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 -11/26 -7/26 -5/26 -9/52 -1/8 -3/26 -1/9 0/1 1/7 2/13 1/6 2/11 1/5 2/9 3/13 1/4 3/11 2/7 3/10 4/13 1/3 14/39 5/13 2/5 5/12 6/13 1/2 7/13 8/13 25/39 2/3 9/13 10/13 11/13 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 0/1 -6/7 -2/1 -1/1 -11/13 -1/1 -5/6 -1/1 -2/3 -9/11 -1/2 0/1 -13/16 -1/1 0/1 -4/5 -1/1 0/1 -7/9 -2/1 -1/1 -10/13 -1/1 -3/4 -1/1 -2/3 -8/11 -1/2 -1/3 -5/7 -1/1 0/1 -7/10 -1/1 0/1 -9/13 -1/1 -2/3 -1/1 -1/2 -9/14 -1/1 -2/3 -25/39 -2/3 -16/25 -2/3 -1/2 -7/11 -2/3 -1/2 -5/8 -2/3 -1/2 -8/13 -1/2 -3/5 -1/2 0/1 -7/12 -1/1 0/1 -4/7 -1/1 -2/3 -5/9 -2/3 -3/5 -6/11 -5/9 -1/2 -7/13 -1/2 -1/2 -1/2 0/1 -6/13 -1/2 -5/11 -1/2 -4/9 -9/20 -3/7 -2/5 -4/9 -1/2 -2/5 -11/25 -2/5 -1/3 -7/16 -2/5 -1/3 -3/7 -2/5 -1/3 -11/26 -1/3 -8/19 -1/3 -2/7 -5/12 -1/3 0/1 -12/29 -1/2 -1/3 -7/17 -2/5 -1/3 -2/5 -1/3 0/1 -5/13 0/1 -3/8 -1/2 0/1 -7/19 -1/1 0/1 -4/11 -1/2 -1/3 -9/25 -2/5 -1/3 -14/39 -1/3 -5/14 -1/3 0/1 -1/3 -1/3 0/1 -4/13 0/1 -3/10 0/1 1/1 -11/37 0/1 1/0 -8/27 0/1 1/0 -5/17 0/1 1/1 -2/7 -1/1 0/1 -7/25 -1/1 -2/3 -12/43 -3/4 -2/3 -5/18 -2/3 -1/2 -3/11 -2/3 -1/2 -7/26 -1/2 -4/15 -1/2 -3/7 -1/4 -1/3 0/1 -3/13 0/1 -2/9 0/1 1/0 -7/32 -1/1 0/1 -5/23 -1/1 0/1 -3/14 -1/1 0/1 -1/5 -1/2 0/1 -5/26 0/1 -4/21 -1/1 0/1 -3/16 -1/1 0/1 -5/27 -1/1 0/1 -2/11 -1/1 -1/2 -3/17 -2/5 -1/3 -4/23 -1/2 -1/3 -9/52 -1/3 -5/29 -1/3 0/1 -1/6 -1/3 0/1 -2/13 0/1 -1/7 -1/1 0/1 -1/8 -1/2 0/1 -2/17 -1/4 0/1 -3/26 0/1 -1/9 -1/1 0/1 0/1 -1/2 0/1 1/7 -1/3 0/1 2/13 0/1 1/6 -1/1 0/1 2/11 -1/2 -1/3 3/16 -1/3 0/1 1/5 -1/2 0/1 2/9 -1/4 0/1 3/13 0/1 1/4 -1/1 0/1 3/11 -1/2 -2/5 2/7 -1/3 0/1 3/10 -1/5 0/1 4/13 0/1 1/3 -1/1 0/1 5/14 -1/1 0/1 14/39 -1/1 9/25 -1/1 -2/3 4/11 -1/1 -1/2 3/8 -1/2 0/1 5/13 0/1 2/5 -1/1 0/1 5/12 -1/1 0/1 3/7 -1/1 -2/3 4/9 -2/3 -1/2 5/11 -4/7 -1/2 6/13 -1/2 1/2 -1/2 0/1 7/13 -1/2 6/11 -1/2 -5/11 11/20 -4/9 -3/7 5/9 -3/7 -2/5 14/25 -1/2 -2/5 9/16 -3/7 -2/5 4/7 -2/5 -1/3 15/26 -1/3 11/19 -1/3 -2/7 7/12 -1/3 0/1 17/29 -1/3 0/1 10/17 -1/4 0/1 3/5 -1/2 0/1 8/13 -1/2 5/8 -1/2 -2/5 12/19 -1/3 0/1 7/11 -1/2 -2/5 16/25 -1/2 -2/5 25/39 -2/5 9/14 -2/5 -1/3 2/3 -1/2 -1/3 9/13 -1/3 7/10 -1/3 0/1 26/37 -1/2 -1/3 19/27 -1/3 0/1 12/17 -1/2 0/1 5/7 -1/3 0/1 18/25 -1/2 0/1 31/43 -1/3 0/1 13/18 -1/2 0/1 8/11 -1/1 -1/2 19/26 -1/2 11/15 -1/2 -4/9 3/4 -2/5 -1/3 10/13 -1/3 7/9 -1/3 -2/7 25/32 -1/3 0/1 18/23 -1/3 -1/4 11/14 -1/3 0/1 4/5 -1/3 0/1 21/26 0/1 17/21 -1/2 0/1 13/16 -1/3 0/1 22/27 -1/2 0/1 9/11 -1/2 0/1 14/17 -1/2 -2/5 19/23 -2/5 -1/3 43/52 -1/3 24/29 -1/2 -1/3 5/6 -2/5 -1/3 11/13 -1/3 6/7 -1/3 -2/7 7/8 -1/4 0/1 15/17 -1/5 0/1 23/26 0/1 8/9 -1/2 0/1 1/1 -1/3 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(51,44,-182,-157) (-1/1,-6/7) -> (-2/7,-7/25) Hyperbolic Matrix(155,132,182,155) (-6/7,-11/13) -> (11/13,6/7) Hyperbolic Matrix(131,110,156,131) (-11/13,-5/6) -> (5/6,11/13) Hyperbolic Matrix(129,106,-286,-235) (-5/6,-9/11) -> (-5/11,-9/20) Hyperbolic Matrix(209,170,-702,-571) (-9/11,-13/16) -> (-3/10,-11/37) Hyperbolic Matrix(79,64,-416,-337) (-13/16,-4/5) -> (-4/21,-3/16) Hyperbolic Matrix(53,42,-130,-103) (-4/5,-7/9) -> (-7/17,-2/5) Hyperbolic Matrix(181,140,234,181) (-7/9,-10/13) -> (10/13,7/9) Hyperbolic Matrix(79,60,104,79) (-10/13,-3/4) -> (3/4,10/13) Hyperbolic Matrix(27,20,-104,-77) (-3/4,-8/11) -> (-4/15,-1/4) Hyperbolic Matrix(105,76,-286,-207) (-8/11,-5/7) -> (-7/19,-4/11) Hyperbolic Matrix(79,56,-182,-129) (-5/7,-7/10) -> (-7/16,-3/7) Hyperbolic Matrix(181,126,260,181) (-7/10,-9/13) -> (9/13,7/10) Hyperbolic Matrix(53,36,78,53) (-9/13,-2/3) -> (2/3,9/13) Hyperbolic Matrix(183,118,-442,-285) (-2/3,-9/14) -> (-5/12,-12/29) Hyperbolic Matrix(701,450,1092,701) (-9/14,-25/39) -> (25/39,9/14) Hyperbolic Matrix(1249,800,1950,1249) (-25/39,-16/25) -> (16/25,25/39) Hyperbolic Matrix(363,232,-1222,-781) (-16/25,-7/11) -> (-11/37,-8/27) Hyperbolic Matrix(79,50,-286,-181) (-7/11,-5/8) -> (-5/18,-3/11) Hyperbolic Matrix(129,80,208,129) (-5/8,-8/13) -> (8/13,5/8) Hyperbolic Matrix(79,48,130,79) (-8/13,-3/5) -> (3/5,8/13) Hyperbolic Matrix(27,16,-130,-77) (-3/5,-7/12) -> (-3/14,-1/5) Hyperbolic Matrix(131,76,-312,-181) (-7/12,-4/7) -> (-8/19,-5/12) Hyperbolic Matrix(53,30,-182,-103) (-4/7,-5/9) -> (-5/17,-2/7) Hyperbolic Matrix(51,28,-286,-157) (-5/9,-6/11) -> (-2/11,-3/17) Hyperbolic Matrix(155,84,286,155) (-6/11,-7/13) -> (7/13,6/11) Hyperbolic Matrix(27,14,52,27) (-7/13,-1/2) -> (1/2,7/13) Hyperbolic Matrix(25,12,52,25) (-1/2,-6/13) -> (6/13,1/2) Hyperbolic Matrix(131,60,286,131) (-6/13,-5/11) -> (5/11,6/13) Hyperbolic Matrix(103,46,-468,-209) (-9/20,-4/9) -> (-2/9,-7/32) Hyperbolic Matrix(363,160,-1300,-573) (-4/9,-11/25) -> (-7/25,-12/43) Hyperbolic Matrix(155,68,-832,-365) (-11/25,-7/16) -> (-3/16,-5/27) Hyperbolic Matrix(391,166,676,287) (-3/7,-11/26) -> (15/26,11/19) Hyperbolic Matrix(389,164,676,285) (-11/26,-8/19) -> (4/7,15/26) Hyperbolic Matrix(155,64,-884,-365) (-12/29,-7/17) -> (-3/17,-4/23) Hyperbolic Matrix(51,20,130,51) (-2/5,-5/13) -> (5/13,2/5) Hyperbolic Matrix(79,30,208,79) (-5/13,-3/8) -> (3/8,5/13) Hyperbolic Matrix(27,10,-208,-77) (-3/8,-7/19) -> (-1/7,-1/8) Hyperbolic Matrix(105,38,-572,-207) (-4/11,-9/25) -> (-5/27,-2/11) Hyperbolic Matrix(701,252,1950,701) (-9/25,-14/39) -> (14/39,9/25) Hyperbolic Matrix(391,140,1092,391) (-14/39,-5/14) -> (5/14,14/39) Hyperbolic Matrix(79,28,-364,-129) (-5/14,-1/3) -> (-5/23,-3/14) Hyperbolic Matrix(25,8,78,25) (-1/3,-4/13) -> (4/13,1/3) Hyperbolic Matrix(79,24,260,79) (-4/13,-3/10) -> (3/10,4/13) Hyperbolic Matrix(27,8,-260,-77) (-8/27,-5/17) -> (-1/9,0/1) Hyperbolic Matrix(79,22,-650,-181) (-12/43,-5/18) -> (-1/8,-2/17) Hyperbolic Matrix(495,134,676,183) (-3/11,-7/26) -> (19/26,11/15) Hyperbolic Matrix(493,132,676,181) (-7/26,-4/15) -> (8/11,19/26) Hyperbolic Matrix(25,6,104,25) (-1/4,-3/13) -> (3/13,1/4) Hyperbolic Matrix(53,12,234,53) (-3/13,-2/9) -> (2/9,3/13) Hyperbolic Matrix(183,40,-1066,-233) (-7/32,-5/23) -> (-5/29,-1/6) Hyperbolic Matrix(547,106,676,131) (-1/5,-5/26) -> (21/26,17/21) Hyperbolic Matrix(545,104,676,129) (-5/26,-4/21) -> (4/5,21/26) Hyperbolic Matrix(2237,388,2704,469) (-4/23,-9/52) -> (43/52,24/29) Hyperbolic Matrix(2235,386,2704,467) (-9/52,-5/29) -> (19/23,43/52) Hyperbolic Matrix(25,4,156,25) (-1/6,-2/13) -> (2/13,1/6) Hyperbolic Matrix(27,4,182,27) (-2/13,-1/7) -> (1/7,2/13) Hyperbolic Matrix(599,70,676,79) (-2/17,-3/26) -> (23/26,8/9) Hyperbolic Matrix(597,68,676,77) (-3/26,-1/9) -> (15/17,23/26) Hyperbolic Matrix(131,-18,182,-25) (0/1,1/7) -> (5/7,18/25) Hyperbolic Matrix(157,-28,286,-51) (1/6,2/11) -> (6/11,11/20) Hyperbolic Matrix(493,-92,702,-131) (2/11,3/16) -> (7/10,26/37) Hyperbolic Matrix(337,-64,416,-79) (3/16,1/5) -> (17/21,13/16) Hyperbolic Matrix(77,-16,130,-27) (1/5,2/9) -> (10/17,3/5) Hyperbolic Matrix(77,-20,104,-27) (1/4,3/11) -> (11/15,3/4) Hyperbolic Matrix(181,-50,286,-79) (3/11,2/7) -> (12/19,7/11) Hyperbolic Matrix(103,-30,182,-53) (2/7,3/10) -> (9/16,4/7) Hyperbolic Matrix(259,-92,442,-157) (1/3,5/14) -> (7/12,17/29) Hyperbolic Matrix(859,-310,1222,-441) (9/25,4/11) -> (26/37,19/27) Hyperbolic Matrix(207,-76,286,-105) (4/11,3/8) -> (13/18,8/11) Hyperbolic Matrix(103,-42,130,-53) (2/5,5/12) -> (11/14,4/5) Hyperbolic Matrix(181,-76,312,-131) (5/12,3/7) -> (11/19,7/12) Hyperbolic Matrix(129,-56,182,-79) (3/7,4/9) -> (12/17,5/7) Hyperbolic Matrix(235,-106,286,-129) (4/9,5/11) -> (9/11,14/17) Hyperbolic Matrix(365,-202,468,-259) (11/20,5/9) -> (7/9,25/32) Hyperbolic Matrix(937,-524,1300,-727) (5/9,14/25) -> (18/25,31/43) Hyperbolic Matrix(677,-380,832,-467) (14/25,9/16) -> (13/16,22/27) Hyperbolic Matrix(729,-428,884,-519) (17/29,10/17) -> (14/17,19/23) Hyperbolic Matrix(181,-114,208,-131) (5/8,12/19) -> (6/7,7/8) Hyperbolic Matrix(467,-298,572,-365) (7/11,16/25) -> (22/27,9/11) Hyperbolic Matrix(285,-184,364,-235) (9/14,2/3) -> (18/23,11/14) Hyperbolic Matrix(233,-164,260,-183) (19/27,12/17) -> (8/9,1/1) Hyperbolic Matrix(571,-412,650,-469) (31/43,13/18) -> (7/8,15/17) Hyperbolic Matrix(883,-690,1066,-833) (25/32,18/23) -> (24/29,5/6) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(51,44,-182,-157) -> Matrix(1,2,-2,-3) Matrix(155,132,182,155) -> Matrix(3,4,-10,-13) Matrix(131,110,156,131) -> Matrix(5,4,-14,-11) Matrix(129,106,-286,-235) -> Matrix(7,4,-16,-9) Matrix(209,170,-702,-571) -> Matrix(1,0,2,1) Matrix(79,64,-416,-337) -> Matrix(1,0,0,1) Matrix(53,42,-130,-103) -> Matrix(1,0,-2,1) Matrix(181,140,234,181) -> Matrix(3,4,-10,-13) Matrix(79,60,104,79) -> Matrix(5,4,-14,-11) Matrix(27,20,-104,-77) -> Matrix(3,2,-8,-5) Matrix(105,76,-286,-207) -> Matrix(1,0,0,1) Matrix(79,56,-182,-129) -> Matrix(3,2,-8,-5) Matrix(181,126,260,181) -> Matrix(1,0,-2,1) Matrix(53,36,78,53) -> Matrix(3,2,-8,-5) Matrix(183,118,-442,-285) -> Matrix(3,2,-8,-5) Matrix(701,450,1092,701) -> Matrix(5,4,-14,-11) Matrix(1249,800,1950,1249) -> Matrix(7,4,-16,-9) Matrix(363,232,-1222,-781) -> Matrix(3,2,-2,-1) Matrix(79,50,-286,-181) -> Matrix(1,0,0,1) Matrix(129,80,208,129) -> Matrix(7,4,-16,-9) Matrix(79,48,130,79) -> Matrix(1,0,0,1) Matrix(27,16,-130,-77) -> Matrix(1,0,0,1) Matrix(131,76,-312,-181) -> Matrix(1,0,-2,1) Matrix(53,30,-182,-103) -> Matrix(3,2,-2,-1) Matrix(51,28,-286,-157) -> Matrix(7,4,-16,-9) Matrix(155,84,286,155) -> Matrix(19,10,-40,-21) Matrix(27,14,52,27) -> Matrix(1,0,0,1) Matrix(25,12,52,25) -> Matrix(1,0,0,1) Matrix(131,60,286,131) -> Matrix(17,8,-32,-15) Matrix(103,46,-468,-209) -> Matrix(5,2,2,1) Matrix(363,160,-1300,-573) -> Matrix(11,4,-14,-5) Matrix(155,68,-832,-365) -> Matrix(5,2,-8,-3) Matrix(391,166,676,287) -> Matrix(11,4,-36,-13) Matrix(389,164,676,285) -> Matrix(13,4,-36,-11) Matrix(155,64,-884,-365) -> Matrix(1,0,0,1) Matrix(51,20,130,51) -> Matrix(1,0,2,1) Matrix(79,30,208,79) -> Matrix(1,0,0,1) Matrix(27,10,-208,-77) -> Matrix(1,0,0,1) Matrix(105,38,-572,-207) -> Matrix(5,2,-8,-3) Matrix(701,252,1950,701) -> Matrix(11,4,-14,-5) Matrix(391,140,1092,391) -> Matrix(1,0,2,1) Matrix(79,28,-364,-129) -> Matrix(1,0,2,1) Matrix(25,8,78,25) -> Matrix(1,0,2,1) Matrix(79,24,260,79) -> Matrix(1,0,-6,1) Matrix(27,8,-260,-77) -> Matrix(1,0,-2,1) Matrix(79,22,-650,-181) -> Matrix(3,2,-8,-5) Matrix(495,134,676,183) -> Matrix(11,6,-24,-13) Matrix(493,132,676,181) -> Matrix(9,4,-16,-7) Matrix(25,6,104,25) -> Matrix(1,0,2,1) Matrix(53,12,234,53) -> Matrix(1,0,-4,1) Matrix(183,40,-1066,-233) -> Matrix(1,0,-2,1) Matrix(547,106,676,131) -> Matrix(1,0,0,1) Matrix(545,104,676,129) -> Matrix(1,0,-2,1) Matrix(2237,388,2704,469) -> Matrix(1,0,0,1) Matrix(2235,386,2704,467) -> Matrix(7,2,-18,-5) Matrix(25,4,156,25) -> Matrix(1,0,2,1) Matrix(27,4,182,27) -> Matrix(1,0,-2,1) Matrix(599,70,676,79) -> Matrix(1,0,2,1) Matrix(597,68,676,77) -> Matrix(1,0,-4,1) Matrix(131,-18,182,-25) -> Matrix(1,0,0,1) Matrix(157,-28,286,-51) -> Matrix(7,4,-16,-9) Matrix(493,-92,702,-131) -> Matrix(1,0,0,1) Matrix(337,-64,416,-79) -> Matrix(1,0,0,1) Matrix(77,-16,130,-27) -> Matrix(1,0,0,1) Matrix(77,-20,104,-27) -> Matrix(3,2,-8,-5) Matrix(181,-50,286,-79) -> Matrix(1,0,0,1) Matrix(103,-30,182,-53) -> Matrix(7,2,-18,-5) Matrix(259,-92,442,-157) -> Matrix(1,0,-2,1) Matrix(859,-310,1222,-441) -> Matrix(3,2,-8,-5) Matrix(207,-76,286,-105) -> Matrix(1,0,0,1) Matrix(103,-42,130,-53) -> Matrix(1,0,-2,1) Matrix(181,-76,312,-131) -> Matrix(1,0,-2,1) Matrix(129,-56,182,-79) -> Matrix(3,2,-8,-5) Matrix(235,-106,286,-129) -> Matrix(7,4,-16,-9) Matrix(365,-202,468,-259) -> Matrix(9,4,-34,-15) Matrix(937,-524,1300,-727) -> Matrix(5,2,-8,-3) Matrix(677,-380,832,-467) -> Matrix(5,2,-8,-3) Matrix(729,-428,884,-519) -> Matrix(7,2,-18,-5) Matrix(181,-114,208,-131) -> Matrix(5,2,-18,-7) Matrix(467,-298,572,-365) -> Matrix(5,2,-8,-3) Matrix(285,-184,364,-235) -> Matrix(5,2,-18,-7) Matrix(233,-164,260,-183) -> Matrix(1,0,0,1) Matrix(571,-412,650,-469) -> Matrix(1,0,-2,1) Matrix(883,-690,1066,-833) -> Matrix(7,2,-18,-5) 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): 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 2/13 1/6 2/11 1/5 2/9 3/13 1/4 2/7 4/13 1/3 14/39 5/13 2/5 6/13 1/2 7/13 8/13 2/3 9/13 10/13 11/13 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 -1/1 0/1 -4/5 -1/1 0/1 -3/4 -1/1 -2/3 -8/11 -1/2 -1/3 -5/7 -1/1 0/1 -7/10 -1/1 0/1 -9/13 -1/1 -2/3 -1/1 -1/2 -3/5 -1/2 0/1 -7/12 -1/1 0/1 -4/7 -1/1 -2/3 -1/2 -1/2 0/1 -4/9 -1/2 -2/5 -7/16 -2/5 -1/3 -3/7 -2/5 -1/3 -2/5 -1/3 0/1 -5/13 0/1 -3/8 -1/2 0/1 -4/11 -1/2 -1/3 -5/14 -1/3 0/1 -1/3 -1/3 0/1 -2/7 -1/1 0/1 -1/4 -1/3 0/1 -3/13 0/1 -2/9 0/1 1/0 -3/14 -1/1 0/1 -1/5 -1/2 0/1 -1/6 -1/3 0/1 0/1 -1/2 0/1 1/7 -1/3 0/1 2/13 0/1 1/6 -1/1 0/1 2/11 -1/2 -1/3 1/5 -1/2 0/1 2/9 -1/4 0/1 3/13 0/1 1/4 -1/1 0/1 3/11 -1/2 -2/5 2/7 -1/3 0/1 3/10 -1/5 0/1 4/13 0/1 1/3 -1/1 0/1 5/14 -1/1 0/1 14/39 -1/1 9/25 -1/1 -2/3 4/11 -1/1 -1/2 3/8 -1/2 0/1 5/13 0/1 2/5 -1/1 0/1 5/12 -1/1 0/1 3/7 -1/1 -2/3 4/9 -2/3 -1/2 5/11 -4/7 -1/2 6/13 -1/2 1/2 -1/2 0/1 7/13 -1/2 6/11 -1/2 -5/11 5/9 -3/7 -2/5 4/7 -2/5 -1/3 15/26 -1/3 11/19 -1/3 -2/7 7/12 -1/3 0/1 17/29 -1/3 0/1 10/17 -1/4 0/1 3/5 -1/2 0/1 8/13 -1/2 5/8 -1/2 -2/5 2/3 -1/2 -1/3 9/13 -1/3 7/10 -1/3 0/1 19/27 -1/3 0/1 12/17 -1/2 0/1 5/7 -1/3 0/1 13/18 -1/2 0/1 8/11 -1/1 -1/2 19/26 -1/2 11/15 -1/2 -4/9 3/4 -2/5 -1/3 10/13 -1/3 7/9 -1/3 -2/7 4/5 -1/3 0/1 5/6 -2/5 -1/3 11/13 -1/3 6/7 -1/3 -2/7 1/1 -1/3 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(27,22,65,53) (-1/1,-4/5) -> (2/5,5/12) Hyperbolic Matrix(51,40,65,51) (-4/5,-3/4) -> (7/9,4/5) Hyperbolic Matrix(79,58,143,105) (-3/4,-8/11) -> (6/11,5/9) Hyperbolic Matrix(25,18,143,103) (-8/11,-5/7) -> (1/6,2/11) Hyperbolic Matrix(79,56,-182,-129) (-5/7,-7/10) -> (-7/16,-3/7) Hyperbolic Matrix(181,126,260,181) (-7/10,-9/13) -> (9/13,7/10) Hyperbolic Matrix(53,36,78,53) (-9/13,-2/3) -> (2/3,9/13) Hyperbolic Matrix(25,16,39,25) (-2/3,-3/5) -> (5/8,2/3) Hyperbolic Matrix(27,16,-130,-77) (-3/5,-7/12) -> (-3/14,-1/5) Hyperbolic Matrix(79,46,91,53) (-7/12,-4/7) -> (6/7,1/1) Hyperbolic Matrix(25,14,91,51) (-4/7,-1/2) -> (3/11,2/7) Hyperbolic Matrix(53,24,117,53) (-1/2,-4/9) -> (4/9,5/11) Hyperbolic Matrix(313,138,533,235) (-4/9,-7/16) -> (17/29,10/17) Hyperbolic Matrix(53,22,65,27) (-3/7,-2/5) -> (4/5,5/6) Hyperbolic Matrix(51,20,130,51) (-2/5,-5/13) -> (5/13,2/5) Hyperbolic Matrix(79,30,208,79) (-5/13,-3/8) -> (3/8,5/13) Hyperbolic Matrix(27,10,143,53) (-3/8,-4/11) -> (2/11,1/5) Hyperbolic Matrix(155,56,429,155) (-4/11,-5/14) -> (9/25,4/11) Hyperbolic Matrix(155,54,221,77) (-5/14,-1/3) -> (7/10,19/27) Hyperbolic Matrix(27,8,91,27) (-1/3,-2/7) -> (2/7,3/10) Hyperbolic Matrix(51,14,91,25) (-2/7,-1/4) -> (5/9,4/7) Hyperbolic Matrix(25,6,104,25) (-1/4,-3/13) -> (3/13,1/4) Hyperbolic Matrix(53,12,234,53) (-3/13,-2/9) -> (2/9,3/13) Hyperbolic Matrix(339,74,481,105) (-2/9,-3/14) -> (19/27,12/17) Hyperbolic Matrix(103,18,143,25) (-1/5,-1/6) -> (5/7,13/18) Hyperbolic Matrix(1,0,13,1) (-1/6,0/1) -> (0/1,1/7) Parabolic Matrix(27,-4,169,-25) (1/7,2/13) -> (2/13,1/6) Parabolic Matrix(77,-16,130,-27) (1/5,2/9) -> (10/17,3/5) Hyperbolic Matrix(77,-20,104,-27) (1/4,3/11) -> (11/15,3/4) Hyperbolic Matrix(53,-16,169,-51) (3/10,4/13) -> (4/13,1/3) Parabolic Matrix(259,-92,442,-157) (1/3,5/14) -> (7/12,17/29) Hyperbolic Matrix(547,-196,1521,-545) (5/14,14/39) -> (14/39,9/25) Parabolic Matrix(207,-76,286,-105) (4/11,3/8) -> (13/18,8/11) Hyperbolic Matrix(181,-76,312,-131) (5/12,3/7) -> (11/19,7/12) Hyperbolic Matrix(129,-56,182,-79) (3/7,4/9) -> (12/17,5/7) Hyperbolic Matrix(79,-36,169,-77) (5/11,6/13) -> (6/13,1/2) Parabolic Matrix(181,-96,247,-131) (1/2,7/13) -> (19/26,11/15) Hyperbolic Matrix(313,-170,429,-233) (7/13,6/11) -> (8/11,19/26) Hyperbolic Matrix(233,-134,273,-157) (4/7,15/26) -> (11/13,6/7) Hyperbolic Matrix(339,-196,403,-233) (15/26,11/19) -> (5/6,11/13) Hyperbolic Matrix(105,-64,169,-103) (3/5,8/13) -> (8/13,5/8) Parabolic Matrix(131,-100,169,-129) (3/4,10/13) -> (10/13,7/9) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(27,22,65,53) -> Matrix(1,0,0,1) Matrix(51,40,65,51) -> Matrix(1,0,-2,1) Matrix(79,58,143,105) -> Matrix(7,4,-16,-9) Matrix(25,18,143,103) -> Matrix(1,0,0,1) Matrix(79,56,-182,-129) -> Matrix(3,2,-8,-5) Matrix(181,126,260,181) -> Matrix(1,0,-2,1) Matrix(53,36,78,53) -> Matrix(3,2,-8,-5) Matrix(25,16,39,25) -> Matrix(3,2,-8,-5) Matrix(27,16,-130,-77) -> Matrix(1,0,0,1) Matrix(79,46,91,53) -> Matrix(1,0,-2,1) Matrix(25,14,91,51) -> Matrix(3,2,-8,-5) Matrix(53,24,117,53) -> Matrix(9,4,-16,-7) Matrix(313,138,533,235) -> Matrix(5,2,-18,-7) Matrix(53,22,65,27) -> Matrix(1,0,0,1) Matrix(51,20,130,51) -> Matrix(1,0,2,1) Matrix(79,30,208,79) -> Matrix(1,0,0,1) Matrix(27,10,143,53) -> Matrix(1,0,0,1) Matrix(155,56,429,155) -> Matrix(5,2,-8,-3) Matrix(155,54,221,77) -> Matrix(1,0,0,1) Matrix(27,8,91,27) -> Matrix(1,0,-2,1) Matrix(51,14,91,25) -> Matrix(3,2,-8,-5) Matrix(25,6,104,25) -> Matrix(1,0,2,1) Matrix(53,12,234,53) -> Matrix(1,0,-4,1) Matrix(339,74,481,105) -> Matrix(1,0,-2,1) Matrix(103,18,143,25) -> Matrix(1,0,0,1) Matrix(1,0,13,1) -> Matrix(1,0,0,1) Matrix(27,-4,169,-25) -> Matrix(1,0,2,1) Matrix(77,-16,130,-27) -> Matrix(1,0,0,1) Matrix(77,-20,104,-27) -> Matrix(3,2,-8,-5) Matrix(53,-16,169,-51) -> Matrix(1,0,4,1) Matrix(259,-92,442,-157) -> Matrix(1,0,-2,1) Matrix(547,-196,1521,-545) -> Matrix(1,2,-2,-3) Matrix(207,-76,286,-105) -> Matrix(1,0,0,1) Matrix(181,-76,312,-131) -> Matrix(1,0,-2,1) Matrix(129,-56,182,-79) -> Matrix(3,2,-8,-5) Matrix(79,-36,169,-77) -> Matrix(7,4,-16,-9) Matrix(181,-96,247,-131) -> Matrix(7,4,-16,-9) Matrix(313,-170,429,-233) -> Matrix(13,6,-24,-11) Matrix(233,-134,273,-157) -> Matrix(11,4,-36,-13) Matrix(339,-196,403,-233) -> Matrix(13,4,-36,-11) Matrix(105,-64,169,-103) -> Matrix(3,2,-8,-5) Matrix(131,-100,169,-129) -> Matrix(11,4,-36,-13) 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): 8 ----------------------------------------------------------------------- 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 (-1/2,0/1) 0 13 2/13 0/1 2 1 1/6 (-1/1,0/1) 0 13 1/5 (-1/2,0/1) 0 13 2/9 (-1/4,0/1) 0 13 3/13 0/1 3 1 1/4 (-1/1,0/1) 0 13 2/7 (-1/3,0/1) 0 13 4/13 0/1 4 1 1/3 (-1/1,0/1) 0 13 5/14 (-1/1,0/1) 0 13 14/39 -1/1 2 1 4/11 (-1/1,-1/2) 0 13 3/8 (-1/2,0/1) 0 13 5/13 0/1 1 1 2/5 (-1/1,0/1) 0 13 3/7 (-1/1,-2/3) 0 13 4/9 (-2/3,-1/2) 0 13 6/13 -1/2 4 1 1/2 (-1/2,0/1) 0 13 5/9 (-3/7,-2/5) 0 13 4/7 (-2/5,-1/3) 0 13 7/12 (-1/3,0/1) 0 13 10/17 (-1/4,0/1) 0 13 3/5 (-1/2,0/1) 0 13 8/13 -1/2 2 1 2/3 (-1/2,-1/3) 0 13 9/13 -1/3 1 1 7/10 (-1/3,0/1) 0 13 19/27 (-1/3,0/1) 0 13 12/17 (-1/2,0/1) 0 13 5/7 (-1/3,0/1) 0 13 13/18 (-1/2,0/1) 0 13 8/11 (-1/1,-1/2) 0 13 19/26 -1/2 5 1 11/15 (-1/2,-4/9) 0 13 3/4 (-2/5,-1/3) 0 13 10/13 -1/3 4 1 4/5 (-1/3,0/1) 0 13 5/6 (-2/5,-1/3) 0 13 11/13 -1/3 4 1 6/7 (-1/3,-2/7) 0 13 1/1 (-1/3,0/1) 0 13 1/0 0/1 1 1 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(1,0,13,-1) (0/1,2/13) -> (0/1,2/13) Reflection Matrix(25,-4,156,-25) (2/13,1/6) -> (2/13,1/6) Reflection Matrix(103,-18,143,-25) (1/6,1/5) -> (5/7,13/18) Glide Reflection Matrix(77,-16,130,-27) (1/5,2/9) -> (10/17,3/5) Hyperbolic Matrix(53,-12,234,-53) (2/9,3/13) -> (2/9,3/13) Reflection Matrix(25,-6,104,-25) (3/13,1/4) -> (3/13,1/4) Reflection Matrix(51,-14,91,-25) (1/4,2/7) -> (5/9,4/7) Glide Reflection Matrix(27,-8,91,-27) (2/7,4/13) -> (2/7,4/13) Reflection Matrix(25,-8,78,-25) (4/13,1/3) -> (4/13,1/3) Reflection Matrix(155,-54,221,-77) (1/3,5/14) -> (7/10,19/27) Glide Reflection Matrix(391,-140,1092,-391) (5/14,14/39) -> (5/14,14/39) Reflection Matrix(155,-56,429,-155) (14/39,4/11) -> (14/39,4/11) Reflection Matrix(207,-76,286,-105) (4/11,3/8) -> (13/18,8/11) Hyperbolic Matrix(79,-30,208,-79) (3/8,5/13) -> (3/8,5/13) Reflection Matrix(51,-20,130,-51) (5/13,2/5) -> (5/13,2/5) Reflection Matrix(53,-22,65,-27) (2/5,3/7) -> (4/5,5/6) Glide Reflection Matrix(129,-56,182,-79) (3/7,4/9) -> (12/17,5/7) Hyperbolic Matrix(53,-24,117,-53) (4/9,6/13) -> (4/9,6/13) Reflection Matrix(25,-12,52,-25) (6/13,1/2) -> (6/13,1/2) Reflection Matrix(105,-58,143,-79) (1/2,5/9) -> (11/15,3/4) Glide Reflection Matrix(79,-46,91,-53) (4/7,7/12) -> (6/7,1/1) Glide Reflection Matrix(467,-274,663,-389) (7/12,10/17) -> (19/27,12/17) Glide Reflection Matrix(79,-48,130,-79) (3/5,8/13) -> (3/5,8/13) Reflection Matrix(25,-16,39,-25) (8/13,2/3) -> (8/13,2/3) Reflection Matrix(53,-36,78,-53) (2/3,9/13) -> (2/3,9/13) Reflection Matrix(181,-126,260,-181) (9/13,7/10) -> (9/13,7/10) Reflection Matrix(417,-304,572,-417) (8/11,19/26) -> (8/11,19/26) Reflection Matrix(571,-418,780,-571) (19/26,11/15) -> (19/26,11/15) Reflection Matrix(79,-60,104,-79) (3/4,10/13) -> (3/4,10/13) Reflection Matrix(51,-40,65,-51) (10/13,4/5) -> (10/13,4/5) Reflection Matrix(131,-110,156,-131) (5/6,11/13) -> (5/6,11/13) Reflection Matrix(155,-132,182,-155) (11/13,6/7) -> (11/13,6/7) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,4,1) (0/1,1/0) -> (-1/2,0/1) Matrix(1,0,13,-1) -> Matrix(-1,0,4,1) (0/1,2/13) -> (-1/2,0/1) Matrix(25,-4,156,-25) -> Matrix(-1,0,2,1) (2/13,1/6) -> (-1/1,0/1) Matrix(103,-18,143,-25) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(77,-16,130,-27) -> Matrix(1,0,0,1) Matrix(53,-12,234,-53) -> Matrix(-1,0,8,1) (2/9,3/13) -> (-1/4,0/1) Matrix(25,-6,104,-25) -> Matrix(-1,0,2,1) (3/13,1/4) -> (-1/1,0/1) Matrix(51,-14,91,-25) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(27,-8,91,-27) -> Matrix(-1,0,6,1) (2/7,4/13) -> (-1/3,0/1) Matrix(25,-8,78,-25) -> Matrix(-1,0,2,1) (4/13,1/3) -> (-1/1,0/1) Matrix(155,-54,221,-77) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(391,-140,1092,-391) -> Matrix(-1,0,2,1) (5/14,14/39) -> (-1/1,0/1) Matrix(155,-56,429,-155) -> Matrix(3,2,-4,-3) (14/39,4/11) -> (-1/1,-1/2) Matrix(207,-76,286,-105) -> Matrix(1,0,0,1) Matrix(79,-30,208,-79) -> Matrix(-1,0,4,1) (3/8,5/13) -> (-1/2,0/1) Matrix(51,-20,130,-51) -> Matrix(-1,0,2,1) (5/13,2/5) -> (-1/1,0/1) Matrix(53,-22,65,-27) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(129,-56,182,-79) -> Matrix(3,2,-8,-5) -1/2 Matrix(53,-24,117,-53) -> Matrix(7,4,-12,-7) (4/9,6/13) -> (-2/3,-1/2) Matrix(25,-12,52,-25) -> Matrix(-1,0,4,1) (6/13,1/2) -> (-1/2,0/1) Matrix(105,-58,143,-79) -> Matrix(9,4,-20,-9) *** -> (-1/2,-2/5) Matrix(79,-46,91,-53) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(467,-274,663,-389) -> Matrix(-1,0,6,1) *** -> (-1/3,0/1) Matrix(79,-48,130,-79) -> Matrix(-1,0,4,1) (3/5,8/13) -> (-1/2,0/1) Matrix(25,-16,39,-25) -> Matrix(5,2,-12,-5) (8/13,2/3) -> (-1/2,-1/3) Matrix(53,-36,78,-53) -> Matrix(5,2,-12,-5) (2/3,9/13) -> (-1/2,-1/3) Matrix(181,-126,260,-181) -> Matrix(-1,0,6,1) (9/13,7/10) -> (-1/3,0/1) Matrix(417,-304,572,-417) -> Matrix(3,2,-4,-3) (8/11,19/26) -> (-1/1,-1/2) Matrix(571,-418,780,-571) -> Matrix(17,8,-36,-17) (19/26,11/15) -> (-1/2,-4/9) Matrix(79,-60,104,-79) -> Matrix(11,4,-30,-11) (3/4,10/13) -> (-2/5,-1/3) Matrix(51,-40,65,-51) -> Matrix(-1,0,6,1) (10/13,4/5) -> (-1/3,0/1) Matrix(131,-110,156,-131) -> Matrix(11,4,-30,-11) (5/6,11/13) -> (-2/5,-1/3) Matrix(155,-132,182,-155) -> Matrix(13,4,-42,-13) (11/13,6/7) -> (-1/3,-2/7) Matrix(-1,2,0,1) -> Matrix(-1,0,6,1) (1/1,1/0) -> (-1/3,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.