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): 504 Minimal number of generators: 85 Number of equivalence classes of cusps: 36 Genus: 25 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 2/11 3/10 4/9 1/2 14/25 5/8 6/7 1/1 7/6 15/11 3/2 8/5 25/14 2/1 9/4 7/3 5/2 8/3 19/7 3/1 10/3 7/2 11/3 4/1 21/5 9/2 43/9 5/1 11/2 6/1 7/1 23/3 8/1 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 1/2 -6/13 0/1 1/0 -5/11 0/1 1/4 -9/20 1/3 2/5 -13/29 1/3 1/2 -4/9 0/1 1/2 -7/16 0/1 1/1 -3/7 0/1 1/3 -8/19 1/3 4/11 -5/12 2/5 1/2 -7/17 5/11 1/2 -2/5 1/2 1/1 -9/23 1/2 2/3 -16/41 2/3 3/4 -7/18 2/3 1/1 -5/13 0/1 1/2 -3/8 1/2 2/3 -7/19 1/2 2/3 -4/11 2/3 3/4 -5/14 4/5 1/1 -6/17 8/9 1/1 -1/3 0/1 1/1 -5/16 1/1 8/7 -9/29 6/5 5/4 -4/13 1/1 4/3 -11/36 4/3 3/2 -7/23 7/5 3/2 -3/10 3/2 2/1 -8/27 2/1 5/2 -5/17 2/1 1/0 -12/41 3/1 1/0 -7/24 1/1 2/1 -2/7 2/1 1/0 -3/11 0/1 1/0 -7/26 0/1 1/2 -4/15 1/1 2/1 -9/34 3/2 2/1 -5/19 2/1 1/0 -1/4 1/1 1/0 -3/13 -5/1 1/0 -11/48 -5/1 -4/1 -8/35 -4/1 1/0 -5/22 -4/1 -3/1 -2/9 -2/1 1/0 -7/32 -2/1 -3/2 -12/55 -2/1 -1/1 -5/23 -2/1 -3/2 -3/14 -4/3 -1/1 -4/19 -1/1 -4/5 -1/5 -1/1 0/1 -2/11 0/1 1/1 -7/39 2/1 1/0 -5/28 -1/1 1/0 -3/17 0/1 1/0 -1/6 0/1 1/0 -4/25 -2/1 1/0 -3/19 -1/1 1/0 -5/32 -2/1 1/0 -2/13 -2/1 -1/1 -3/20 -1/1 -2/3 -4/27 -3/5 -1/2 -5/34 -1/2 -3/7 -1/7 -1/2 0/1 -1/8 0/1 1/2 -1/9 0/1 1/0 -2/19 -1/1 0/1 -1/10 0/1 1/1 0/1 0/1 1/0 1/6 -1/2 0/1 2/11 0/1 3/16 0/1 1/6 1/5 0/1 1/2 2/9 1/1 2/1 3/13 1/1 1/0 1/4 0/1 1/0 2/7 -1/1 0/1 3/10 0/1 4/13 0/1 1/3 1/3 0/1 1/1 3/8 1/1 4/3 2/5 2/1 1/0 3/7 5/1 1/0 4/9 1/0 5/11 -7/1 1/0 1/2 -1/1 1/0 5/9 -2/1 1/0 14/25 -2/1 23/41 -2/1 -7/4 9/16 -2/1 -3/2 4/7 -2/1 -1/1 3/5 0/1 1/0 5/8 1/0 7/11 -4/1 1/0 2/3 -2/1 1/0 5/7 -2/1 1/0 3/4 -2/1 -3/2 4/5 -4/3 -1/1 5/6 -8/7 -1/1 6/7 -1/1 7/8 -1/1 -8/9 1/1 -1/1 0/1 7/6 -1/1 13/11 -1/1 -16/17 6/5 -1/1 -8/9 11/9 -6/7 -5/6 5/4 -1/1 -4/5 14/11 -4/5 -3/4 9/7 -7/9 -3/4 4/3 -3/4 -2/3 15/11 -2/3 26/19 -2/3 -13/20 11/8 -2/3 -5/8 7/5 -2/3 -1/2 17/12 -3/5 -1/2 10/7 -1/1 -2/3 3/2 -2/3 -1/2 8/5 -1/2 13/8 -1/2 -2/5 5/3 -1/2 0/1 12/7 0/1 1/0 7/4 -1/1 -2/3 16/9 -3/4 -2/3 25/14 -2/3 34/19 -2/3 -5/8 9/5 -2/3 -1/2 2/1 -1/1 -1/2 9/4 -1/2 16/7 -1/2 -11/23 7/3 -1/2 -5/11 26/11 -5/11 -4/9 19/8 -1/2 -4/9 12/5 -4/9 -3/7 5/2 -1/2 -2/5 18/7 -2/5 -3/8 31/12 -2/5 -1/3 13/5 -2/5 -3/8 8/3 -4/11 -1/3 19/7 -1/3 30/11 -1/3 -12/37 11/4 -1/3 -4/13 3/1 -1/3 0/1 10/3 0/1 17/5 0/1 1/1 7/2 -1/1 0/1 25/7 -2/3 -1/2 18/5 -1/2 -1/3 11/3 -1/2 0/1 4/1 -1/2 0/1 21/5 -1/2 38/9 -1/2 -4/9 17/4 -1/2 -2/5 13/3 -1/2 -1/3 22/5 -1/2 -2/5 9/2 -2/5 -1/3 14/3 -1/3 -2/7 19/4 -3/11 -1/4 43/9 -1/4 67/14 -1/4 -9/37 24/5 -1/4 -3/13 5/1 -1/4 0/1 11/2 0/1 17/3 0/1 1/4 6/1 0/1 1/0 7/1 -1/2 0/1 15/2 -1/3 0/1 23/3 0/1 31/4 0/1 1/1 8/1 -1/1 0/1 1/0 -1/2 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(95,44,-434,-201) (-1/2,-6/13) -> (-2/9,-7/32) Hyperbolic Matrix(287,132,50,23) (-6/13,-5/11) -> (17/3,6/1) Hyperbolic Matrix(235,106,-756,-341) (-5/11,-9/20) -> (-5/16,-9/29) Hyperbolic Matrix(379,170,-1652,-741) (-9/20,-13/29) -> (-3/13,-11/48) Hyperbolic Matrix(143,64,-896,-401) (-13/29,-4/9) -> (-4/25,-3/19) Hyperbolic Matrix(95,42,-328,-145) (-4/9,-7/16) -> (-7/24,-2/7) Hyperbolic Matrix(321,140,94,41) (-7/16,-3/7) -> (17/5,7/2) Hyperbolic Matrix(47,20,-228,-97) (-3/7,-8/19) -> (-4/19,-1/5) Hyperbolic Matrix(181,76,-674,-283) (-8/19,-5/12) -> (-7/26,-4/15) Hyperbolic Matrix(135,56,-446,-185) (-5/12,-7/17) -> (-7/23,-3/10) Hyperbolic Matrix(307,126,134,55) (-7/17,-2/5) -> (16/7,7/3) Hyperbolic Matrix(301,118,-1028,-403) (-2/5,-9/23) -> (-5/17,-12/41) Hyperbolic Matrix(1151,450,642,251) (-9/23,-16/41) -> (34/19,9/5) Hyperbolic Matrix(595,232,-2598,-1013) (-16/41,-7/18) -> (-11/48,-8/35) Hyperbolic Matrix(129,50,-596,-231) (-7/18,-5/13) -> (-5/23,-3/14) Hyperbolic Matrix(209,80,128,49) (-5/13,-3/8) -> (13/8,5/3) Hyperbolic Matrix(43,16,-250,-93) (-3/8,-7/19) -> (-3/17,-1/6) Hyperbolic Matrix(207,76,-700,-257) (-7/19,-4/11) -> (-8/27,-5/17) Hyperbolic Matrix(83,30,-368,-133) (-4/11,-5/14) -> (-5/22,-2/9) Hyperbolic Matrix(79,28,-522,-185) (-5/14,-6/17) -> (-2/13,-3/20) Hyperbolic Matrix(239,84,202,71) (-6/17,-1/3) -> (13/11,6/5) Hyperbolic Matrix(37,12,40,13) (-1/3,-5/16) -> (7/8,1/1) Hyperbolic Matrix(149,46,-826,-255) (-9/29,-4/13) -> (-2/11,-7/39) Hyperbolic Matrix(523,160,-2396,-733) (-4/13,-11/36) -> (-7/32,-12/55) Hyperbolic Matrix(223,68,-1420,-433) (-11/36,-7/23) -> (-3/19,-5/32) Hyperbolic Matrix(557,166,406,121) (-3/10,-8/27) -> (26/19,11/8) Hyperbolic Matrix(219,64,-1468,-429) (-12/41,-7/24) -> (-3/20,-4/27) Hyperbolic Matrix(71,20,110,31) (-2/7,-3/11) -> (7/11,2/3) Hyperbolic Matrix(37,10,-322,-87) (-3/11,-7/26) -> (-1/8,-1/9) Hyperbolic Matrix(143,38,-922,-245) (-4/15,-9/34) -> (-5/32,-2/13) Hyperbolic Matrix(953,252,1698,449) (-9/34,-5/19) -> (23/41,9/16) Hyperbolic Matrix(107,28,-600,-157) (-5/19,-1/4) -> (-5/28,-3/17) Hyperbolic Matrix(33,8,70,17) (-1/4,-3/13) -> (5/11,1/2) Hyperbolic Matrix(35,8,-372,-85) (-8/35,-5/22) -> (-1/10,0/1) Hyperbolic Matrix(101,22,-932,-203) (-12/55,-5/23) -> (-1/9,-2/19) Hyperbolic Matrix(629,134,230,49) (-3/14,-4/19) -> (30/11,11/4) Hyperbolic Matrix(31,6,98,19) (-1/5,-2/11) -> (4/13,1/3) Hyperbolic Matrix(223,40,-1522,-273) (-7/39,-5/28) -> (-5/34,-1/7) Hyperbolic Matrix(653,106,154,25) (-1/6,-4/25) -> (38/9,17/4) Hyperbolic Matrix(2625,388,548,81) (-4/27,-5/34) -> (67/14,24/5) Hyperbolic Matrix(29,4,152,21) (-1/7,-1/8) -> (3/16,1/5) Hyperbolic Matrix(669,70,86,9) (-2/19,-1/10) -> (31/4,8/1) Hyperbolic Matrix(113,-18,44,-7) (0/1,1/6) -> (5/2,18/7) Hyperbolic Matrix(45,-8,242,-43) (1/6,2/11) -> (2/11,3/16) Parabolic Matrix(129,-28,106,-23) (1/5,2/9) -> (6/5,11/9) Hyperbolic Matrix(401,-92,170,-39) (2/9,3/13) -> (7/3,26/11) Hyperbolic Matrix(273,-64,64,-15) (3/13,1/4) -> (17/4,13/3) Hyperbolic Matrix(61,-16,42,-11) (1/4,2/7) -> (10/7,3/2) Hyperbolic Matrix(61,-18,200,-59) (2/7,3/10) -> (3/10,4/13) Parabolic Matrix(57,-20,20,-7) (1/3,3/8) -> (11/4,3/1) Hyperbolic Matrix(131,-50,76,-29) (3/8,2/5) -> (12/7,7/4) Hyperbolic Matrix(73,-30,56,-23) (2/5,3/7) -> (9/7,4/3) Hyperbolic Matrix(73,-32,162,-71) (3/7,4/9) -> (4/9,5/11) Parabolic Matrix(167,-92,118,-65) (1/2,5/9) -> (7/5,17/12) Hyperbolic Matrix(701,-392,1250,-699) (5/9,14/25) -> (14/25,23/41) Parabolic Matrix(549,-310,232,-131) (9/16,4/7) -> (26/11,19/8) Hyperbolic Matrix(131,-76,50,-29) (4/7,3/5) -> (13/5,8/3) Hyperbolic Matrix(81,-50,128,-79) (3/5,5/8) -> (5/8,7/11) Parabolic Matrix(61,-42,16,-11) (2/3,5/7) -> (11/3,4/1) Hyperbolic Matrix(105,-76,76,-55) (5/7,3/4) -> (11/8,7/5) Hyperbolic Matrix(73,-56,30,-23) (3/4,4/5) -> (12/5,5/2) Hyperbolic Matrix(129,-106,28,-23) (4/5,5/6) -> (9/2,14/3) Hyperbolic Matrix(85,-72,98,-83) (5/6,6/7) -> (6/7,7/8) Parabolic Matrix(85,-98,72,-83) (1/1,7/6) -> (7/6,13/11) Parabolic Matrix(163,-202,46,-57) (11/9,5/4) -> (7/2,25/7) Hyperbolic Matrix(413,-524,160,-203) (5/4,14/11) -> (18/7,31/12) Hyperbolic Matrix(297,-380,68,-87) (14/11,9/7) -> (13/3,22/5) Hyperbolic Matrix(331,-450,242,-329) (4/3,15/11) -> (15/11,26/19) Parabolic Matrix(301,-428,64,-91) (17/12,10/7) -> (14/3,19/4) Hyperbolic Matrix(81,-128,50,-79) (3/2,8/5) -> (8/5,13/8) Parabolic Matrix(67,-114,10,-17) (5/3,12/7) -> (6/1,7/1) Hyperbolic Matrix(169,-298,38,-67) (7/4,16/9) -> (22/5,9/2) Hyperbolic Matrix(701,-1250,392,-699) (16/9,25/14) -> (25/14,34/19) Parabolic Matrix(101,-184,28,-51) (9/5,2/1) -> (18/5,11/3) Hyperbolic Matrix(73,-162,32,-71) (2/1,9/4) -> (9/4,16/7) Parabolic Matrix(69,-164,8,-19) (19/8,12/5) -> (8/1,1/0) Hyperbolic Matrix(159,-412,22,-57) (31/12,13/5) -> (7/1,15/2) Hyperbolic Matrix(267,-722,98,-265) (8/3,19/7) -> (19/7,30/11) Parabolic Matrix(61,-200,18,-59) (3/1,10/3) -> (10/3,17/5) Parabolic Matrix(193,-690,40,-143) (25/7,18/5) -> (24/5,5/1) Hyperbolic Matrix(211,-882,50,-209) (4/1,21/5) -> (21/5,38/9) Parabolic Matrix(775,-3698,162,-773) (19/4,43/9) -> (43/9,67/14) Parabolic Matrix(45,-242,8,-43) (5/1,11/2) -> (11/2,17/3) Parabolic Matrix(139,-1058,18,-137) (15/2,23/3) -> (23/3,31/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(95,44,-434,-201) -> Matrix(1,-2,0,1) Matrix(287,132,50,23) -> Matrix(1,0,0,1) Matrix(235,106,-756,-341) -> Matrix(19,-6,16,-5) Matrix(379,170,-1652,-741) -> Matrix(13,-6,-2,1) Matrix(143,64,-896,-401) -> Matrix(5,-2,-2,1) Matrix(95,42,-328,-145) -> Matrix(3,-2,2,-1) Matrix(321,140,94,41) -> Matrix(1,0,-2,1) Matrix(47,20,-228,-97) -> Matrix(1,0,-4,1) Matrix(181,76,-674,-283) -> Matrix(5,-2,8,-3) Matrix(135,56,-446,-185) -> Matrix(19,-8,12,-5) Matrix(307,126,134,55) -> Matrix(21,-10,-44,21) Matrix(301,118,-1028,-403) -> Matrix(7,-4,2,-1) Matrix(1151,450,642,251) -> Matrix(7,-4,-12,7) Matrix(595,232,-2598,-1013) -> Matrix(19,-14,-4,3) Matrix(129,50,-596,-231) -> Matrix(1,-2,0,1) Matrix(209,80,128,49) -> Matrix(1,0,-4,1) Matrix(43,16,-250,-93) -> Matrix(3,-2,2,-1) Matrix(207,76,-700,-257) -> Matrix(7,-4,2,-1) Matrix(83,30,-368,-133) -> Matrix(11,-8,-4,3) Matrix(79,28,-522,-185) -> Matrix(7,-6,-8,7) Matrix(239,84,202,71) -> Matrix(17,-16,-18,17) Matrix(37,12,40,13) -> Matrix(1,0,-2,1) Matrix(149,46,-826,-255) -> Matrix(3,-4,4,-5) Matrix(523,160,-2396,-733) -> Matrix(5,-6,-4,5) Matrix(223,68,-1420,-433) -> Matrix(7,-10,-2,3) Matrix(557,166,406,121) -> Matrix(9,-16,-14,25) Matrix(219,64,-1468,-429) -> Matrix(1,0,-2,1) Matrix(71,20,110,31) -> Matrix(1,-4,0,1) Matrix(37,10,-322,-87) -> Matrix(1,0,0,1) Matrix(143,38,-922,-245) -> Matrix(3,-4,-2,3) Matrix(953,252,1698,449) -> Matrix(7,-12,-4,7) Matrix(107,28,-600,-157) -> Matrix(1,-2,0,1) Matrix(33,8,70,17) -> Matrix(1,-2,0,1) Matrix(35,8,-372,-85) -> Matrix(1,4,0,1) Matrix(101,22,-932,-203) -> Matrix(1,2,-2,-3) Matrix(629,134,230,49) -> Matrix(7,8,-22,-25) Matrix(31,6,98,19) -> Matrix(1,0,2,1) Matrix(223,40,-1522,-273) -> Matrix(1,-2,-2,5) Matrix(653,106,154,25) -> Matrix(1,-2,-2,5) Matrix(2625,388,548,81) -> Matrix(11,6,-46,-25) Matrix(29,4,152,21) -> Matrix(1,0,4,1) Matrix(669,70,86,9) -> Matrix(1,0,0,1) Matrix(113,-18,44,-7) -> Matrix(3,2,-8,-5) Matrix(45,-8,242,-43) -> Matrix(1,0,8,1) Matrix(129,-28,106,-23) -> Matrix(7,-6,-8,7) Matrix(401,-92,170,-39) -> Matrix(1,-6,-2,13) Matrix(273,-64,64,-15) -> Matrix(1,-2,-2,5) Matrix(61,-16,42,-11) -> Matrix(1,2,-2,-3) Matrix(61,-18,200,-59) -> Matrix(1,0,4,1) Matrix(57,-20,20,-7) -> Matrix(1,0,-4,1) Matrix(131,-50,76,-29) -> Matrix(1,-2,0,1) Matrix(73,-30,56,-23) -> Matrix(3,-8,-4,11) Matrix(73,-32,162,-71) -> Matrix(1,-12,0,1) Matrix(167,-92,118,-65) -> Matrix(1,4,-2,-7) Matrix(701,-392,1250,-699) -> Matrix(7,16,-4,-9) Matrix(549,-310,232,-131) -> Matrix(9,14,-20,-31) Matrix(131,-76,50,-29) -> Matrix(3,2,-8,-5) Matrix(81,-50,128,-79) -> Matrix(1,-4,0,1) Matrix(61,-42,16,-11) -> Matrix(1,2,-2,-3) Matrix(105,-76,76,-55) -> Matrix(1,4,-2,-7) Matrix(73,-56,30,-23) -> Matrix(5,8,-12,-19) Matrix(129,-106,28,-23) -> Matrix(5,6,-16,-19) Matrix(85,-72,98,-83) -> Matrix(15,16,-16,-17) Matrix(85,-98,72,-83) -> Matrix(15,16,-16,-17) Matrix(163,-202,46,-57) -> Matrix(5,4,-4,-3) Matrix(413,-524,160,-203) -> Matrix(7,6,-20,-17) Matrix(297,-380,68,-87) -> Matrix(13,10,-30,-23) Matrix(331,-450,242,-329) -> Matrix(47,32,-72,-49) Matrix(301,-428,64,-91) -> Matrix(1,0,-2,1) Matrix(81,-128,50,-79) -> Matrix(7,4,-16,-9) Matrix(67,-114,10,-17) -> Matrix(1,0,0,1) Matrix(169,-298,38,-67) -> Matrix(5,4,-14,-11) Matrix(701,-1250,392,-699) -> Matrix(23,16,-36,-25) Matrix(101,-184,28,-51) -> Matrix(3,2,-8,-5) Matrix(73,-162,32,-71) -> Matrix(23,12,-48,-25) Matrix(69,-164,8,-19) -> Matrix(9,4,-16,-7) Matrix(159,-412,22,-57) -> Matrix(5,2,-18,-7) Matrix(267,-722,98,-265) -> Matrix(47,16,-144,-49) Matrix(61,-200,18,-59) -> Matrix(1,0,4,1) Matrix(193,-690,40,-143) -> Matrix(3,2,-14,-9) Matrix(211,-882,50,-209) -> Matrix(7,4,-16,-9) Matrix(775,-3698,162,-773) -> Matrix(47,12,-192,-49) Matrix(45,-242,8,-43) -> Matrix(1,0,8,1) Matrix(139,-1058,18,-137) -> Matrix(1,0,4,1) 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): 24 Degree of the the map X: 24 Degree of the the map Y: 84 ----------------------------------------------------------------------- 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): 84 Minimal number of generators: 15 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: 12 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 1/1 8/5 2/1 9/4 19/7 3/1 10/3 4/1 5/1 11/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 0/1 1/0 1/4 0/1 1/0 1/3 0/1 1/1 2/5 2/1 1/0 1/2 -1/1 1/0 2/3 -2/1 1/0 5/7 -2/1 1/0 3/4 -2/1 -3/2 4/5 -4/3 -1/1 1/1 -1/1 0/1 4/3 -3/4 -2/3 3/2 -2/3 -1/2 8/5 -1/2 5/3 -1/2 0/1 2/1 -1/1 -1/2 9/4 -1/2 7/3 -1/2 -5/11 5/2 -1/2 -2/5 8/3 -4/11 -1/3 19/7 -1/3 11/4 -1/3 -4/13 3/1 -1/3 0/1 10/3 0/1 7/2 -1/1 0/1 4/1 -1/2 0/1 5/1 -1/4 0/1 11/2 0/1 6/1 0/1 1/0 1/0 -1/2 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(22,-5,31,-7) (0/1,1/4) -> (2/3,5/7) Hyperbolic Matrix(40,-11,11,-3) (1/4,1/3) -> (7/2,4/1) Hyperbolic Matrix(29,-11,37,-14) (1/3,2/5) -> (3/4,4/5) Hyperbolic Matrix(45,-19,19,-8) (2/5,1/2) -> (7/3,5/2) Hyperbolic Matrix(16,-9,9,-5) (1/2,2/3) -> (5/3,2/1) Hyperbolic Matrix(46,-33,7,-5) (5/7,3/4) -> (6/1,1/0) Hyperbolic Matrix(58,-47,21,-17) (4/5,1/1) -> (11/4,3/1) Hyperbolic Matrix(29,-37,11,-14) (1/1,4/3) -> (5/2,8/3) Hyperbolic Matrix(22,-31,5,-7) (4/3,3/2) -> (4/1,5/1) Hyperbolic Matrix(41,-64,25,-39) (3/2,8/5) -> (8/5,5/3) Parabolic Matrix(37,-81,16,-35) (2/1,9/4) -> (9/4,7/3) Parabolic Matrix(134,-361,49,-132) (8/3,19/7) -> (19/7,11/4) Parabolic Matrix(31,-100,9,-29) (3/1,10/3) -> (10/3,7/2) Parabolic Matrix(23,-121,4,-21) (5/1,11/2) -> (11/2,6/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,2,1) Matrix(22,-5,31,-7) -> Matrix(1,-2,0,1) Matrix(40,-11,11,-3) -> Matrix(1,0,-2,1) Matrix(29,-11,37,-14) -> Matrix(3,-4,-2,3) Matrix(45,-19,19,-8) -> Matrix(1,-4,-2,9) Matrix(16,-9,9,-5) -> Matrix(1,2,-2,-3) Matrix(46,-33,7,-5) -> Matrix(1,2,-2,-3) Matrix(58,-47,21,-17) -> Matrix(3,4,-10,-13) Matrix(29,-37,11,-14) -> Matrix(5,4,-14,-11) Matrix(22,-31,5,-7) -> Matrix(3,2,-8,-5) Matrix(41,-64,25,-39) -> Matrix(3,2,-8,-5) Matrix(37,-81,16,-35) -> Matrix(11,6,-24,-13) Matrix(134,-361,49,-132) -> Matrix(23,8,-72,-25) Matrix(31,-100,9,-29) -> Matrix(1,0,2,1) Matrix(23,-121,4,-21) -> Matrix(1,0,4,1) 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): 4 ----------------------------------------------------------------------- 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 2 1 1/1 (-1/1,0/1) 0 13 4/3 (-3/4,-2/3) 0 13 3/2 (-2/3,-1/2) 0 13 8/5 -1/2 2 1 2/1 (-1/1,-1/2) 0 13 9/4 -1/2 6 1 5/2 (-1/2,-2/5) 0 13 8/3 (-4/11,-1/3) 0 13 19/7 -1/3 8 1 3/1 (-1/3,0/1) 0 13 10/3 0/1 2 1 4/1 (-1/2,0/1) 0 13 5/1 (-1/4,0/1) 0 13 11/2 0/1 4 1 1/0 (-1/2,0/1) 0 13 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(29,-37,11,-14) (1/1,4/3) -> (5/2,8/3) Hyperbolic Matrix(22,-31,5,-7) (4/3,3/2) -> (4/1,5/1) Hyperbolic Matrix(31,-48,20,-31) (3/2,8/5) -> (3/2,8/5) Reflection Matrix(9,-16,5,-9) (8/5,2/1) -> (8/5,2/1) Reflection Matrix(17,-36,8,-17) (2/1,9/4) -> (2/1,9/4) Reflection Matrix(19,-45,8,-19) (9/4,5/2) -> (9/4,5/2) Reflection Matrix(113,-304,42,-113) (8/3,19/7) -> (8/3,19/7) Reflection Matrix(20,-57,7,-20) (19/7,3/1) -> (19/7,3/1) Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) Reflection Matrix(11,-40,3,-11) (10/3,4/1) -> (10/3,4/1) Reflection Matrix(21,-110,4,-21) (5/1,11/2) -> (5/1,11/2) Reflection Matrix(-1,11,0,1) (11/2,1/0) -> (11/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,4,1) (-1/1,1/0) -> (-1/2,0/1) Matrix(0,1,1,0) -> Matrix(-1,0,2,1) (-1/1,1/1) -> (-1/1,0/1) Matrix(29,-37,11,-14) -> Matrix(5,4,-14,-11) Matrix(22,-31,5,-7) -> Matrix(3,2,-8,-5) -1/2 Matrix(31,-48,20,-31) -> Matrix(7,4,-12,-7) (3/2,8/5) -> (-2/3,-1/2) Matrix(9,-16,5,-9) -> Matrix(3,2,-4,-3) (8/5,2/1) -> (-1/1,-1/2) Matrix(17,-36,8,-17) -> Matrix(3,2,-4,-3) (2/1,9/4) -> (-1/1,-1/2) Matrix(19,-45,8,-19) -> Matrix(9,4,-20,-9) (9/4,5/2) -> (-1/2,-2/5) Matrix(113,-304,42,-113) -> Matrix(23,8,-66,-23) (8/3,19/7) -> (-4/11,-1/3) Matrix(20,-57,7,-20) -> Matrix(-1,0,6,1) (19/7,3/1) -> (-1/3,0/1) Matrix(19,-60,6,-19) -> Matrix(-1,0,6,1) (3/1,10/3) -> (-1/3,0/1) Matrix(11,-40,3,-11) -> Matrix(-1,0,4,1) (10/3,4/1) -> (-1/2,0/1) Matrix(21,-110,4,-21) -> Matrix(-1,0,8,1) (5/1,11/2) -> (-1/4,0/1) Matrix(-1,11,0,1) -> Matrix(-1,0,4,1) (11/2,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.