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 1/1 2/1 1/0 -6/1 0/1 2/1 -5/1 1/1 -14/3 8/7 6/5 -9/2 1/1 5/4 4/3 -4/1 4/3 2/1 -11/3 3/2 5/3 2/1 -7/2 3/2 5/3 2/1 -10/3 2/1 -3/1 1/1 3/2 2/1 -14/5 0/1 2/1 -11/4 1/1 4/3 3/2 -30/11 3/2 -19/7 3/2 5/3 2/1 -8/3 8/5 2/1 -5/2 2/1 -12/5 2/1 8/3 -19/8 3/1 4/1 1/0 -7/3 2/1 3/1 1/0 -9/4 1/1 2/1 1/0 -20/9 2/1 -11/5 2/1 3/1 1/0 -2/1 0/1 2/1 -11/6 1/1 3/2 2/1 -20/11 2/1 -9/5 1/1 2/1 1/0 -7/4 1/1 3/2 2/1 -5/3 2/1 -13/8 2/1 7/3 5/2 -21/13 7/3 12/5 5/2 -8/5 2/1 8/3 -19/12 2/1 5/2 3/1 -30/19 3/1 -11/7 3/1 4/1 1/0 -14/9 0/1 2/1 -17/11 1/1 3/2 2/1 -20/13 2/1 -3/2 2/1 3/1 1/0 -10/7 2/1 -7/5 2/1 5/2 3/1 -11/8 2/1 5/2 3/1 -4/3 2/1 4/1 -9/7 4/1 5/1 1/0 -5/4 1/0 -11/9 -2/1 -1/1 1/0 -6/5 0/1 2/1 -13/11 1/1 2/1 1/0 -20/17 2/1 -7/6 1/1 2/1 1/0 -1/1 1/1 2/1 1/0 -7/8 0/1 1/1 1/0 -6/7 0/1 2/1 -5/6 1/1 -14/17 4/3 2/1 -9/11 1/1 3/2 2/1 -4/5 0/1 2/1 -11/14 1/1 3/2 2/1 -7/9 1/1 2/1 1/0 -10/13 2/1 -3/4 1/1 2/1 1/0 -14/19 2/1 4/1 -11/15 2/1 5/2 3/1 -30/41 3/1 -19/26 3/1 4/1 1/0 -8/11 2/1 4/1 -5/7 1/0 -12/17 -2/1 0/1 -19/27 -1/1 0/1 1/0 -7/10 -1/1 0/1 1/0 -9/13 -1/1 0/1 1/0 -2/3 0/1 2/1 -9/14 2/1 3/1 1/0 -7/11 2/1 3/1 1/0 -5/8 1/0 -13/21 -2/1 -1/1 1/0 -21/34 -2/1 -3/2 -1/1 -8/13 -2/1 0/1 -19/31 1/1 2/1 1/0 -30/49 1/0 -11/18 -2/1 -1/1 1/0 -14/23 -2/3 0/1 -17/28 -1/2 -1/3 0/1 -20/33 0/1 -3/5 0/1 1/1 1/0 -10/17 0/1 -7/12 0/1 1/1 1/0 -11/19 0/1 1/2 1/1 -4/7 0/1 2/3 -9/16 2/3 3/4 1/1 -5/9 1/1 -11/20 1/1 6/5 5/4 -6/11 4/3 2/1 -13/24 1/1 3/2 2/1 -20/37 2/1 -7/13 1/1 2/1 1/0 -1/2 1/1 2/1 1/0 0/1 0/1 2/1 1/2 1/1 2/1 1/0 7/13 1/1 3/2 2/1 6/11 4/3 2/1 5/9 2/1 4/7 0/1 2/1 7/12 1/1 3/2 2/1 3/5 1/1 2/1 1/0 11/18 2/1 3/1 1/0 8/13 0/1 2/1 5/8 3/2 1/0 12/19 0/1 2/1 19/30 1/1 2/1 1/0 7/11 1/1 2/1 1/0 9/14 1/1 3/2 2/1 2/3 0/1 2/1 7/10 1/1 2/1 1/0 5/7 1/1 3/1 13/18 1/1 2/1 1/0 8/11 0/1 2/1 19/26 0/1 1/2 1/1 11/15 1/1 3/2 2/1 14/19 4/3 2/1 3/4 1/1 2/1 1/0 7/9 2/1 3/1 1/0 4/5 0/1 2/1 9/11 1/1 2/1 1/0 5/6 2/1 11/13 2/1 5/2 3/1 6/7 2/1 4/1 1/1 1/1 2/1 1/0 7/6 1/1 2/1 1/0 6/5 2/1 4/1 5/4 1/0 14/11 -6/1 -4/1 9/7 -3/1 -2/1 1/0 4/3 -2/1 0/1 15/11 0/1 11/8 -1/1 0/1 1/0 7/5 0/1 1/1 1/0 17/12 -1/1 0/1 1/0 10/7 0/1 3/2 0/1 1/1 1/0 14/9 0/1 2/5 11/7 1/2 2/3 1/1 41/26 4/5 5/6 1/1 30/19 1/1 19/12 1/1 2/1 1/0 8/5 0/1 2/3 5/3 1/1 12/7 4/3 2/1 31/18 4/3 7/5 3/2 19/11 3/2 5/3 2/1 26/15 4/3 2/1 33/19 1/1 3/2 2/1 7/4 1/1 3/2 2/1 16/9 0/1 2/1 9/5 1/1 3/2 2/1 20/11 2/1 11/6 2/1 3/1 1/0 2/1 0/1 2/1 11/5 -1/1 0/1 1/0 20/9 0/1 9/4 0/1 1/2 1/1 7/3 0/1 1/2 1/1 5/2 1/1 13/5 1/1 4/3 3/2 21/8 1/1 4/3 3/2 8/3 4/3 2/1 27/10 1/1 4/3 3/2 19/7 1/1 4/3 3/2 49/18 10/7 13/9 3/2 30/11 3/2 11/4 3/2 5/3 2/1 25/9 2/1 14/5 4/3 2/1 17/6 5/3 7/4 2/1 20/7 2/1 3/1 1/1 2/1 1/0 13/4 1/1 3/2 2/1 10/3 2/1 7/2 1/1 2/1 1/0 18/5 0/1 2/1 11/3 2/1 3/1 1/0 4/1 0/1 2/1 13/3 2/1 3/1 1/0 9/2 2/1 3/1 1/0 5/1 1/0 11/2 0/1 1/1 1/0 17/3 -1/1 0/1 1/0 40/7 0/1 23/4 0/1 1/2 1/1 6/1 0/1 2/1 13/2 -1/1 0/1 1/0 20/3 0/1 27/4 0/1 1/1 1/0 7/1 0/1 1/1 1/0 1/0 1/1 2/1 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(3,-2,2,-1) Matrix(19,120,-16,-101) -> Matrix(1,0,0,1) Matrix(19,100,-4,-21) -> Matrix(7,-6,6,-5) Matrix(61,280,-100,-459) -> Matrix(5,-6,-4,5) Matrix(19,80,-24,-101) -> Matrix(1,-2,2,-3) Matrix(21,80,16,61) -> Matrix(3,-4,-2,3) Matrix(39,140,-56,-201) -> Matrix(1,-2,2,-3) Matrix(41,140,12,41) -> Matrix(5,-8,2,-3) Matrix(19,60,-32,-101) -> Matrix(1,-2,2,-3) Matrix(99,280,-64,-181) -> Matrix(1,0,0,1) Matrix(79,220,-144,-401) -> Matrix(3,-2,2,-1) Matrix(241,660,88,241) -> Matrix(13,-18,8,-11) Matrix(419,1140,-684,-1861) -> Matrix(5,-8,2,-3) Matrix(141,380,-200,-539) -> Matrix(1,-2,2,-3) Matrix(39,100,-16,-41) -> Matrix(9,-16,4,-7) Matrix(159,380,100,239) -> Matrix(1,-2,0,1) Matrix(59,140,8,19) -> Matrix(1,-2,0,1) Matrix(61,140,44,101) -> Matrix(1,-2,0,1) Matrix(161,360,72,161) -> Matrix(1,-2,2,-3) Matrix(181,400,100,221) -> Matrix(3,-8,2,-5) Matrix(101,220,28,61) -> Matrix(1,0,0,1) Matrix(21,40,32,61) -> Matrix(1,0,0,1) Matrix(241,440,132,241) -> Matrix(5,-8,2,-3) Matrix(221,400,100,181) -> Matrix(1,-2,0,1) Matrix(79,140,-136,-241) -> Matrix(1,-2,2,-3) Matrix(59,100,-36,-61) -> Matrix(9,-16,4,-7) Matrix(99,160,-112,-181) -> Matrix(1,-2,-2,5) Matrix(199,320,-324,-521) -> Matrix(1,-2,-2,5) Matrix(201,320,76,121) -> Matrix(5,-14,4,-11) Matrix(721,1140,456,721) -> Matrix(3,-8,2,-5) Matrix(419,660,-572,-901) -> Matrix(5,-18,2,-7) Matrix(179,280,140,219) -> Matrix(1,-6,0,1) Matrix(259,400,-428,-661) -> Matrix(1,-2,2,-3) Matrix(261,400,92,141) -> Matrix(7,-16,4,-9) Matrix(41,60,28,41) -> Matrix(1,-2,0,1) Matrix(99,140,-128,-181) -> Matrix(3,-8,2,-5) Matrix(101,140,44,61) -> Matrix(1,-2,0,1) Matrix(59,80,-104,-141) -> Matrix(1,-4,2,-7) Matrix(61,80,16,21) -> Matrix(1,-2,0,1) Matrix(79,100,-64,-81) -> Matrix(1,-6,0,1) Matrix(181,220,116,141) -> Matrix(1,0,2,1) Matrix(339,400,-628,-741) -> Matrix(1,0,0,1) Matrix(341,400,52,61) -> Matrix(1,-2,0,1) Matrix(121,140,-172,-199) -> Matrix(1,-2,0,1) Matrix(139,120,-256,-221) -> Matrix(3,-2,2,-1) Matrix(119,100,-144,-121) -> Matrix(3,-2,2,-1) Matrix(341,280,464,381) -> Matrix(1,0,0,1) Matrix(221,180,124,101) -> Matrix(1,0,0,1) Matrix(179,140,280,219) -> Matrix(1,0,0,1) Matrix(261,200,184,141) -> Matrix(1,-2,0,1) Matrix(379,280,-624,-461) -> Matrix(1,-2,-2,5) Matrix(299,220,352,259) -> Matrix(1,0,0,1) Matrix(2461,1800,1560,1141) -> Matrix(5,-16,6,-19) Matrix(521,380,824,601) -> Matrix(1,-2,0,1) Matrix(139,100,-196,-141) -> Matrix(1,-4,0,1) Matrix(59,40,28,19) -> Matrix(1,0,0,1) Matrix(61,40,32,21) -> Matrix(1,0,0,1) Matrix(281,180,64,41) -> Matrix(1,0,0,1) Matrix(159,100,-256,-161) -> Matrix(1,-4,0,1) Matrix(259,160,484,299) -> Matrix(3,4,2,3) Matrix(519,320,712,439) -> Matrix(1,2,0,1) Matrix(2941,1800,1080,661) -> Matrix(3,16,2,11) Matrix(1879,1140,328,199) -> Matrix(1,0,4,1) Matrix(341,200,104,61) -> Matrix(3,-2,2,-1) Matrix(139,80,172,99) -> Matrix(3,-2,2,-1) Matrix(179,100,-324,-181) -> Matrix(9,-8,8,-7) Matrix(1479,800,220,119) -> Matrix(1,-2,2,-3) Matrix(261,140,412,221) -> Matrix(1,0,0,1) Matrix(1,0,4,1) -> Matrix(1,0,0,1) Matrix(701,-380,404,-219) -> Matrix(1,0,0,1) Matrix(401,-220,144,-79) -> Matrix(1,0,0,1) Matrix(141,-80,104,-59) -> Matrix(1,-2,0,1) 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,-2,0,1) Matrix(619,-380,360,-221) -> Matrix(3,-2,2,-1) Matrix(161,-100,256,-159) -> Matrix(1,0,0,1) 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(3,-2,2,-1) Matrix(901,-660,572,-419) -> Matrix(3,-4,4,-5) Matrix(461,-340,80,-59) -> Matrix(1,-2,2,-3) Matrix(181,-140,128,-99) -> Matrix(1,-2,0,1) Matrix(101,-80,24,-19) -> Matrix(1,0,0,1) Matrix(121,-100,144,-119) -> Matrix(5,-8,2,-3) Matrix(159,-140,92,-81) -> Matrix(3,-8,2,-5) Matrix(141,-160,52,-59) -> Matrix(3,-2,2,-1) Matrix(101,-120,16,-19) -> Matrix(1,-2,0,1) Matrix(81,-100,64,-79) -> Matrix(1,-8,0,1) Matrix(321,-440,116,-159) -> Matrix(3,-2,2,-1) Matrix(181,-280,64,-99) -> Matrix(7,-2,4,-1) Matrix(61,-100,36,-59) -> Matrix(5,-4,4,-3) Matrix(881,-1520,324,-559) -> Matrix(11,-18,8,-13) Matrix(541,-940,80,-139) -> Matrix(1,-2,2,-3) Matrix(41,-100,16,-39) -> Matrix(5,-4,4,-3) Matrix(159,-460,28,-81) -> Matrix(1,-2,0,1) Matrix(21,-100,4,-19) -> Matrix(1,-2,0,1) 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): 17 Degree of the the map X: 17 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. ----------------------------------------------------------------------- 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 -7/1 0 10 -20/3 2/1 1 2 -6/1 (0/1,2/1) 0 10 -5/1 1/1 3 2 -14/3 (8/7,6/5) 0 10 -9/2 0 10 -4/1 0 10 -11/3 0 10 -7/2 0 10 -10/3 2/1 1 2 -3/1 0 10 -20/7 2/1 5 2 -14/5 (0/1,2/1) 0 10 -11/4 0 10 -30/11 3/2 5 2 -19/7 0 10 -8/3 0 10 -5/2 2/1 2 2 -12/5 0 10 -19/8 0 10 -7/3 0 10 -9/4 0 10 -20/9 2/1 3 2 -11/5 0 10 -2/1 (0/1,2/1) 0 10 0/1 (0/1,2/1) 0 2 2/3 (0/1,2/1) 0 10 7/10 0 10 5/7 0 2 13/18 0 10 8/11 0 10 11/15 0 10 3/4 0 10 7/9 0 10 4/5 0 10 5/6 2/1 1 2 6/7 (2/1,4/1) 0 10 1/1 0 10 7/6 0 10 6/5 (2/1,4/1) 0 10 5/4 1/0 4 2 14/11 (-6/1,-4/1) 0 10 9/7 0 10 4/3 0 10 15/11 0/1 1 2 11/8 0 10 7/5 0 10 10/7 0/1 1 2 3/2 0 10 14/9 (0/1,2/5) 0 10 11/7 0 10 30/19 1/1 5 2 19/12 0 10 8/5 0 10 5/3 1/1 2 2 12/7 0 10 31/18 0 10 19/11 0 10 26/15 (4/3,2/1) 0 10 7/4 0 10 9/5 0 10 20/11 2/1 3 2 2/1 (0/1,2/1) 0 10 20/9 0/1 3 2 9/4 0 10 7/3 0 10 5/2 1/1 2 2 13/5 0 10 21/8 0 10 8/3 0 10 27/10 0 10 19/7 0 10 49/18 0 10 30/11 3/2 5 2 11/4 0 10 25/9 2/1 1 2 14/5 (4/3,2/1) 0 10 17/6 0 10 20/7 2/1 5 2 3/1 0 10 13/4 0 10 10/3 2/1 1 2 7/2 0 10 18/5 (0/1,2/1) 0 10 11/3 0 10 4/1 0 10 13/3 0 10 9/2 0 10 5/1 1/0 1 2 11/2 0 10 17/3 0 10 40/7 0/1 5 2 6/1 (0/1,2/1) 0 10 13/2 0 10 20/3 0/1 1 2 7/1 0 10 1/0 0 10 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(59,400,9,61) (-7/1,-20/3) -> (13/2,20/3) Glide Reflection Matrix(19,120,-3,-19) (-20/3,-6/1) -> (-20/3,-6/1) Reflection Matrix(19,100,-4,-21) (-6/1,-5/1) -> (-5/1,-14/3) Parabolic Matrix(61,280,39,179) (-14/3,-9/2) -> (14/9,11/7) Glide Reflection Matrix(19,80,5,21) (-9/2,-4/1) -> (11/3,4/1) Glide Reflection Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(39,140,17,61) (-11/3,-7/2) -> (9/4,7/3) Glide Reflection Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(19,60,13,41) (-10/3,-3/1) -> (10/7,3/2) Glide Reflection Matrix(139,400,49,141) (-3/1,-20/7) -> (17/6,20/7) Glide Reflection Matrix(99,280,-35,-99) (-20/7,-14/5) -> (-20/7,-14/5) Reflection Matrix(101,280,79,219) (-14/5,-11/4) -> (14/11,9/7) Glide Reflection Matrix(241,660,88,241) (-11/4,-30/11) -> (30/11,11/4) Hyperbolic Matrix(419,1140,265,721) (-30/11,-19/7) -> (30/19,19/12) Glide Reflection Matrix(119,320,45,121) (-19/7,-8/3) -> (21/8,8/3) Glide Reflection 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(-1,0,1,1) (-2/1,0/1) -> (-2/1,0/1) Reflection Matrix(1,0,3,-1) (0/1,2/3) -> (0/1,2/3) Reflection 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(521,-380,303,-221) (8/11,11/15) -> (12/7,31/18) Glide Reflection Matrix(299,-220,53,-39) (11/15,3/4) -> (11/2,17/3) Glide Reflection Matrix(79,-60,25,-19) (3/4,7/9) -> (3/1,13/4) Glide Reflection Matrix(101,-80,24,-19) (7/9,4/5) -> (4/1,13/3) Hyperbolic Matrix(99,-80,73,-59) (4/5,5/6) -> (4/3,15/11) Glide Reflection Matrix(259,-220,93,-79) (5/6,6/7) -> (25/9,14/5) Glide Reflection 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(141,-200,43,-61) (7/5,10/7) -> (13/4,10/3) Glide Reflection Matrix(181,-280,64,-99) (3/2,14/9) -> (14/5,17/6) Hyperbolic Matrix(1141,-1800,419,-661) (11/7,30/19) -> (49/18,30/11) Glide Reflection 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(599,-1040,345,-599) (26/15,40/23) -> (26/15,40/23) Reflection Matrix(241,-420,35,-61) (33/19,7/4) -> (27/4,7/1) Glide Reflection Matrix(101,-180,23,-41) (7/4,9/5) -> (13/3,9/2) Glide Reflection Matrix(21,-40,11,-21) (20/11,2/1) -> (20/11,2/1) Reflection Matrix(19,-40,9,-19) (2/1,20/9) -> (2/1,20/9) Reflection 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 Matrix(41,-240,7,-41) (40/7,6/1) -> (40/7,6/1) Reflection Matrix(101,-680,15,-101) (20/3,34/5) -> (20/3,34/5) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(21,160,8,61) -> Matrix(3,-2,2,-1) 1/1 Matrix(59,400,9,61) -> Matrix(1,-2,-1,1) Matrix(19,120,-3,-19) -> Matrix(1,0,1,-1) (-20/3,-6/1) -> (0/1,2/1) Matrix(19,100,-4,-21) -> Matrix(7,-6,6,-5) 1/1 Matrix(61,280,39,179) -> Matrix(5,-6,9,-11) Matrix(19,80,5,21) -> Matrix(1,-2,-1,1) Matrix(21,80,16,61) -> Matrix(3,-4,-2,3) Matrix(39,140,17,61) -> Matrix(1,-2,-1,1) Matrix(41,140,12,41) -> Matrix(5,-8,2,-3) 2/1 Matrix(19,60,13,41) -> Matrix(1,-2,-1,1) Matrix(139,400,49,141) -> Matrix(9,-16,5,-9) *** -> (8/5,2/1) Matrix(99,280,-35,-99) -> Matrix(1,0,1,-1) (-20/7,-14/5) -> (0/1,2/1) Matrix(101,280,79,219) -> Matrix(5,-6,-1,1) Matrix(241,660,88,241) -> Matrix(13,-18,8,-11) 3/2 Matrix(419,1140,265,721) -> Matrix(5,-8,3,-5) *** -> (4/3,2/1) Matrix(119,320,45,121) -> Matrix(9,-14,7,-11) Matrix(39,100,-16,-41) -> Matrix(9,-16,4,-7) 2/1 Matrix(159,380,100,239) -> Matrix(1,-2,0,1) 1/0 Matrix(59,140,8,19) -> Matrix(1,-2,0,1) 1/0 Matrix(61,140,44,101) -> Matrix(1,-2,0,1) 1/0 Matrix(161,360,72,161) -> Matrix(1,-2,2,-3) 1/1 Matrix(181,400,100,221) -> Matrix(3,-8,2,-5) 2/1 Matrix(101,220,28,61) -> Matrix(1,0,0,1) Matrix(-1,0,1,1) -> Matrix(1,0,1,-1) (-2/1,0/1) -> (0/1,2/1) Matrix(1,0,3,-1) -> Matrix(1,0,1,-1) (0/1,2/3) -> (0/1,2/1) 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(3,-2,2,-1) 1/1 Matrix(521,-380,303,-221) -> Matrix(1,2,1,1) Matrix(299,-220,53,-39) -> Matrix(1,-2,-1,1) Matrix(79,-60,25,-19) -> Matrix(1,0,1,-1) *** -> (0/1,2/1) Matrix(101,-80,24,-19) -> Matrix(1,0,0,1) Matrix(99,-80,73,-59) -> Matrix(1,-2,-1,1) Matrix(259,-220,93,-79) -> Matrix(1,0,1,-1) *** -> (0/1,2/1) Matrix(159,-140,92,-81) -> Matrix(3,-8,2,-5) 2/1 Matrix(141,-160,52,-59) -> Matrix(3,-2,2,-1) 1/1 Matrix(101,-120,16,-19) -> Matrix(1,-2,0,1) 1/0 Matrix(81,-100,64,-79) -> Matrix(1,-8,0,1) 1/0 Matrix(321,-440,116,-159) -> Matrix(3,-2,2,-1) 1/1 Matrix(141,-200,43,-61) -> Matrix(1,2,1,1) Matrix(181,-280,64,-99) -> Matrix(7,-2,4,-1) Matrix(1141,-1800,419,-661) -> Matrix(19,-16,13,-11) Matrix(61,-100,36,-59) -> Matrix(5,-4,4,-3) 1/1 Matrix(881,-1520,324,-559) -> Matrix(11,-18,8,-13) 3/2 Matrix(599,-1040,345,-599) -> Matrix(5,-8,3,-5) (26/15,40/23) -> (4/3,2/1) Matrix(241,-420,35,-61) -> Matrix(1,-2,-1,1) Matrix(101,-180,23,-41) -> Matrix(1,0,1,-1) *** -> (0/1,2/1) Matrix(21,-40,11,-21) -> Matrix(1,0,1,-1) (20/11,2/1) -> (0/1,2/1) Matrix(19,-40,9,-19) -> Matrix(1,0,1,-1) (2/1,20/9) -> (0/1,2/1) Matrix(41,-100,16,-39) -> Matrix(5,-4,4,-3) 1/1 Matrix(159,-460,28,-81) -> Matrix(1,-2,0,1) 1/0 Matrix(21,-100,4,-19) -> Matrix(1,-2,0,1) 1/0 Matrix(41,-240,7,-41) -> Matrix(1,0,1,-1) (40/7,6/1) -> (0/1,2/1) Matrix(101,-680,15,-101) -> Matrix(1,0,1,-1) (20/3,34/5) -> (0/1,2/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.