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 -13/2 0/1 -6/1 1/1 -11/2 0/1 1/0 -16/3 -1/1 -5/1 1/1 -9/2 1/1 1/0 -13/3 1/0 -4/1 -1/1 -11/3 -1/1 -7/2 -1/1 1/0 -10/3 -1/1 -13/4 -1/1 -3/1 -1/1 -14/5 -1/1 -39/14 -1/2 -25/9 -1/3 -11/4 -1/2 0/1 -8/3 -1/3 -13/5 0/1 -5/2 0/1 1/0 -12/5 1/1 -7/3 -1/1 -9/4 -1/1 1/0 -11/5 -1/1 -13/6 -1/1 -2/1 -1/1 -13/7 -1/2 -11/6 -1/2 0/1 -20/11 -1/1 -9/5 -1/1 -25/14 -1/2 0/1 -16/9 -1/1 -7/4 -1/1 -1/2 -26/15 -1/2 -19/11 -1/3 -12/7 -1/1 -29/17 -1/1 -17/10 -1/1 -1/2 -5/3 -1/1 -13/8 -1/2 -8/5 -1/3 -19/12 -1/2 -1/3 -11/7 -1/3 -25/16 -1/4 0/1 -39/25 0/1 -14/9 -1/1 -3/2 -1/2 0/1 -13/9 -1/2 -10/7 -1/3 -37/26 -1/2 0/1 -27/19 -1/3 -17/12 -1/2 -1/3 -7/5 -1/3 -25/18 -1/2 0/1 -43/31 -1/1 -18/13 -1/3 -11/8 -1/2 0/1 -26/19 -1/2 -15/11 -1/3 -4/3 -1/3 -13/10 0/1 -9/7 -1/1 -32/25 -1/1 -23/18 -1/2 0/1 -14/11 -1/1 -5/4 -1/2 0/1 -26/21 0/1 -21/17 1/1 -16/13 -1/1 -27/22 -1/2 0/1 -11/9 -1/1 -17/14 -1/1 -1/2 -23/19 -1/1 -52/43 -1/2 -29/24 -1/2 0/1 -6/5 -1/1 -13/11 -1/2 -7/6 -1/2 -1/3 -8/7 -1/3 -17/15 -1/1 -26/23 -1/2 -9/8 -1/2 -1/3 -1/1 -1/3 0/1 0/1 1/1 1/3 7/6 1/3 1/2 13/11 1/2 6/5 1/1 11/9 1/1 16/13 1/1 5/4 0/1 1/2 9/7 1/1 13/10 0/1 4/3 1/3 11/8 0/1 1/2 7/5 1/3 10/7 1/3 13/9 1/2 3/2 0/1 1/2 14/9 1/1 39/25 0/1 25/16 0/1 1/4 11/7 1/3 8/5 1/3 13/8 1/2 5/3 1/1 12/7 1/1 7/4 1/2 1/1 9/5 1/1 11/6 0/1 1/2 13/7 1/2 2/1 1/1 13/6 1/1 11/5 1/1 20/9 1/1 9/4 1/1 1/0 25/11 1/1 16/7 1/1 7/3 1/1 26/11 1/0 19/8 -1/1 1/0 12/5 -1/1 29/12 0/1 1/0 17/7 -1/1 5/2 0/1 1/0 13/5 0/1 8/3 1/3 19/7 1/3 11/4 0/1 1/2 25/9 1/3 39/14 1/2 14/5 1/1 3/1 1/1 13/4 1/1 10/3 1/1 37/11 1/1 27/8 2/1 1/0 17/5 1/1 7/2 1/1 1/0 25/7 1/1 43/12 1/1 1/0 18/5 1/1 11/3 1/1 26/7 1/0 15/4 0/1 1/0 4/1 1/1 13/3 1/0 9/2 -1/1 1/0 32/7 -1/1 23/5 -1/1 14/3 -1/1 5/1 -1/1 26/5 0/1 21/4 0/1 1/2 16/3 1/1 27/5 1/1 11/2 0/1 1/0 17/3 -1/1 23/4 0/1 1/0 52/9 1/0 29/5 -1/1 6/1 -1/1 13/2 0/1 7/1 1/1 8/1 1/1 17/2 1/1 1/0 26/3 1/0 9/1 -1/1 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(1,0,-2,1) Matrix(27,182,4,27) -> Matrix(1,0,2,1) Matrix(25,156,4,25) -> Matrix(1,0,-2,1) Matrix(51,286,-28,-157) -> Matrix(1,0,-2,1) Matrix(131,702,-92,-493) -> Matrix(1,0,-2,1) Matrix(79,416,-64,-337) -> Matrix(1,0,0,1) Matrix(27,130,-16,-77) -> Matrix(1,0,-2,1) Matrix(53,234,12,53) -> Matrix(1,-2,0,1) Matrix(25,104,6,25) -> Matrix(1,2,0,1) Matrix(27,104,-20,-77) -> Matrix(1,0,-2,1) Matrix(79,286,-50,-181) -> Matrix(1,0,-2,1) Matrix(53,182,-30,-103) -> Matrix(1,2,-2,-3) Matrix(79,260,24,79) -> Matrix(3,4,2,3) Matrix(25,78,8,25) -> Matrix(1,0,2,1) Matrix(157,442,-92,-259) -> Matrix(1,0,0,1) Matrix(391,1092,140,391) -> Matrix(3,2,4,3) Matrix(701,1950,252,701) -> Matrix(5,2,12,5) Matrix(441,1222,-310,-859) -> Matrix(1,0,0,1) Matrix(105,286,-76,-207) -> Matrix(1,0,0,1) Matrix(79,208,30,79) -> Matrix(1,0,6,1) Matrix(51,130,20,51) -> Matrix(1,0,0,1) Matrix(53,130,-42,-103) -> Matrix(1,0,-2,1) Matrix(131,312,-76,-181) -> Matrix(1,0,-2,1) Matrix(79,182,-56,-129) -> Matrix(1,0,-2,1) Matrix(129,286,-106,-235) -> Matrix(1,2,-2,-3) Matrix(131,286,60,131) -> Matrix(3,4,2,3) Matrix(25,52,12,25) -> Matrix(1,0,2,1) Matrix(27,52,14,27) -> Matrix(3,2,4,3) Matrix(155,286,84,155) -> Matrix(1,0,4,1) Matrix(259,468,-202,-365) -> Matrix(1,0,0,1) Matrix(727,1300,-524,-937) -> Matrix(1,0,0,1) Matrix(467,832,-380,-677) -> Matrix(1,0,0,1) Matrix(389,676,164,285) -> Matrix(3,2,-2,-1) Matrix(391,676,166,287) -> Matrix(5,2,2,1) Matrix(519,884,-428,-729) -> Matrix(1,0,0,1) Matrix(79,130,48,79) -> Matrix(3,2,4,3) Matrix(129,208,80,129) -> Matrix(5,2,12,5) Matrix(131,208,-114,-181) -> Matrix(1,0,0,1) Matrix(365,572,-298,-467) -> Matrix(1,0,2,1) Matrix(1249,1950,800,1249) -> Matrix(1,0,8,1) Matrix(701,1092,450,701) -> Matrix(1,0,2,1) Matrix(235,364,-184,-285) -> Matrix(1,0,0,1) Matrix(53,78,36,53) -> Matrix(1,0,4,1) Matrix(181,260,126,181) -> Matrix(5,2,12,5) Matrix(183,260,-164,-233) -> Matrix(1,0,0,1) Matrix(469,650,-412,-571) -> Matrix(1,0,0,1) Matrix(493,676,132,181) -> Matrix(1,0,2,1) Matrix(495,676,134,183) -> Matrix(5,2,2,1) Matrix(79,104,60,79) -> Matrix(1,0,6,1) Matrix(181,234,140,181) -> Matrix(1,0,2,1) Matrix(833,1066,-690,-883) -> Matrix(1,0,0,1) Matrix(545,676,104,129) -> Matrix(1,0,4,1) Matrix(547,676,106,131) -> Matrix(1,0,-2,1) Matrix(2235,2704,386,467) -> Matrix(3,2,-2,-1) Matrix(2237,2704,388,469) -> Matrix(1,0,2,1) Matrix(131,156,110,131) -> Matrix(3,2,4,3) Matrix(155,182,132,155) -> Matrix(5,2,12,5) Matrix(597,676,68,77) -> Matrix(3,2,-2,-1) Matrix(599,676,70,79) -> Matrix(5,2,2,1) Matrix(1,0,2,1) -> Matrix(1,0,6,1) Matrix(157,-182,44,-51) -> Matrix(1,0,-2,1) Matrix(235,-286,106,-129) -> Matrix(3,-2,2,-1) Matrix(571,-702,170,-209) -> Matrix(3,-2,2,-1) Matrix(337,-416,64,-79) -> Matrix(1,0,0,1) Matrix(103,-130,42,-53) -> Matrix(1,0,-2,1) Matrix(77,-104,20,-27) -> Matrix(1,0,-2,1) Matrix(207,-286,76,-105) -> Matrix(1,0,0,1) Matrix(129,-182,56,-79) -> Matrix(1,0,-2,1) Matrix(285,-442,118,-183) -> Matrix(1,0,-2,1) Matrix(781,-1222,232,-363) -> Matrix(7,-2,4,-1) Matrix(181,-286,50,-79) -> Matrix(1,0,-2,1) Matrix(77,-130,16,-27) -> Matrix(1,0,-2,1) Matrix(181,-312,76,-131) -> Matrix(1,0,-2,1) Matrix(103,-182,30,-53) -> Matrix(3,-2,2,-1) Matrix(157,-286,28,-51) -> Matrix(1,0,-2,1) Matrix(209,-468,46,-103) -> Matrix(1,-2,0,1) Matrix(573,-1300,160,-363) -> Matrix(1,0,0,1) Matrix(365,-832,68,-155) -> Matrix(1,0,0,1) Matrix(365,-884,64,-155) -> Matrix(1,0,0,1) Matrix(77,-208,10,-27) -> Matrix(1,0,-2,1) Matrix(207,-572,38,-105) -> Matrix(1,0,-2,1) Matrix(129,-364,28,-79) -> Matrix(1,0,-2,1) Matrix(77,-260,8,-27) -> Matrix(1,-2,0,1) Matrix(181,-650,22,-79) -> Matrix(1,0,0,1) Matrix(233,-1066,40,-183) -> Matrix(1,0,0,1) 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): 16 Degree of the the map X: 16 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): 126 Minimal number of generators: 25 Number of equivalence classes of elliptic points of order 2: 6 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 12 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 13/10 13/9 3/2 13/8 2/1 13/4 7/2 4/1 13/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 1/1 -9/2 1/1 1/0 -13/3 1/0 -4/1 -1/1 -7/2 -1/1 1/0 -3/1 -1/1 -2/1 -1/1 -7/4 -1/1 -1/2 -12/7 -1/1 -5/3 -1/1 -3/2 -1/2 0/1 -13/9 -1/2 -10/7 -1/3 -7/5 -1/3 -4/3 -1/3 -5/4 -1/2 0/1 -1/1 -1/3 0/1 0/1 1/1 1/3 5/4 0/1 1/2 9/7 1/1 13/10 0/1 4/3 1/3 7/5 1/3 10/7 1/3 13/9 1/2 3/2 0/1 1/2 8/5 1/3 13/8 1/2 5/3 1/1 12/7 1/1 7/4 1/2 1/1 9/5 1/1 2/1 1/1 3/1 1/1 13/4 1/1 10/3 1/1 7/2 1/1 1/0 11/3 1/1 4/1 1/1 13/3 1/0 9/2 -1/1 1/0 5/1 -1/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,26,4,21) (-5/1,1/0) -> (1/1,5/4) Hyperbolic Matrix(31,143,-18,-83) (-5/1,-9/2) -> (-7/4,-12/7) 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(25,91,14,51) (-4/1,-7/2) -> (7/4,9/5) Hyperbolic Matrix(27,91,8,27) (-7/2,-3/1) -> (10/3,7/2) Hyperbolic Matrix(5,13,-2,-5) (-3/1,-2/1) -> (-3/1,-2/1) Elliptic Matrix(51,91,14,25) (-2/1,-7/4) -> (7/2,11/3) Hyperbolic Matrix(99,169,-58,-99) (-12/7,-5/3) -> (-12/7,-5/3) Elliptic Matrix(25,39,16,25) (-5/3,-3/2) -> (3/2,8/5) 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(73,104,40,57) (-10/7,-7/5) -> (9/5,2/1) Hyperbolic Matrix(47,65,-34,-47) (-7/5,-4/3) -> (-7/5,-4/3) Elliptic Matrix(51,65,40,51) (-4/3,-5/4) -> (5/4,9/7) Hyperbolic Matrix(21,26,4,5) (-5/4,-1/1) -> (5/1,1/0) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(131,-169,100,-129) (9/7,13/10) -> (13/10,4/3) Parabolic Matrix(47,-65,34,-47) (4/3,7/5) -> (4/3,7/5) Elliptic Matrix(83,-117,22,-31) (7/5,10/7) -> (11/3,4/1) Hyperbolic Matrix(105,-169,64,-103) (8/5,13/8) -> (13/8,5/3) Parabolic Matrix(99,-169,58,-99) (5/3,12/7) -> (5/3,12/7) Elliptic Matrix(83,-143,18,-31) (12/7,7/4) -> (9/2,5/1) Hyperbolic Matrix(5,-13,2,-5) (2/1,3/1) -> (2/1,3/1) Elliptic Matrix(53,-169,16,-51) (3/1,13/4) -> (13/4,10/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,26,4,21) -> Matrix(1,0,2,1) Matrix(31,143,-18,-83) -> Matrix(1,0,-2,1) Matrix(53,234,12,53) -> Matrix(1,-2,0,1) Matrix(25,104,6,25) -> Matrix(1,2,0,1) Matrix(25,91,14,51) -> Matrix(1,0,2,1) Matrix(27,91,8,27) -> Matrix(1,2,0,1) Matrix(5,13,-2,-5) -> Matrix(1,0,0,1) Matrix(51,91,14,25) -> Matrix(1,0,2,1) Matrix(99,169,-58,-99) -> Matrix(1,0,0,1) Matrix(25,39,16,25) -> Matrix(1,0,4,1) Matrix(53,78,36,53) -> Matrix(1,0,4,1) Matrix(181,260,126,181) -> Matrix(5,2,12,5) Matrix(73,104,40,57) -> Matrix(1,0,4,1) Matrix(47,65,-34,-47) -> Matrix(1,0,0,1) Matrix(51,65,40,51) -> Matrix(1,0,4,1) Matrix(21,26,4,5) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,6,1) Matrix(131,-169,100,-129) -> Matrix(1,0,2,1) Matrix(47,-65,34,-47) -> Matrix(1,0,0,1) Matrix(83,-117,22,-31) -> Matrix(1,0,-2,1) Matrix(105,-169,64,-103) -> Matrix(5,-2,8,-3) Matrix(99,-169,58,-99) -> Matrix(1,0,0,1) Matrix(83,-143,18,-31) -> Matrix(1,0,-2,1) Matrix(5,-13,2,-5) -> Matrix(1,0,0,1) Matrix(53,-169,16,-51) -> Matrix(3,-2,2,-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 0/1 0/1 3 1 1/1 1/3 1 13 5/4 (0/1,1/2) 0 13 13/10 0/1 2 1 4/3 1/3 1 13 7/5 1/3 1 13 10/7 1/3 1 13 13/9 1/2 1 1 3/2 (0/1,1/2) 0 13 13/8 1/2 2 1 5/3 1/1 1 13 12/7 1/1 1 13 7/4 (1/2,1/1) 0 13 2/1 1/1 1 13 3/1 1/1 1 13 13/4 1/1 2 1 7/2 (1/1,1/0) 0 13 11/3 1/1 1 13 4/1 1/1 1 13 13/3 1/0 2 1 9/2 (-1/1,1/0) 0 13 5/1 -1/1 1 13 1/0 (0/1,1/0) 0 13 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(21,-26,4,-5) (1/1,5/4) -> (5/1,1/0) Glide Reflection Matrix(51,-65,40,-51) (5/4,13/10) -> (5/4,13/10) Reflection Matrix(79,-104,60,-79) (13/10,4/3) -> (13/10,4/3) Reflection Matrix(47,-65,34,-47) (4/3,7/5) -> (4/3,7/5) Elliptic Matrix(83,-117,22,-31) (7/5,10/7) -> (11/3,4/1) Hyperbolic Matrix(181,-260,126,-181) (10/7,13/9) -> (10/7,13/9) Reflection Matrix(53,-78,36,-53) (13/9,3/2) -> (13/9,3/2) Reflection Matrix(25,-39,16,-25) (3/2,13/8) -> (3/2,13/8) Reflection Matrix(79,-130,48,-79) (13/8,5/3) -> (13/8,5/3) Reflection Matrix(99,-169,58,-99) (5/3,12/7) -> (5/3,12/7) Elliptic Matrix(83,-143,18,-31) (12/7,7/4) -> (9/2,5/1) Hyperbolic Matrix(51,-91,14,-25) (7/4,2/1) -> (7/2,11/3) Glide Reflection Matrix(5,-13,2,-5) (2/1,3/1) -> (2/1,3/1) Elliptic Matrix(25,-78,8,-25) (3/1,13/4) -> (3/1,13/4) Reflection Matrix(27,-91,8,-27) (13/4,7/2) -> (13/4,7/2) 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 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(1,0,2,-1) -> Matrix(1,0,6,-1) (0/1,1/1) -> (0/1,1/3) Matrix(21,-26,4,-5) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(51,-65,40,-51) -> Matrix(1,0,4,-1) (5/4,13/10) -> (0/1,1/2) Matrix(79,-104,60,-79) -> Matrix(1,0,6,-1) (13/10,4/3) -> (0/1,1/3) Matrix(47,-65,34,-47) -> Matrix(1,0,0,1) Matrix(83,-117,22,-31) -> Matrix(1,0,-2,1) 0/1 Matrix(181,-260,126,-181) -> Matrix(5,-2,12,-5) (10/7,13/9) -> (1/3,1/2) Matrix(53,-78,36,-53) -> Matrix(1,0,4,-1) (13/9,3/2) -> (0/1,1/2) Matrix(25,-39,16,-25) -> Matrix(1,0,4,-1) (3/2,13/8) -> (0/1,1/2) Matrix(79,-130,48,-79) -> Matrix(3,-2,4,-3) (13/8,5/3) -> (1/2,1/1) Matrix(99,-169,58,-99) -> Matrix(1,0,0,1) Matrix(83,-143,18,-31) -> Matrix(1,0,-2,1) 0/1 Matrix(51,-91,14,-25) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(5,-13,2,-5) -> Matrix(1,0,0,1) Matrix(25,-78,8,-25) -> Matrix(1,0,2,-1) (3/1,13/4) -> (0/1,1/1) Matrix(27,-91,8,-27) -> Matrix(-1,2,0,1) (13/4,7/2) -> (1/1,1/0) Matrix(25,-104,6,-25) -> Matrix(-1,2,0,1) (4/1,13/3) -> (1/1,1/0) Matrix(53,-234,12,-53) -> Matrix(1,2,0,-1) (13/3,9/2) -> (-1/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.