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): 576 Minimal number of generators: 97 Number of equivalence classes of cusps: 48 Genus: 25 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -7/1 -6/1 -5/1 -4/1 -11/3 -3/1 -8/3 -5/2 -2/1 -5/3 -5/4 -1/1 -5/6 -5/7 -5/8 -5/9 0/1 1/2 5/9 5/8 5/7 3/4 5/6 1/1 5/4 10/7 3/2 30/19 5/3 7/4 2/1 20/9 9/4 5/2 8/3 30/11 20/7 3/1 10/3 7/2 11/3 4/1 9/2 5/1 6/1 20/3 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -3/1 -6/1 -2/1 -5/1 -3/1 -1/1 -14/3 -2/1 -9/2 -2/1 -4/1 1/0 -11/3 -3/1 -7/2 -2/1 -10/3 -2/1 -3/1 -1/1 -14/5 -2/1 -11/4 1/0 -30/11 -1/1 -19/7 -1/1 -8/3 1/0 -5/2 1/0 -12/5 1/0 -19/8 1/0 -7/3 -3/1 -9/4 -5/2 -20/9 -2/1 -11/5 -1/1 -2/1 -2/1 -11/6 -2/1 -20/11 -2/1 -9/5 -5/3 -7/4 -3/2 -5/3 -1/1 -13/8 1/0 -21/13 -1/1 -8/5 1/0 -19/12 1/0 -30/19 1/0 -11/7 -3/1 -14/9 -2/1 -17/11 -1/1 -20/13 1/0 -3/2 -2/1 -10/7 -1/1 -7/5 -1/1 -11/8 -3/2 -4/3 1/0 -9/7 -1/1 -5/4 -2/1 -11/9 -5/3 -6/5 -2/1 -13/11 -1/1 -20/17 -1/1 -7/6 -2/1 -1/1 -1/1 -7/8 1/0 -6/7 0/1 -5/6 1/0 -14/17 -2/1 -9/11 -3/1 -4/5 1/0 -11/14 -2/1 -7/9 -1/1 -10/13 -1/1 -3/4 1/0 -14/19 -2/1 -11/15 -1/1 -30/41 1/0 -19/26 -4/1 -8/11 1/0 -5/7 -3/1 -1/1 -12/17 1/0 -19/27 -3/1 -7/10 -2/1 -9/13 -3/1 -2/3 -2/1 -9/14 -2/1 -7/11 -1/1 -5/8 -2/1 -13/21 -5/3 -21/34 -8/5 -8/13 -3/2 -19/31 -1/1 -30/49 -1/1 -11/18 -2/1 -14/23 -2/1 -17/28 -3/2 -20/33 -3/2 -3/5 -1/1 -10/17 -2/1 -7/12 -3/2 -11/19 -5/3 -4/7 -3/2 -9/16 -5/4 -5/9 -1/1 -11/20 -1/2 -6/11 0/1 -13/24 1/0 -20/37 -1/1 -7/13 -1/1 -1/2 -2/1 0/1 -1/1 1/2 0/1 7/13 -1/1 6/11 -2/1 5/9 -1/1 4/7 -1/2 7/12 -1/2 3/5 -1/1 11/18 0/1 8/13 -1/2 5/8 0/1 12/19 1/0 19/30 -2/1 7/11 -1/1 9/14 0/1 2/3 0/1 7/10 0/1 5/7 -1/1 1/1 13/18 0/1 8/11 1/0 19/26 2/1 11/15 -1/1 14/19 0/1 3/4 1/0 7/9 -1/1 4/5 1/0 9/11 1/1 5/6 1/0 11/13 -1/1 6/7 -2/1 1/1 -1/1 7/6 0/1 6/5 0/1 5/4 0/1 14/11 0/1 9/7 -1/1 4/3 1/0 15/11 -1/1 11/8 -1/2 7/5 -1/1 17/12 1/0 10/7 -1/1 3/2 0/1 14/9 0/1 11/7 1/1 41/26 4/1 30/19 1/0 19/12 1/0 8/5 1/0 5/3 -1/1 12/7 -1/2 31/18 0/1 19/11 -1/1 26/15 0/1 33/19 -1/1 7/4 -1/2 16/9 -1/2 9/5 -1/3 20/11 0/1 11/6 0/1 2/1 0/1 11/5 -1/1 20/9 0/1 9/4 1/2 7/3 1/1 5/2 1/0 13/5 -1/1 21/8 1/0 8/3 1/0 27/10 -2/1 19/7 -1/1 49/18 -2/1 30/11 -1/1 11/4 1/0 25/9 -1/1 14/5 0/1 17/6 0/1 20/7 1/0 3/1 -1/1 13/4 -1/2 10/3 0/1 7/2 0/1 18/5 0/1 11/3 1/1 4/1 1/0 13/3 -1/1 9/2 0/1 5/1 -1/1 1/1 11/2 0/1 17/3 1/1 40/7 1/0 23/4 1/0 6/1 0/1 13/2 0/1 20/3 1/1 27/4 1/0 7/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(21,160,8,61) (-7/1,1/0) -> (13/5,21/8) Hyperbolic Matrix(19,120,-16,-101) (-7/1,-6/1) -> (-6/5,-13/11) Hyperbolic Matrix(19,100,-4,-21) (-6/1,-5/1) -> (-5/1,-14/3) Parabolic Matrix(61,280,-100,-459) (-14/3,-9/2) -> (-11/18,-14/23) Hyperbolic Matrix(19,80,-24,-101) (-9/2,-4/1) -> (-4/5,-11/14) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(39,140,-56,-201) (-11/3,-7/2) -> (-7/10,-9/13) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(19,60,-32,-101) (-10/3,-3/1) -> (-3/5,-10/17) Hyperbolic Matrix(99,280,-64,-181) (-3/1,-14/5) -> (-14/9,-17/11) Hyperbolic Matrix(79,220,-144,-401) (-14/5,-11/4) -> (-11/20,-6/11) Hyperbolic Matrix(241,660,88,241) (-11/4,-30/11) -> (30/11,11/4) Hyperbolic Matrix(419,1140,-684,-1861) (-30/11,-19/7) -> (-19/31,-30/49) Hyperbolic Matrix(141,380,-200,-539) (-19/7,-8/3) -> (-12/17,-19/27) Hyperbolic Matrix(39,100,-16,-41) (-8/3,-5/2) -> (-5/2,-12/5) Parabolic Matrix(159,380,100,239) (-12/5,-19/8) -> (19/12,8/5) Hyperbolic Matrix(59,140,8,19) (-19/8,-7/3) -> (7/1,1/0) Hyperbolic Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(161,360,72,161) (-9/4,-20/9) -> (20/9,9/4) Hyperbolic Matrix(181,400,100,221) (-20/9,-11/5) -> (9/5,20/11) Hyperbolic Matrix(101,220,28,61) (-11/5,-2/1) -> (18/5,11/3) Hyperbolic Matrix(21,40,32,61) (-2/1,-11/6) -> (9/14,2/3) Hyperbolic Matrix(241,440,132,241) (-11/6,-20/11) -> (20/11,11/6) Hyperbolic Matrix(221,400,100,181) (-20/11,-9/5) -> (11/5,20/9) Hyperbolic Matrix(79,140,-136,-241) (-9/5,-7/4) -> (-7/12,-11/19) Hyperbolic Matrix(59,100,-36,-61) (-7/4,-5/3) -> (-5/3,-13/8) Parabolic Matrix(99,160,-112,-181) (-13/8,-21/13) -> (-1/1,-7/8) Hyperbolic Matrix(199,320,-324,-521) (-21/13,-8/5) -> (-8/13,-19/31) Hyperbolic Matrix(201,320,76,121) (-8/5,-19/12) -> (21/8,8/3) Hyperbolic Matrix(721,1140,456,721) (-19/12,-30/19) -> (30/19,19/12) Hyperbolic Matrix(419,660,-572,-901) (-30/19,-11/7) -> (-11/15,-30/41) Hyperbolic Matrix(179,280,140,219) (-11/7,-14/9) -> (14/11,9/7) Hyperbolic Matrix(259,400,-428,-661) (-17/11,-20/13) -> (-20/33,-3/5) Hyperbolic Matrix(261,400,92,141) (-20/13,-3/2) -> (17/6,20/7) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(99,140,-128,-181) (-10/7,-7/5) -> (-7/9,-10/13) Hyperbolic Matrix(101,140,44,61) (-7/5,-11/8) -> (9/4,7/3) Hyperbolic Matrix(59,80,-104,-141) (-11/8,-4/3) -> (-4/7,-9/16) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(79,100,-64,-81) (-9/7,-5/4) -> (-5/4,-11/9) Parabolic Matrix(181,220,116,141) (-11/9,-6/5) -> (14/9,11/7) 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(121,140,-172,-199) (-7/6,-1/1) -> (-19/27,-7/10) Hyperbolic Matrix(139,120,-256,-221) (-7/8,-6/7) -> (-6/11,-13/24) Hyperbolic Matrix(119,100,-144,-121) (-6/7,-5/6) -> (-5/6,-14/17) Parabolic Matrix(341,280,464,381) (-14/17,-9/11) -> (11/15,14/19) Hyperbolic Matrix(221,180,124,101) (-9/11,-4/5) -> (16/9,9/5) Hyperbolic Matrix(179,140,280,219) (-11/14,-7/9) -> (7/11,9/14) Hyperbolic Matrix(261,200,184,141) (-10/13,-3/4) -> (17/12,10/7) Hyperbolic Matrix(379,280,-624,-461) (-3/4,-14/19) -> (-14/23,-17/28) Hyperbolic Matrix(299,220,352,259) (-14/19,-11/15) -> (11/13,6/7) Hyperbolic Matrix(2461,1800,1560,1141) (-30/41,-19/26) -> (41/26,30/19) Hyperbolic Matrix(521,380,824,601) (-19/26,-8/11) -> (12/19,19/30) Hyperbolic Matrix(139,100,-196,-141) (-8/11,-5/7) -> (-5/7,-12/17) Parabolic Matrix(59,40,28,19) (-9/13,-2/3) -> (2/1,11/5) Hyperbolic Matrix(61,40,32,21) (-2/3,-9/14) -> (11/6,2/1) Hyperbolic Matrix(281,180,64,41) (-9/14,-7/11) -> (13/3,9/2) Hyperbolic Matrix(159,100,-256,-161) (-7/11,-5/8) -> (-5/8,-13/21) Parabolic Matrix(259,160,484,299) (-13/21,-21/34) -> (1/2,7/13) Hyperbolic Matrix(519,320,712,439) (-21/34,-8/13) -> (8/11,19/26) Hyperbolic Matrix(2941,1800,1080,661) (-30/49,-11/18) -> (49/18,30/11) Hyperbolic Matrix(1879,1140,328,199) (-17/28,-20/33) -> (40/7,23/4) Hyperbolic Matrix(341,200,104,61) (-10/17,-7/12) -> (13/4,10/3) Hyperbolic Matrix(139,80,172,99) (-11/19,-4/7) -> (4/5,9/11) Hyperbolic Matrix(179,100,-324,-181) (-9/16,-5/9) -> (-5/9,-11/20) Parabolic Matrix(1479,800,220,119) (-13/24,-20/37) -> (20/3,27/4) Hyperbolic Matrix(261,140,412,221) (-7/13,-1/2) -> (19/30,7/11) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(701,-380,404,-219) (7/13,6/11) -> (26/15,33/19) Hyperbolic Matrix(401,-220,144,-79) (6/11,5/9) -> (25/9,14/5) Hyperbolic Matrix(141,-80,104,-59) (5/9,4/7) -> (4/3,15/11) Hyperbolic Matrix(241,-140,136,-79) (4/7,7/12) -> (7/4,16/9) Hyperbolic Matrix(101,-60,32,-19) (7/12,3/5) -> (3/1,13/4) 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(161,-100,256,-159) (8/13,5/8) -> (5/8,12/19) Parabolic Matrix(201,-140,56,-39) (2/3,7/10) -> (7/2,18/5) Hyperbolic Matrix(141,-100,196,-139) (7/10,5/7) -> (5/7,13/18) Parabolic Matrix(441,-320,164,-119) (13/18,8/11) -> (8/3,27/10) Hyperbolic Matrix(901,-660,572,-419) (19/26,11/15) -> (11/7,41/26) Hyperbolic Matrix(461,-340,80,-59) (14/19,3/4) -> (23/4,6/1) Hyperbolic Matrix(181,-140,128,-99) (3/4,7/9) -> (7/5,17/12) Hyperbolic Matrix(101,-80,24,-19) (7/9,4/5) -> (4/1,13/3) Hyperbolic Matrix(121,-100,144,-119) (9/11,5/6) -> (5/6,11/13) Parabolic Matrix(159,-140,92,-81) (6/7,1/1) -> (19/11,26/15) Hyperbolic Matrix(141,-160,52,-59) (1/1,7/6) -> (27/10,19/7) Hyperbolic Matrix(101,-120,16,-19) (7/6,6/5) -> (6/1,13/2) Hyperbolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(321,-440,116,-159) (15/11,11/8) -> (11/4,25/9) Hyperbolic Matrix(181,-280,64,-99) (3/2,14/9) -> (14/5,17/6) 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(541,-940,80,-139) (33/19,7/4) -> (27/4,7/1) Hyperbolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(159,-460,28,-81) (20/7,3/1) -> (17/3,40/7) Hyperbolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(21,160,8,61) -> Matrix(1,2,0,1) Matrix(19,120,-16,-101) -> Matrix(3,8,-2,-5) Matrix(19,100,-4,-21) -> Matrix(1,0,0,1) Matrix(61,280,-100,-459) -> Matrix(3,8,-2,-5) Matrix(19,80,-24,-101) -> Matrix(1,0,0,1) Matrix(21,80,16,61) -> Matrix(1,2,0,1) Matrix(39,140,-56,-201) -> Matrix(1,0,0,1) Matrix(41,140,12,41) -> Matrix(1,2,0,1) Matrix(19,60,-32,-101) -> Matrix(1,0,0,1) Matrix(99,280,-64,-181) -> Matrix(1,0,0,1) Matrix(79,220,-144,-401) -> Matrix(1,2,-2,-3) Matrix(241,660,88,241) -> Matrix(1,0,0,1) Matrix(419,1140,-684,-1861) -> Matrix(3,2,-2,-1) Matrix(141,380,-200,-539) -> Matrix(1,-2,0,1) Matrix(39,100,-16,-41) -> Matrix(1,-2,0,1) Matrix(159,380,100,239) -> Matrix(1,2,0,1) Matrix(59,140,8,19) -> Matrix(1,4,0,1) Matrix(61,140,44,101) -> Matrix(1,2,0,1) Matrix(161,360,72,161) -> Matrix(1,2,4,9) Matrix(181,400,100,221) -> Matrix(1,2,-4,-7) Matrix(101,220,28,61) -> Matrix(1,2,0,1) Matrix(21,40,32,61) -> Matrix(1,2,-2,-3) Matrix(241,440,132,241) -> Matrix(1,2,0,1) Matrix(221,400,100,181) -> Matrix(1,2,-4,-7) Matrix(79,140,-136,-241) -> Matrix(1,0,0,1) Matrix(59,100,-36,-61) -> Matrix(1,2,-2,-3) Matrix(99,160,-112,-181) -> Matrix(1,0,0,1) Matrix(199,320,-324,-521) -> Matrix(3,2,-2,-1) Matrix(201,320,76,121) -> Matrix(1,0,0,1) Matrix(721,1140,456,721) -> Matrix(1,-2,0,1) Matrix(419,660,-572,-901) -> Matrix(1,2,0,1) Matrix(179,280,140,219) -> Matrix(1,2,0,1) Matrix(259,400,-428,-661) -> Matrix(3,2,-2,-1) Matrix(261,400,92,141) -> Matrix(1,2,0,1) Matrix(41,60,28,41) -> Matrix(1,2,-2,-3) Matrix(99,140,-128,-181) -> Matrix(1,0,0,1) Matrix(101,140,44,61) -> Matrix(1,2,0,1) Matrix(59,80,-104,-141) -> Matrix(3,2,-2,-1) Matrix(61,80,16,21) -> Matrix(1,2,0,1) Matrix(79,100,-64,-81) -> Matrix(3,8,-2,-5) Matrix(181,220,116,141) -> Matrix(1,2,-2,-3) Matrix(339,400,-628,-741) -> Matrix(1,2,-2,-3) Matrix(341,400,52,61) -> Matrix(1,2,0,1) Matrix(121,140,-172,-199) -> Matrix(5,8,-2,-3) Matrix(139,120,-256,-221) -> Matrix(1,0,0,1) Matrix(119,100,-144,-121) -> Matrix(1,-2,0,1) Matrix(341,280,464,381) -> Matrix(1,2,0,1) Matrix(221,180,124,101) -> Matrix(1,2,-2,-3) Matrix(179,140,280,219) -> Matrix(1,2,-2,-3) Matrix(261,200,184,141) -> Matrix(1,0,0,1) Matrix(379,280,-624,-461) -> Matrix(3,8,-2,-5) Matrix(299,220,352,259) -> Matrix(1,0,0,1) Matrix(2461,1800,1560,1141) -> Matrix(1,8,0,1) Matrix(521,380,824,601) -> Matrix(1,2,0,1) Matrix(139,100,-196,-141) -> Matrix(1,0,0,1) Matrix(59,40,28,19) -> Matrix(1,2,0,1) Matrix(61,40,32,21) -> Matrix(1,2,-2,-3) Matrix(281,180,64,41) -> Matrix(1,2,-2,-3) Matrix(159,100,-256,-161) -> Matrix(3,8,-2,-5) Matrix(259,160,484,299) -> Matrix(5,8,-2,-3) Matrix(519,320,712,439) -> Matrix(1,2,-2,-3) Matrix(2941,1800,1080,661) -> Matrix(1,0,0,1) Matrix(1879,1140,328,199) -> Matrix(1,2,-2,-3) Matrix(341,200,104,61) -> Matrix(1,2,-4,-7) Matrix(139,80,172,99) -> Matrix(1,2,-2,-3) Matrix(179,100,-324,-181) -> Matrix(5,6,-6,-7) Matrix(1479,800,220,119) -> Matrix(1,2,0,1) Matrix(261,140,412,221) -> Matrix(1,0,0,1) Matrix(1,0,4,1) -> Matrix(1,2,-2,-3) Matrix(701,-380,404,-219) -> Matrix(1,2,-2,-3) Matrix(401,-220,144,-79) -> Matrix(1,2,-2,-3) Matrix(141,-80,104,-59) -> Matrix(3,2,-2,-1) Matrix(241,-140,136,-79) -> Matrix(1,0,0,1) Matrix(101,-60,32,-19) -> Matrix(1,0,0,1) Matrix(361,-220,64,-39) -> Matrix(1,0,2,1) Matrix(619,-380,360,-221) -> Matrix(1,0,0,1) Matrix(161,-100,256,-159) -> Matrix(1,0,2,1) Matrix(201,-140,56,-39) -> Matrix(1,0,0,1) Matrix(141,-100,196,-139) -> Matrix(1,0,0,1) Matrix(441,-320,164,-119) -> Matrix(1,-2,0,1) Matrix(901,-660,572,-419) -> Matrix(1,2,0,1) Matrix(461,-340,80,-59) -> Matrix(1,0,0,1) Matrix(181,-140,128,-99) -> Matrix(1,0,0,1) Matrix(101,-80,24,-19) -> Matrix(1,0,0,1) Matrix(121,-100,144,-119) -> Matrix(1,-2,0,1) Matrix(159,-140,92,-81) -> Matrix(1,2,-2,-3) Matrix(141,-160,52,-59) -> Matrix(3,2,-2,-1) Matrix(101,-120,16,-19) -> Matrix(1,0,2,1) Matrix(81,-100,64,-79) -> Matrix(1,0,2,1) Matrix(321,-440,116,-159) -> Matrix(3,2,-2,-1) Matrix(181,-280,64,-99) -> Matrix(1,0,0,1) Matrix(61,-100,36,-59) -> Matrix(1,2,-2,-3) Matrix(881,-1520,324,-559) -> Matrix(3,2,-2,-1) Matrix(541,-940,80,-139) -> Matrix(1,0,2,1) Matrix(41,-100,16,-39) -> Matrix(1,-2,0,1) Matrix(159,-460,28,-81) -> Matrix(1,2,0,1) Matrix(21,-100,4,-19) -> 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): 20 Degree of the the map X: 20 Degree of the the map Y: 96 ----------------------------------------------------------------------- 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): 288 Minimal number of generators: 49 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: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -5/1 -4/1 -11/3 -3/1 -5/2 -2/1 -5/3 -5/4 -1/1 -5/7 -5/9 0/1 1/2 5/9 5/7 1/1 5/4 10/7 3/2 5/3 2/1 20/9 7/3 5/2 3/1 10/3 7/2 11/3 4/1 5/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 -3/1 -1/1 -9/2 -2/1 -4/1 1/0 -11/3 -3/1 -7/2 -2/1 -10/3 -2/1 -3/1 -1/1 -5/2 1/0 -7/3 -3/1 -9/4 -5/2 -20/9 -2/1 -11/5 -1/1 -2/1 -2/1 -5/3 -1/1 -8/5 1/0 -19/12 1/0 -30/19 1/0 -11/7 -3/1 -3/2 -2/1 -10/7 -1/1 -7/5 -1/1 -11/8 -3/2 -4/3 1/0 -9/7 -1/1 -5/4 -2/1 -1/1 -1/1 -5/6 1/0 -9/11 -3/1 -4/5 1/0 -11/14 -2/1 -7/9 -1/1 -10/13 -1/1 -3/4 1/0 -5/7 -3/1 -1/1 -7/10 -2/1 -9/13 -3/1 -2/3 -2/1 -9/14 -2/1 -7/11 -1/1 -5/8 -2/1 -3/5 -1/1 -10/17 -2/1 -7/12 -3/2 -11/19 -5/3 -4/7 -3/2 -5/9 -1/1 -6/11 0/1 -1/2 -2/1 0/1 -1/1 1/2 0/1 5/9 -1/1 4/7 -1/2 7/12 -1/2 3/5 -1/1 2/3 0/1 5/7 -1/1 1/1 8/11 1/0 3/4 1/0 7/9 -1/1 4/5 1/0 1/1 -1/1 6/5 0/1 5/4 0/1 14/11 0/1 9/7 -1/1 4/3 1/0 15/11 -1/1 11/8 -1/2 7/5 -1/1 17/12 1/0 10/7 -1/1 3/2 0/1 5/3 -1/1 7/4 -1/2 16/9 -1/2 9/5 -1/3 20/11 0/1 11/6 0/1 2/1 0/1 11/5 -1/1 20/9 0/1 9/4 1/2 7/3 1/1 5/2 1/0 13/5 -1/1 21/8 1/0 8/3 1/0 3/1 -1/1 13/4 -1/2 10/3 0/1 7/2 0/1 18/5 0/1 11/3 1/1 4/1 1/0 5/1 -1/1 1/1 6/1 0/1 7/1 1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(9,50,-2,-11) (-5/1,1/0) -> (-5/1,-9/2) Parabolic Matrix(19,80,-24,-101) (-9/2,-4/1) -> (-4/5,-11/14) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(39,140,-56,-201) (-11/3,-7/2) -> (-7/10,-9/13) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(19,60,-32,-101) (-10/3,-3/1) -> (-3/5,-10/17) Hyperbolic Matrix(11,30,-18,-49) (-3/1,-5/2) -> (-5/8,-3/5) Hyperbolic Matrix(29,70,-46,-111) (-5/2,-7/3) -> (-7/11,-5/8) Hyperbolic Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(161,360,72,161) (-9/4,-20/9) -> (20/9,9/4) Hyperbolic Matrix(181,400,100,221) (-20/9,-11/5) -> (9/5,20/11) Hyperbolic Matrix(101,220,28,61) (-11/5,-2/1) -> (18/5,11/3) Hyperbolic Matrix(29,50,-18,-31) (-2/1,-5/3) -> (-5/3,-8/5) Parabolic Matrix(201,320,76,121) (-8/5,-19/12) -> (21/8,8/3) Hyperbolic Matrix(449,710,246,389) (-19/12,-30/19) -> (20/11,11/6) Hyperbolic Matrix(349,550,158,249) (-30/19,-11/7) -> (11/5,20/9) Hyperbolic Matrix(71,110,-122,-189) (-11/7,-3/2) -> (-7/12,-11/19) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(99,140,-128,-181) (-10/7,-7/5) -> (-7/9,-10/13) Hyperbolic Matrix(101,140,44,61) (-7/5,-11/8) -> (9/4,7/3) Hyperbolic Matrix(51,70,-94,-129) (-11/8,-4/3) -> (-6/11,-1/2) Hyperbolic Matrix(61,80,16,21) (-4/3,-9/7) -> (11/3,4/1) Hyperbolic Matrix(71,90,-86,-109) (-9/7,-5/4) -> (-5/6,-9/11) Hyperbolic Matrix(9,10,-10,-11) (-5/4,-1/1) -> (-1/1,-5/6) Parabolic Matrix(221,180,124,101) (-9/11,-4/5) -> (16/9,9/5) Hyperbolic Matrix(371,290,142,111) (-11/14,-7/9) -> (13/5,21/8) Hyperbolic Matrix(261,200,184,141) (-10/13,-3/4) -> (17/12,10/7) Hyperbolic Matrix(69,50,-98,-71) (-3/4,-5/7) -> (-5/7,-7/10) Parabolic Matrix(59,40,28,19) (-9/13,-2/3) -> (2/1,11/5) Hyperbolic Matrix(61,40,32,21) (-2/3,-9/14) -> (11/6,2/1) Hyperbolic Matrix(109,70,14,9) (-9/14,-7/11) -> (7/1,1/0) Hyperbolic Matrix(341,200,104,61) (-10/17,-7/12) -> (13/4,10/3) Hyperbolic Matrix(329,190,258,149) (-11/19,-4/7) -> (14/11,9/7) Hyperbolic Matrix(89,50,-162,-91) (-4/7,-5/9) -> (-5/9,-6/11) Parabolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(129,-70,94,-51) (1/2,5/9) -> (15/11,11/8) Hyperbolic Matrix(141,-80,104,-59) (5/9,4/7) -> (4/3,15/11) Hyperbolic Matrix(241,-140,136,-79) (4/7,7/12) -> (7/4,16/9) Hyperbolic Matrix(101,-60,32,-19) (7/12,3/5) -> (3/1,13/4) Hyperbolic Matrix(49,-30,18,-11) (3/5,2/3) -> (8/3,3/1) Hyperbolic Matrix(71,-50,98,-69) (2/3,5/7) -> (5/7,8/11) Parabolic Matrix(149,-110,42,-31) (8/11,3/4) -> (7/2,18/5) Hyperbolic Matrix(181,-140,128,-99) (3/4,7/9) -> (7/5,17/12) Hyperbolic Matrix(89,-70,14,-11) (7/9,4/5) -> (6/1,7/1) Hyperbolic Matrix(11,-10,10,-9) (4/5,1/1) -> (1/1,6/5) Parabolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(9,50,-2,-11) -> Matrix(2,5,-1,-2) Matrix(19,80,-24,-101) -> Matrix(1,0,0,1) Matrix(21,80,16,61) -> Matrix(1,2,0,1) Matrix(39,140,-56,-201) -> Matrix(1,0,0,1) Matrix(41,140,12,41) -> Matrix(1,2,0,1) Matrix(19,60,-32,-101) -> Matrix(1,0,0,1) Matrix(11,30,-18,-49) -> Matrix(2,1,-1,0) Matrix(29,70,-46,-111) -> Matrix(2,5,-1,-2) Matrix(61,140,44,101) -> Matrix(1,2,0,1) Matrix(161,360,72,161) -> Matrix(1,2,4,9) Matrix(181,400,100,221) -> Matrix(1,2,-4,-7) Matrix(101,220,28,61) -> Matrix(1,2,0,1) Matrix(29,50,-18,-31) -> Matrix(0,-1,1,2) Matrix(201,320,76,121) -> Matrix(1,0,0,1) Matrix(449,710,246,389) -> Matrix(0,-1,1,0) Matrix(349,550,158,249) -> Matrix(0,-1,1,4) Matrix(71,110,-122,-189) -> Matrix(2,1,-1,0) Matrix(41,60,28,41) -> Matrix(1,2,-2,-3) Matrix(99,140,-128,-181) -> Matrix(1,0,0,1) Matrix(101,140,44,61) -> Matrix(1,2,0,1) Matrix(51,70,-94,-129) -> Matrix(0,-1,1,2) Matrix(61,80,16,21) -> Matrix(1,2,0,1) Matrix(71,90,-86,-109) -> Matrix(2,5,-1,-2) Matrix(9,10,-10,-11) -> Matrix(0,-1,1,2) Matrix(221,180,124,101) -> Matrix(1,2,-2,-3) Matrix(371,290,142,111) -> Matrix(0,-1,1,2) Matrix(261,200,184,141) -> Matrix(1,0,0,1) Matrix(69,50,-98,-71) -> Matrix(2,5,-1,-2) Matrix(59,40,28,19) -> Matrix(1,2,0,1) Matrix(61,40,32,21) -> Matrix(1,2,-2,-3) Matrix(109,70,14,9) -> Matrix(2,3,1,2) Matrix(341,200,104,61) -> Matrix(1,2,-4,-7) Matrix(329,190,258,149) -> Matrix(2,3,1,2) Matrix(89,50,-162,-91) -> Matrix(2,3,-3,-4) Matrix(1,0,4,1) -> Matrix(1,2,-2,-3) Matrix(129,-70,94,-51) -> Matrix(0,-1,1,2) Matrix(141,-80,104,-59) -> Matrix(3,2,-2,-1) Matrix(241,-140,136,-79) -> Matrix(1,0,0,1) Matrix(101,-60,32,-19) -> Matrix(1,0,0,1) Matrix(49,-30,18,-11) -> Matrix(2,1,-1,0) Matrix(71,-50,98,-69) -> Matrix(0,-1,1,0) Matrix(149,-110,42,-31) -> Matrix(0,-1,1,0) Matrix(181,-140,128,-99) -> Matrix(1,0,0,1) Matrix(89,-70,14,-11) -> Matrix(0,-1,1,0) Matrix(11,-10,10,-9) -> Matrix(0,-1,1,2) Matrix(81,-100,64,-79) -> Matrix(1,0,2,1) Matrix(31,-50,18,-29) -> Matrix(0,-1,1,2) Matrix(41,-100,16,-39) -> Matrix(1,-2,0,1) Matrix(11,-50,2,-9) -> Matrix(0,-1,1,0) 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): 20 ----------------------------------------------------------------------- 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 -1/1 1 2 1/2 0/1 1 10 5/9 -1/1 3 2 4/7 -1/2 1 10 7/12 -1/2 1 10 3/5 -1/1 1 10 2/3 0/1 1 10 5/7 (-1/1,1/1).(0/1,1/0) 0 2 8/11 1/0 1 10 3/4 1/0 1 10 7/9 -1/1 1 10 4/5 1/0 1 10 1/1 -1/1 1 10 6/5 0/1 1 10 5/4 0/1 1 2 14/11 0/1 1 10 9/7 -1/1 1 10 4/3 1/0 1 10 15/11 -1/1 3 2 11/8 -1/2 1 10 7/5 -1/1 1 10 17/12 1/0 1 10 10/7 -1/1 1 2 3/2 0/1 1 10 5/3 -1/1 1 2 7/4 -1/2 1 10 16/9 -1/2 1 10 9/5 -1/3 1 10 20/11 0/1 6 2 11/6 0/1 1 10 2/1 0/1 1 10 11/5 -1/1 1 10 20/9 0/1 6 2 9/4 1/2 1 10 7/3 1/1 1 10 5/2 1/0 1 2 13/5 -1/1 1 10 21/8 1/0 1 10 8/3 1/0 1 10 3/1 -1/1 1 10 13/4 -1/2 1 10 10/3 0/1 2 2 7/2 0/1 1 10 18/5 0/1 1 10 11/3 1/1 1 10 4/1 1/0 1 10 5/1 (-1/1,1/1).(0/1,1/0) 0 2 6/1 0/1 1 10 7/1 1/1 1 10 1/0 1/0 1 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(129,-70,94,-51) (1/2,5/9) -> (15/11,11/8) Hyperbolic Matrix(141,-80,104,-59) (5/9,4/7) -> (4/3,15/11) Hyperbolic Matrix(241,-140,136,-79) (4/7,7/12) -> (7/4,16/9) Hyperbolic Matrix(101,-60,32,-19) (7/12,3/5) -> (3/1,13/4) Hyperbolic Matrix(49,-30,18,-11) (3/5,2/3) -> (8/3,3/1) Hyperbolic Matrix(71,-50,98,-69) (2/3,5/7) -> (5/7,8/11) Parabolic Matrix(149,-110,42,-31) (8/11,3/4) -> (7/2,18/5) Hyperbolic Matrix(181,-140,128,-99) (3/4,7/9) -> (7/5,17/12) Hyperbolic Matrix(89,-70,14,-11) (7/9,4/5) -> (6/1,7/1) Hyperbolic Matrix(11,-10,10,-9) (4/5,1/1) -> (1/1,6/5) Parabolic Matrix(81,-100,64,-79) (6/5,5/4) -> (5/4,14/11) Parabolic Matrix(211,-270,118,-151) (14/11,9/7) -> (16/9,9/5) Glide Reflection Matrix(61,-80,16,-21) (9/7,4/3) -> (11/3,4/1) Glide Reflection Matrix(101,-140,44,-61) (11/8,7/5) -> (9/4,7/3) Glide Reflection Matrix(239,-340,168,-239) (17/12,10/7) -> (17/12,10/7) Reflection Matrix(41,-60,28,-41) (10/7,3/2) -> (10/7,3/2) Reflection Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(221,-400,100,-181) (9/5,20/11) -> (11/5,20/9) Glide Reflection Matrix(241,-440,132,-241) (20/11,11/6) -> (20/11,11/6) Reflection Matrix(69,-130,26,-49) (11/6,2/1) -> (21/8,8/3) Glide Reflection Matrix(101,-220,28,-61) (2/1,11/5) -> (18/5,11/3) Glide Reflection Matrix(161,-360,72,-161) (20/9,9/4) -> (20/9,9/4) Reflection Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(61,-160,8,-21) (13/5,21/8) -> (7/1,1/0) Glide 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(11,-50,2,-9) (4/1,5/1) -> (5/1,6/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,2,0,-1) (0/1,1/0) -> (-1/1,1/0) Matrix(1,0,4,-1) -> Matrix(-1,0,2,1) (0/1,1/2) -> (-1/1,0/1) Matrix(129,-70,94,-51) -> Matrix(0,-1,1,2) -1/1 Matrix(141,-80,104,-59) -> Matrix(3,2,-2,-1) -1/1 Matrix(241,-140,136,-79) -> Matrix(1,0,0,1) Matrix(101,-60,32,-19) -> Matrix(1,0,0,1) Matrix(49,-30,18,-11) -> Matrix(2,1,-1,0) -1/1 Matrix(71,-50,98,-69) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) Matrix(149,-110,42,-31) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) Matrix(181,-140,128,-99) -> Matrix(1,0,0,1) Matrix(89,-70,14,-11) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) Matrix(11,-10,10,-9) -> Matrix(0,-1,1,2) -1/1 Matrix(81,-100,64,-79) -> Matrix(1,0,2,1) 0/1 Matrix(211,-270,118,-151) -> Matrix(0,1,1,-2) Matrix(61,-80,16,-21) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(101,-140,44,-61) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(239,-340,168,-239) -> Matrix(1,2,0,-1) (17/12,10/7) -> (-1/1,1/0) Matrix(41,-60,28,-41) -> Matrix(-1,0,2,1) (10/7,3/2) -> (-1/1,0/1) Matrix(31,-50,18,-29) -> Matrix(0,-1,1,2) -1/1 Matrix(221,-400,100,-181) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(241,-440,132,-241) -> Matrix(1,0,0,-1) (20/11,11/6) -> (0/1,1/0) Matrix(69,-130,26,-49) -> Matrix(0,1,1,0) *** -> (-1/1,1/1) Matrix(101,-220,28,-61) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(161,-360,72,-161) -> Matrix(1,0,4,-1) (20/9,9/4) -> (0/1,1/2) Matrix(41,-100,16,-39) -> Matrix(1,-2,0,1) 1/0 Matrix(61,-160,8,-21) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(79,-260,24,-79) -> Matrix(-1,0,4,1) (13/4,10/3) -> (-1/2,0/1) Matrix(41,-140,12,-41) -> Matrix(1,0,0,-1) (10/3,7/2) -> (0/1,1/0) Matrix(11,-50,2,-9) -> Matrix(0,-1,1,0) (-1/1,1/1).(0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.