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 -6/1 -5/1 -9/2 -4/1 -10/3 -3/1 -11/4 -12/5 -2/1 -1/1 -12/19 -10/17 -1/2 -6/13 -4/11 -2/9 0/1 1/6 3/11 8/27 2/5 1/2 8/13 3/4 1/1 4/3 3/2 8/5 13/8 5/3 9/5 2/1 5/2 8/3 14/5 3/1 43/13 10/3 7/2 11/3 4/1 9/2 23/5 14/3 5/1 6/1 8/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 0/1 -11/2 -1/1 0/1 1/0 -5/1 -1/1 0/1 1/0 -19/4 -2/1 -1/1 1/0 -14/3 -1/1 1/0 -9/2 -1/1 -4/1 -1/1 0/1 -15/4 -1/1 -1/2 0/1 -26/7 -1/1 0/1 -11/3 -1/1 -1/2 0/1 -18/5 -1/1 0/1 -7/2 -1/1 -1/2 0/1 -10/3 0/1 -13/4 0/1 1/1 1/0 -3/1 -1/1 0/1 1/0 -17/6 -1/1 0/1 1/0 -14/5 -1/1 0/1 -11/4 0/1 -8/3 0/1 1/0 -13/5 -1/1 0/1 1/0 -18/7 1/1 1/0 -5/2 -2/1 -1/1 1/0 -12/5 -1/1 -19/8 -1/1 -3/4 -2/3 -7/3 -1/1 -1/2 0/1 -23/10 -1/1 -2/3 -1/2 -16/7 -1/2 0/1 -9/4 -1/1 0/1 1/0 -11/5 -1/1 0/1 1/0 -2/1 -1/1 0/1 -1/1 0/1 -2/3 0/1 1/1 -9/14 0/1 1/1 1/0 -16/25 0/1 1/2 -7/11 0/1 1/2 1/1 -12/19 1/1 -17/27 1/1 4/3 3/2 -5/8 1/1 2/1 1/0 -8/13 0/1 1/0 -11/18 0/1 -14/23 0/1 1/1 -3/5 0/1 1/1 1/0 -10/17 0/1 -17/29 0/1 1/3 1/2 -7/12 0/1 1/2 1/1 -11/19 0/1 1/2 1/1 -15/26 0/1 1/2 1/1 -4/7 0/1 1/1 -9/16 1/1 -14/25 1/1 1/0 -5/9 0/1 1/1 1/0 -1/2 0/1 1/1 1/0 -6/13 1/0 -5/11 -1/1 0/1 1/0 -4/9 0/1 1/0 -7/16 0/1 1/1 1/0 -10/23 0/1 1/1 -3/7 1/1 2/1 1/0 -2/5 -1/1 1/0 -7/18 -1/1 0/1 1/0 -5/13 -2/1 -1/1 1/0 -8/21 -1/1 0/1 -3/8 -1/1 0/1 1/0 -4/11 -1/1 1/1 -5/14 -1/1 0/1 1/0 -1/3 -1/1 0/1 1/0 -2/7 0/1 1/1 -5/18 0/1 1/2 1/1 -8/29 1/1 -3/11 1/1 2/1 1/0 -7/26 1/1 2/1 1/0 -4/15 2/1 1/0 -1/4 -1/1 0/1 1/0 -2/9 0/1 -3/14 0/1 1/2 1/1 -1/5 0/1 1/1 1/0 -2/11 -1/1 0/1 -1/6 0/1 1/1 1/0 -1/7 -1/1 0/1 1/0 0/1 0/1 1/0 1/7 -1/1 0/1 1/0 1/6 0/1 2/11 0/1 1/1 1/5 0/1 1/1 1/0 2/9 -1/1 1/0 5/22 -2/1 -1/1 1/0 3/13 -1/1 0/1 1/0 1/4 -1/1 0/1 1/0 4/15 -1/2 0/1 3/11 0/1 8/29 0/1 1/4 5/18 0/1 1/3 1/2 2/7 0/1 1/1 7/24 0/1 1/2 1/1 5/17 0/1 1/2 1/1 8/27 1/1 3/10 1/1 2/1 1/0 1/3 -1/1 0/1 1/0 2/5 0/1 3/7 0/1 1/2 1/1 7/16 0/1 1/2 1/1 4/9 0/1 1/1 5/11 0/1 1/1 1/0 6/13 0/1 1/1 1/2 0/1 1/1 1/0 6/11 3/1 1/0 5/9 1/0 9/16 -5/1 -4/1 1/0 4/7 -2/1 1/0 11/19 -3/1 -2/1 1/0 7/12 -2/1 -1/1 1/0 10/17 -2/1 -1/1 3/5 -2/1 -1/1 1/0 8/13 -1/1 13/21 -1/1 -3/4 -2/3 5/8 -1/1 -1/2 0/1 7/11 -1/1 -1/2 0/1 2/3 -1/1 0/1 5/7 -1/2 -1/3 0/1 3/4 0/1 7/9 0/1 1/4 1/3 4/5 0/1 1/2 9/11 1/2 2/3 1/1 5/6 0/1 1/2 1/1 11/13 1/1 2/1 1/0 6/7 0/1 1/1 1/1 0/1 1/1 1/0 4/3 -1/1 1/1 7/5 0/1 1/1 1/0 24/17 0/1 1/1 41/29 1/1 17/12 1/1 2/1 1/0 27/19 0/1 1/1 1/0 10/7 1/1 1/0 13/9 -1/1 0/1 1/0 3/2 0/1 1/1 1/0 14/9 0/1 1/1 11/7 0/1 1/1 1/0 19/12 0/1 1/2 1/1 27/17 0/1 1/2 1/1 35/22 1/2 2/3 1/1 8/5 0/1 1/1 21/13 1/2 2/3 1/1 13/8 1/1 31/19 1/1 3/2 2/1 18/11 1/1 2/1 5/3 1/1 2/1 1/0 22/13 1/0 17/10 -2/1 -1/1 1/0 29/17 -1/1 0/1 1/0 12/7 0/1 1/0 19/11 0/1 1/2 1/1 7/4 0/1 1/1 1/0 16/9 0/1 1/1 9/5 1/1 20/11 1/1 2/1 11/6 1/1 2/1 1/0 2/1 1/1 1/0 7/3 3/1 4/1 1/0 5/2 1/0 13/5 -7/1 -6/1 1/0 21/8 -5/1 -4/1 1/0 29/11 -5/1 -4/1 1/0 8/3 -4/1 1/0 27/10 -5/1 -4/1 1/0 19/7 -4/1 -3/1 1/0 30/11 -4/1 -3/1 41/15 -3/1 52/19 -3/1 -2/1 11/4 -4/1 -3/1 1/0 25/9 -3/1 -5/2 -2/1 14/5 -3/1 -2/1 3/1 -2/1 -1/1 1/0 13/4 -1/1 -1/2 0/1 23/7 -1/1 0/1 1/0 33/10 -1/1 0/1 1/0 43/13 -1/1 10/3 -1/1 0/1 27/8 0/1 17/5 -1/1 0/1 1/0 7/2 -1/1 0/1 1/0 18/5 -1/1 0/1 11/3 0/1 26/7 0/1 1/1 15/4 0/1 1/1 1/0 4/1 0/1 1/0 9/2 1/1 2/1 1/0 23/5 1/0 14/3 -1/1 1/0 5/1 0/1 1/1 1/0 6/1 1/0 7/1 -4/1 -3/1 1/0 8/1 -2/1 1/0 1/0 -1/1 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,32,-18,-115) (-6/1,1/0) -> (-2/7,-5/18) Hyperbolic Matrix(43,240,12,67) (-6/1,-11/2) -> (7/2,18/5) Hyperbolic Matrix(13,68,30,157) (-11/2,-5/1) -> (3/7,7/16) Hyperbolic Matrix(133,636,78,373) (-5/1,-19/4) -> (17/10,29/17) Hyperbolic Matrix(23,108,102,479) (-19/4,-14/3) -> (2/9,5/22) Hyperbolic Matrix(55,252,-98,-449) (-14/3,-9/2) -> (-9/16,-14/25) Hyperbolic Matrix(17,72,-30,-127) (-9/2,-4/1) -> (-4/7,-9/16) Hyperbolic Matrix(71,268,40,151) (-4/1,-15/4) -> (7/4,16/9) Hyperbolic Matrix(31,116,66,247) (-15/4,-26/7) -> (6/13,1/2) Hyperbolic Matrix(217,804,78,289) (-26/7,-11/3) -> (25/9,14/5) Hyperbolic Matrix(11,40,58,211) (-11/3,-18/5) -> (2/11,1/5) Hyperbolic Matrix(55,196,-126,-449) (-18/5,-7/2) -> (-7/16,-10/23) Hyperbolic Matrix(59,200,-18,-61) (-7/2,-10/3) -> (-10/3,-13/4) Parabolic Matrix(51,164,88,283) (-13/4,-3/1) -> (11/19,7/12) Hyperbolic Matrix(181,516,114,325) (-3/1,-17/6) -> (19/12,27/17) Hyperbolic Matrix(47,132,162,455) (-17/6,-14/5) -> (2/7,7/24) Hyperbolic Matrix(111,308,-182,-505) (-14/5,-11/4) -> (-11/18,-14/23) Hyperbolic Matrix(65,176,-106,-287) (-11/4,-8/3) -> (-8/13,-11/18) Hyperbolic Matrix(35,92,-78,-205) (-8/3,-13/5) -> (-5/11,-4/9) Hyperbolic Matrix(141,364,98,253) (-13/5,-18/7) -> (10/7,13/9) Hyperbolic Matrix(19,48,36,91) (-18/7,-5/2) -> (1/2,6/11) Hyperbolic Matrix(119,288,-50,-121) (-5/2,-12/5) -> (-12/5,-19/8) Parabolic Matrix(63,148,20,47) (-19/8,-7/3) -> (3/1,13/4) Hyperbolic Matrix(321,740,226,521) (-7/3,-23/10) -> (17/12,27/19) Hyperbolic Matrix(87,200,10,23) (-23/10,-16/7) -> (8/1,1/0) Hyperbolic Matrix(23,52,88,199) (-16/7,-9/4) -> (1/4,4/15) Hyperbolic Matrix(47,104,-174,-385) (-9/4,-11/5) -> (-3/11,-7/26) Hyperbolic Matrix(95,208,58,127) (-11/5,-2/1) -> (18/11,5/3) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(161,104,274,177) (-2/3,-9/14) -> (7/12,10/17) Hyperbolic Matrix(237,152,856,549) (-9/14,-16/25) -> (8/29,5/18) Hyperbolic Matrix(857,548,502,321) (-16/25,-7/11) -> (29/17,12/7) Hyperbolic Matrix(455,288,-722,-457) (-7/11,-12/19) -> (-12/19,-17/27) Parabolic Matrix(223,140,266,167) (-17/27,-5/8) -> (5/6,11/13) Hyperbolic Matrix(149,92,34,21) (-5/8,-8/13) -> (4/1,9/2) Hyperbolic Matrix(145,88,318,193) (-14/23,-3/5) -> (5/11,6/13) Hyperbolic Matrix(339,200,-578,-341) (-3/5,-10/17) -> (-10/17,-17/29) Parabolic Matrix(1011,592,374,219) (-17/29,-7/12) -> (27/10,19/7) Hyperbolic Matrix(337,196,98,57) (-7/12,-11/19) -> (17/5,7/2) Hyperbolic Matrix(263,152,898,519) (-11/19,-15/26) -> (7/24,5/17) Hyperbolic Matrix(597,344,328,189) (-15/26,-4/7) -> (20/11,11/6) Hyperbolic Matrix(765,428,538,301) (-14/25,-5/9) -> (27/19,10/7) Hyperbolic Matrix(59,32,94,51) (-5/9,-1/2) -> (5/8,7/11) Hyperbolic Matrix(265,124,156,73) (-1/2,-6/13) -> (22/13,17/10) Hyperbolic Matrix(307,140,182,83) (-6/13,-5/11) -> (5/3,22/13) Hyperbolic Matrix(199,88,52,23) (-4/9,-7/16) -> (15/4,4/1) Hyperbolic Matrix(295,128,348,151) (-10/23,-3/7) -> (11/13,6/7) Hyperbolic Matrix(49,20,22,9) (-3/7,-2/5) -> (2/1,7/3) Hyperbolic Matrix(91,36,48,19) (-2/5,-7/18) -> (11/6,2/1) Hyperbolic Matrix(227,88,276,107) (-7/18,-5/13) -> (9/11,5/6) Hyperbolic Matrix(545,208,338,129) (-5/13,-8/21) -> (8/5,21/13) Hyperbolic Matrix(179,68,408,155) (-8/21,-3/8) -> (7/16,4/9) Hyperbolic Matrix(87,32,-242,-89) (-3/8,-4/11) -> (-4/11,-5/14) Parabolic Matrix(45,16,194,69) (-5/14,-1/3) -> (3/13,1/4) Hyperbolic Matrix(41,12,58,17) (-1/3,-2/7) -> (2/3,5/7) Hyperbolic Matrix(231,64,776,215) (-5/18,-8/29) -> (8/27,3/10) Hyperbolic Matrix(233,64,790,217) (-8/29,-3/11) -> (5/17,8/27) Hyperbolic Matrix(613,164,228,61) (-7/26,-4/15) -> (8/3,27/10) Hyperbolic Matrix(151,40,268,71) (-4/15,-1/4) -> (9/16,4/7) Hyperbolic Matrix(35,8,-162,-37) (-1/4,-2/9) -> (-2/9,-3/14) Parabolic Matrix(249,52,158,33) (-3/14,-1/5) -> (11/7,19/12) Hyperbolic Matrix(103,20,36,7) (-1/5,-2/11) -> (14/5,3/1) Hyperbolic Matrix(67,12,240,43) (-2/11,-1/6) -> (5/18,2/7) Hyperbolic Matrix(99,16,68,11) (-1/6,-1/7) -> (13/9,3/2) Hyperbolic Matrix(1,0,14,1) (-1/7,0/1) -> (0/1,1/7) Parabolic Matrix(291,-44,86,-13) (1/7,1/6) -> (27/8,17/5) Hyperbolic Matrix(357,-64,106,-19) (1/6,2/11) -> (10/3,27/8) Hyperbolic Matrix(115,-24,24,-5) (1/5,2/9) -> (14/3,5/1) Hyperbolic Matrix(1049,-240,660,-151) (5/22,3/13) -> (27/17,35/22) Hyperbolic Matrix(133,-36,484,-131) (4/15,3/11) -> (3/11,8/29) Parabolic Matrix(193,-60,74,-23) (3/10,1/3) -> (13/5,21/8) Hyperbolic Matrix(21,-8,50,-19) (1/3,2/5) -> (2/5,3/7) Parabolic Matrix(327,-148,232,-105) (4/9,5/11) -> (7/5,24/17) Hyperbolic Matrix(379,-208,82,-45) (6/11,5/9) -> (23/5,14/3) Hyperbolic Matrix(449,-252,98,-55) (5/9,9/16) -> (9/2,23/5) Hyperbolic Matrix(355,-204,134,-77) (4/7,11/19) -> (29/11,8/3) Hyperbolic Matrix(617,-364,378,-223) (10/17,3/5) -> (31/19,18/11) Hyperbolic Matrix(209,-128,338,-207) (3/5,8/13) -> (8/13,13/21) Parabolic Matrix(431,-268,156,-97) (13/21,5/8) -> (11/4,25/9) Hyperbolic Matrix(187,-120,120,-77) (7/11,2/3) -> (14/9,11/7) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(203,-160,118,-93) (7/9,4/5) -> (12/7,19/11) Hyperbolic Matrix(123,-100,16,-13) (4/5,9/11) -> (7/1,8/1) Hyperbolic Matrix(163,-144,60,-53) (6/7,1/1) -> (19/7,30/11) Hyperbolic Matrix(25,-32,18,-23) (1/1,4/3) -> (4/3,7/5) Parabolic Matrix(2205,-3116,806,-1139) (24/17,41/29) -> (41/15,52/19) Hyperbolic Matrix(1473,-2084,446,-631) (41/29,17/12) -> (33/10,43/13) Hyperbolic Matrix(187,-288,50,-77) (3/2,14/9) -> (26/7,15/4) Hyperbolic Matrix(1199,-1908,438,-697) (35/22,8/5) -> (52/19,11/4) Hyperbolic Matrix(417,-676,256,-415) (21/13,13/8) -> (13/8,31/19) Parabolic Matrix(235,-408,72,-125) (19/11,7/4) -> (13/4,23/7) Hyperbolic Matrix(181,-324,100,-179) (16/9,9/5) -> (9/5,20/11) Parabolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(547,-1440,166,-437) (21/8,29/11) -> (23/7,33/10) Hyperbolic Matrix(623,-1700,188,-513) (30/11,41/15) -> (43/13,10/3) Hyperbolic Matrix(133,-484,36,-131) (18/5,11/3) -> (11/3,26/7) Parabolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,32,-18,-115) -> Matrix(1,0,2,1) Matrix(43,240,12,67) -> Matrix(1,0,0,1) Matrix(13,68,30,157) -> Matrix(1,0,2,1) Matrix(133,636,78,373) -> Matrix(1,0,0,1) Matrix(23,108,102,479) -> Matrix(1,0,0,1) Matrix(55,252,-98,-449) -> Matrix(1,2,0,1) Matrix(17,72,-30,-127) -> Matrix(1,0,2,1) Matrix(71,268,40,151) -> Matrix(1,0,2,1) Matrix(31,116,66,247) -> Matrix(1,0,2,1) Matrix(217,804,78,289) -> Matrix(1,-2,0,1) Matrix(11,40,58,211) -> Matrix(1,0,2,1) Matrix(55,196,-126,-449) -> Matrix(1,0,2,1) Matrix(59,200,-18,-61) -> Matrix(1,0,2,1) Matrix(51,164,88,283) -> Matrix(1,-2,0,1) Matrix(181,516,114,325) -> Matrix(1,0,2,1) Matrix(47,132,162,455) -> Matrix(1,0,2,1) Matrix(111,308,-182,-505) -> Matrix(1,0,2,1) Matrix(65,176,-106,-287) -> Matrix(1,0,0,1) Matrix(35,92,-78,-205) -> Matrix(1,0,0,1) Matrix(141,364,98,253) -> Matrix(1,0,0,1) Matrix(19,48,36,91) -> Matrix(1,2,0,1) Matrix(119,288,-50,-121) -> Matrix(3,4,-4,-5) Matrix(63,148,20,47) -> Matrix(3,2,-2,-1) Matrix(321,740,226,521) -> Matrix(1,0,2,1) Matrix(87,200,10,23) -> Matrix(3,2,-2,-1) Matrix(23,52,88,199) -> Matrix(1,0,0,1) Matrix(47,104,-174,-385) -> Matrix(1,2,0,1) Matrix(95,208,58,127) -> Matrix(1,2,0,1) Matrix(3,4,-4,-5) -> Matrix(1,0,2,1) Matrix(161,104,274,177) -> Matrix(1,-2,0,1) Matrix(237,152,856,549) -> Matrix(1,0,2,1) Matrix(857,548,502,321) -> Matrix(1,0,-2,1) Matrix(455,288,-722,-457) -> Matrix(5,-4,4,-3) Matrix(223,140,266,167) -> Matrix(1,-2,2,-3) Matrix(149,92,34,21) -> Matrix(1,0,0,1) Matrix(145,88,318,193) -> Matrix(1,0,0,1) Matrix(339,200,-578,-341) -> Matrix(1,0,2,1) Matrix(1011,592,374,219) -> Matrix(9,-4,-2,1) Matrix(337,196,98,57) -> Matrix(1,0,-2,1) Matrix(263,152,898,519) -> Matrix(1,0,0,1) Matrix(597,344,328,189) -> Matrix(3,-2,2,-1) Matrix(765,428,538,301) -> Matrix(1,0,0,1) Matrix(59,32,94,51) -> Matrix(1,0,-2,1) Matrix(265,124,156,73) -> Matrix(1,-2,0,1) Matrix(307,140,182,83) -> Matrix(1,2,0,1) Matrix(199,88,52,23) -> Matrix(1,0,0,1) Matrix(295,128,348,151) -> Matrix(1,0,0,1) Matrix(49,20,22,9) -> Matrix(1,2,0,1) Matrix(91,36,48,19) -> Matrix(1,2,0,1) Matrix(227,88,276,107) -> Matrix(1,0,2,1) Matrix(545,208,338,129) -> Matrix(1,0,2,1) Matrix(179,68,408,155) -> Matrix(1,0,2,1) Matrix(87,32,-242,-89) -> Matrix(1,0,0,1) Matrix(45,16,194,69) -> Matrix(1,0,0,1) Matrix(41,12,58,17) -> Matrix(1,0,-2,1) Matrix(231,64,776,215) -> Matrix(3,-2,2,-1) Matrix(233,64,790,217) -> Matrix(1,-2,2,-3) Matrix(613,164,228,61) -> Matrix(1,-6,0,1) Matrix(151,40,268,71) -> Matrix(1,-4,0,1) Matrix(35,8,-162,-37) -> Matrix(1,0,2,1) Matrix(249,52,158,33) -> Matrix(1,0,0,1) Matrix(103,20,36,7) -> Matrix(1,-2,0,1) Matrix(67,12,240,43) -> Matrix(1,0,2,1) Matrix(99,16,68,11) -> Matrix(1,0,0,1) Matrix(1,0,14,1) -> Matrix(1,0,0,1) Matrix(291,-44,86,-13) -> Matrix(1,0,0,1) Matrix(357,-64,106,-19) -> Matrix(1,0,-2,1) Matrix(115,-24,24,-5) -> Matrix(1,0,0,1) Matrix(1049,-240,660,-151) -> Matrix(1,0,2,1) Matrix(133,-36,484,-131) -> Matrix(1,0,6,1) Matrix(193,-60,74,-23) -> Matrix(1,-6,0,1) Matrix(21,-8,50,-19) -> Matrix(1,0,2,1) Matrix(327,-148,232,-105) -> Matrix(1,0,0,1) Matrix(379,-208,82,-45) -> Matrix(1,-4,0,1) Matrix(449,-252,98,-55) -> Matrix(1,6,0,1) Matrix(355,-204,134,-77) -> Matrix(1,-2,0,1) Matrix(617,-364,378,-223) -> Matrix(3,4,2,3) Matrix(209,-128,338,-207) -> Matrix(3,4,-4,-5) Matrix(431,-268,156,-97) -> Matrix(7,4,-2,-1) Matrix(187,-120,120,-77) -> Matrix(1,0,2,1) Matrix(49,-36,64,-47) -> Matrix(1,0,6,1) Matrix(203,-160,118,-93) -> Matrix(1,0,-2,1) Matrix(123,-100,16,-13) -> Matrix(5,-2,-2,1) Matrix(163,-144,60,-53) -> Matrix(1,-4,0,1) Matrix(25,-32,18,-23) -> Matrix(1,0,0,1) Matrix(2205,-3116,806,-1139) -> Matrix(5,-2,-2,1) Matrix(1473,-2084,446,-631) -> Matrix(1,-2,0,1) Matrix(187,-288,50,-77) -> Matrix(1,0,0,1) Matrix(1199,-1908,438,-697) -> Matrix(5,-2,-2,1) Matrix(417,-676,256,-415) -> Matrix(5,-4,4,-3) Matrix(235,-408,72,-125) -> Matrix(1,0,-2,1) Matrix(181,-324,100,-179) -> Matrix(3,-2,2,-1) Matrix(41,-100,16,-39) -> Matrix(1,-10,0,1) Matrix(547,-1440,166,-437) -> Matrix(1,4,0,1) Matrix(623,-1700,188,-513) -> Matrix(1,4,-2,-7) Matrix(133,-484,36,-131) -> Matrix(1,0,2,1) Matrix(13,-72,2,-11) -> Matrix(1,-4,0,1) 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): 14 Degree of the the map X: 14 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: 30 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -5/1 -9/2 -4/1 -10/3 -3/1 -12/5 -2/1 -1/1 -12/19 -4/11 0/1 8/27 2/5 8/13 3/4 4/5 1/1 4/3 8/5 13/8 2/1 5/2 8/3 3/1 11/3 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 0/1 -11/2 -1/1 0/1 1/0 -5/1 -1/1 0/1 1/0 -9/2 -1/1 -4/1 -1/1 0/1 -7/2 -1/1 -1/2 0/1 -10/3 0/1 -13/4 0/1 1/1 1/0 -3/1 -1/1 0/1 1/0 -5/2 -2/1 -1/1 1/0 -12/5 -1/1 -7/3 -1/1 -1/2 0/1 -9/4 -1/1 0/1 1/0 -11/5 -1/1 0/1 1/0 -2/1 -1/1 0/1 -1/1 0/1 -2/3 0/1 1/1 -7/11 0/1 1/2 1/1 -12/19 1/1 -5/8 1/1 2/1 1/0 -8/13 0/1 1/0 -3/5 0/1 1/1 1/0 -4/7 0/1 1/1 -5/9 0/1 1/1 1/0 -1/2 0/1 1/1 1/0 -3/7 1/1 2/1 1/0 -2/5 -1/1 1/0 -7/18 -1/1 0/1 1/0 -5/13 -2/1 -1/1 1/0 -8/21 -1/1 0/1 -3/8 -1/1 0/1 1/0 -4/11 -1/1 1/1 -1/3 -1/1 0/1 1/0 0/1 0/1 1/0 1/4 -1/1 0/1 1/0 2/7 0/1 1/1 5/17 0/1 1/2 1/1 8/27 1/1 3/10 1/1 2/1 1/0 1/3 -1/1 0/1 1/0 2/5 0/1 3/7 0/1 1/2 1/1 7/16 0/1 1/2 1/1 4/9 0/1 1/1 1/2 0/1 1/1 1/0 4/7 -2/1 1/0 11/19 -3/1 -2/1 1/0 7/12 -2/1 -1/1 1/0 10/17 -2/1 -1/1 3/5 -2/1 -1/1 1/0 8/13 -1/1 5/8 -1/1 -1/2 0/1 7/11 -1/1 -1/2 0/1 2/3 -1/1 0/1 5/7 -1/2 -1/3 0/1 3/4 0/1 7/9 0/1 1/4 1/3 4/5 0/1 1/2 9/11 1/2 2/3 1/1 5/6 0/1 1/2 1/1 6/7 0/1 1/1 1/1 0/1 1/1 1/0 4/3 -1/1 1/1 3/2 0/1 1/1 1/0 14/9 0/1 1/1 11/7 0/1 1/1 1/0 8/5 0/1 1/1 21/13 1/2 2/3 1/1 13/8 1/1 31/19 1/1 3/2 2/1 18/11 1/1 2/1 5/3 1/1 2/1 1/0 17/10 -2/1 -1/1 1/0 12/7 0/1 1/0 19/11 0/1 1/2 1/1 7/4 0/1 1/1 1/0 16/9 0/1 1/1 9/5 1/1 11/6 1/1 2/1 1/0 2/1 1/1 1/0 7/3 3/1 4/1 1/0 5/2 1/0 13/5 -7/1 -6/1 1/0 21/8 -5/1 -4/1 1/0 29/11 -5/1 -4/1 1/0 8/3 -4/1 1/0 3/1 -2/1 -1/1 1/0 7/2 -1/1 0/1 1/0 18/5 -1/1 0/1 11/3 0/1 26/7 0/1 1/1 15/4 0/1 1/1 1/0 4/1 0/1 1/0 9/2 1/1 2/1 1/0 5/1 0/1 1/1 1/0 6/1 1/0 7/1 -4/1 -3/1 1/0 1/0 -1/1 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,32,17,109) (-6/1,1/0) -> (2/7,5/17) Hyperbolic Matrix(43,240,12,67) (-6/1,-11/2) -> (7/2,18/5) Hyperbolic Matrix(13,68,30,157) (-11/2,-5/1) -> (3/7,7/16) Hyperbolic Matrix(31,144,17,79) (-5/1,-9/2) -> (9/5,11/6) Hyperbolic Matrix(41,180,23,101) (-9/2,-4/1) -> (16/9,9/5) Hyperbolic Matrix(11,40,-19,-69) (-4/1,-7/2) -> (-3/5,-4/7) Hyperbolic Matrix(59,200,-18,-61) (-7/2,-10/3) -> (-10/3,-13/4) Parabolic Matrix(51,164,88,283) (-13/4,-3/1) -> (11/19,7/12) Hyperbolic Matrix(17,44,5,13) (-3/1,-5/2) -> (3/1,7/2) Hyperbolic Matrix(59,144,-25,-61) (-5/2,-12/5) -> (-12/5,-7/3) Parabolic Matrix(85,196,49,113) (-7/3,-9/4) -> (19/11,7/4) Hyperbolic Matrix(229,508,87,193) (-9/4,-11/5) -> (21/8,29/11) Hyperbolic Matrix(95,208,58,127) (-11/5,-2/1) -> (18/11,5/3) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(81,52,95,61) (-2/3,-7/11) -> (5/6,6/7) Hyperbolic Matrix(227,144,-361,-229) (-7/11,-12/19) -> (-12/19,-5/8) Parabolic Matrix(149,92,34,21) (-5/8,-8/13) -> (4/1,9/2) Hyperbolic Matrix(215,132,57,35) (-8/13,-3/5) -> (15/4,4/1) Hyperbolic Matrix(193,108,109,61) (-4/7,-5/9) -> (7/4,16/9) Hyperbolic Matrix(59,32,94,51) (-5/9,-1/2) -> (5/8,7/11) Hyperbolic Matrix(53,24,11,5) (-1/2,-3/7) -> (9/2,5/1) Hyperbolic Matrix(49,20,22,9) (-3/7,-2/5) -> (2/1,7/3) Hyperbolic Matrix(91,36,48,19) (-2/5,-7/18) -> (11/6,2/1) Hyperbolic Matrix(227,88,276,107) (-7/18,-5/13) -> (9/11,5/6) Hyperbolic Matrix(545,208,338,129) (-5/13,-8/21) -> (8/5,21/13) Hyperbolic Matrix(179,68,408,155) (-8/21,-3/8) -> (7/16,4/9) Hyperbolic Matrix(43,16,-121,-45) (-3/8,-4/11) -> (-4/11,-1/3) Parabolic Matrix(1,0,7,1) (-1/3,0/1) -> (0/1,1/4) Parabolic Matrix(43,-12,61,-17) (1/4,2/7) -> (2/3,5/7) Hyperbolic Matrix(217,-64,729,-215) (5/17,8/27) -> (8/27,3/10) Parabolic Matrix(193,-60,74,-23) (3/10,1/3) -> (13/5,21/8) Hyperbolic Matrix(21,-8,50,-19) (1/3,2/5) -> (2/5,3/7) Parabolic Matrix(115,-52,73,-33) (4/9,1/2) -> (11/7,8/5) Hyperbolic Matrix(37,-20,13,-7) (1/2,4/7) -> (8/3,3/1) Hyperbolic Matrix(355,-204,134,-77) (4/7,11/19) -> (29/11,8/3) Hyperbolic Matrix(89,-52,101,-59) (7/12,10/17) -> (6/7,1/1) Hyperbolic Matrix(617,-364,378,-223) (10/17,3/5) -> (31/19,18/11) Hyperbolic Matrix(105,-64,169,-103) (3/5,8/13) -> (8/13,5/8) Parabolic Matrix(187,-120,120,-77) (7/11,2/3) -> (14/9,11/7) Hyperbolic Matrix(49,-36,64,-47) (5/7,3/4) -> (3/4,7/9) Parabolic Matrix(203,-160,118,-93) (7/9,4/5) -> (12/7,19/11) Hyperbolic Matrix(217,-176,127,-103) (4/5,9/11) -> (17/10,12/7) Hyperbolic Matrix(13,-16,9,-11) (1/1,4/3) -> (4/3,3/2) Parabolic Matrix(187,-288,50,-77) (3/2,14/9) -> (26/7,15/4) Hyperbolic Matrix(417,-676,256,-415) (21/13,13/8) -> (13/8,31/19) Parabolic Matrix(31,-52,3,-5) (5/3,17/10) -> (7/1,1/0) Hyperbolic Matrix(41,-100,16,-39) (7/3,5/2) -> (5/2,13/5) Parabolic Matrix(133,-484,36,-131) (18/5,11/3) -> (11/3,26/7) Parabolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,32,17,109) -> Matrix(1,0,2,1) Matrix(43,240,12,67) -> Matrix(1,0,0,1) Matrix(13,68,30,157) -> Matrix(1,0,2,1) Matrix(31,144,17,79) -> Matrix(1,2,0,1) Matrix(41,180,23,101) -> Matrix(1,0,2,1) Matrix(11,40,-19,-69) -> Matrix(1,0,2,1) Matrix(59,200,-18,-61) -> Matrix(1,0,2,1) Matrix(51,164,88,283) -> Matrix(1,-2,0,1) Matrix(17,44,5,13) -> Matrix(1,0,0,1) Matrix(59,144,-25,-61) -> Matrix(1,2,-2,-3) Matrix(85,196,49,113) -> Matrix(1,0,2,1) Matrix(229,508,87,193) -> Matrix(1,-4,0,1) Matrix(95,208,58,127) -> Matrix(1,2,0,1) Matrix(3,4,-4,-5) -> Matrix(1,0,2,1) Matrix(81,52,95,61) -> Matrix(1,0,0,1) Matrix(227,144,-361,-229) -> Matrix(3,-2,2,-1) Matrix(149,92,34,21) -> Matrix(1,0,0,1) Matrix(215,132,57,35) -> Matrix(1,0,0,1) Matrix(193,108,109,61) -> Matrix(1,0,0,1) Matrix(59,32,94,51) -> Matrix(1,0,-2,1) Matrix(53,24,11,5) -> Matrix(1,0,0,1) Matrix(49,20,22,9) -> Matrix(1,2,0,1) Matrix(91,36,48,19) -> Matrix(1,2,0,1) Matrix(227,88,276,107) -> Matrix(1,0,2,1) Matrix(545,208,338,129) -> Matrix(1,0,2,1) Matrix(179,68,408,155) -> Matrix(1,0,2,1) Matrix(43,16,-121,-45) -> Matrix(1,0,0,1) Matrix(1,0,7,1) -> Matrix(1,0,0,1) Matrix(43,-12,61,-17) -> Matrix(1,0,-2,1) Matrix(217,-64,729,-215) -> Matrix(3,-2,2,-1) Matrix(193,-60,74,-23) -> Matrix(1,-6,0,1) Matrix(21,-8,50,-19) -> Matrix(1,0,2,1) Matrix(115,-52,73,-33) -> Matrix(1,0,0,1) Matrix(37,-20,13,-7) -> Matrix(1,-2,0,1) Matrix(355,-204,134,-77) -> Matrix(1,-2,0,1) Matrix(89,-52,101,-59) -> Matrix(1,2,0,1) Matrix(617,-364,378,-223) -> Matrix(3,4,2,3) Matrix(105,-64,169,-103) -> Matrix(1,2,-2,-3) Matrix(187,-120,120,-77) -> Matrix(1,0,2,1) Matrix(49,-36,64,-47) -> Matrix(1,0,6,1) Matrix(203,-160,118,-93) -> Matrix(1,0,-2,1) Matrix(217,-176,127,-103) -> Matrix(1,0,-2,1) Matrix(13,-16,9,-11) -> Matrix(1,0,0,1) Matrix(187,-288,50,-77) -> Matrix(1,0,0,1) Matrix(417,-676,256,-415) -> Matrix(5,-4,4,-3) Matrix(31,-52,3,-5) -> Matrix(1,-2,0,1) Matrix(41,-100,16,-39) -> Matrix(1,-10,0,1) Matrix(133,-484,36,-131) -> Matrix(1,0,2,1) Matrix(13,-72,2,-11) -> Matrix(1,-4,0,1) 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 7 ----------------------------------------------------------------------- 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 -5/1 0 14 -9/2 -1/1 1 2 -4/1 (-1/1,0/1) 0 14 -10/3 0/1 2 2 -3/1 0 14 -5/2 0 14 -12/5 -1/1 4 2 -7/3 0 14 -9/4 0 14 -2/1 (-1/1,0/1) 0 14 -1/1 0/1 1 2 0/1 (0/1,1/0) 0 14 2/5 0/1 2 2 4/9 (0/1,1/1) 0 14 1/2 0 14 4/7 (-2/1,1/0) 0 14 7/12 0 14 10/17 (-2/1,-1/1) 0 14 3/5 0 14 8/13 -1/1 4 2 5/8 0 14 7/11 0 14 2/3 (-1/1,0/1) 0 14 3/4 0/1 3 2 4/5 (0/1,1/2) 0 14 5/6 0 14 1/1 0 14 4/3 0 2 3/2 0 14 14/9 (0/1,1/1) 0 14 11/7 0 14 8/5 (0/1,1/1) 0 14 13/8 1/1 2 2 18/11 (1/1,2/1) 0 14 5/3 0 14 12/7 (0/1,1/0) 0 14 7/4 0 14 9/5 1/1 1 2 2/1 (1/1,1/0) 0 14 5/2 1/0 5 2 8/3 (-4/1,1/0) 0 14 3/1 0 14 7/2 0 14 18/5 (-1/1,0/1) 0 14 11/3 0/1 1 2 4/1 (0/1,1/0) 0 14 6/1 1/0 4 2 1/0 0 14 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,32,8,51) (-5/1,1/0) -> (5/8,7/11) Glide Reflection Matrix(23,108,13,61) (-5/1,-9/2) -> (7/4,9/5) Glide Reflection Matrix(17,72,-4,-17) (-9/2,-4/1) -> (-9/2,-4/1) Reflection Matrix(11,40,-3,-11) (-4/1,-10/3) -> (-4/1,-10/3) Reflection Matrix(49,160,-15,-49) (-10/3,-16/5) -> (-10/3,-16/5) Reflection Matrix(39,124,67,213) (-13/4,-3/1) -> (11/19,7/12) Glide Reflection Matrix(17,44,5,13) (-3/1,-5/2) -> (3/1,7/2) Hyperbolic Matrix(59,144,-25,-61) (-5/2,-12/5) -> (-12/5,-7/3) Parabolic Matrix(23,52,27,61) (-7/3,-9/4) -> (5/6,1/1) Glide Reflection Matrix(47,104,80,177) (-9/4,-2/1) -> (7/12,10/17) Glide Reflection Matrix(3,4,-2,-3) (-2/1,-1/1) -> (-2/1,-1/1) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,5,-1) (0/1,2/5) -> (0/1,2/5) Reflection Matrix(19,-8,45,-19) (2/5,4/9) -> (2/5,4/9) Reflection Matrix(115,-52,73,-33) (4/9,1/2) -> (11/7,8/5) Hyperbolic Matrix(37,-20,13,-7) (1/2,4/7) -> (8/3,3/1) Hyperbolic Matrix(125,-72,217,-125) (4/7,18/31) -> (4/7,18/31) Reflection Matrix(175,-104,106,-63) (10/17,3/5) -> (18/11,5/3) Glide Reflection Matrix(105,-64,169,-103) (3/5,8/13) -> (8/13,5/8) Parabolic Matrix(187,-120,120,-77) (7/11,2/3) -> (14/9,11/7) Hyperbolic Matrix(17,-12,24,-17) (2/3,3/4) -> (2/3,3/4) Reflection Matrix(31,-24,40,-31) (3/4,4/5) -> (3/4,4/5) Reflection Matrix(107,-88,62,-51) (4/5,5/6) -> (12/7,7/4) Glide Reflection Matrix(13,-16,9,-11) (1/1,4/3) -> (4/3,3/2) Parabolic Matrix(99,-152,28,-43) (3/2,14/9) -> (7/2,18/5) Glide Reflection Matrix(129,-208,80,-129) (8/5,13/8) -> (8/5,13/8) Reflection Matrix(287,-468,176,-287) (13/8,18/11) -> (13/8,18/11) Reflection Matrix(31,-52,3,-5) (5/3,17/10) -> (7/1,1/0) Hyperbolic Matrix(155,-264,91,-155) (22/13,12/7) -> (22/13,12/7) Reflection Matrix(19,-36,10,-19) (9/5,2/1) -> (9/5,2/1) Reflection Matrix(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(31,-80,12,-31) (5/2,8/3) -> (5/2,8/3) Reflection Matrix(109,-396,30,-109) (18/5,11/3) -> (18/5,11/3) Reflection Matrix(23,-88,6,-23) (11/3,4/1) -> (11/3,4/1) Reflection Matrix(5,-24,1,-5) (4/1,6/1) -> (4/1,6/1) Reflection Matrix(7,-48,1,-7) (6/1,8/1) -> (6/1,8/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(5,32,8,51) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(23,108,13,61) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(17,72,-4,-17) -> Matrix(-1,0,2,1) (-9/2,-4/1) -> (-1/1,0/1) Matrix(11,40,-3,-11) -> Matrix(-1,0,2,1) (-4/1,-10/3) -> (-1/1,0/1) Matrix(49,160,-15,-49) -> Matrix(1,0,0,-1) (-10/3,-16/5) -> (0/1,1/0) Matrix(39,124,67,213) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(17,44,5,13) -> Matrix(1,0,0,1) Matrix(59,144,-25,-61) -> Matrix(1,2,-2,-3) -1/1 Matrix(23,52,27,61) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(47,104,80,177) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(3,4,-2,-3) -> Matrix(-1,0,2,1) (-2/1,-1/1) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,5,-1) -> Matrix(1,0,0,-1) (0/1,2/5) -> (0/1,1/0) Matrix(19,-8,45,-19) -> Matrix(1,0,2,-1) (2/5,4/9) -> (0/1,1/1) Matrix(115,-52,73,-33) -> Matrix(1,0,0,1) Matrix(37,-20,13,-7) -> Matrix(1,-2,0,1) 1/0 Matrix(125,-72,217,-125) -> Matrix(1,4,0,-1) (4/7,18/31) -> (-2/1,1/0) Matrix(175,-104,106,-63) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(105,-64,169,-103) -> Matrix(1,2,-2,-3) -1/1 Matrix(187,-120,120,-77) -> Matrix(1,0,2,1) 0/1 Matrix(17,-12,24,-17) -> Matrix(-1,0,2,1) (2/3,3/4) -> (-1/1,0/1) Matrix(31,-24,40,-31) -> Matrix(1,0,4,-1) (3/4,4/5) -> (0/1,1/2) Matrix(107,-88,62,-51) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(13,-16,9,-11) -> Matrix(1,0,0,1) Matrix(99,-152,28,-43) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(129,-208,80,-129) -> Matrix(1,0,2,-1) (8/5,13/8) -> (0/1,1/1) Matrix(287,-468,176,-287) -> Matrix(3,-4,2,-3) (13/8,18/11) -> (1/1,2/1) Matrix(31,-52,3,-5) -> Matrix(1,-2,0,1) 1/0 Matrix(155,-264,91,-155) -> Matrix(1,0,0,-1) (22/13,12/7) -> (0/1,1/0) Matrix(19,-36,10,-19) -> Matrix(-1,2,0,1) (9/5,2/1) -> (1/1,1/0) Matrix(9,-20,4,-9) -> Matrix(-1,2,0,1) (2/1,5/2) -> (1/1,1/0) Matrix(31,-80,12,-31) -> Matrix(1,8,0,-1) (5/2,8/3) -> (-4/1,1/0) Matrix(109,-396,30,-109) -> Matrix(-1,0,2,1) (18/5,11/3) -> (-1/1,0/1) Matrix(23,-88,6,-23) -> Matrix(1,0,0,-1) (11/3,4/1) -> (0/1,1/0) Matrix(5,-24,1,-5) -> Matrix(1,0,0,-1) (4/1,6/1) -> (0/1,1/0) Matrix(7,-48,1,-7) -> Matrix(1,4,0,-1) (6/1,8/1) -> (-2/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.