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 0/1 1/1 13/11 13/10 13/9 3/2 39/25 13/8 13/7 2/1 13/6 26/11 5/2 13/5 39/14 3/1 13/4 10/3 7/2 11/3 26/7 4/1 13/3 9/2 14/3 5/1 26/5 11/2 52/9 6/1 13/2 7/1 8/1 26/3 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 -1/1 0/1 -13/2 0/1 -6/1 1/0 -11/2 -1/2 0/1 -16/3 -1/4 -5/1 0/1 1/2 -9/2 1/1 1/0 -13/3 1/0 -4/1 1/0 -11/3 -2/1 1/0 -7/2 -2/1 -1/1 -10/3 -3/2 -13/4 -1/1 -3/1 -1/1 0/1 -14/5 -3/4 -39/14 -2/3 -25/9 -2/3 -1/2 -11/4 -2/3 -1/2 -8/3 -1/2 -13/5 -1/2 -5/2 -1/2 0/1 -12/5 1/0 -7/3 -1/1 -2/3 -9/4 -1/1 -1/2 -11/5 -2/3 -1/2 -13/6 -1/2 -2/1 -1/2 -13/7 -1/2 -11/6 -1/2 -2/5 -20/11 -3/8 -9/5 -1/2 -1/3 -25/14 -1/2 -2/5 -16/9 -3/8 -7/4 -2/5 -1/3 -26/15 -1/3 -19/11 -1/3 -4/13 -12/7 -1/4 -29/17 -2/5 -1/3 -17/10 -1/3 -1/4 -5/3 -1/2 0/1 -13/8 -1/2 -8/5 -1/2 -19/12 -2/5 -1/3 -11/7 -1/2 -2/5 -25/16 -1/2 -2/5 -39/25 -2/5 -14/9 -3/8 -3/2 -1/3 0/1 -13/9 -1/3 -10/7 -3/10 -37/26 -2/7 -1/4 -27/19 -2/7 -1/4 -17/12 -1/3 -1/4 -7/5 -1/3 -2/7 -25/18 -2/7 -1/4 -43/31 -3/11 -1/4 -18/13 -1/4 -11/8 -2/7 -1/4 -26/19 -1/4 -15/11 -1/4 0/1 -4/3 -1/4 -13/10 -1/4 -9/7 -1/4 -1/5 -32/25 -3/14 -23/18 -1/5 -2/11 -14/11 -1/4 -5/4 -1/6 0/1 -26/21 0/1 -21/17 0/1 1/6 -16/13 1/0 -27/22 -1/2 0/1 -11/9 -1/2 0/1 -17/14 -1/2 -1/3 -23/19 -4/11 -1/3 -52/43 -1/3 -29/24 -1/3 -6/19 -6/5 -1/4 -13/11 0/1 -7/6 -1/3 0/1 -8/7 -1/4 -17/15 -1/3 -1/4 -26/23 -1/4 -9/8 -1/4 -1/5 -1/1 -1/4 0/1 0/1 0/1 1/1 0/1 1/4 7/6 0/1 1/3 13/11 0/1 6/5 1/4 11/9 0/1 1/2 16/13 1/0 5/4 0/1 1/6 9/7 1/5 1/4 13/10 1/4 4/3 1/4 11/8 1/4 2/7 7/5 2/7 1/3 10/7 3/10 13/9 1/3 3/2 0/1 1/3 14/9 3/8 39/25 2/5 25/16 2/5 1/2 11/7 2/5 1/2 8/5 1/2 13/8 1/2 5/3 0/1 1/2 12/7 1/4 7/4 1/3 2/5 9/5 1/3 1/2 11/6 2/5 1/2 13/7 1/2 2/1 1/2 13/6 1/2 11/5 1/2 2/3 20/9 3/4 9/4 1/2 1/1 25/11 1/2 2/3 16/7 3/4 7/3 2/3 1/1 26/11 1/1 19/8 1/1 4/3 12/5 1/0 29/12 2/3 1/1 17/7 1/1 1/0 5/2 0/1 1/2 13/5 1/2 8/3 1/2 19/7 2/3 1/1 11/4 1/2 2/3 25/9 1/2 2/3 39/14 2/3 14/5 3/4 3/1 0/1 1/1 13/4 1/1 10/3 3/2 37/11 2/1 1/0 27/8 2/1 1/0 17/5 1/1 1/0 7/2 1/1 2/1 25/7 2/1 1/0 43/12 3/1 1/0 18/5 1/0 11/3 2/1 1/0 26/7 1/0 15/4 0/1 1/0 4/1 1/0 13/3 1/0 9/2 -1/1 1/0 32/7 -3/2 23/5 -1/1 -2/3 14/3 1/0 5/1 -1/2 0/1 26/5 0/1 21/4 0/1 1/10 16/3 1/4 27/5 0/1 1/2 11/2 0/1 1/2 17/3 1/2 1/1 23/4 4/5 1/1 52/9 1/1 29/5 1/1 6/5 6/1 1/0 13/2 0/1 7/1 0/1 1/1 8/1 1/0 17/2 1/1 1/0 26/3 1/0 9/1 -1/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(25,182,-18,-131) (-7/1,1/0) -> (-7/5,-25/18) Hyperbolic Matrix(27,182,4,27) (-7/1,-13/2) -> (13/2,7/1) Hyperbolic Matrix(25,156,4,25) (-13/2,-6/1) -> (6/1,13/2) Hyperbolic Matrix(51,286,-28,-157) (-6/1,-11/2) -> (-11/6,-20/11) Hyperbolic Matrix(131,702,-92,-493) (-11/2,-16/3) -> (-10/7,-37/26) Hyperbolic Matrix(79,416,-64,-337) (-16/3,-5/1) -> (-21/17,-16/13) Hyperbolic Matrix(27,130,-16,-77) (-5/1,-9/2) -> (-17/10,-5/3) Hyperbolic Matrix(53,234,12,53) (-9/2,-13/3) -> (13/3,9/2) Hyperbolic Matrix(25,104,6,25) (-13/3,-4/1) -> (4/1,13/3) Hyperbolic Matrix(27,104,-20,-77) (-4/1,-11/3) -> (-15/11,-4/3) Hyperbolic Matrix(79,286,-50,-181) (-11/3,-7/2) -> (-19/12,-11/7) Hyperbolic Matrix(53,182,-30,-103) (-7/2,-10/3) -> (-16/9,-7/4) Hyperbolic Matrix(79,260,24,79) (-10/3,-13/4) -> (13/4,10/3) Hyperbolic Matrix(25,78,8,25) (-13/4,-3/1) -> (3/1,13/4) Hyperbolic Matrix(157,442,-92,-259) (-3/1,-14/5) -> (-12/7,-29/17) Hyperbolic Matrix(391,1092,140,391) (-14/5,-39/14) -> (39/14,14/5) Hyperbolic Matrix(701,1950,252,701) (-39/14,-25/9) -> (25/9,39/14) Hyperbolic Matrix(441,1222,-310,-859) (-25/9,-11/4) -> (-37/26,-27/19) Hyperbolic Matrix(105,286,-76,-207) (-11/4,-8/3) -> (-18/13,-11/8) Hyperbolic Matrix(79,208,30,79) (-8/3,-13/5) -> (13/5,8/3) Hyperbolic Matrix(51,130,20,51) (-13/5,-5/2) -> (5/2,13/5) Hyperbolic Matrix(53,130,-42,-103) (-5/2,-12/5) -> (-14/11,-5/4) Hyperbolic Matrix(131,312,-76,-181) (-12/5,-7/3) -> (-19/11,-12/7) Hyperbolic Matrix(79,182,-56,-129) (-7/3,-9/4) -> (-17/12,-7/5) Hyperbolic Matrix(129,286,-106,-235) (-9/4,-11/5) -> (-11/9,-17/14) Hyperbolic Matrix(131,286,60,131) (-11/5,-13/6) -> (13/6,11/5) Hyperbolic Matrix(25,52,12,25) (-13/6,-2/1) -> (2/1,13/6) Hyperbolic Matrix(27,52,14,27) (-2/1,-13/7) -> (13/7,2/1) Hyperbolic Matrix(155,286,84,155) (-13/7,-11/6) -> (11/6,13/7) Hyperbolic Matrix(259,468,-202,-365) (-20/11,-9/5) -> (-9/7,-32/25) Hyperbolic Matrix(727,1300,-524,-937) (-9/5,-25/14) -> (-25/18,-43/31) Hyperbolic Matrix(467,832,-380,-677) (-25/14,-16/9) -> (-16/13,-27/22) Hyperbolic Matrix(389,676,164,285) (-7/4,-26/15) -> (26/11,19/8) Hyperbolic Matrix(391,676,166,287) (-26/15,-19/11) -> (7/3,26/11) Hyperbolic Matrix(519,884,-428,-729) (-29/17,-17/10) -> (-17/14,-23/19) Hyperbolic Matrix(79,130,48,79) (-5/3,-13/8) -> (13/8,5/3) Hyperbolic Matrix(129,208,80,129) (-13/8,-8/5) -> (8/5,13/8) Hyperbolic Matrix(131,208,-114,-181) (-8/5,-19/12) -> (-7/6,-8/7) Hyperbolic Matrix(365,572,-298,-467) (-11/7,-25/16) -> (-27/22,-11/9) Hyperbolic Matrix(1249,1950,800,1249) (-25/16,-39/25) -> (39/25,25/16) Hyperbolic Matrix(701,1092,450,701) (-39/25,-14/9) -> (14/9,39/25) Hyperbolic Matrix(235,364,-184,-285) (-14/9,-3/2) -> (-23/18,-14/11) Hyperbolic Matrix(53,78,36,53) (-3/2,-13/9) -> (13/9,3/2) Hyperbolic Matrix(181,260,126,181) (-13/9,-10/7) -> (10/7,13/9) Hyperbolic Matrix(183,260,-164,-233) (-27/19,-17/12) -> (-9/8,-1/1) Hyperbolic Matrix(469,650,-412,-571) (-43/31,-18/13) -> (-8/7,-17/15) Hyperbolic Matrix(493,676,132,181) (-11/8,-26/19) -> (26/7,15/4) Hyperbolic Matrix(495,676,134,183) (-26/19,-15/11) -> (11/3,26/7) Hyperbolic Matrix(79,104,60,79) (-4/3,-13/10) -> (13/10,4/3) Hyperbolic Matrix(181,234,140,181) (-13/10,-9/7) -> (9/7,13/10) Hyperbolic Matrix(833,1066,-690,-883) (-32/25,-23/18) -> (-29/24,-6/5) Hyperbolic Matrix(545,676,104,129) (-5/4,-26/21) -> (26/5,21/4) Hyperbolic Matrix(547,676,106,131) (-26/21,-21/17) -> (5/1,26/5) Hyperbolic Matrix(2235,2704,386,467) (-23/19,-52/43) -> (52/9,29/5) Hyperbolic Matrix(2237,2704,388,469) (-52/43,-29/24) -> (23/4,52/9) Hyperbolic Matrix(131,156,110,131) (-6/5,-13/11) -> (13/11,6/5) Hyperbolic Matrix(155,182,132,155) (-13/11,-7/6) -> (7/6,13/11) Hyperbolic Matrix(597,676,68,77) (-17/15,-26/23) -> (26/3,9/1) Hyperbolic Matrix(599,676,70,79) (-26/23,-9/8) -> (17/2,26/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(157,-182,44,-51) (1/1,7/6) -> (7/2,25/7) Hyperbolic Matrix(235,-286,106,-129) (6/5,11/9) -> (11/5,20/9) Hyperbolic Matrix(571,-702,170,-209) (11/9,16/13) -> (10/3,37/11) Hyperbolic Matrix(337,-416,64,-79) (16/13,5/4) -> (21/4,16/3) Hyperbolic Matrix(103,-130,42,-53) (5/4,9/7) -> (17/7,5/2) Hyperbolic Matrix(77,-104,20,-27) (4/3,11/8) -> (15/4,4/1) Hyperbolic Matrix(207,-286,76,-105) (11/8,7/5) -> (19/7,11/4) Hyperbolic Matrix(129,-182,56,-79) (7/5,10/7) -> (16/7,7/3) Hyperbolic Matrix(285,-442,118,-183) (3/2,14/9) -> (12/5,29/12) Hyperbolic Matrix(781,-1222,232,-363) (25/16,11/7) -> (37/11,27/8) Hyperbolic Matrix(181,-286,50,-79) (11/7,8/5) -> (18/5,11/3) Hyperbolic Matrix(77,-130,16,-27) (5/3,12/7) -> (14/3,5/1) Hyperbolic Matrix(181,-312,76,-131) (12/7,7/4) -> (19/8,12/5) Hyperbolic Matrix(103,-182,30,-53) (7/4,9/5) -> (17/5,7/2) Hyperbolic Matrix(157,-286,28,-51) (9/5,11/6) -> (11/2,17/3) Hyperbolic Matrix(209,-468,46,-103) (20/9,9/4) -> (9/2,32/7) Hyperbolic Matrix(573,-1300,160,-363) (9/4,25/11) -> (25/7,43/12) Hyperbolic Matrix(365,-832,68,-155) (25/11,16/7) -> (16/3,27/5) Hyperbolic Matrix(365,-884,64,-155) (29/12,17/7) -> (17/3,23/4) Hyperbolic Matrix(77,-208,10,-27) (8/3,19/7) -> (7/1,8/1) Hyperbolic Matrix(207,-572,38,-105) (11/4,25/9) -> (27/5,11/2) Hyperbolic Matrix(129,-364,28,-79) (14/5,3/1) -> (23/5,14/3) Hyperbolic Matrix(77,-260,8,-27) (27/8,17/5) -> (9/1,1/0) Hyperbolic Matrix(181,-650,22,-79) (43/12,18/5) -> (8/1,17/2) Hyperbolic Matrix(233,-1066,40,-183) (32/7,23/5) -> (29/5,6/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(25,182,-18,-131) -> Matrix(1,2,-4,-7) Matrix(27,182,4,27) -> Matrix(1,0,2,1) Matrix(25,156,4,25) -> Matrix(1,0,0,1) Matrix(51,286,-28,-157) -> Matrix(3,2,-8,-5) Matrix(131,702,-92,-493) -> Matrix(5,2,-18,-7) Matrix(79,416,-64,-337) -> Matrix(1,0,4,1) Matrix(27,130,-16,-77) -> Matrix(1,0,-4,1) Matrix(53,234,12,53) -> Matrix(1,-2,0,1) Matrix(25,104,6,25) -> Matrix(1,0,0,1) Matrix(27,104,-20,-77) -> Matrix(1,2,-4,-7) Matrix(79,286,-50,-181) -> Matrix(1,0,-2,1) Matrix(53,182,-30,-103) -> Matrix(1,0,-2,1) Matrix(79,260,24,79) -> Matrix(5,6,4,5) Matrix(25,78,8,25) -> Matrix(1,0,2,1) Matrix(157,442,-92,-259) -> Matrix(3,2,-8,-5) Matrix(391,1092,140,391) -> Matrix(17,12,24,17) Matrix(701,1950,252,701) -> Matrix(7,4,12,7) Matrix(441,1222,-310,-859) -> Matrix(1,0,-2,1) Matrix(105,286,-76,-207) -> Matrix(1,0,-2,1) Matrix(79,208,30,79) -> Matrix(3,2,4,3) Matrix(51,130,20,51) -> Matrix(1,0,4,1) Matrix(53,130,-42,-103) -> Matrix(1,0,-4,1) Matrix(131,312,-76,-181) -> Matrix(1,2,-4,-7) Matrix(79,182,-56,-129) -> Matrix(1,0,-2,1) Matrix(129,286,-106,-235) -> Matrix(3,2,-8,-5) Matrix(131,286,60,131) -> Matrix(7,4,12,7) Matrix(25,52,12,25) -> Matrix(3,2,4,3) Matrix(27,52,14,27) -> Matrix(5,2,12,5) Matrix(155,286,84,155) -> Matrix(9,4,20,9) Matrix(259,468,-202,-365) -> Matrix(1,0,-2,1) Matrix(727,1300,-524,-937) -> Matrix(9,4,-34,-15) Matrix(467,832,-380,-677) -> Matrix(5,2,-8,-3) Matrix(389,676,164,285) -> Matrix(17,6,14,5) Matrix(391,676,166,287) -> Matrix(19,6,22,7) Matrix(519,884,-428,-729) -> Matrix(7,2,-18,-5) Matrix(79,130,48,79) -> Matrix(1,0,4,1) Matrix(129,208,80,129) -> Matrix(5,2,12,5) Matrix(131,208,-114,-181) -> Matrix(5,2,-18,-7) Matrix(365,572,-298,-467) -> Matrix(5,2,-8,-3) Matrix(1249,1950,800,1249) -> Matrix(9,4,20,9) Matrix(701,1092,450,701) -> Matrix(31,12,80,31) Matrix(235,364,-184,-285) -> Matrix(5,2,-28,-11) Matrix(53,78,36,53) -> Matrix(1,0,6,1) Matrix(181,260,126,181) -> Matrix(19,6,60,19) Matrix(183,260,-164,-233) -> Matrix(7,2,-32,-9) Matrix(469,650,-412,-571) -> Matrix(7,2,-32,-9) Matrix(493,676,132,181) -> Matrix(7,2,-4,-1) Matrix(495,676,134,183) -> Matrix(9,2,4,1) Matrix(79,104,60,79) -> Matrix(1,0,8,1) Matrix(181,234,140,181) -> Matrix(9,2,40,9) Matrix(833,1066,-690,-883) -> Matrix(19,4,-62,-13) Matrix(545,676,104,129) -> Matrix(1,0,16,1) Matrix(547,676,106,131) -> Matrix(1,0,-8,1) Matrix(2235,2704,386,467) -> Matrix(29,10,26,9) Matrix(2237,2704,388,469) -> Matrix(31,10,34,11) Matrix(131,156,110,131) -> Matrix(1,0,8,1) Matrix(155,182,132,155) -> Matrix(1,0,6,1) Matrix(597,676,68,77) -> Matrix(7,2,-4,-1) Matrix(599,676,70,79) -> Matrix(9,2,4,1) Matrix(1,0,2,1) -> Matrix(1,0,8,1) Matrix(157,-182,44,-51) -> Matrix(7,-2,4,-1) Matrix(235,-286,106,-129) -> Matrix(5,-2,8,-3) Matrix(571,-702,170,-209) -> Matrix(3,-2,2,-1) Matrix(337,-416,64,-79) -> Matrix(1,0,4,1) Matrix(103,-130,42,-53) -> Matrix(1,0,-4,1) Matrix(77,-104,20,-27) -> Matrix(7,-2,4,-1) Matrix(207,-286,76,-105) -> Matrix(1,0,-2,1) Matrix(129,-182,56,-79) -> Matrix(1,0,-2,1) Matrix(285,-442,118,-183) -> Matrix(5,-2,8,-3) Matrix(781,-1222,232,-363) -> Matrix(1,0,-2,1) Matrix(181,-286,50,-79) -> Matrix(1,0,-2,1) Matrix(77,-130,16,-27) -> Matrix(1,0,-4,1) Matrix(181,-312,76,-131) -> Matrix(7,-2,4,-1) Matrix(103,-182,30,-53) -> Matrix(1,0,-2,1) Matrix(157,-286,28,-51) -> Matrix(5,-2,8,-3) Matrix(209,-468,46,-103) -> Matrix(1,0,-2,1) Matrix(573,-1300,160,-363) -> Matrix(7,-4,2,-1) Matrix(365,-832,68,-155) -> Matrix(3,-2,8,-5) Matrix(365,-884,64,-155) -> Matrix(1,-2,2,-3) Matrix(77,-208,10,-27) -> Matrix(3,-2,2,-1) Matrix(207,-572,38,-105) -> Matrix(3,-2,8,-5) Matrix(129,-364,28,-79) -> Matrix(3,-2,-4,3) Matrix(77,-260,8,-27) -> Matrix(1,-2,0,1) Matrix(181,-650,22,-79) -> Matrix(1,-2,0,1) Matrix(233,-1066,40,-183) -> Matrix(3,4,2,3) 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): 22 Degree of the the map X: 22 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): 252 Minimal number of generators: 43 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: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 13/6 26/11 13/5 39/14 3/1 13/4 10/3 26/7 4/1 13/3 14/3 5/1 26/5 11/2 52/9 6/1 13/2 7/1 8/1 26/3 9/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 0/1 1/4 7/6 0/1 1/3 13/11 0/1 6/5 1/4 11/9 0/1 1/2 16/13 1/0 5/4 0/1 1/6 9/7 1/5 1/4 13/10 1/4 4/3 1/4 11/8 1/4 2/7 7/5 2/7 1/3 10/7 3/10 13/9 1/3 3/2 0/1 1/3 14/9 3/8 39/25 2/5 25/16 2/5 1/2 11/7 2/5 1/2 8/5 1/2 13/8 1/2 5/3 0/1 1/2 12/7 1/4 7/4 1/3 2/5 9/5 1/3 1/2 11/6 2/5 1/2 13/7 1/2 2/1 1/2 13/6 1/2 11/5 1/2 2/3 20/9 3/4 9/4 1/2 1/1 25/11 1/2 2/3 16/7 3/4 7/3 2/3 1/1 26/11 1/1 19/8 1/1 4/3 12/5 1/0 29/12 2/3 1/1 17/7 1/1 1/0 5/2 0/1 1/2 13/5 1/2 8/3 1/2 19/7 2/3 1/1 11/4 1/2 2/3 25/9 1/2 2/3 39/14 2/3 14/5 3/4 3/1 0/1 1/1 13/4 1/1 10/3 3/2 37/11 2/1 1/0 27/8 2/1 1/0 17/5 1/1 1/0 7/2 1/1 2/1 25/7 2/1 1/0 43/12 3/1 1/0 18/5 1/0 11/3 2/1 1/0 26/7 1/0 15/4 0/1 1/0 4/1 1/0 13/3 1/0 9/2 -1/1 1/0 32/7 -3/2 23/5 -1/1 -2/3 14/3 1/0 5/1 -1/2 0/1 26/5 0/1 21/4 0/1 1/10 16/3 1/4 27/5 0/1 1/2 11/2 0/1 1/2 17/3 1/2 1/1 23/4 4/5 1/1 52/9 1/1 29/5 1/1 6/5 6/1 1/0 13/2 0/1 7/1 0/1 1/1 8/1 1/0 17/2 1/1 1/0 26/3 1/0 9/1 -1/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(157,-182,44,-51) (1/1,7/6) -> (7/2,25/7) Hyperbolic Matrix(155,-182,23,-27) (7/6,13/11) -> (13/2,7/1) Hyperbolic Matrix(131,-156,21,-25) (13/11,6/5) -> (6/1,13/2) Hyperbolic Matrix(235,-286,106,-129) (6/5,11/9) -> (11/5,20/9) Hyperbolic Matrix(571,-702,170,-209) (11/9,16/13) -> (10/3,37/11) Hyperbolic Matrix(337,-416,64,-79) (16/13,5/4) -> (21/4,16/3) Hyperbolic Matrix(103,-130,42,-53) (5/4,9/7) -> (17/7,5/2) Hyperbolic Matrix(181,-234,41,-53) (9/7,13/10) -> (13/3,9/2) Hyperbolic Matrix(79,-104,19,-25) (13/10,4/3) -> (4/1,13/3) Hyperbolic Matrix(77,-104,20,-27) (4/3,11/8) -> (15/4,4/1) Hyperbolic Matrix(207,-286,76,-105) (11/8,7/5) -> (19/7,11/4) Hyperbolic Matrix(129,-182,56,-79) (7/5,10/7) -> (16/7,7/3) Hyperbolic Matrix(181,-260,55,-79) (10/7,13/9) -> (13/4,10/3) Hyperbolic Matrix(53,-78,17,-25) (13/9,3/2) -> (3/1,13/4) Hyperbolic Matrix(285,-442,118,-183) (3/2,14/9) -> (12/5,29/12) Hyperbolic Matrix(701,-1092,251,-391) (14/9,39/25) -> (39/14,14/5) Hyperbolic Matrix(1249,-1950,449,-701) (39/25,25/16) -> (25/9,39/14) Hyperbolic Matrix(781,-1222,232,-363) (25/16,11/7) -> (37/11,27/8) Hyperbolic Matrix(181,-286,50,-79) (11/7,8/5) -> (18/5,11/3) Hyperbolic Matrix(129,-208,49,-79) (8/5,13/8) -> (13/5,8/3) Hyperbolic Matrix(79,-130,31,-51) (13/8,5/3) -> (5/2,13/5) Hyperbolic Matrix(77,-130,16,-27) (5/3,12/7) -> (14/3,5/1) Hyperbolic Matrix(181,-312,76,-131) (12/7,7/4) -> (19/8,12/5) Hyperbolic Matrix(103,-182,30,-53) (7/4,9/5) -> (17/5,7/2) Hyperbolic Matrix(157,-286,28,-51) (9/5,11/6) -> (11/2,17/3) Hyperbolic Matrix(155,-286,71,-131) (11/6,13/7) -> (13/6,11/5) Hyperbolic Matrix(27,-52,13,-25) (13/7,2/1) -> (2/1,13/6) Parabolic Matrix(209,-468,46,-103) (20/9,9/4) -> (9/2,32/7) Hyperbolic Matrix(573,-1300,160,-363) (9/4,25/11) -> (25/7,43/12) Hyperbolic Matrix(365,-832,68,-155) (25/11,16/7) -> (16/3,27/5) Hyperbolic Matrix(287,-676,121,-285) (7/3,26/11) -> (26/11,19/8) Parabolic Matrix(365,-884,64,-155) (29/12,17/7) -> (17/3,23/4) Hyperbolic Matrix(77,-208,10,-27) (8/3,19/7) -> (7/1,8/1) Hyperbolic Matrix(207,-572,38,-105) (11/4,25/9) -> (27/5,11/2) Hyperbolic Matrix(129,-364,28,-79) (14/5,3/1) -> (23/5,14/3) Hyperbolic Matrix(77,-260,8,-27) (27/8,17/5) -> (9/1,1/0) Hyperbolic Matrix(181,-650,22,-79) (43/12,18/5) -> (8/1,17/2) Hyperbolic Matrix(183,-676,49,-181) (11/3,26/7) -> (26/7,15/4) Parabolic Matrix(233,-1066,40,-183) (32/7,23/5) -> (29/5,6/1) Hyperbolic Matrix(131,-676,25,-129) (5/1,26/5) -> (26/5,21/4) Parabolic Matrix(469,-2704,81,-467) (23/4,52/9) -> (52/9,29/5) Parabolic Matrix(79,-676,9,-77) (17/2,26/3) -> (26/3,9/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,4,1) Matrix(157,-182,44,-51) -> Matrix(7,-2,4,-1) Matrix(155,-182,23,-27) -> Matrix(1,0,-2,1) Matrix(131,-156,21,-25) -> Matrix(1,0,-4,1) Matrix(235,-286,106,-129) -> Matrix(5,-2,8,-3) Matrix(571,-702,170,-209) -> Matrix(3,-2,2,-1) Matrix(337,-416,64,-79) -> Matrix(1,0,4,1) Matrix(103,-130,42,-53) -> Matrix(1,0,-4,1) Matrix(181,-234,41,-53) -> Matrix(9,-2,-4,1) Matrix(79,-104,19,-25) -> Matrix(1,0,-4,1) Matrix(77,-104,20,-27) -> Matrix(7,-2,4,-1) Matrix(207,-286,76,-105) -> Matrix(1,0,-2,1) Matrix(129,-182,56,-79) -> Matrix(1,0,-2,1) Matrix(181,-260,55,-79) -> Matrix(19,-6,16,-5) Matrix(53,-78,17,-25) -> Matrix(1,0,-2,1) Matrix(285,-442,118,-183) -> Matrix(5,-2,8,-3) Matrix(701,-1092,251,-391) -> Matrix(31,-12,44,-17) Matrix(1249,-1950,449,-701) -> Matrix(9,-4,16,-7) Matrix(781,-1222,232,-363) -> Matrix(1,0,-2,1) Matrix(181,-286,50,-79) -> Matrix(1,0,-2,1) Matrix(129,-208,49,-79) -> Matrix(5,-2,8,-3) Matrix(79,-130,31,-51) -> Matrix(1,0,0,1) Matrix(77,-130,16,-27) -> Matrix(1,0,-4,1) Matrix(181,-312,76,-131) -> Matrix(7,-2,4,-1) Matrix(103,-182,30,-53) -> Matrix(1,0,-2,1) Matrix(157,-286,28,-51) -> Matrix(5,-2,8,-3) Matrix(155,-286,71,-131) -> Matrix(9,-4,16,-7) Matrix(27,-52,13,-25) -> Matrix(5,-2,8,-3) Matrix(209,-468,46,-103) -> Matrix(1,0,-2,1) Matrix(573,-1300,160,-363) -> Matrix(7,-4,2,-1) Matrix(365,-832,68,-155) -> Matrix(3,-2,8,-5) Matrix(287,-676,121,-285) -> Matrix(7,-6,6,-5) Matrix(365,-884,64,-155) -> Matrix(1,-2,2,-3) Matrix(77,-208,10,-27) -> Matrix(3,-2,2,-1) Matrix(207,-572,38,-105) -> Matrix(3,-2,8,-5) Matrix(129,-364,28,-79) -> Matrix(3,-2,-4,3) Matrix(77,-260,8,-27) -> Matrix(1,-2,0,1) Matrix(181,-650,22,-79) -> Matrix(1,-2,0,1) Matrix(183,-676,49,-181) -> Matrix(1,-2,0,1) Matrix(233,-1066,40,-183) -> Matrix(3,4,2,3) Matrix(131,-676,25,-129) -> Matrix(1,0,12,1) Matrix(469,-2704,81,-467) -> Matrix(11,-10,10,-9) Matrix(79,-676,9,-77) -> 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 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 0/1 0/1 4 1 2/1 1/2 2 13 13/6 1/2 2 2 11/5 (1/2,2/3) 0 26 9/4 (1/2,1/1) 0 26 7/3 (2/3,1/1) 0 26 26/11 1/1 6 1 12/5 1/0 2 13 5/2 (0/1,1/2) 0 26 13/5 1/2 2 2 8/3 1/2 2 13 11/4 (1/2,2/3) 0 26 25/9 (1/2,2/3) 0 26 39/14 2/3 4 2 14/5 3/4 2 13 3/1 (0/1,1/1) 0 26 13/4 1/1 6 2 10/3 3/2 2 13 27/8 (2/1,1/0) 0 26 17/5 (1/1,1/0) 0 26 7/2 (1/1,2/1) 0 26 18/5 1/0 2 13 11/3 (2/1,1/0) 0 26 26/7 1/0 2 1 4/1 1/0 2 13 13/3 1/0 2 2 9/2 (-1/1,1/0) 0 26 23/5 (-1/1,-2/3) 0 26 14/3 1/0 2 13 5/1 (-1/2,0/1) 0 26 26/5 0/1 12 1 16/3 1/4 2 13 27/5 (0/1,1/2) 0 26 11/2 (0/1,1/2) 0 26 17/3 (1/2,1/1) 0 26 23/4 (4/5,1/1) 0 26 52/9 1/1 10 1 6/1 1/0 2 13 13/2 0/1 2 2 7/1 (0/1,1/1) 0 26 8/1 1/0 2 13 26/3 1/0 2 1 9/1 (-1/1,1/0) 0 26 1/0 (0/1,1/0) 0 26 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(25,-52,12,-25) (2/1,13/6) -> (2/1,13/6) Reflection Matrix(131,-286,60,-131) (13/6,11/5) -> (13/6,11/5) Reflection Matrix(129,-286,23,-51) (11/5,9/4) -> (11/2,17/3) Glide Reflection Matrix(79,-182,23,-53) (9/4,7/3) -> (17/5,7/2) Glide Reflection Matrix(155,-364,66,-155) (7/3,26/11) -> (7/3,26/11) Reflection Matrix(131,-312,55,-131) (26/11,12/5) -> (26/11,12/5) Reflection Matrix(53,-130,11,-27) (12/5,5/2) -> (14/3,5/1) Glide Reflection Matrix(51,-130,20,-51) (5/2,13/5) -> (5/2,13/5) Reflection Matrix(79,-208,30,-79) (13/5,8/3) -> (13/5,8/3) Reflection Matrix(105,-286,29,-79) (8/3,11/4) -> (18/5,11/3) Glide Reflection Matrix(207,-572,38,-105) (11/4,25/9) -> (27/5,11/2) Hyperbolic Matrix(701,-1950,252,-701) (25/9,39/14) -> (25/9,39/14) Reflection Matrix(391,-1092,140,-391) (39/14,14/5) -> (39/14,14/5) Reflection Matrix(129,-364,28,-79) (14/5,3/1) -> (23/5,14/3) Hyperbolic Matrix(25,-78,8,-25) (3/1,13/4) -> (3/1,13/4) Reflection Matrix(79,-260,24,-79) (13/4,10/3) -> (13/4,10/3) Reflection Matrix(209,-702,39,-131) (10/3,27/8) -> (16/3,27/5) Glide Reflection Matrix(77,-260,8,-27) (27/8,17/5) -> (9/1,1/0) Hyperbolic Matrix(51,-182,7,-25) (7/2,18/5) -> (7/1,8/1) Glide Reflection Matrix(155,-572,42,-155) (11/3,26/7) -> (11/3,26/7) Reflection Matrix(27,-104,7,-27) (26/7,4/1) -> (26/7,4/1) Reflection 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(131,-598,23,-105) (9/2,23/5) -> (17/3,23/4) Glide Reflection Matrix(51,-260,10,-51) (5/1,26/5) -> (5/1,26/5) Reflection Matrix(79,-416,15,-79) (26/5,16/3) -> (26/5,16/3) Reflection Matrix(415,-2392,72,-415) (23/4,52/9) -> (23/4,52/9) Reflection Matrix(53,-312,9,-53) (52/9,6/1) -> (52/9,6/1) Reflection Matrix(25,-156,4,-25) (6/1,13/2) -> (6/1,13/2) Reflection Matrix(27,-182,4,-27) (13/2,7/1) -> (13/2,7/1) Reflection Matrix(25,-208,3,-25) (8/1,26/3) -> (8/1,26/3) Reflection Matrix(53,-468,6,-53) (26/3,9/1) -> (26/3,9/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,4,-1) (0/1,2/1) -> (0/1,1/2) Matrix(25,-52,12,-25) -> Matrix(3,-2,4,-3) (2/1,13/6) -> (1/2,1/1) Matrix(131,-286,60,-131) -> Matrix(7,-4,12,-7) (13/6,11/5) -> (1/2,2/3) Matrix(129,-286,23,-51) -> Matrix(3,-2,4,-3) *** -> (1/2,1/1) Matrix(79,-182,23,-53) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(155,-364,66,-155) -> Matrix(5,-4,6,-5) (7/3,26/11) -> (2/3,1/1) Matrix(131,-312,55,-131) -> Matrix(-1,2,0,1) (26/11,12/5) -> (1/1,1/0) Matrix(53,-130,11,-27) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(51,-130,20,-51) -> Matrix(1,0,4,-1) (5/2,13/5) -> (0/1,1/2) Matrix(79,-208,30,-79) -> Matrix(3,-2,4,-3) (13/5,8/3) -> (1/2,1/1) Matrix(105,-286,29,-79) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(207,-572,38,-105) -> Matrix(3,-2,8,-5) 1/2 Matrix(701,-1950,252,-701) -> Matrix(7,-4,12,-7) (25/9,39/14) -> (1/2,2/3) Matrix(391,-1092,140,-391) -> Matrix(17,-12,24,-17) (39/14,14/5) -> (2/3,3/4) Matrix(129,-364,28,-79) -> Matrix(3,-2,-4,3) Matrix(25,-78,8,-25) -> Matrix(1,0,2,-1) (3/1,13/4) -> (0/1,1/1) Matrix(79,-260,24,-79) -> Matrix(5,-6,4,-5) (13/4,10/3) -> (1/1,3/2) Matrix(209,-702,39,-131) -> Matrix(1,-2,2,-5) Matrix(77,-260,8,-27) -> Matrix(1,-2,0,1) 1/0 Matrix(51,-182,7,-25) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(155,-572,42,-155) -> Matrix(-1,4,0,1) (11/3,26/7) -> (2/1,1/0) Matrix(27,-104,7,-27) -> Matrix(-1,2,0,1) (26/7,4/1) -> (1/1,1/0) Matrix(25,-104,6,-25) -> Matrix(1,0,0,-1) (4/1,13/3) -> (0/1,1/0) Matrix(53,-234,12,-53) -> Matrix(1,2,0,-1) (13/3,9/2) -> (-1/1,1/0) Matrix(131,-598,23,-105) -> Matrix(1,2,2,3) Matrix(51,-260,10,-51) -> Matrix(-1,0,4,1) (5/1,26/5) -> (-1/2,0/1) Matrix(79,-416,15,-79) -> Matrix(1,0,8,-1) (26/5,16/3) -> (0/1,1/4) Matrix(415,-2392,72,-415) -> Matrix(9,-8,10,-9) (23/4,52/9) -> (4/5,1/1) Matrix(53,-312,9,-53) -> Matrix(-1,2,0,1) (52/9,6/1) -> (1/1,1/0) Matrix(25,-156,4,-25) -> Matrix(1,0,0,-1) (6/1,13/2) -> (0/1,1/0) Matrix(27,-182,4,-27) -> Matrix(1,0,2,-1) (13/2,7/1) -> (0/1,1/1) Matrix(25,-208,3,-25) -> Matrix(1,0,0,-1) (8/1,26/3) -> (0/1,1/0) Matrix(53,-468,6,-53) -> Matrix(1,2,0,-1) (26/3,9/1) -> (-1/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.