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 -5/1 -3/1 -5/3 -1/1 0/1 1/2 5/8 3/4 5/6 1/1 5/4 3/2 5/3 2/1 5/2 8/3 30/11 3/1 10/3 4/1 5/1 6/1 20/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 0/1 -5/1 -1/2 1/0 -4/1 -1/1 -3/1 -1/2 1/0 -14/5 0/1 -11/4 -1/1 0/1 -8/3 -1/1 -5/2 -1/2 -2/1 0/1 -5/3 -1/2 1/0 -8/5 -1/1 -19/12 -1/1 0/1 -30/19 -1/1 -11/7 -3/4 -1/2 -3/2 -1/2 -10/7 0/1 -7/5 -1/2 1/0 -4/3 -1/1 -5/4 -1/2 -6/5 -2/5 -13/11 -1/2 -3/8 -20/17 -1/3 -7/6 -1/2 -1/1 -1/2 -1/4 -6/7 0/1 -5/6 -1/2 -1/4 -4/5 -1/3 -3/4 -1/3 0/1 -14/19 0/1 -11/15 -1/2 -1/4 -8/11 -1/3 -5/7 -1/2 -1/4 -2/3 0/1 -5/8 -1/2 -1/4 -8/13 -1/3 -19/31 -1/2 -1/4 -30/49 -1/3 -11/18 -1/4 -3/5 -1/2 -1/4 -10/17 0/1 -7/12 -1/3 0/1 -4/7 -1/3 -5/9 -1/2 -1/4 -6/11 0/1 -13/24 -1/3 0/1 -20/37 -1/3 -7/13 -1/2 -1/4 -1/2 -1/4 0/1 0/1 1/2 1/2 4/7 1/1 3/5 1/2 1/0 11/18 1/2 8/13 1/1 5/8 1/2 1/0 2/3 0/1 3/4 0/1 1/1 7/9 1/2 1/0 4/5 1/1 5/6 1/2 1/0 6/7 0/1 1/1 1/2 1/0 6/5 2/1 5/4 1/0 4/3 -1/1 3/2 1/0 14/9 0/1 11/7 -3/2 1/0 8/5 -1/1 5/3 -1/2 1/0 12/7 -1/1 31/18 1/0 19/11 -1/2 1/0 26/15 0/1 7/4 -1/1 0/1 2/1 0/1 5/2 1/0 8/3 -1/1 19/7 -3/2 1/0 49/18 -3/2 30/11 -1/1 11/4 -1/1 0/1 3/1 -1/2 1/0 13/4 -1/1 0/1 10/3 0/1 7/2 1/0 4/1 -1/1 9/2 1/0 5/1 -1/2 1/0 11/2 1/0 17/3 -1/2 1/0 23/4 -1/1 0/1 6/1 0/1 13/2 1/0 20/3 -1/1 27/4 -1/1 0/1 7/1 -1/2 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(11,80,-4,-29) (-6/1,1/0) -> (-14/5,-11/4) Hyperbolic Matrix(11,60,-20,-109) (-6/1,-5/1) -> (-5/9,-6/11) Hyperbolic Matrix(9,40,-16,-71) (-5/1,-4/1) -> (-4/7,-5/9) Hyperbolic Matrix(11,40,-8,-29) (-4/1,-3/1) -> (-7/5,-4/3) Hyperbolic Matrix(71,200,-60,-169) (-3/1,-14/5) -> (-6/5,-13/11) Hyperbolic Matrix(89,240,-56,-151) (-11/4,-8/3) -> (-8/5,-19/12) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(11,20,-16,-29) (-2/1,-5/3) -> (-5/7,-2/3) Hyperbolic Matrix(49,80,-68,-111) (-5/3,-8/5) -> (-8/11,-5/7) Hyperbolic Matrix(569,900,208,329) (-19/12,-30/19) -> (30/11,11/4) Hyperbolic Matrix(571,900,-932,-1469) (-30/19,-11/7) -> (-19/31,-30/49) Hyperbolic Matrix(51,80,-44,-69) (-11/7,-3/2) -> (-7/6,-1/1) Hyperbolic Matrix(69,100,20,29) (-3/2,-10/7) -> (10/3,7/2) Hyperbolic Matrix(71,100,-120,-169) (-10/7,-7/5) -> (-3/5,-10/17) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(339,400,-628,-741) (-13/11,-20/17) -> (-20/37,-7/13) Hyperbolic Matrix(341,400,52,61) (-20/17,-7/6) -> (13/2,20/3) Hyperbolic Matrix(91,80,-124,-109) (-1/1,-6/7) -> (-14/19,-11/15) Hyperbolic Matrix(71,60,84,71) (-6/7,-5/6) -> (5/6,6/7) Hyperbolic Matrix(49,40,60,49) (-5/6,-4/5) -> (4/5,5/6) Hyperbolic Matrix(51,40,-88,-69) (-4/5,-3/4) -> (-7/12,-4/7) Hyperbolic Matrix(271,200,-500,-369) (-3/4,-14/19) -> (-6/11,-13/24) Hyperbolic Matrix(329,240,-536,-391) (-11/15,-8/11) -> (-8/13,-19/31) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(129,80,208,129) (-5/8,-8/13) -> (8/13,5/8) Hyperbolic Matrix(2941,1800,1080,661) (-30/49,-11/18) -> (49/18,30/11) Hyperbolic Matrix(131,80,-244,-149) (-11/18,-3/5) -> (-7/13,-1/2) Hyperbolic Matrix(341,200,104,61) (-10/17,-7/12) -> (13/4,10/3) Hyperbolic Matrix(1479,800,220,119) (-13/24,-20/37) -> (20/3,27/4) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(71,-40,16,-9) (1/2,4/7) -> (4/1,9/2) Hyperbolic Matrix(69,-40,88,-51) (4/7,3/5) -> (7/9,4/5) Hyperbolic Matrix(361,-220,64,-39) (3/5,11/18) -> (11/2,17/3) Hyperbolic Matrix(619,-380,360,-221) (11/18,8/13) -> (12/7,31/18) Hyperbolic Matrix(29,-20,16,-11) (2/3,3/4) -> (7/4,2/1) Hyperbolic Matrix(129,-100,40,-31) (3/4,7/9) -> (3/1,13/4) Hyperbolic Matrix(159,-140,92,-81) (6/7,1/1) -> (19/11,26/15) Hyperbolic Matrix(69,-80,44,-51) (1/1,6/5) -> (14/9,11/7) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(129,-200,20,-31) (3/2,14/9) -> (6/1,13/2) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(881,-1520,324,-559) (31/18,19/11) -> (19/7,49/18) Hyperbolic Matrix(369,-640,64,-111) (26/15,7/4) -> (23/4,6/1) Hyperbolic Matrix(29,-80,4,-11) (11/4,3/1) -> (7/1,1/0) Hyperbolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic Matrix(109,-620,16,-91) (17/3,23/4) -> (27/4,7/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,80,-4,-29) -> Matrix(1,0,0,1) Matrix(11,60,-20,-109) -> Matrix(1,0,-2,1) Matrix(9,40,-16,-71) -> Matrix(1,0,-2,1) Matrix(11,40,-8,-29) -> Matrix(1,0,0,1) Matrix(71,200,-60,-169) -> Matrix(3,2,-8,-5) Matrix(89,240,-56,-151) -> Matrix(1,0,0,1) Matrix(31,80,12,31) -> Matrix(3,2,-2,-1) Matrix(9,20,4,9) -> Matrix(1,0,2,1) Matrix(11,20,-16,-29) -> Matrix(1,0,-2,1) Matrix(49,80,-68,-111) -> Matrix(1,0,-2,1) Matrix(569,900,208,329) -> Matrix(1,0,0,1) Matrix(571,900,-932,-1469) -> Matrix(3,2,-8,-5) Matrix(51,80,-44,-69) -> Matrix(3,2,-8,-5) Matrix(69,100,20,29) -> Matrix(1,0,2,1) Matrix(71,100,-120,-169) -> Matrix(1,0,-2,1) Matrix(31,40,24,31) -> Matrix(3,2,-2,-1) Matrix(49,60,40,49) -> Matrix(9,4,2,1) Matrix(339,400,-628,-741) -> Matrix(5,2,-18,-7) Matrix(341,400,52,61) -> Matrix(1,0,2,1) Matrix(91,80,-124,-109) -> Matrix(1,0,0,1) Matrix(71,60,84,71) -> Matrix(1,0,4,1) Matrix(49,40,60,49) -> Matrix(1,0,4,1) Matrix(51,40,-88,-69) -> Matrix(1,0,0,1) Matrix(271,200,-500,-369) -> Matrix(1,0,0,1) Matrix(329,240,-536,-391) -> Matrix(1,0,0,1) Matrix(31,20,48,31) -> Matrix(1,0,4,1) Matrix(129,80,208,129) -> Matrix(1,0,4,1) Matrix(2941,1800,1080,661) -> Matrix(13,4,-10,-3) Matrix(131,80,-244,-149) -> Matrix(1,0,0,1) Matrix(341,200,104,61) -> Matrix(1,0,2,1) Matrix(1479,800,220,119) -> Matrix(1,0,2,1) Matrix(1,0,4,1) -> Matrix(1,0,6,1) Matrix(71,-40,16,-9) -> Matrix(1,0,-2,1) Matrix(69,-40,88,-51) -> Matrix(1,0,0,1) Matrix(361,-220,64,-39) -> Matrix(1,0,-2,1) Matrix(619,-380,360,-221) -> Matrix(1,0,-2,1) Matrix(29,-20,16,-11) -> Matrix(1,0,-2,1) Matrix(129,-100,40,-31) -> Matrix(1,0,-2,1) Matrix(159,-140,92,-81) -> Matrix(1,0,-2,1) Matrix(69,-80,44,-51) -> Matrix(1,-2,0,1) Matrix(29,-40,8,-11) -> Matrix(1,0,0,1) Matrix(129,-200,20,-31) -> Matrix(1,0,0,1) Matrix(151,-240,56,-89) -> Matrix(1,0,0,1) Matrix(61,-100,36,-59) -> Matrix(1,0,0,1) Matrix(881,-1520,324,-559) -> Matrix(3,2,-2,-1) Matrix(369,-640,64,-111) -> Matrix(1,0,0,1) Matrix(29,-80,4,-11) -> Matrix(1,0,0,1) Matrix(21,-100,4,-19) -> Matrix(1,0,0,1) Matrix(109,-620,16,-91) -> Matrix(1,0,0,1) 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): 6 Degree of the the map X: 6 Degree of the the map Y: 48 Permutation triple for Y: ((1,2)(3,10,8,7,20,6,16,5,4,11)(9,18,14,13,21,19,26,23,22,12)(15,35,17,36,39,24,41,25,29,30)(27,43,31,40,42,38,46,33,32,37)(28,34)(44,47)(45,48); (1,5,18,37,35,47,36,38,19,6)(2,8,26,42,41,44,29,27,9,3)(4,14,34,21,20,39,46,45,32,15)(7,23,28,12,11,30,43,48,40,24)(10,25)(13,33)(16,17)(22,31); (1,3,12,31,30,44,35,32,13,4)(2,6,21,33,39,47,41,40,22,7)(5,17,37,45,43,29,10,9,28,14)(8,25,42,48,46,36,16,19,34,23)(11,15)(18,27)(20,24)(26,38)) ----------------------------------------------------------------------- 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: 16 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/2 5/8 3/4 5/6 1/1 5/4 3/2 5/3 2/1 5/2 3/1 10/3 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -1/2 1/0 -4/1 -1/1 -3/1 -1/2 1/0 -5/2 -1/2 -2/1 0/1 -5/3 -1/2 1/0 -3/2 -1/2 -10/7 0/1 -7/5 -1/2 1/0 -4/3 -1/1 -5/4 -1/2 -1/1 -1/2 -1/4 -5/6 -1/2 -1/4 -4/5 -1/3 -3/4 -1/3 0/1 -5/7 -1/2 -1/4 -2/3 0/1 -5/8 -1/2 -1/4 -3/5 -1/2 -1/4 -10/17 0/1 -7/12 -1/3 0/1 -4/7 -1/3 -5/9 -1/2 -1/4 -1/2 -1/4 0/1 0/1 1/2 1/2 4/7 1/1 3/5 1/2 1/0 5/8 1/2 1/0 2/3 0/1 3/4 0/1 1/1 7/9 1/2 1/0 4/5 1/1 5/6 1/2 1/0 1/1 1/2 1/0 5/4 1/0 4/3 -1/1 3/2 1/0 5/3 -1/2 1/0 7/4 -1/1 0/1 2/1 0/1 5/2 1/0 3/1 -1/2 1/0 13/4 -1/1 0/1 10/3 0/1 7/2 1/0 4/1 -1/1 9/2 1/0 5/1 -1/2 1/0 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,10,0,1) (-5/1,1/0) -> (5/1,1/0) Parabolic Matrix(9,40,-16,-71) (-5/1,-4/1) -> (-4/7,-5/9) Hyperbolic Matrix(11,40,-8,-29) (-4/1,-3/1) -> (-7/5,-4/3) Hyperbolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(11,20,-16,-29) (-2/1,-5/3) -> (-5/7,-2/3) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(69,100,20,29) (-3/2,-10/7) -> (10/3,7/2) Hyperbolic Matrix(71,100,-120,-169) (-10/7,-7/5) -> (-3/5,-10/17) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(9,10,8,9) (-5/4,-1/1) -> (1/1,5/4) Hyperbolic Matrix(11,10,12,11) (-1/1,-5/6) -> (5/6,1/1) Hyperbolic Matrix(49,40,60,49) (-5/6,-4/5) -> (4/5,5/6) Hyperbolic Matrix(51,40,-88,-69) (-4/5,-3/4) -> (-7/12,-4/7) Hyperbolic Matrix(69,50,40,29) (-3/4,-5/7) -> (5/3,7/4) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(341,200,104,61) (-10/17,-7/12) -> (13/4,10/3) Hyperbolic Matrix(91,50,20,11) (-5/9,-1/2) -> (9/2,5/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(71,-40,16,-9) (1/2,4/7) -> (4/1,9/2) Hyperbolic Matrix(69,-40,88,-51) (4/7,3/5) -> (7/9,4/5) Hyperbolic Matrix(29,-20,16,-11) (2/3,3/4) -> (7/4,2/1) Hyperbolic Matrix(129,-100,40,-31) (3/4,7/9) -> (3/1,13/4) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,10,0,1) -> Matrix(1,1,-2,-1) Matrix(9,40,-16,-71) -> Matrix(1,0,-2,1) Matrix(11,40,-8,-29) -> Matrix(1,0,0,1) Matrix(11,30,4,11) -> Matrix(1,1,-2,-1) Matrix(9,20,4,9) -> Matrix(1,0,2,1) Matrix(11,20,-16,-29) -> Matrix(1,0,-2,1) Matrix(19,30,12,19) -> Matrix(1,1,-2,-1) Matrix(69,100,20,29) -> Matrix(1,0,2,1) Matrix(71,100,-120,-169) -> Matrix(1,0,-2,1) Matrix(31,40,24,31) -> Matrix(3,2,-2,-1) Matrix(9,10,8,9) -> Matrix(3,1,2,1) Matrix(11,10,12,11) -> Matrix(3,1,2,1) Matrix(49,40,60,49) -> Matrix(1,0,4,1) Matrix(51,40,-88,-69) -> Matrix(1,0,0,1) Matrix(69,50,40,29) -> Matrix(3,1,-4,-1) Matrix(31,20,48,31) -> Matrix(1,0,4,1) Matrix(49,30,80,49) -> Matrix(3,1,2,1) Matrix(341,200,104,61) -> Matrix(1,0,2,1) Matrix(91,50,20,11) -> Matrix(3,1,-4,-1) Matrix(1,0,4,1) -> Matrix(1,0,6,1) Matrix(71,-40,16,-9) -> Matrix(1,0,-2,1) Matrix(69,-40,88,-51) -> Matrix(1,0,0,1) Matrix(29,-20,16,-11) -> Matrix(1,0,-2,1) Matrix(129,-100,40,-31) -> Matrix(1,0,-2,1) Matrix(29,-40,8,-11) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 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 0/1 0/1 3 2 1/2 1/2 1 10 4/7 1/1 1 10 3/5 (1/2,1/0) 0 10 5/8 (0/1,1/1).(1/2,1/0) 0 2 2/3 0/1 1 10 3/4 (0/1,1/1).(1/2,1/0) 0 10 7/9 (1/2,1/0) 0 10 4/5 1/1 1 10 5/6 (0/1,1/1).(1/2,1/0) 0 2 1/1 (1/2,1/0) 0 10 5/4 1/0 3 2 4/3 -1/1 1 10 3/2 1/0 1 10 5/3 (-1/2,1/0) 0 2 7/4 (-1/1,0/1).(-1/2,1/0) 0 10 2/1 0/1 1 10 5/2 1/0 1 2 3/1 (-1/2,1/0) 0 10 13/4 (-1/1,0/1).(-1/2,1/0) 0 10 10/3 0/1 1 2 7/2 1/0 1 10 4/1 -1/1 1 10 9/2 1/0 1 10 5/1 (-1/2,1/0) 0 2 1/0 (-1/1,0/1).(-1/2,1/0) 0 10 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(71,-40,16,-9) (1/2,4/7) -> (4/1,9/2) Hyperbolic Matrix(69,-40,88,-51) (4/7,3/5) -> (7/9,4/5) Hyperbolic Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(31,-20,48,-31) (5/8,2/3) -> (5/8,2/3) Reflection Matrix(29,-20,16,-11) (2/3,3/4) -> (7/4,2/1) Hyperbolic Matrix(129,-100,40,-31) (3/4,7/9) -> (3/1,13/4) Hyperbolic Matrix(49,-40,60,-49) (4/5,5/6) -> (4/5,5/6) Reflection Matrix(11,-10,12,-11) (5/6,1/1) -> (5/6,1/1) Reflection Matrix(9,-10,8,-9) (1/1,5/4) -> (1/1,5/4) Reflection Matrix(31,-40,24,-31) (5/4,4/3) -> (5/4,4/3) Reflection Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(41,-70,24,-41) (5/3,7/4) -> (5/3,7/4) Reflection Matrix(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(79,-260,24,-79) (13/4,10/3) -> (13/4,10/3) Reflection Matrix(41,-140,12,-41) (10/3,7/2) -> (10/3,7/2) Reflection Matrix(19,-90,4,-19) (9/2,5/1) -> (9/2,5/1) Reflection Matrix(-1,10,0,1) (5/1,1/0) -> (5/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,4,-1) -> Matrix(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Matrix(71,-40,16,-9) -> Matrix(1,0,-2,1) 0/1 Matrix(69,-40,88,-51) -> Matrix(1,0,0,1) Matrix(49,-30,80,-49) -> Matrix(-1,1,0,1) (3/5,5/8) -> (1/2,1/0) Matrix(31,-20,48,-31) -> Matrix(1,0,2,-1) (5/8,2/3) -> (0/1,1/1) Matrix(29,-20,16,-11) -> Matrix(1,0,-2,1) 0/1 Matrix(129,-100,40,-31) -> Matrix(1,0,-2,1) 0/1 Matrix(49,-40,60,-49) -> Matrix(1,0,2,-1) (4/5,5/6) -> (0/1,1/1) Matrix(11,-10,12,-11) -> Matrix(-1,1,0,1) (5/6,1/1) -> (1/2,1/0) Matrix(9,-10,8,-9) -> Matrix(-1,1,0,1) (1/1,5/4) -> (1/2,1/0) Matrix(31,-40,24,-31) -> Matrix(1,2,0,-1) (5/4,4/3) -> (-1/1,1/0) Matrix(29,-40,8,-11) -> Matrix(1,0,0,1) Matrix(19,-30,12,-19) -> Matrix(1,1,0,-1) (3/2,5/3) -> (-1/2,1/0) Matrix(41,-70,24,-41) -> Matrix(1,1,0,-1) (5/3,7/4) -> (-1/2,1/0) Matrix(9,-20,4,-9) -> Matrix(1,0,0,-1) (2/1,5/2) -> (0/1,1/0) Matrix(11,-30,4,-11) -> Matrix(1,1,0,-1) (5/2,3/1) -> (-1/2,1/0) Matrix(79,-260,24,-79) -> Matrix(-1,0,2,1) (13/4,10/3) -> (-1/1,0/1) Matrix(41,-140,12,-41) -> Matrix(1,0,0,-1) (10/3,7/2) -> (0/1,1/0) Matrix(19,-90,4,-19) -> Matrix(1,1,0,-1) (9/2,5/1) -> (-1/2,1/0) Matrix(-1,10,0,1) -> Matrix(1,1,0,-1) (5/1,1/0) -> (-1/2,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.