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: 24 Genus: 13 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 1/6 2/5 1/2 3/4 1/1 4/3 3/2 17/11 9/5 2/1 29/13 7/3 5/2 3/1 11/3 4/1 13/3 23/5 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/11 -5/11 1/9 -4/9 2/19 1/9 -3/7 1/9 1/8 -2/5 2/15 1/7 -5/13 1/7 3/20 -8/21 1/7 2/13 -3/8 2/13 3/19 -1/3 1/5 -2/7 3/11 2/7 -5/18 2/7 1/3 -3/11 3/10 1/3 -4/15 8/25 1/3 -1/4 1/3 2/5 -1/5 1/2 1/1 -2/11 4/5 1/1 -1/6 1/1 2/1 -1/7 1/1 0/1 -1/1 0/1 1/6 -1/2 1/5 -1/3 2/9 -1/1 0/1 1/4 -2/5 -1/3 1/3 -1/3 -1/4 3/8 -3/11 -4/15 2/5 -1/4 5/12 -5/21 -4/17 3/7 -3/13 7/16 -2/9 -5/23 4/9 -2/9 -1/5 1/2 -2/9 -1/5 3/5 -1/5 -3/16 8/13 -3/16 5/8 -1/5 -2/11 7/11 -5/27 2/3 -2/11 -3/17 9/13 -3/17 7/10 -4/23 -5/29 12/17 -4/23 -5/29 5/7 -5/29 -1/6 3/4 -1/6 7/9 -1/6 -7/43 4/5 -5/31 -4/25 1/1 -1/7 5/4 -4/31 -5/39 4/3 -1/8 11/8 -8/65 -7/57 18/13 -6/49 -5/41 7/5 -1/8 -5/41 10/7 -5/41 -4/33 23/16 -4/33 -11/91 13/9 -3/25 3/2 -3/25 -2/17 17/11 -2/17 14/9 -2/17 -13/111 11/7 -5/43 19/12 -2/17 -1/9 8/5 -2/17 -1/9 21/13 -5/43 13/8 -3/26 18/11 -4/35 -1/9 5/3 -3/26 -1/9 7/4 -8/71 -1/9 9/5 -1/9 11/6 -1/9 -12/109 2/1 -1/9 -2/19 11/5 -1/9 -1/10 20/9 -1/9 0/1 29/13 -1/9 9/4 -1/9 -2/19 16/7 -5/47 -2/19 39/17 -2/19 23/10 -2/19 -11/105 7/3 -3/29 5/2 -1/10 13/5 -5/51 21/8 -4/41 -3/31 29/11 -1/10 -3/31 8/3 -4/41 -3/31 3/1 -1/10 -1/11 7/2 -4/43 -1/11 11/3 -1/11 15/4 -1/11 -8/89 4/1 -1/11 -2/23 13/3 -1/12 -1/13 35/8 -2/25 -1/13 57/13 -1/13 22/5 -1/13 0/1 9/2 -1/13 0/1 23/5 0/1 37/8 -1/5 0/1 14/3 -1/9 0/1 5/1 -1/11 11/2 -2/23 -3/35 6/1 -1/12 7/1 -1/12 -1/13 1/0 -1/13 0/1 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(131,60,24,11) (-1/2,-5/11) -> (5/1,11/2) Hyperbolic Matrix(103,46,150,67) (-5/11,-4/9) -> (2/3,9/13) Hyperbolic Matrix(23,10,-122,-53) (-4/9,-3/7) -> (-1/5,-2/11) Hyperbolic Matrix(71,30,26,11) (-3/7,-2/5) -> (8/3,3/1) Hyperbolic Matrix(67,26,-250,-97) (-2/5,-5/13) -> (-3/11,-4/15) Hyperbolic Matrix(339,130,206,79) (-5/13,-8/21) -> (18/11,5/3) Hyperbolic Matrix(179,68,408,155) (-8/21,-3/8) -> (7/16,4/9) Hyperbolic Matrix(23,8,20,7) (-3/8,-1/3) -> (1/1,5/4) Hyperbolic Matrix(19,6,22,7) (-1/3,-2/7) -> (4/5,1/1) Hyperbolic Matrix(351,98,154,43) (-2/7,-5/18) -> (9/4,16/7) Hyperbolic Matrix(137,38,18,5) (-5/18,-3/11) -> (7/1,1/0) Hyperbolic Matrix(173,46,94,25) (-4/15,-1/4) -> (11/6,2/1) Hyperbolic Matrix(19,4,52,11) (-1/4,-1/5) -> (1/3,3/8) Hyperbolic Matrix(189,34,50,9) (-2/11,-1/6) -> (15/4,4/1) Hyperbolic Matrix(99,16,68,11) (-1/6,-1/7) -> (13/9,3/2) Hyperbolic Matrix(77,8,48,5) (-1/7,0/1) -> (8/5,21/13) Hyperbolic Matrix(121,-18,74,-11) (0/1,1/6) -> (13/8,18/11) Hyperbolic Matrix(191,-34,118,-21) (1/6,1/5) -> (21/13,13/8) Hyperbolic Matrix(115,-24,24,-5) (1/5,2/9) -> (14/3,5/1) Hyperbolic Matrix(171,-40,124,-29) (2/9,1/4) -> (11/8,18/13) Hyperbolic Matrix(33,-10,10,-3) (1/4,1/3) -> (3/1,7/2) Hyperbolic Matrix(41,-16,100,-39) (3/8,2/5) -> (2/5,5/12) Parabolic Matrix(265,-112,168,-71) (5/12,3/7) -> (11/7,19/12) Hyperbolic Matrix(273,-118,118,-51) (3/7,7/16) -> (23/10,7/3) Hyperbolic Matrix(87,-40,124,-57) (4/9,1/2) -> (7/10,12/17) Hyperbolic Matrix(45,-26,26,-15) (1/2,3/5) -> (5/3,7/4) Hyperbolic Matrix(121,-74,18,-11) (3/5,8/13) -> (6/1,7/1) Hyperbolic Matrix(191,-118,34,-21) (8/13,5/8) -> (11/2,6/1) Hyperbolic Matrix(335,-212,128,-81) (5/8,7/11) -> (13/5,21/8) Hyperbolic Matrix(187,-120,120,-77) (7/11,2/3) -> (14/9,11/7) Hyperbolic Matrix(429,-298,298,-207) (9/13,7/10) -> (23/16,13/9) Hyperbolic Matrix(327,-232,148,-105) (12/17,5/7) -> (11/5,20/9) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(101,-80,24,-19) (7/9,4/5) -> (4/1,13/3) Hyperbolic Matrix(49,-64,36,-47) (5/4,4/3) -> (4/3,11/8) Parabolic Matrix(417,-578,158,-219) (18/13,7/5) -> (29/11,8/3) Hyperbolic Matrix(87,-124,40,-57) (7/5,10/7) -> (2/1,11/5) Hyperbolic Matrix(483,-692,104,-149) (10/7,23/16) -> (37/8,14/3) Hyperbolic Matrix(331,-508,144,-221) (3/2,17/11) -> (39/17,23/10) Hyperbolic Matrix(527,-818,230,-357) (17/11,14/9) -> (16/7,39/17) Hyperbolic Matrix(309,-490,70,-111) (19/12,8/5) -> (22/5,9/2) Hyperbolic Matrix(91,-162,50,-89) (7/4,9/5) -> (9/5,11/6) Parabolic Matrix(799,-1778,182,-405) (20/9,29/13) -> (57/13,22/5) Hyperbolic Matrix(683,-1528,156,-349) (29/13,9/4) -> (35/8,57/13) Hyperbolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(489,-1288,112,-295) (21/8,29/11) -> (13/3,35/8) Hyperbolic Matrix(67,-242,18,-65) (7/2,11/3) -> (11/3,15/4) Parabolic Matrix(231,-1058,50,-229) (9/2,23/5) -> (23/5,37/8) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,24,1) Matrix(131,60,24,11) -> Matrix(19,-2,-218,23) Matrix(103,46,150,67) -> Matrix(39,-4,-224,23) Matrix(23,10,-122,-53) -> Matrix(17,-2,26,-3) Matrix(71,30,26,11) -> Matrix(17,-2,-178,21) Matrix(67,26,-250,-97) -> Matrix(41,-6,130,-19) Matrix(339,130,206,79) -> Matrix(41,-6,-362,53) Matrix(179,68,408,155) -> Matrix(27,-4,-128,19) Matrix(23,8,20,7) -> Matrix(11,-2,-82,15) Matrix(19,6,22,7) -> Matrix(9,-2,-58,13) Matrix(351,98,154,43) -> Matrix(13,-4,-120,37) Matrix(137,38,18,5) -> Matrix(7,-2,-94,27) Matrix(173,46,94,25) -> Matrix(31,-10,-282,91) Matrix(19,4,52,11) -> Matrix(3,-2,-10,7) Matrix(189,34,50,9) -> Matrix(7,-6,-78,67) Matrix(99,16,68,11) -> Matrix(1,-4,-8,33) Matrix(77,8,48,5) -> Matrix(3,2,-26,-17) Matrix(121,-18,74,-11) -> Matrix(5,4,-44,-35) Matrix(191,-34,118,-21) -> Matrix(19,8,-164,-69) Matrix(115,-24,24,-5) -> Matrix(1,0,-8,1) Matrix(171,-40,124,-29) -> Matrix(11,6,-90,-49) Matrix(33,-10,10,-3) -> Matrix(7,2,-74,-21) Matrix(41,-16,100,-39) -> Matrix(31,8,-128,-33) Matrix(265,-112,168,-71) -> Matrix(59,14,-510,-121) Matrix(273,-118,118,-51) -> Matrix(53,12,-508,-115) Matrix(87,-40,124,-57) -> Matrix(25,6,-146,-35) Matrix(45,-26,26,-15) -> Matrix(31,6,-274,-53) Matrix(121,-74,18,-11) -> Matrix(21,4,-268,-51) Matrix(191,-118,34,-21) -> Matrix(43,8,-500,-93) Matrix(335,-212,128,-81) -> Matrix(53,10,-546,-103) Matrix(187,-120,120,-77) -> Matrix(109,20,-932,-171) Matrix(429,-298,298,-207) -> Matrix(137,24,-1136,-199) Matrix(327,-232,148,-105) -> Matrix(23,4,-236,-41) Matrix(49,-36,64,-47) -> Matrix(71,12,-432,-73) Matrix(101,-80,24,-19) -> Matrix(37,6,-438,-71) Matrix(49,-64,36,-47) -> Matrix(95,12,-768,-97) Matrix(417,-578,158,-219) -> Matrix(17,2,-162,-19) Matrix(87,-124,40,-57) -> Matrix(49,6,-482,-59) Matrix(483,-692,104,-149) -> Matrix(33,4,-256,-31) Matrix(331,-508,144,-221) -> Matrix(237,28,-2260,-267) Matrix(527,-818,230,-357) -> Matrix(307,36,-2908,-341) Matrix(309,-490,70,-111) -> Matrix(17,2,-230,-27) Matrix(91,-162,50,-89) -> Matrix(179,20,-1620,-181) Matrix(799,-1778,182,-405) -> Matrix(1,0,-4,1) Matrix(683,-1528,156,-349) -> Matrix(37,4,-472,-51) Matrix(41,-100,16,-39) -> Matrix(79,8,-800,-81) Matrix(489,-1288,112,-295) -> Matrix(21,2,-242,-23) Matrix(67,-242,18,-65) -> Matrix(131,12,-1452,-133) Matrix(231,-1058,50,-229) -> Matrix(1,0,8,1) 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): 12 Degree of the the map X: 24 Degree of the the map Y: 48 Permutation triple for Y: ((1,6,22,36,44,42,48,46,38,14,37,23,7,2)(3,12,34,40,45,29,41,32,11,8,25,35,13,4)(5,18,43,26,31,10,9,30,17,24,33,47,39,19)(15,16)(20,21)(27,28); (1,4,16,8,7,17,5)(3,10,6,21,23,26,11)(9,28,43,40,37,22,29)(12,32,39,38,20,44,33)(13,19,27,24,25,36,14)(15,34,42,18,30,46,41); (1,2,8,26,28,19,32,46,48,34,33,27,9,3)(4,14,39,47,44,25,16,41,22,10,31,23,40,15)(5,13,35,24,7,21,38,30,29,45,43,42,20,6)(11,12)(17,18)(36,37)) ----------------------------------------------------------------------- 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): 144 Minimal number of generators: 25 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: 18 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/1 4/3 7/5 17/11 5/3 9/5 2/1 29/13 7/3 5/2 3/1 11/3 4/1 23/5 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 -1/1 0/1 1/6 -1/2 1/5 -1/3 2/9 -1/1 0/1 1/4 -2/5 -1/3 1/3 -1/3 -1/4 2/5 -1/4 3/7 -3/13 4/9 -2/9 -1/5 1/2 -2/9 -1/5 3/5 -1/5 -3/16 8/13 -3/16 5/8 -1/5 -2/11 7/11 -5/27 2/3 -2/11 -3/17 7/10 -4/23 -5/29 12/17 -4/23 -5/29 5/7 -5/29 -1/6 3/4 -1/6 1/1 -1/7 4/3 -1/8 7/5 -1/8 -5/41 10/7 -5/41 -4/33 3/2 -3/25 -2/17 17/11 -2/17 14/9 -2/17 -13/111 11/7 -5/43 8/5 -2/17 -1/9 13/8 -3/26 5/3 -3/26 -1/9 7/4 -8/71 -1/9 9/5 -1/9 2/1 -1/9 -2/19 11/5 -1/9 -1/10 20/9 -1/9 0/1 29/13 -1/9 9/4 -1/9 -2/19 7/3 -3/29 5/2 -1/10 3/1 -1/10 -1/11 7/2 -4/43 -1/11 11/3 -1/11 4/1 -1/11 -2/23 9/2 -1/13 0/1 23/5 0/1 14/3 -1/9 0/1 5/1 -1/11 6/1 -1/12 1/0 -1/13 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(38,-5,61,-8) (0/1,1/6) -> (8/13,5/8) Hyperbolic Matrix(60,-11,11,-2) (1/6,1/5) -> (5/1,6/1) Hyperbolic Matrix(115,-24,24,-5) (1/5,2/9) -> (14/3,5/1) Hyperbolic Matrix(46,-11,67,-16) (2/9,1/4) -> (2/3,7/10) Hyperbolic Matrix(33,-10,10,-3) (1/4,1/3) -> (3/1,7/2) Hyperbolic Matrix(30,-11,11,-4) (1/3,2/5) -> (5/2,3/1) Hyperbolic Matrix(70,-29,29,-12) (2/5,3/7) -> (7/3,5/2) Hyperbolic Matrix(98,-43,155,-68) (3/7,4/9) -> (5/8,7/11) Hyperbolic Matrix(87,-40,124,-57) (4/9,1/2) -> (7/10,12/17) Hyperbolic Matrix(45,-26,26,-15) (1/2,3/5) -> (5/3,7/4) Hyperbolic Matrix(130,-79,79,-48) (3/5,8/13) -> (13/8,5/3) Hyperbolic Matrix(187,-120,120,-77) (7/11,2/3) -> (14/9,11/7) Hyperbolic Matrix(327,-232,148,-105) (12/17,5/7) -> (11/5,20/9) Hyperbolic Matrix(56,-41,41,-30) (5/7,3/4) -> (4/3,7/5) Hyperbolic Matrix(8,-7,7,-6) (3/4,1/1) -> (1/1,4/3) Parabolic Matrix(87,-124,40,-57) (7/5,10/7) -> (2/1,11/5) Hyperbolic Matrix(46,-67,11,-16) (10/7,3/2) -> (4/1,9/2) Hyperbolic Matrix(188,-289,121,-186) (3/2,17/11) -> (17/11,14/9) Parabolic Matrix(98,-155,43,-68) (11/7,8/5) -> (9/4,7/3) Hyperbolic Matrix(38,-61,5,-8) (8/5,13/8) -> (6/1,1/0) Hyperbolic Matrix(46,-81,25,-44) (7/4,9/5) -> (9/5,2/1) Parabolic Matrix(378,-841,169,-376) (20/9,29/13) -> (29/13,9/4) Parabolic Matrix(34,-121,9,-32) (7/2,11/3) -> (11/3,4/1) Parabolic Matrix(116,-529,25,-114) (9/2,23/5) -> (23/5,14/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,12,1) Matrix(38,-5,61,-8) -> Matrix(1,2,-6,-11) Matrix(60,-11,11,-2) -> Matrix(5,2,-58,-23) Matrix(115,-24,24,-5) -> Matrix(1,0,-8,1) Matrix(46,-11,67,-16) -> Matrix(9,4,-52,-23) Matrix(33,-10,10,-3) -> Matrix(7,2,-74,-21) Matrix(30,-11,11,-4) -> Matrix(7,2,-74,-21) Matrix(70,-29,29,-12) -> Matrix(25,6,-246,-59) Matrix(98,-43,155,-68) -> Matrix(19,4,-100,-21) Matrix(87,-40,124,-57) -> Matrix(25,6,-146,-35) Matrix(45,-26,26,-15) -> Matrix(31,6,-274,-53) Matrix(130,-79,79,-48) -> Matrix(31,6,-274,-53) Matrix(187,-120,120,-77) -> Matrix(109,20,-932,-171) Matrix(327,-232,148,-105) -> Matrix(23,4,-236,-41) Matrix(56,-41,41,-30) -> Matrix(59,10,-478,-81) Matrix(8,-7,7,-6) -> Matrix(13,2,-98,-15) Matrix(87,-124,40,-57) -> Matrix(49,6,-482,-59) Matrix(46,-67,11,-16) -> Matrix(33,4,-388,-47) Matrix(188,-289,121,-186) -> Matrix(271,32,-2312,-273) Matrix(98,-155,43,-68) -> Matrix(35,4,-324,-37) Matrix(38,-61,5,-8) -> Matrix(17,2,-230,-27) Matrix(46,-81,25,-44) -> Matrix(89,10,-810,-91) Matrix(378,-841,169,-376) -> Matrix(17,2,-162,-19) Matrix(34,-121,9,-32) -> Matrix(65,6,-726,-67) Matrix(116,-529,25,-114) -> Matrix(1,0,4,1) 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): 6 ----------------------------------------------------------------------- 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 12 1 1/1 -1/7 2 7 4/3 -1/8 6 2 7/5 (-1/8,-5/41) 0 7 10/7 (-5/41,-4/33) 0 14 3/2 (-3/25,-2/17) 0 14 17/11 -2/17 8 1 11/7 -5/43 2 7 8/5 (-2/17,-1/9) 0 14 13/8 -3/26 2 2 5/3 (-3/26,-1/9) 0 7 9/5 -1/9 10 1 2/1 (-1/9,-2/19) 0 14 11/5 (-1/9,-1/10) 0 7 29/13 -1/9 2 1 9/4 (-1/9,-2/19) 0 14 7/3 -3/29 2 7 5/2 -1/10 4 2 3/1 (-1/10,-1/11) 0 7 11/3 -1/11 6 1 4/1 (-1/11,-2/23) 0 14 9/2 (-1/13,0/1) 0 14 23/5 0/1 4 1 5/1 -1/11 2 7 6/1 -1/12 2 2 1/0 (-1/13,0/1) 0 14 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(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(7,-8,6,-7) (1/1,4/3) -> (1/1,4/3) Reflection Matrix(41,-56,30,-41) (4/3,7/5) -> (4/3,7/5) Reflection Matrix(87,-124,40,-57) (7/5,10/7) -> (2/1,11/5) Hyperbolic Matrix(46,-67,11,-16) (10/7,3/2) -> (4/1,9/2) Hyperbolic Matrix(67,-102,44,-67) (3/2,17/11) -> (3/2,17/11) Reflection Matrix(120,-187,77,-120) (17/11,11/7) -> (17/11,11/7) Reflection Matrix(98,-155,43,-68) (11/7,8/5) -> (9/4,7/3) Hyperbolic Matrix(38,-61,5,-8) (8/5,13/8) -> (6/1,1/0) Hyperbolic Matrix(79,-130,48,-79) (13/8,5/3) -> (13/8,5/3) Reflection Matrix(26,-45,15,-26) (5/3,9/5) -> (5/3,9/5) Reflection Matrix(19,-36,10,-19) (9/5,2/1) -> (9/5,2/1) Reflection Matrix(144,-319,65,-144) (11/5,29/13) -> (11/5,29/13) Reflection Matrix(233,-522,104,-233) (29/13,9/4) -> (29/13,9/4) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(10,-33,3,-10) (3/1,11/3) -> (3/1,11/3) Reflection Matrix(23,-88,6,-23) (11/3,4/1) -> (11/3,4/1) Reflection Matrix(91,-414,20,-91) (9/2,23/5) -> (9/2,23/5) Reflection Matrix(24,-115,5,-24) (23/5,5/1) -> (23/5,5/1) Reflection Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,26,1) (-1/1,1/0) -> (-1/13,0/1) Matrix(0,1,1,0) -> Matrix(-1,0,14,1) (-1/1,1/1) -> (-1/7,0/1) Matrix(7,-8,6,-7) -> Matrix(15,2,-112,-15) (1/1,4/3) -> (-1/7,-1/8) Matrix(41,-56,30,-41) -> Matrix(81,10,-656,-81) (4/3,7/5) -> (-1/8,-5/41) Matrix(87,-124,40,-57) -> Matrix(49,6,-482,-59) Matrix(46,-67,11,-16) -> Matrix(33,4,-388,-47) Matrix(67,-102,44,-67) -> Matrix(101,12,-850,-101) (3/2,17/11) -> (-3/25,-2/17) Matrix(120,-187,77,-120) -> Matrix(171,20,-1462,-171) (17/11,11/7) -> (-2/17,-5/43) Matrix(98,-155,43,-68) -> Matrix(35,4,-324,-37) -1/9 Matrix(38,-61,5,-8) -> Matrix(17,2,-230,-27) Matrix(79,-130,48,-79) -> Matrix(53,6,-468,-53) (13/8,5/3) -> (-3/26,-1/9) Matrix(26,-45,15,-26) -> Matrix(53,6,-468,-53) (5/3,9/5) -> (-3/26,-1/9) Matrix(19,-36,10,-19) -> Matrix(37,4,-342,-37) (9/5,2/1) -> (-1/9,-2/19) Matrix(144,-319,65,-144) -> Matrix(19,2,-180,-19) (11/5,29/13) -> (-1/9,-1/10) Matrix(233,-522,104,-233) -> Matrix(37,4,-342,-37) (29/13,9/4) -> (-1/9,-2/19) Matrix(29,-70,12,-29) -> Matrix(59,6,-580,-59) (7/3,5/2) -> (-3/29,-1/10) Matrix(11,-30,4,-11) -> Matrix(21,2,-220,-21) (5/2,3/1) -> (-1/10,-1/11) Matrix(10,-33,3,-10) -> Matrix(21,2,-220,-21) (3/1,11/3) -> (-1/10,-1/11) Matrix(23,-88,6,-23) -> Matrix(45,4,-506,-45) (11/3,4/1) -> (-1/11,-2/23) Matrix(91,-414,20,-91) -> Matrix(-1,0,26,1) (9/2,23/5) -> (-1/13,0/1) Matrix(24,-115,5,-24) -> Matrix(-1,0,22,1) (23/5,5/1) -> (-1/11,0/1) Matrix(11,-60,2,-11) -> Matrix(23,2,-264,-23) (5/1,6/1) -> (-1/11,-1/12) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.