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): 384 Minimal number of generators: 65 Number of equivalence classes of cusps: 32 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -3/7 -1/3 -1/5 -1/7 0/1 3/13 1/3 5/11 1/2 3/5 7/9 1/1 9/7 3/2 5/3 31/17 2/1 11/5 17/7 5/2 3/1 7/2 4/1 13/3 9/2 5/1 11/2 6/1 13/2 7/1 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 1/4 -6/13 1/2 -5/11 1/4 -9/20 0/1 1/0 -4/9 1/6 -7/16 0/1 1/6 -10/23 3/14 -3/7 1/4 -5/12 1/3 1/2 -7/17 1/0 -9/22 -1/4 0/1 -2/5 1/6 -3/8 1/4 1/3 -4/11 1/2 -5/14 0/1 1/6 -1/3 1/4 -4/13 3/10 -3/10 1/3 1/2 -5/17 1/4 -7/24 1/4 1/3 -2/7 1/2 -1/4 1/4 1/3 -4/17 9/26 -7/30 6/17 5/14 -3/13 3/8 -5/22 1/3 3/8 -2/9 1/2 -3/14 3/8 2/5 -7/33 5/12 -4/19 1/2 -1/5 1/2 -3/16 0/1 1/0 -5/27 1/4 -2/11 1/2 -1/6 1/4 1/3 -2/13 5/14 -3/20 1/3 3/8 -1/7 3/8 -1/8 3/7 1/2 0/1 1/2 1/5 3/4 2/9 5/6 3/13 1/1 4/17 7/6 1/4 1/1 1/0 1/3 1/2 4/11 1/2 3/8 2/3 3/4 2/5 1/2 7/17 3/4 5/12 3/4 1/1 3/7 3/4 4/9 9/10 5/11 1/1 6/13 11/10 1/2 1/1 1/0 4/7 3/2 3/5 1/0 8/13 -1/2 5/8 0/1 1/0 17/27 1/2 12/19 1/2 7/11 1/0 9/14 0/1 1/2 2/3 1/2 3/4 1/2 1/1 7/9 1/1 11/14 1/1 7/6 4/5 3/2 5/6 2/1 1/0 1/1 1/0 5/4 -1/2 0/1 9/7 0/1 13/10 0/1 1/6 4/3 1/2 11/8 0/1 1/4 18/13 1/2 7/5 1/2 3/2 1/1 1/0 8/5 5/2 5/3 1/0 12/7 -7/2 19/11 1/0 7/4 -2/1 1/0 16/9 -3/2 9/5 1/0 20/11 -3/2 31/17 -1/1 11/6 -1/1 1/0 13/7 1/0 2/1 -1/2 11/5 0/1 20/9 1/18 9/4 0/1 1/8 7/3 1/4 12/5 1/2 41/17 5/12 29/12 4/9 1/2 17/7 1/2 22/9 1/2 5/2 1/2 1/1 8/3 1/2 19/7 1/0 11/4 1/1 1/0 3/1 1/0 13/4 -1/1 1/0 23/7 1/0 10/3 -3/2 17/5 -3/4 7/2 -1/2 0/1 11/3 1/0 37/10 -1/1 1/0 63/17 -1/1 26/7 -1/2 15/4 0/1 1/0 4/1 -1/2 13/3 0/1 22/5 1/10 9/2 0/1 1/4 14/3 1/2 33/7 1/2 19/4 1/2 2/3 5/1 1/0 11/2 0/1 1/4 6/1 1/2 13/2 1/1 1/0 7/1 1/0 8/1 -1/2 1/0 0/1 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(29,14,2,1) (-1/2,-6/13) -> (8/1,1/0) Hyperbolic Matrix(87,40,-472,-217) (-6/13,-5/11) -> (-5/27,-2/11) Hyperbolic Matrix(173,78,-814,-367) (-5/11,-9/20) -> (-3/14,-7/33) Hyperbolic Matrix(521,234,118,53) (-9/20,-4/9) -> (22/5,9/2) Hyperbolic Matrix(113,50,174,77) (-4/9,-7/16) -> (9/14,2/3) Hyperbolic Matrix(225,98,-962,-419) (-7/16,-10/23) -> (-4/17,-7/30) Hyperbolic Matrix(313,136,168,73) (-10/23,-3/7) -> (13/7,2/1) Hyperbolic Matrix(81,34,-274,-115) (-3/7,-5/12) -> (-3/10,-5/17) Hyperbolic Matrix(29,12,-220,-91) (-5/12,-7/17) -> (-1/7,-1/8) Hyperbolic Matrix(161,66,-866,-355) (-7/17,-9/22) -> (-3/16,-5/27) Hyperbolic Matrix(485,198,218,89) (-9/22,-2/5) -> (20/9,9/4) Hyperbolic Matrix(79,30,50,19) (-2/5,-3/8) -> (3/2,8/5) Hyperbolic Matrix(27,10,-154,-57) (-3/8,-4/11) -> (-2/11,-1/6) Hyperbolic Matrix(199,72,152,55) (-4/11,-5/14) -> (13/10,4/3) Hyperbolic Matrix(125,44,196,69) (-5/14,-1/3) -> (7/11,9/14) Hyperbolic Matrix(165,52,92,29) (-1/3,-4/13) -> (16/9,9/5) Hyperbolic Matrix(183,56,232,71) (-4/13,-3/10) -> (11/14,4/5) Hyperbolic Matrix(499,146,270,79) (-5/17,-7/24) -> (11/6,13/7) Hyperbolic Matrix(69,20,-452,-131) (-7/24,-2/7) -> (-2/13,-3/20) Hyperbolic Matrix(23,6,42,11) (-2/7,-1/4) -> (1/2,4/7) Hyperbolic Matrix(41,10,86,21) (-1/4,-4/17) -> (6/13,1/2) Hyperbolic Matrix(1607,374,666,155) (-7/30,-3/13) -> (41/17,29/12) Hyperbolic Matrix(61,14,-414,-95) (-3/13,-5/22) -> (-3/20,-1/7) Hyperbolic Matrix(249,56,40,9) (-5/22,-2/9) -> (6/1,13/2) Hyperbolic Matrix(287,62,162,35) (-2/9,-3/14) -> (7/4,16/9) Hyperbolic Matrix(1175,248,488,103) (-7/33,-4/19) -> (12/5,41/17) Hyperbolic Matrix(383,80,608,127) (-4/19,-1/5) -> (17/27,12/19) Hyperbolic Matrix(297,56,472,89) (-1/5,-3/16) -> (5/8,17/27) Hyperbolic Matrix(37,6,154,25) (-1/6,-2/13) -> (4/17,1/4) Hyperbolic Matrix(181,22,74,9) (-1/8,0/1) -> (22/9,5/2) Hyperbolic Matrix(67,-12,28,-5) (0/1,1/5) -> (7/3,12/5) Hyperbolic Matrix(211,-46,78,-17) (1/5,2/9) -> (8/3,19/7) Hyperbolic Matrix(79,-18,338,-77) (2/9,3/13) -> (3/13,4/17) Parabolic Matrix(37,-10,26,-7) (1/4,1/3) -> (7/5,3/2) Hyperbolic Matrix(131,-46,94,-33) (1/3,4/11) -> (18/13,7/5) Hyperbolic Matrix(211,-78,46,-17) (4/11,3/8) -> (9/2,14/3) Hyperbolic Matrix(57,-22,70,-27) (3/8,2/5) -> (4/5,5/6) Hyperbolic Matrix(171,-70,22,-9) (2/5,7/17) -> (7/1,8/1) Hyperbolic Matrix(305,-126,46,-19) (7/17,5/12) -> (13/2,7/1) Hyperbolic Matrix(305,-128,112,-47) (5/12,3/7) -> (19/7,11/4) Hyperbolic Matrix(223,-98,66,-29) (3/7,4/9) -> (10/3,17/5) Hyperbolic Matrix(111,-50,242,-109) (4/9,5/11) -> (5/11,6/13) Parabolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(227,-140,60,-37) (8/13,5/8) -> (15/4,4/1) Hyperbolic Matrix(391,-248,216,-137) (12/19,7/11) -> (9/5,20/11) Hyperbolic Matrix(47,-34,18,-13) (2/3,3/4) -> (5/2,8/3) Hyperbolic Matrix(127,-98,162,-125) (3/4,7/9) -> (7/9,11/14) Parabolic Matrix(121,-102,70,-59) (5/6,1/1) -> (19/11,7/4) Hyperbolic Matrix(51,-62,14,-17) (1/1,5/4) -> (7/2,11/3) Hyperbolic Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(81,-110,14,-19) (4/3,11/8) -> (11/2,6/1) Hyperbolic Matrix(403,-556,108,-149) (11/8,18/13) -> (26/7,15/4) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(271,-466,82,-141) (12/7,19/11) -> (23/7,10/3) Hyperbolic Matrix(1135,-2066,306,-557) (20/11,31/17) -> (63/17,26/7) Hyperbolic Matrix(1007,-1840,272,-497) (31/17,11/6) -> (37/10,63/17) Hyperbolic Matrix(111,-242,50,-109) (2/1,11/5) -> (11/5,20/9) Parabolic Matrix(53,-122,10,-23) (9/4,7/3) -> (5/1,11/2) Hyperbolic Matrix(529,-1280,112,-271) (29/12,17/7) -> (33/7,19/4) Hyperbolic Matrix(395,-964,84,-205) (17/7,22/9) -> (14/3,33/7) Hyperbolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(303,-994,82,-269) (13/4,23/7) -> (11/3,37/10) Hyperbolic Matrix(105,-358,22,-75) (17/5,7/2) -> (19/4,5/1) Hyperbolic Matrix(79,-338,18,-77) (4/1,13/3) -> (13/3,22/5) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(29,14,2,1) -> Matrix(1,0,-4,1) Matrix(87,40,-472,-217) -> Matrix(1,0,0,1) Matrix(173,78,-814,-367) -> Matrix(3,-2,8,-5) Matrix(521,234,118,53) -> Matrix(1,0,4,1) Matrix(113,50,174,77) -> Matrix(1,0,-4,1) Matrix(225,98,-962,-419) -> Matrix(31,-6,88,-17) Matrix(313,136,168,73) -> Matrix(9,-2,-4,1) Matrix(81,34,-274,-115) -> Matrix(1,0,0,1) Matrix(29,12,-220,-91) -> Matrix(3,-2,8,-5) Matrix(161,66,-866,-355) -> Matrix(1,0,4,1) Matrix(485,198,218,89) -> Matrix(1,0,12,1) Matrix(79,30,50,19) -> Matrix(7,-2,4,-1) Matrix(27,10,-154,-57) -> Matrix(1,0,0,1) Matrix(199,72,152,55) -> Matrix(1,0,0,1) Matrix(125,44,196,69) -> Matrix(1,0,-4,1) Matrix(165,52,92,29) -> Matrix(1,0,-4,1) Matrix(183,56,232,71) -> Matrix(19,-6,16,-5) Matrix(499,146,270,79) -> Matrix(1,0,-4,1) Matrix(69,20,-452,-131) -> Matrix(13,-4,36,-11) Matrix(23,6,42,11) -> Matrix(7,-2,4,-1) Matrix(41,10,86,21) -> Matrix(7,-2,4,-1) Matrix(1607,374,666,155) -> Matrix(39,-14,92,-33) Matrix(61,14,-414,-95) -> Matrix(1,0,0,1) Matrix(249,56,40,9) -> Matrix(5,-2,8,-3) Matrix(287,62,162,35) -> Matrix(11,-4,-8,3) Matrix(1175,248,488,103) -> Matrix(1,0,0,1) Matrix(383,80,608,127) -> Matrix(5,-2,8,-3) Matrix(297,56,472,89) -> Matrix(1,0,0,1) Matrix(37,6,154,25) -> Matrix(7,-2,4,-1) Matrix(181,22,74,9) -> Matrix(9,-4,16,-7) Matrix(67,-12,28,-5) -> Matrix(3,-2,8,-5) Matrix(211,-46,78,-17) -> Matrix(5,-4,4,-3) Matrix(79,-18,338,-77) -> Matrix(13,-12,12,-11) Matrix(37,-10,26,-7) -> Matrix(1,0,0,1) Matrix(131,-46,94,-33) -> Matrix(3,-2,8,-5) Matrix(211,-78,46,-17) -> Matrix(3,-2,8,-5) Matrix(57,-22,70,-27) -> Matrix(5,-4,4,-3) Matrix(171,-70,22,-9) -> Matrix(3,-2,-4,3) Matrix(305,-126,46,-19) -> Matrix(5,-4,4,-3) Matrix(305,-128,112,-47) -> Matrix(5,-4,4,-3) Matrix(223,-98,66,-29) -> Matrix(7,-6,-8,7) Matrix(111,-50,242,-109) -> Matrix(21,-20,20,-19) Matrix(61,-36,100,-59) -> Matrix(1,-2,0,1) Matrix(227,-140,60,-37) -> Matrix(1,0,0,1) Matrix(391,-248,216,-137) -> Matrix(1,-2,0,1) Matrix(47,-34,18,-13) -> Matrix(1,0,0,1) Matrix(127,-98,162,-125) -> Matrix(9,-8,8,-7) Matrix(121,-102,70,-59) -> Matrix(1,-4,0,1) Matrix(51,-62,14,-17) -> Matrix(1,0,0,1) Matrix(127,-162,98,-125) -> Matrix(1,0,8,1) Matrix(81,-110,14,-19) -> Matrix(1,0,0,1) Matrix(403,-556,108,-149) -> Matrix(1,0,-4,1) Matrix(61,-100,36,-59) -> Matrix(1,-6,0,1) Matrix(271,-466,82,-141) -> Matrix(1,2,0,1) Matrix(1135,-2066,306,-557) -> Matrix(3,4,-4,-5) Matrix(1007,-1840,272,-497) -> Matrix(1,0,0,1) Matrix(111,-242,50,-109) -> Matrix(1,0,20,1) Matrix(53,-122,10,-23) -> Matrix(1,0,-4,1) Matrix(529,-1280,112,-271) -> Matrix(13,-6,24,-11) Matrix(395,-964,84,-205) -> Matrix(7,-4,16,-9) Matrix(25,-72,8,-23) -> Matrix(1,-2,0,1) Matrix(303,-994,82,-269) -> Matrix(1,0,0,1) Matrix(105,-358,22,-75) -> Matrix(3,2,4,3) Matrix(79,-338,18,-77) -> Matrix(1,0,12,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 11 Degree of the the map X: 22 Degree of the the map Y: 64 Permutation triple for Y: ((1,6,22,37,63,53,46,64,62,42,31,61,50,23,7,2)(3,12,38,57,27,8,14,41,35,34,47,55,58,39,13,4)(5,11,36,49,51,56,25,19,33,10,9,32,59,28,45,18)(15,30,29,54,24,40,44,17,21,20,48,60,52,26,43,16); (1,4,16,33,62,41,17,5)(3,10,35,11)(6,21,42,15)(7,25,40,39,46,45,26,8)(9,30,12,22,36,20,34,31)(13,14)(18,19)(23,52,53,24)(27,56,58,28)(29,48)(37,61)(50,57,60,59,63,55,54,51); (1,2,8,28,60,29,9,3)(4,14,26,23,51,49,22,15)(5,19,7,24,55,47,20,6)(10,16,43,45,58,63,61,34)(11,17,44,25,27,50,37,12)(13,40,53,59,32,31,21,41)(18,46,52,57,38,30,42,33)(35,62,64,39,56,54,48,36)) ----------------------------------------------------------------------- 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): 192 Minimal number of generators: 33 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: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -3/7 -1/3 -1/5 -1/7 0/1 1/3 1/2 3/5 1/1 9/7 3/2 5/3 2/1 11/5 17/7 5/2 3/1 4/1 13/3 5/1 6/1 7/1 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 1/4 -5/11 1/4 -4/9 1/6 -3/7 1/4 -5/12 1/3 1/2 -7/17 1/0 -2/5 1/6 -3/8 1/4 1/3 -4/11 1/2 -1/3 1/4 -4/13 3/10 -3/10 1/3 1/2 -5/17 1/4 -2/7 1/2 -1/4 1/4 1/3 -4/17 9/26 -3/13 3/8 -5/22 1/3 3/8 -2/9 1/2 -1/5 1/2 -2/11 1/2 -1/6 1/4 1/3 -2/13 5/14 -1/7 3/8 -1/8 3/7 1/2 0/1 1/2 1/5 3/4 1/4 1/1 1/0 1/3 1/2 2/5 1/2 3/7 3/4 1/2 1/1 1/0 3/5 1/0 5/8 0/1 1/0 2/3 1/2 3/4 1/2 1/1 1/1 1/0 5/4 -1/2 0/1 9/7 0/1 13/10 0/1 1/6 4/3 1/2 7/5 1/2 3/2 1/1 1/0 5/3 1/0 7/4 -2/1 1/0 2/1 -1/2 11/5 0/1 20/9 1/18 9/4 0/1 1/8 7/3 1/4 12/5 1/2 17/7 1/2 22/9 1/2 5/2 1/2 1/1 3/1 1/0 7/2 -1/2 0/1 4/1 -1/2 13/3 0/1 22/5 1/10 9/2 0/1 1/4 5/1 1/0 6/1 1/2 7/1 1/0 8/1 -1/2 1/0 0/1 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(61,28,-268,-123) (-1/2,-5/11) -> (-3/13,-5/22) Hyperbolic Matrix(31,14,-206,-93) (-5/11,-4/9) -> (-2/13,-1/7) Hyperbolic Matrix(59,26,-202,-89) (-4/9,-3/7) -> (-5/17,-2/7) Hyperbolic Matrix(81,34,-274,-115) (-3/7,-5/12) -> (-3/10,-5/17) Hyperbolic Matrix(29,12,-220,-91) (-5/12,-7/17) -> (-1/7,-1/8) Hyperbolic Matrix(83,34,-354,-145) (-7/17,-2/5) -> (-4/17,-3/13) Hyperbolic Matrix(51,20,28,11) (-2/5,-3/8) -> (7/4,2/1) Hyperbolic Matrix(27,10,-154,-57) (-3/8,-4/11) -> (-2/11,-1/6) Hyperbolic Matrix(23,8,-72,-25) (-4/11,-1/3) -> (-1/3,-4/13) Parabolic Matrix(209,64,160,49) (-4/13,-3/10) -> (13/10,4/3) Hyperbolic Matrix(43,12,68,19) (-2/7,-1/4) -> (5/8,2/3) Hyperbolic Matrix(233,56,104,25) (-1/4,-4/17) -> (20/9,9/4) Hyperbolic Matrix(185,42,22,5) (-5/22,-2/9) -> (8/1,1/0) Hyperbolic Matrix(19,4,-100,-21) (-2/9,-1/5) -> (-1/5,-2/11) Parabolic Matrix(249,40,56,9) (-1/6,-2/13) -> (22/5,9/2) Hyperbolic Matrix(181,22,74,9) (-1/8,0/1) -> (22/9,5/2) Hyperbolic Matrix(67,-12,28,-5) (0/1,1/5) -> (7/3,12/5) Hyperbolic Matrix(65,-14,14,-3) (1/5,1/4) -> (9/2,5/1) Hyperbolic Matrix(37,-10,26,-7) (1/4,1/3) -> (7/5,3/2) Hyperbolic Matrix(47,-18,34,-13) (1/3,2/5) -> (4/3,7/5) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(77,-34,34,-15) (3/7,1/2) -> (9/4,7/3) Hyperbolic Matrix(31,-18,50,-29) (1/2,3/5) -> (3/5,5/8) Parabolic Matrix(37,-26,10,-7) (2/3,3/4) -> (7/2,4/1) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(127,-162,98,-125) (5/4,9/7) -> (9/7,13/10) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic Matrix(111,-242,50,-109) (2/1,11/5) -> (11/5,20/9) Parabolic Matrix(239,-578,98,-237) (12/5,17/7) -> (17/7,22/9) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(79,-338,18,-77) (4/1,13/3) -> (13/3,22/5) Parabolic Matrix(15,-98,2,-13) (6/1,7/1) -> (7/1,8/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(61,28,-268,-123) -> Matrix(1,-1,4,-3) Matrix(31,14,-206,-93) -> Matrix(1,-1,4,-3) Matrix(59,26,-202,-89) -> Matrix(5,-1,16,-3) Matrix(81,34,-274,-115) -> Matrix(1,0,0,1) Matrix(29,12,-220,-91) -> Matrix(3,-2,8,-5) Matrix(83,34,-354,-145) -> Matrix(3,1,8,3) Matrix(51,20,28,11) -> Matrix(5,-1,-4,1) Matrix(27,10,-154,-57) -> Matrix(1,0,0,1) Matrix(23,8,-72,-25) -> Matrix(5,-1,16,-3) Matrix(209,64,160,49) -> Matrix(3,-1,16,-5) Matrix(43,12,68,19) -> Matrix(3,-1,4,-1) Matrix(233,56,104,25) -> Matrix(3,-1,28,-9) Matrix(185,42,22,5) -> Matrix(3,-1,-8,3) Matrix(19,4,-100,-21) -> Matrix(3,-1,4,-1) Matrix(249,40,56,9) -> Matrix(3,-1,16,-5) Matrix(181,22,74,9) -> Matrix(9,-4,16,-7) Matrix(67,-12,28,-5) -> Matrix(3,-2,8,-5) Matrix(65,-14,14,-3) -> Matrix(1,-1,4,-3) Matrix(37,-10,26,-7) -> Matrix(1,0,0,1) Matrix(47,-18,34,-13) -> Matrix(1,-1,4,-3) Matrix(67,-28,12,-5) -> Matrix(1,-1,4,-3) Matrix(77,-34,34,-15) -> Matrix(1,-1,8,-7) Matrix(31,-18,50,-29) -> Matrix(1,-1,0,1) Matrix(37,-26,10,-7) -> Matrix(1,-1,0,1) Matrix(9,-8,8,-7) -> Matrix(1,-1,0,1) Matrix(127,-162,98,-125) -> Matrix(1,0,8,1) Matrix(31,-50,18,-29) -> Matrix(1,-3,0,1) Matrix(111,-242,50,-109) -> Matrix(1,0,20,1) Matrix(239,-578,98,-237) -> Matrix(11,-5,20,-9) Matrix(13,-36,4,-11) -> Matrix(1,-1,0,1) Matrix(79,-338,18,-77) -> Matrix(1,0,12,1) Matrix(15,-98,2,-13) -> Matrix(1,-1,0,1) 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): 11 ----------------------------------------------------------------------- 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 0/1 1/2 1 16 1/5 3/4 1 8 1/4 (1/1,1/0) 0 16 1/3 1/2 1 4 2/5 1/2 1 16 3/7 3/4 1 8 1/2 (1/1,1/0) 0 16 3/5 1/0 1 2 2/3 1/2 1 16 3/4 (1/2,1/1) 0 16 1/1 1/0 1 8 5/4 (-1/2,0/1) 0 16 9/7 0/1 4 1 4/3 1/2 1 16 7/5 1/2 1 4 3/2 (1/1,1/0) 0 16 5/3 1/0 3 2 2/1 -1/2 1 16 11/5 0/1 10 1 9/4 (0/1,1/8) 0 16 7/3 1/4 1 8 12/5 1/2 1 16 17/7 1/2 5 2 5/2 (1/2,1/1) 0 16 3/1 1/0 1 4 7/2 (-1/2,0/1) 0 16 4/1 -1/2 1 16 13/3 0/1 6 1 9/2 (0/1,1/4) 0 16 5/1 1/0 1 8 6/1 1/2 1 16 7/1 1/0 1 2 1/0 (0/1,1/0) 0 16 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(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(67,-12,28,-5) (0/1,1/5) -> (7/3,12/5) Hyperbolic Matrix(65,-14,14,-3) (1/5,1/4) -> (9/2,5/1) Hyperbolic Matrix(37,-10,26,-7) (1/4,1/3) -> (7/5,3/2) Hyperbolic Matrix(47,-18,34,-13) (1/3,2/5) -> (4/3,7/5) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(77,-34,34,-15) (3/7,1/2) -> (9/4,7/3) Hyperbolic Matrix(11,-6,20,-11) (1/2,3/5) -> (1/2,3/5) Reflection Matrix(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(37,-26,10,-7) (2/3,3/4) -> (7/2,4/1) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) Reflection Matrix(21,-44,10,-21) (2/1,11/5) -> (2/1,11/5) Reflection Matrix(89,-198,40,-89) (11/5,9/4) -> (11/5,9/4) Reflection Matrix(169,-408,70,-169) (12/5,17/7) -> (12/5,17/7) Reflection Matrix(69,-170,28,-69) (17/7,5/2) -> (17/7,5/2) Reflection Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(25,-104,6,-25) (4/1,13/3) -> (4/1,13/3) Reflection Matrix(53,-234,12,-53) (13/3,9/2) -> (13/3,9/2) Reflection Matrix(13,-84,2,-13) (6/1,7/1) -> (6/1,7/1) Reflection Matrix(-1,14,0,1) (7/1,1/0) -> (7/1,1/0) 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(-1,0,2,1) -> Matrix(1,0,4,-1) (-1/1,0/1) -> (0/1,1/2) Matrix(67,-12,28,-5) -> Matrix(3,-2,8,-5) 1/2 Matrix(65,-14,14,-3) -> Matrix(1,-1,4,-3) 1/2 Matrix(37,-10,26,-7) -> Matrix(1,0,0,1) Matrix(47,-18,34,-13) -> Matrix(1,-1,4,-3) 1/2 Matrix(67,-28,12,-5) -> Matrix(1,-1,4,-3) 1/2 Matrix(77,-34,34,-15) -> Matrix(1,-1,8,-7) Matrix(11,-6,20,-11) -> Matrix(-1,2,0,1) (1/2,3/5) -> (1/1,1/0) Matrix(19,-12,30,-19) -> Matrix(-1,1,0,1) (3/5,2/3) -> (1/2,1/0) Matrix(37,-26,10,-7) -> Matrix(1,-1,0,1) 1/0 Matrix(9,-8,8,-7) -> Matrix(1,-1,0,1) 1/0 Matrix(71,-90,56,-71) -> Matrix(-1,0,4,1) (5/4,9/7) -> (-1/2,0/1) Matrix(55,-72,42,-55) -> Matrix(1,0,4,-1) (9/7,4/3) -> (0/1,1/2) Matrix(19,-30,12,-19) -> Matrix(-1,2,0,1) (3/2,5/3) -> (1/1,1/0) Matrix(11,-20,6,-11) -> Matrix(1,1,0,-1) (5/3,2/1) -> (-1/2,1/0) Matrix(21,-44,10,-21) -> Matrix(-1,0,4,1) (2/1,11/5) -> (-1/2,0/1) Matrix(89,-198,40,-89) -> Matrix(1,0,16,-1) (11/5,9/4) -> (0/1,1/8) Matrix(169,-408,70,-169) -> Matrix(7,-3,16,-7) (12/5,17/7) -> (3/8,1/2) Matrix(69,-170,28,-69) -> Matrix(3,-2,4,-3) (17/7,5/2) -> (1/2,1/1) Matrix(13,-36,4,-11) -> Matrix(1,-1,0,1) 1/0 Matrix(25,-104,6,-25) -> Matrix(-1,0,4,1) (4/1,13/3) -> (-1/2,0/1) Matrix(53,-234,12,-53) -> Matrix(1,0,8,-1) (13/3,9/2) -> (0/1,1/4) Matrix(13,-84,2,-13) -> Matrix(-1,1,0,1) (6/1,7/1) -> (1/2,1/0) Matrix(-1,14,0,1) -> Matrix(1,0,0,-1) (7/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.