Hyperbolic Matrix(1669,-1032,600,-371) (21/34,13/21) -> (25/9,39/14) Hyperbolic Matrix(155,-96,134,-83) (13/21,5/8) -> (1/1,7/6) Glide Reflection Matrix(111,-70,176,-111) (5/8,7/11) -> (5/8,7/11) Reflection Matrix(43,-28,66,-43) (7/11,2/3) -> (7/11,2/3) Reflection Matrix(43,-30,10,-7) (2/3,5/7) -> (4/1,5/1) Glide Reflection Matrix(147,-106,104,-75) (5/7,8/11) -> (7/5,10/7) Glide Reflection Matrix(103,-76,42,-31) (8/11,3/4) -> (12/5,5/2) Glide Reflection Matrix(41,-32,50,-39) (3/4,4/5) -> (4/5,5/6) Parabolic Matrix(155,-134,96,-83) (6/7,1/1) -> (8/5,21/13) Glide Reflection Matrix(49,-60,40,-49) (6/5,5/4) -> (6/5,5/4) Reflection Matrix(31,-40,24,-31) (5/4,4/3) -> (5/4,4/3) Reflection Matrix(45,-62,8,-11) (4/3,7/5) -> (5/1,6/1) Hyperbolic Matrix(147,-212,52,-75) (10/7,13/9) -> (14/5,3/1) Glide Reflection Matrix(43,-66,28,-43) (3/2,11/7) -> (3/2,11/7) Reflection Matrix(111,-176,70,-111) (11/7,8/5) -> (11/7,8/5) Reflection Matrix(343,-556,124,-201) (21/13,13/8) -> (11/4,25/9) Hyperbolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(417,-988,176,-417) (26/11,19/8) -> (26/11,19/8) Reflection Matrix(191,-456,80,-191) (19/8,12/5) -> (19/8,12/5) Reflection Matrix(51,-130,20,-51) (5/2,13/5) -> (5/2,13/5) Reflection Matrix(79,-208,30,-79) (13/5,8/3) -> (13/5,8/3) Reflection Matrix(1483,-4134,532,-1483) (39/14,53/19) -> (39/14,53/19) Reflection Matrix(531,-1484,190,-531) (53/19,14/5) -> (53/19,14/5) Reflection Matrix(41,-140,12,-41) (10/3,7/2) -> (10/3,7/2) Reflection Matrix(15,-56,4,-15) (7/2,4/1) -> (7/2,4/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,2,1) (-1/1,1/0) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(189,-26,80,-11) -> Matrix(1,2,0,1) 1/0 Matrix(95,-14,156,-23) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(91,-16,108,-19) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(303,-56,92,-17) -> Matrix(1,0,0,1) Matrix(119,-26,206,-45) -> Matrix(1,-2,0,1) 1/0 Matrix(87,-20,74,-17) -> Matrix(1,0,-2,1) 0/1 Matrix(29,-8,98,-27) -> Matrix(1,-2,0,1) 1/0 Matrix(139,-42,96,-29) -> Matrix(1,0,0,1) Matrix(81,-28,26,-9) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(343,-124,556,-201) -> Matrix(1,0,2,1) 0/1 Matrix(185,-68,302,-111) -> Matrix(1,0,0,1) Matrix(79,-30,208,-79) -> Matrix(-1,0,6,1) (3/8,5/13) -> (-1/3,0/1) Matrix(51,-20,130,-51) -> Matrix(1,0,4,-1) (5/13,2/5) -> (0/1,1/2) Matrix(77,-32,12,-5) -> Matrix(3,-2,-2,1) Matrix(253,-106,432,-181) -> Matrix(5,-4,-4,3) Matrix(127,-54,214,-91) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(101,-44,62,-27) -> Matrix(1,0,-2,1) 0/1 Matrix(93,-52,34,-19) -> Matrix(1,0,0,1) Matrix(523,-304,160,-93) -> Matrix(1,2,-2,-3) -1/1 Matrix(1669,-1032,600,-371) -> Matrix(1,0,0,1) Matrix(155,-96,134,-83) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(111,-70,176,-111) -> Matrix(-1,0,2,1) (5/8,7/11) -> (-1/1,0/1) Matrix(43,-28,66,-43) -> Matrix(1,0,0,-1) (7/11,2/3) -> (0/1,1/0) Matrix(43,-30,10,-7) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(147,-106,104,-75) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(103,-76,42,-31) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(41,-32,50,-39) -> Matrix(1,-2,0,1) 1/0 Matrix(155,-134,96,-83) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(49,-60,40,-49) -> Matrix(-1,0,4,1) (6/5,5/4) -> (-1/2,0/1) Matrix(31,-40,24,-31) -> Matrix(1,0,0,-1) (5/4,4/3) -> (0/1,1/0) Matrix(45,-62,8,-11) -> Matrix(1,-2,0,1) 1/0 Matrix(147,-212,52,-75) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(43,-66,28,-43) -> Matrix(1,2,0,-1) (3/2,11/7) -> (-1/1,1/0) Matrix(111,-176,70,-111) -> Matrix(5,4,-6,-5) (11/7,8/5) -> (-1/1,-2/3) Matrix(343,-556,124,-201) -> Matrix(3,2,-2,-1) -1/1 Matrix(13,-24,6,-11) -> Matrix(1,0,2,1) 0/1 Matrix(417,-988,176,-417) -> Matrix(-1,4,0,1) (26/11,19/8) -> (2/1,1/0) Matrix(191,-456,80,-191) -> Matrix(-1,2,0,1) (19/8,12/5) -> (1/1,1/0) Matrix(51,-130,20,-51) -> Matrix(-1,2,0,1) (5/2,13/5) -> (1/1,1/0) Matrix(79,-208,30,-79) -> Matrix(1,4,0,-1) (13/5,8/3) -> (-2/1,1/0) Matrix(1483,-4134,532,-1483) -> Matrix(-1,2,0,1) (39/14,53/19) -> (1/1,1/0) Matrix(531,-1484,190,-531) -> Matrix(1,2,0,-1) (53/19,14/5) -> (-1/1,1/0) Matrix(41,-140,12,-41) -> Matrix(-1,0,2,1) (10/3,7/2) -> (-1/1,0/1) Matrix(15,-56,4,-15) -> Matrix(1,0,0,-1) (7/2,4/1) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.