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 1/22 -6/13 5/101 -5/11 4/79 -9/20 1/20 -13/29 2/39 -4/9 1/19 -7/16 5/96 -3/7 2/37 -8/19 1/19 -5/12 3/56 -7/17 4/73 -2/5 1/17 -9/23 4/73 -16/41 7/125 -7/18 1/18 -5/13 2/35 -3/8 1/16 -7/19 2/29 -4/11 1/19 -5/14 1/18 -6/17 1/17 -1/3 0/1 -5/16 1/16 -9/29 2/31 -4/13 1/15 -11/36 3/52 -7/23 2/33 -3/10 1/14 -8/27 1/23 -5/17 2/37 -12/41 3/53 -7/24 3/52 -2/7 1/17 -3/11 2/31 -7/26 1/14 -4/15 1/15 -9/34 7/106 -5/19 4/59 -1/4 1/16 -3/13 4/59 -11/48 3/44 -8/35 11/161 -5/22 1/14 -2/9 3/43 -7/32 3/40 -12/55 1/11 -5/23 0/1 -3/14 1/14 -4/19 1/15 -1/5 2/29 -2/11 5/69 -7/39 6/83 -5/28 7/96 -3/17 8/109 -1/6 1/14 -4/25 5/67 -3/19 2/27 -5/32 7/92 -2/13 1/13 -3/20 3/40 -4/27 3/41 -5/34 7/94 -1/7 4/53 -1/8 5/64 -1/9 2/25 -2/19 9/113 -1/10 5/62 0/1 1/11 1/6 5/46 2/11 1/9 3/16 13/116 1/5 4/35 2/9 1/9 3/13 2/17 1/4 1/8 2/7 5/41 3/10 1/8 4/13 9/71 1/3 2/15 3/8 1/8 2/5 3/23 3/7 4/29 4/9 1/7 5/11 6/41 1/2 1/6 5/9 4/29 14/25 1/7 23/41 18/125 9/16 7/48 4/7 1/7 3/5 2/13 5/8 1/6 7/11 4/23 2/3 1/5 5/7 2/7 3/4 1/8 4/5 1/7 5/6 1/6 6/7 1/5 7/8 1/4 1/1 0/1 7/6 1/6 13/11 2/11 6/5 1/5 11/9 2/9 5/4 1/4 14/11 3/19 9/7 2/11 4/3 1/3 15/11 0/1 26/19 1/21 11/8 1/12 7/5 2/15 17/12 3/20 10/7 3/19 3/2 1/6 8/5 1/5 13/8 5/24 5/3 2/9 12/7 1/3 7/4 1/4 16/9 7/29 25/14 1/4 34/19 15/59 9/5 4/15 2/1 1/5 9/4 1/4 16/7 9/35 7/3 4/15 26/11 3/11 19/8 11/40 12/5 1/3 5/2 3/10 18/7 3/7 31/12 1/0 13/5 0/1 8/3 1/3 19/7 0/1 30/11 1/5 11/4 1/4 3/1 2/7 10/3 1/3 17/5 12/35 7/2 5/14 25/7 6/17 18/5 7/19 11/3 8/21 4/1 1/3 21/5 2/5 38/9 11/27 17/4 5/12 13/3 2/5 22/5 7/15 9/2 1/2 14/3 3/7 19/4 3/8 43/9 2/5 67/14 17/42 24/5 7/17 5/1 4/9 11/2 1/2 17/3 14/27 6/1 5/9 7/1 2/3 15/2 9/14 23/3 2/3 31/4 19/28 8/1 5/7 1/0 1/0 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(1,0,22,1) Matrix(95,44,-434,-201) -> Matrix(41,-2,554,-27) Matrix(287,132,50,23) -> Matrix(201,-10,382,-19) Matrix(235,106,-756,-341) -> Matrix(39,-2,644,-33) Matrix(379,170,-1652,-741) -> Matrix(197,-10,2896,-147) Matrix(143,64,-896,-401) -> Matrix(157,-8,2100,-107) Matrix(95,42,-328,-145) -> Matrix(39,-2,644,-33) Matrix(321,140,94,41) -> Matrix(191,-10,554,-29) Matrix(47,20,-228,-97) -> Matrix(1,0,-4,1) Matrix(181,76,-674,-283) -> Matrix(37,-2,574,-31) Matrix(135,56,-446,-185) -> Matrix(37,-2,574,-31) Matrix(307,126,134,55) -> Matrix(145,-8,562,-31) Matrix(301,118,-1028,-403) -> Matrix(37,-2,648,-35) Matrix(1151,450,642,251) -> Matrix(145,-8,562,-31) Matrix(595,232,-2598,-1013) -> Matrix(177,-10,2602,-147) Matrix(129,50,-596,-231) -> Matrix(35,-2,508,-29) Matrix(209,80,128,49) -> Matrix(69,-4,328,-19) Matrix(43,16,-250,-93) -> Matrix(33,-2,446,-27) Matrix(207,76,-700,-257) -> Matrix(1,0,4,1) Matrix(83,30,-368,-133) -> Matrix(35,-2,508,-29) Matrix(79,28,-522,-185) -> Matrix(33,-2,446,-27) Matrix(239,84,202,71) -> Matrix(33,-2,182,-11) Matrix(37,12,40,13) -> Matrix(1,0,-12,1) Matrix(149,46,-826,-255) -> Matrix(65,-4,894,-55) Matrix(523,160,-2396,-733) -> Matrix(1,0,-4,1) Matrix(223,68,-1420,-433) -> Matrix(67,-4,888,-53) Matrix(557,166,406,121) -> Matrix(1,0,-2,1) Matrix(219,64,-1468,-429) -> Matrix(1,0,-4,1) Matrix(71,20,110,31) -> Matrix(33,-2,182,-11) Matrix(37,10,-322,-87) -> Matrix(61,-4,778,-51) Matrix(143,38,-922,-245) -> Matrix(1,0,-2,1) Matrix(953,252,1698,449) -> Matrix(211,-14,1462,-97) Matrix(107,28,-600,-157) -> Matrix(57,-4,784,-55) Matrix(33,8,70,17) -> Matrix(31,-2,202,-13) Matrix(35,8,-372,-85) -> Matrix(117,-8,1448,-99) Matrix(101,22,-932,-203) -> Matrix(31,-2,388,-25) Matrix(629,134,230,49) -> Matrix(1,0,-10,1) Matrix(31,6,98,19) -> Matrix(57,-4,442,-31) Matrix(223,40,-1522,-273) -> Matrix(193,-14,2578,-187) Matrix(653,106,154,25) -> Matrix(163,-12,394,-29) Matrix(2625,388,548,81) -> Matrix(271,-20,664,-49) Matrix(29,4,152,21) -> Matrix(105,-8,932,-71) Matrix(669,70,86,9) -> Matrix(351,-28,514,-41) Matrix(113,-18,44,-7) -> Matrix(19,-2,48,-5) Matrix(45,-8,242,-43) -> Matrix(163,-18,1458,-161) Matrix(129,-28,106,-23) -> Matrix(17,-2,94,-11) Matrix(401,-92,170,-39) -> Matrix(87,-10,322,-37) Matrix(273,-64,64,-15) -> Matrix(69,-8,164,-19) Matrix(61,-16,42,-11) -> Matrix(17,-2,94,-11) Matrix(61,-18,200,-59) -> Matrix(113,-14,896,-111) Matrix(57,-20,20,-7) -> Matrix(1,0,-4,1) Matrix(131,-50,76,-29) -> Matrix(15,-2,68,-9) Matrix(73,-30,56,-23) -> Matrix(15,-2,68,-9) Matrix(73,-32,162,-71) -> Matrix(71,-10,490,-69) Matrix(167,-92,118,-65) -> Matrix(15,-2,98,-13) Matrix(701,-392,1250,-699) -> Matrix(155,-22,1078,-153) Matrix(549,-310,232,-131) -> Matrix(67,-10,248,-37) Matrix(131,-76,50,-29) -> Matrix(13,-2,46,-7) Matrix(81,-50,128,-79) -> Matrix(37,-6,216,-35) Matrix(61,-42,16,-11) -> Matrix(11,-2,28,-5) Matrix(105,-76,76,-55) -> Matrix(1,0,4,1) Matrix(73,-56,30,-23) -> Matrix(13,-2,46,-7) Matrix(129,-106,28,-23) -> Matrix(11,-2,28,-5) Matrix(85,-72,98,-83) -> Matrix(11,-2,50,-9) Matrix(85,-98,72,-83) -> Matrix(13,-2,72,-11) Matrix(163,-202,46,-57) -> Matrix(21,-4,58,-11) Matrix(413,-524,160,-203) -> Matrix(1,0,-4,1) Matrix(297,-380,68,-87) -> Matrix(23,-4,52,-9) Matrix(331,-450,242,-329) -> Matrix(1,0,18,1) Matrix(301,-428,64,-91) -> Matrix(1,0,-4,1) Matrix(81,-128,50,-79) -> Matrix(31,-6,150,-29) Matrix(67,-114,10,-17) -> Matrix(17,-4,30,-7) Matrix(169,-298,38,-67) -> Matrix(1,0,-2,1) Matrix(701,-1250,392,-699) -> Matrix(89,-22,352,-87) Matrix(101,-184,28,-51) -> Matrix(13,-4,36,-11) Matrix(73,-162,32,-71) -> Matrix(41,-10,160,-39) Matrix(69,-164,8,-19) -> Matrix(29,-8,40,-11) Matrix(159,-412,22,-57) -> Matrix(9,-2,14,-3) Matrix(267,-722,98,-265) -> Matrix(1,0,2,1) Matrix(61,-200,18,-59) -> Matrix(43,-14,126,-41) Matrix(193,-690,40,-143) -> Matrix(39,-14,92,-33) Matrix(211,-882,50,-209) -> Matrix(61,-24,150,-59) Matrix(775,-3698,162,-773) -> Matrix(101,-40,250,-99) Matrix(45,-242,8,-43) -> Matrix(37,-18,72,-35) Matrix(139,-1058,18,-137) -> Matrix(85,-56,126,-83) 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): 42 Degree of the the map X: 42 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 1/11 1/4 1/8 1/3 2/15 2/5 3/23 1/2 1/6 2/3 1/5 5/7 2/7 3/4 1/8 4/5 1/7 1/1 0/1 4/3 1/3 3/2 1/6 8/5 1/5 5/3 2/9 2/1 1/5 9/4 1/4 7/3 4/15 5/2 3/10 8/3 1/3 19/7 0/1 11/4 1/4 3/1 2/7 10/3 1/3 7/2 5/14 4/1 1/3 5/1 4/9 11/2 1/2 6/1 5/9 1/0 1/0 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(1,0,11,1) Matrix(22,-5,31,-7) -> Matrix(9,-1,37,-4) Matrix(40,-11,11,-3) -> Matrix(25,-3,67,-8) Matrix(29,-11,37,-14) -> Matrix(8,-1,41,-5) Matrix(45,-19,19,-8) -> Matrix(22,-3,81,-11) Matrix(16,-9,9,-5) -> Matrix(7,-1,29,-4) Matrix(46,-33,7,-5) -> Matrix(3,-1,7,-2) Matrix(58,-47,21,-17) -> Matrix(5,-1,21,-4) Matrix(29,-37,11,-14) -> Matrix(6,-1,19,-3) Matrix(22,-31,5,-7) -> Matrix(7,-1,15,-2) Matrix(41,-64,25,-39) -> Matrix(16,-3,75,-14) Matrix(37,-81,16,-35) -> Matrix(21,-5,80,-19) Matrix(134,-361,49,-132) -> Matrix(1,0,1,1) Matrix(31,-100,9,-29) -> Matrix(22,-7,63,-20) Matrix(23,-121,4,-21) -> Matrix(19,-9,36,-17) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 1 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: 1 Number of equivalence classes of cusps: 1 Genus: 0 Degree of H/liftables -> H/(image of liftables): 42 ----------------------------------------------------------------------- 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 11 1 1/1 0/1 1 13 4/3 1/3 1 13 3/2 1/6 1 13 8/5 1/5 3 1 2/1 1/5 1 13 9/4 1/4 5 1 5/2 3/10 1 13 8/3 1/3 1 13 19/7 0/1 1 1 3/1 2/7 1 13 10/3 1/3 7 1 4/1 1/3 1 13 5/1 4/9 1 13 11/2 1/2 9 1 1/0 1/0 1 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(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(0,1,1,0) -> Matrix(1,0,11,-1) (-1/1,1/1) -> (0/1,2/11) Matrix(29,-37,11,-14) -> Matrix(6,-1,19,-3) Matrix(22,-31,5,-7) -> Matrix(7,-1,15,-2) Matrix(31,-48,20,-31) -> Matrix(11,-2,60,-11) (3/2,8/5) -> (1/6,1/5) Matrix(9,-16,5,-9) -> Matrix(4,-1,15,-4) (8/5,2/1) -> (1/5,1/3) Matrix(17,-36,8,-17) -> Matrix(9,-2,40,-9) (2/1,9/4) -> (1/5,1/4) Matrix(19,-45,8,-19) -> Matrix(11,-3,40,-11) (9/4,5/2) -> (1/4,3/10) Matrix(113,-304,42,-113) -> Matrix(1,0,6,-1) (8/3,19/7) -> (0/1,1/3) Matrix(20,-57,7,-20) -> Matrix(1,0,7,-1) (19/7,3/1) -> (0/1,2/7) Matrix(19,-60,6,-19) -> Matrix(13,-4,42,-13) (3/1,10/3) -> (2/7,1/3) Matrix(11,-40,3,-11) -> Matrix(8,-3,21,-8) (10/3,4/1) -> (1/3,3/7) Matrix(21,-110,4,-21) -> Matrix(17,-8,36,-17) (5/1,11/2) -> (4/9,1/2) Matrix(-1,11,0,1) -> Matrix(-1,1,0,1) (11/2,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.