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 1/12 -13/2 1/11 -6/1 1/11 1/10 -11/2 1/10 2/19 -16/3 1/8 1/7 -5/1 1/12 -9/2 2/21 1/10 -13/3 1/10 -4/1 1/10 1/9 -11/3 1/12 -7/2 1/10 2/19 -10/3 3/28 1/9 -13/4 1/9 -3/1 1/8 -14/5 3/28 1/9 -39/14 1/9 -25/9 9/80 -11/4 3/26 2/17 -8/3 3/25 1/8 -13/5 1/8 -5/2 1/8 2/15 -12/5 1/8 1/7 -7/3 1/8 -9/4 2/15 3/22 -11/5 5/36 -13/6 1/7 -2/1 1/7 1/6 -13/7 1/6 -11/6 1/6 6/35 -20/11 1/6 3/17 -9/5 5/28 -25/14 2/11 3/16 -16/9 3/16 1/5 -7/4 1/6 2/11 -26/15 2/11 -19/11 3/16 -12/7 3/16 1/5 -29/17 1/4 -17/10 0/1 1/6 -5/3 3/16 -13/8 1/5 -8/5 1/5 5/24 -19/12 3/14 8/37 -11/7 1/4 -25/16 2/9 1/4 -39/25 1/4 -14/9 1/5 1/4 -3/2 2/9 1/4 -13/9 1/4 -10/7 1/4 7/27 -37/26 11/42 16/61 -27/19 19/72 -17/12 4/15 7/26 -7/5 1/4 -25/18 1/4 2/7 -43/31 9/32 -18/13 1/4 1/3 -11/8 5/18 2/7 -26/19 2/7 -15/11 7/24 -4/3 3/10 1/3 -13/10 1/3 -9/7 7/20 -32/25 1/3 5/14 -23/18 4/11 3/8 -14/11 1/3 3/8 -5/4 3/8 2/5 -26/21 2/5 -21/17 13/32 -16/13 5/12 3/7 -27/22 2/5 5/12 -11/9 5/12 -17/14 4/9 1/2 -23/19 7/16 -52/43 4/9 -29/24 4/9 9/20 -6/5 5/11 1/2 -13/11 1/2 -7/6 1/2 6/11 -8/7 3/5 5/8 -17/15 13/20 -26/23 2/3 -9/8 2/3 7/10 -1/1 1/0 0/1 0/1 1/1 1/24 7/6 6/133 1/22 13/11 1/22 6/5 1/22 5/109 11/9 5/108 16/13 3/65 5/108 5/4 2/43 3/64 9/7 7/148 13/10 1/21 4/3 1/21 3/62 11/8 2/41 5/102 7/5 1/20 10/7 7/141 1/20 13/9 1/20 3/2 1/20 2/39 14/9 1/20 1/19 39/25 1/20 25/16 1/20 2/39 11/7 1/20 8/5 5/96 1/19 13/8 1/19 5/3 3/56 12/7 1/19 3/56 7/4 2/37 1/18 9/5 5/92 11/6 6/109 1/18 13/7 1/18 2/1 1/18 1/17 13/6 1/17 11/5 5/84 20/9 3/50 5/83 9/4 3/50 2/33 25/11 11/180 16/7 5/81 1/16 7/3 1/16 26/11 0/1 19/8 0/1 1/18 12/5 1/17 1/16 29/12 0/1 1/16 17/7 3/52 5/2 2/33 1/16 13/5 1/16 8/3 1/16 3/47 19/7 1/16 11/4 2/31 3/46 25/9 9/136 39/14 1/15 14/5 1/15 3/44 3/1 1/16 13/4 1/15 10/3 1/15 3/44 37/11 3/44 27/8 3/44 2/29 17/5 5/72 7/2 2/29 1/14 25/7 7/96 43/12 9/122 2/27 18/5 3/40 1/13 11/3 1/12 26/7 0/1 15/4 0/1 1/18 4/1 1/15 1/14 13/3 1/14 9/2 1/14 2/27 32/7 1/14 3/41 23/5 3/40 14/3 1/13 1/12 5/1 1/12 26/5 0/1 21/4 0/1 1/20 16/3 1/17 1/16 27/5 5/76 11/2 2/29 1/14 17/3 3/40 23/4 0/1 1/12 52/9 0/1 29/5 1/16 6/1 1/14 1/13 13/2 1/13 7/1 1/12 8/1 1/12 1/11 17/2 0/1 1/6 26/3 0/1 9/1 1/16 1/0 0/1 1/12 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(25,-2,88,-7) Matrix(27,182,4,27) -> Matrix(23,-2,288,-25) Matrix(25,156,4,25) -> Matrix(21,-2,284,-27) Matrix(51,286,-28,-157) -> Matrix(41,-4,236,-23) Matrix(131,702,-92,-493) -> Matrix(49,-6,188,-23) Matrix(79,416,-64,-337) -> Matrix(11,-2,28,-5) Matrix(27,130,-16,-77) -> Matrix(21,-2,116,-11) Matrix(53,234,12,53) -> Matrix(41,-4,564,-55) Matrix(25,104,6,25) -> Matrix(19,-2,276,-29) Matrix(27,104,-20,-77) -> Matrix(17,-2,60,-7) Matrix(79,286,-50,-181) -> Matrix(23,-2,104,-9) Matrix(53,182,-30,-103) -> Matrix(1,0,-4,1) Matrix(79,260,24,79) -> Matrix(55,-6,816,-89) Matrix(25,78,8,25) -> Matrix(17,-2,264,-31) Matrix(157,442,-92,-259) -> Matrix(1,0,-4,1) Matrix(391,1092,140,391) -> Matrix(55,-6,816,-89) Matrix(701,1950,252,701) -> Matrix(161,-18,2424,-271) Matrix(441,1222,-310,-859) -> Matrix(229,-26,872,-99) Matrix(105,286,-76,-207) -> Matrix(33,-4,124,-15) Matrix(79,208,30,79) -> Matrix(49,-6,776,-95) Matrix(51,130,20,51) -> Matrix(31,-4,504,-65) Matrix(53,130,-42,-103) -> Matrix(29,-4,80,-11) Matrix(131,312,-76,-181) -> Matrix(13,-2,72,-11) Matrix(79,182,-56,-129) -> Matrix(17,-2,60,-7) Matrix(129,286,-106,-235) -> Matrix(73,-10,168,-23) Matrix(131,286,60,131) -> Matrix(71,-10,1200,-169) Matrix(25,52,12,25) -> Matrix(13,-2,228,-35) Matrix(27,52,14,27) -> Matrix(13,-2,228,-35) Matrix(155,286,84,155) -> Matrix(71,-12,1284,-217) Matrix(259,468,-202,-365) -> Matrix(91,-16,256,-45) Matrix(727,1300,-524,-937) -> Matrix(43,-8,156,-29) Matrix(467,832,-380,-677) -> Matrix(23,-4,52,-9) Matrix(389,676,164,285) -> Matrix(11,-2,204,-37) Matrix(391,676,166,287) -> Matrix(11,-2,160,-29) Matrix(519,884,-428,-729) -> Matrix(23,-4,52,-9) Matrix(79,130,48,79) -> Matrix(31,-6,584,-113) Matrix(129,208,80,129) -> Matrix(49,-10,936,-191) Matrix(131,208,-114,-181) -> Matrix(47,-10,80,-17) Matrix(365,572,-298,-467) -> Matrix(37,-8,88,-19) Matrix(1249,1950,800,1249) -> Matrix(17,-4,336,-79) Matrix(701,1092,450,701) -> Matrix(9,-2,176,-39) Matrix(235,364,-184,-285) -> Matrix(11,-2,28,-5) Matrix(53,78,36,53) -> Matrix(17,-4,336,-79) Matrix(181,260,126,181) -> Matrix(55,-14,1104,-281) Matrix(183,260,-164,-233) -> Matrix(53,-14,72,-19) Matrix(469,650,-412,-571) -> Matrix(27,-8,44,-13) Matrix(493,676,132,181) -> Matrix(7,-2,144,-41) Matrix(495,676,134,183) -> Matrix(7,-2,60,-17) Matrix(79,104,60,79) -> Matrix(19,-6,396,-125) Matrix(181,234,140,181) -> Matrix(41,-14,864,-295) Matrix(833,1066,-690,-883) -> Matrix(67,-24,148,-53) Matrix(545,676,104,129) -> Matrix(5,-2,108,-43) Matrix(547,676,106,131) -> Matrix(5,-2,28,-11) Matrix(2235,2704,386,467) -> Matrix(9,-4,160,-71) Matrix(2237,2704,388,469) -> Matrix(9,-4,88,-39) Matrix(131,156,110,131) -> Matrix(21,-10,460,-219) Matrix(155,182,132,155) -> Matrix(23,-12,508,-265) Matrix(597,676,68,77) -> Matrix(3,-2,68,-45) Matrix(599,676,70,79) -> Matrix(3,-2,8,-5) Matrix(1,0,2,1) -> Matrix(1,0,24,1) Matrix(157,-182,44,-51) -> Matrix(89,-4,1224,-55) Matrix(235,-286,106,-129) -> Matrix(217,-10,3624,-167) Matrix(571,-702,170,-209) -> Matrix(87,-4,1240,-57) Matrix(337,-416,64,-79) -> Matrix(43,-2,796,-37) Matrix(103,-130,42,-53) -> Matrix(85,-4,1424,-67) Matrix(77,-104,20,-27) -> Matrix(41,-2,636,-31) Matrix(207,-286,76,-105) -> Matrix(81,-4,1276,-63) Matrix(129,-182,56,-79) -> Matrix(41,-2,636,-31) Matrix(285,-442,118,-183) -> Matrix(39,-2,644,-33) Matrix(781,-1222,232,-363) -> Matrix(77,-4,1136,-59) Matrix(181,-286,50,-79) -> Matrix(39,-2,488,-25) Matrix(77,-130,16,-27) -> Matrix(37,-2,500,-27) Matrix(181,-312,76,-131) -> Matrix(37,-2,648,-35) Matrix(103,-182,30,-53) -> Matrix(1,0,-4,1) Matrix(157,-286,28,-51) -> Matrix(73,-4,1004,-55) Matrix(209,-468,46,-103) -> Matrix(67,-4,888,-53) Matrix(573,-1300,160,-363) -> Matrix(197,-12,2676,-163) Matrix(365,-832,68,-155) -> Matrix(65,-4,1024,-63) Matrix(365,-884,64,-155) -> Matrix(1,0,-4,1) Matrix(77,-208,10,-27) -> Matrix(31,-2,388,-25) Matrix(207,-572,38,-105) -> Matrix(61,-4,900,-59) Matrix(129,-364,28,-79) -> Matrix(29,-2,392,-27) Matrix(77,-260,8,-27) -> Matrix(29,-2,392,-27) Matrix(181,-650,22,-79) -> Matrix(27,-2,284,-21) Matrix(233,-1066,40,-183) -> Matrix(27,-2,392,-29) 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): 21 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): 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 1/1 13/11 13/10 13/9 3/2 39/25 13/8 9/5 13/7 2/1 13/6 5/2 13/5 3/1 13/4 11/3 4/1 13/3 5/1 6/1 13/2 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 1/12 -4/1 1/10 1/9 -11/3 1/12 -7/2 1/10 2/19 -10/3 3/28 1/9 -13/4 1/9 -3/1 1/8 -5/2 1/8 2/15 -12/5 1/8 1/7 -7/3 1/8 -2/1 1/7 1/6 -9/5 5/28 -16/9 3/16 1/5 -7/4 1/6 2/11 -5/3 3/16 -13/8 1/5 -8/5 1/5 5/24 -11/7 1/4 -14/9 1/5 1/4 -3/2 2/9 1/4 -7/5 1/4 -4/3 3/10 1/3 -13/10 1/3 -9/7 7/20 -14/11 1/3 3/8 -5/4 3/8 2/5 -6/5 5/11 1/2 -1/1 1/0 0/1 0/1 1/1 1/24 7/6 6/133 1/22 13/11 1/22 6/5 1/22 5/109 11/9 5/108 5/4 2/43 3/64 9/7 7/148 13/10 1/21 4/3 1/21 3/62 11/8 2/41 5/102 7/5 1/20 10/7 7/141 1/20 13/9 1/20 3/2 1/20 2/39 14/9 1/20 1/19 39/25 1/20 25/16 1/20 2/39 11/7 1/20 8/5 5/96 1/19 13/8 1/19 5/3 3/56 12/7 1/19 3/56 7/4 2/37 1/18 9/5 5/92 11/6 6/109 1/18 13/7 1/18 2/1 1/18 1/17 13/6 1/17 11/5 5/84 9/4 3/50 2/33 7/3 1/16 26/11 0/1 19/8 0/1 1/18 12/5 1/17 1/16 29/12 0/1 1/16 17/7 3/52 5/2 2/33 1/16 13/5 1/16 8/3 1/16 3/47 3/1 1/16 13/4 1/15 10/3 1/15 3/44 27/8 3/44 2/29 17/5 5/72 7/2 2/29 1/14 18/5 3/40 1/13 11/3 1/12 26/7 0/1 15/4 0/1 1/18 4/1 1/15 1/14 13/3 1/14 9/2 1/14 2/27 5/1 1/12 6/1 1/14 1/13 13/2 1/13 7/1 1/12 1/0 0/1 1/12 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(12,65,7,38) (-5/1,1/0) -> (5/3,12/7) Hyperbolic Matrix(14,65,3,14) (-5/1,-4/1) -> (9/2,5/1) Hyperbolic Matrix(38,143,17,64) (-4/1,-11/3) -> (11/5,9/4) Hyperbolic Matrix(40,143,33,118) (-11/3,-7/2) -> (6/5,11/9) 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(14,39,5,14) (-3/1,-5/2) -> (8/3,3/1) Hyperbolic Matrix(53,130,-42,-103) (-5/2,-12/5) -> (-14/11,-5/4) Hyperbolic Matrix(38,91,5,12) (-12/5,-7/3) -> (7/1,1/0) Hyperbolic Matrix(40,91,29,66) (-7/3,-2/1) -> (11/8,7/5) Hyperbolic Matrix(64,117,35,64) (-2/1,-9/5) -> (9/5,11/6) Hyperbolic Matrix(298,533,123,220) (-9/5,-16/9) -> (29/12,17/7) Hyperbolic Matrix(38,65,7,12) (-7/4,-5/3) -> (5/1,6/1) 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(90,143,73,116) (-8/5,-11/7) -> (11/9,5/4) Hyperbolic Matrix(274,429,175,274) (-11/7,-14/9) -> (25/16,11/7) Hyperbolic Matrix(144,221,43,66) (-14/9,-3/2) -> (10/3,27/8) Hyperbolic Matrix(64,91,45,64) (-3/2,-7/5) -> (7/5,10/7) Hyperbolic Matrix(66,91,29,40) (-7/5,-4/3) -> (9/4,7/3) 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(376,481,111,142) (-9/7,-14/11) -> (27/8,17/5) Hyperbolic Matrix(118,143,33,40) (-5/4,-6/5) -> (7/2,18/5) Hyperbolic Matrix(12,13,11,12) (-6/5,-1/1) -> (1/1,7/6) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(144,-169,121,-142) (7/6,13/11) -> (13/11,6/5) Parabolic 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(118,-169,81,-116) (10/7,13/9) -> (13/9,3/2) Parabolic Matrix(285,-442,118,-183) (3/2,14/9) -> (12/5,29/12) Hyperbolic Matrix(976,-1521,625,-974) (14/9,39/25) -> (39/25,25/16) Parabolic Matrix(181,-286,50,-79) (11/7,8/5) -> (18/5,11/3) 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(92,-169,49,-90) (11/6,13/7) -> (13/7,2/1) Parabolic Matrix(116,-247,31,-66) (2/1,13/6) -> (26/7,15/4) Hyperbolic Matrix(196,-429,53,-116) (13/6,11/5) -> (11/3,26/7) Hyperbolic Matrix(116,-273,17,-40) (7/3,26/11) -> (13/2,7/1) Hyperbolic Matrix(170,-403,27,-64) (26/11,19/8) -> (6/1,13/2) Hyperbolic Matrix(66,-169,25,-64) (5/2,13/5) -> (13/5,8/3) Parabolic Matrix(40,-169,9,-38) (4/1,13/3) -> (13/3,9/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(12,65,7,38) -> Matrix(9,-1,172,-19) Matrix(14,65,3,14) -> Matrix(11,-1,144,-13) Matrix(38,143,17,64) -> Matrix(7,-1,120,-17) Matrix(40,143,33,118) -> Matrix(31,-3,672,-65) Matrix(53,182,-30,-103) -> Matrix(1,0,-4,1) Matrix(79,260,24,79) -> Matrix(55,-6,816,-89) Matrix(25,78,8,25) -> Matrix(17,-2,264,-31) Matrix(14,39,5,14) -> Matrix(9,-1,136,-15) Matrix(53,130,-42,-103) -> Matrix(29,-4,80,-11) Matrix(38,91,5,12) -> Matrix(7,-1,92,-13) Matrix(40,91,29,66) -> Matrix(23,-3,468,-61) Matrix(64,117,35,64) -> Matrix(29,-5,528,-91) Matrix(298,533,123,220) -> Matrix(5,-1,96,-19) Matrix(38,65,7,12) -> Matrix(5,-1,76,-15) Matrix(79,130,48,79) -> Matrix(31,-6,584,-113) Matrix(129,208,80,129) -> Matrix(49,-10,936,-191) Matrix(90,143,73,116) -> Matrix(33,-7,712,-151) Matrix(274,429,175,274) -> Matrix(13,-3,256,-59) Matrix(144,221,43,66) -> Matrix(23,-5,336,-73) Matrix(64,91,45,64) -> Matrix(19,-5,384,-101) Matrix(66,91,29,40) -> Matrix(11,-3,180,-49) Matrix(79,104,60,79) -> Matrix(19,-6,396,-125) Matrix(181,234,140,181) -> Matrix(41,-14,864,-295) Matrix(376,481,111,142) -> Matrix(25,-9,364,-131) Matrix(118,143,33,40) -> Matrix(7,-3,96,-41) Matrix(12,13,11,12) -> Matrix(1,-1,24,-23) Matrix(1,0,2,1) -> Matrix(1,0,24,1) Matrix(144,-169,121,-142) -> Matrix(243,-11,5324,-241) Matrix(103,-130,42,-53) -> Matrix(85,-4,1424,-67) Matrix(77,-104,20,-27) -> Matrix(41,-2,636,-31) Matrix(118,-169,81,-116) -> Matrix(181,-9,3600,-179) Matrix(285,-442,118,-183) -> Matrix(39,-2,644,-33) Matrix(976,-1521,625,-974) -> Matrix(21,-1,400,-19) Matrix(181,-286,50,-79) -> Matrix(39,-2,488,-25) Matrix(181,-312,76,-131) -> Matrix(37,-2,648,-35) Matrix(103,-182,30,-53) -> Matrix(1,0,-4,1) Matrix(92,-169,49,-90) -> Matrix(127,-7,2268,-125) Matrix(116,-247,31,-66) -> Matrix(17,-1,324,-19) Matrix(196,-429,53,-116) -> Matrix(17,-1,120,-7) Matrix(116,-273,17,-40) -> Matrix(15,-1,196,-13) Matrix(170,-403,27,-64) -> Matrix(19,-1,248,-13) Matrix(66,-169,25,-64) -> Matrix(81,-5,1280,-79) Matrix(40,-169,9,-38) -> Matrix(43,-3,588,-41) 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): 21 ----------------------------------------------------------------------- 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 12 1 1/1 1/24 1 13 13/11 1/22 11 1 6/5 (1/22,5/109) 0 13 5/4 (2/43,3/64) 0 13 9/7 7/148 1 13 13/10 1/21 10 1 4/3 (1/21,3/62) 0 13 7/5 1/20 1 13 13/9 1/20 9 1 3/2 (1/20,2/39) 0 13 14/9 (1/20,1/19) 0 13 39/25 1/20 1 1 11/7 1/20 1 13 8/5 (5/96,1/19) 0 13 13/8 1/19 8 1 5/3 3/56 1 13 7/4 (2/37,1/18) 0 13 9/5 5/92 1 13 13/7 1/18 7 1 2/1 (1/18,1/17) 0 13 9/4 (3/50,2/33) 0 13 7/3 1/16 1 13 12/5 (1/17,1/16) 0 13 17/7 3/52 1 13 5/2 (2/33,1/16) 0 13 13/5 1/16 5 1 3/1 1/16 1 13 13/4 1/15 4 1 10/3 (1/15,3/44) 0 13 27/8 (3/44,2/29) 0 13 17/5 5/72 1 13 7/2 (2/29,1/14) 0 13 18/5 (3/40,1/13) 0 13 11/3 1/12 1 13 26/7 0/1 6 1 15/4 (0/1,1/18) 0 13 4/1 (1/15,1/14) 0 13 13/3 1/14 3 1 5/1 1/12 1 13 6/1 (1/14,1/13) 0 13 13/2 1/13 2 1 7/1 1/12 1 13 1/0 (0/1,1/12) 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(12,-13,11,-12) (1/1,13/11) -> (1/1,13/11) Reflection Matrix(131,-156,110,-131) (13/11,6/5) -> (13/11,6/5) Reflection Matrix(118,-143,33,-40) (6/5,5/4) -> (7/2,18/5) Glide Reflection Matrix(103,-130,42,-53) (5/4,9/7) -> (17/7,5/2) Hyperbolic Matrix(181,-234,140,-181) (9/7,13/10) -> (9/7,13/10) Reflection Matrix(79,-104,60,-79) (13/10,4/3) -> (13/10,4/3) Reflection Matrix(66,-91,29,-40) (4/3,7/5) -> (9/4,7/3) Glide Reflection Matrix(64,-91,45,-64) (7/5,13/9) -> (7/5,13/9) Reflection Matrix(53,-78,36,-53) (13/9,3/2) -> (13/9,3/2) Reflection Matrix(144,-221,43,-66) (3/2,14/9) -> (10/3,27/8) Glide Reflection Matrix(701,-1092,450,-701) (14/9,39/25) -> (14/9,39/25) Reflection Matrix(274,-429,175,-274) (39/25,11/7) -> (39/25,11/7) Reflection Matrix(181,-286,50,-79) (11/7,8/5) -> (18/5,11/3) Hyperbolic Matrix(129,-208,80,-129) (8/5,13/8) -> (8/5,13/8) Reflection Matrix(79,-130,48,-79) (13/8,5/3) -> (13/8,5/3) Reflection Matrix(38,-65,7,-12) (5/3,7/4) -> (5/1,6/1) Glide Reflection Matrix(103,-182,30,-53) (7/4,9/5) -> (17/5,7/2) Hyperbolic Matrix(64,-117,35,-64) (9/5,13/7) -> (9/5,13/7) Reflection Matrix(27,-52,14,-27) (13/7,2/1) -> (13/7,2/1) Reflection Matrix(64,-143,17,-38) (2/1,9/4) -> (15/4,4/1) Glide Reflection Matrix(38,-91,5,-12) (7/3,12/5) -> (7/1,1/0) Glide Reflection Matrix(274,-663,81,-196) (12/5,17/7) -> (27/8,17/5) Glide Reflection Matrix(51,-130,20,-51) (5/2,13/5) -> (5/2,13/5) Reflection Matrix(14,-39,5,-14) (13/5,3/1) -> (13/5,3/1) Reflection 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(155,-572,42,-155) (11/3,26/7) -> (11/3,26/7) Reflection Matrix(209,-780,56,-209) (26/7,15/4) -> (26/7,15/4) Reflection Matrix(25,-104,6,-25) (4/1,13/3) -> (4/1,13/3) Reflection Matrix(14,-65,3,-14) (13/3,5/1) -> (13/3,5/1) 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 IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,24,-1) (0/1,1/0) -> (0/1,1/12) Matrix(1,0,2,-1) -> Matrix(1,0,48,-1) (0/1,1/1) -> (0/1,1/24) Matrix(12,-13,11,-12) -> Matrix(23,-1,528,-23) (1/1,13/11) -> (1/24,1/22) Matrix(131,-156,110,-131) -> Matrix(219,-10,4796,-219) (13/11,6/5) -> (1/22,5/109) Matrix(118,-143,33,-40) -> Matrix(65,-3,888,-41) Matrix(103,-130,42,-53) -> Matrix(85,-4,1424,-67) Matrix(181,-234,140,-181) -> Matrix(295,-14,6216,-295) (9/7,13/10) -> (7/148,1/21) Matrix(79,-104,60,-79) -> Matrix(125,-6,2604,-125) (13/10,4/3) -> (1/21,3/62) Matrix(66,-91,29,-40) -> Matrix(61,-3,996,-49) Matrix(64,-91,45,-64) -> Matrix(101,-5,2040,-101) (7/5,13/9) -> (5/102,1/20) Matrix(53,-78,36,-53) -> Matrix(79,-4,1560,-79) (13/9,3/2) -> (1/20,2/39) Matrix(144,-221,43,-66) -> Matrix(97,-5,1416,-73) Matrix(701,-1092,450,-701) -> Matrix(39,-2,760,-39) (14/9,39/25) -> (1/20,1/19) Matrix(274,-429,175,-274) -> Matrix(59,-3,1160,-59) (39/25,11/7) -> (1/20,3/58) Matrix(181,-286,50,-79) -> Matrix(39,-2,488,-25) Matrix(129,-208,80,-129) -> Matrix(191,-10,3648,-191) (8/5,13/8) -> (5/96,1/19) Matrix(79,-130,48,-79) -> Matrix(113,-6,2128,-113) (13/8,5/3) -> (1/19,3/56) Matrix(38,-65,7,-12) -> Matrix(19,-1,284,-15) Matrix(103,-182,30,-53) -> Matrix(1,0,-4,1) 0/1 Matrix(64,-117,35,-64) -> Matrix(91,-5,1656,-91) (9/5,13/7) -> (5/92,1/18) Matrix(27,-52,14,-27) -> Matrix(35,-2,612,-35) (13/7,2/1) -> (1/18,1/17) Matrix(64,-143,17,-38) -> Matrix(17,-1,288,-17) *** -> (1/18,1/16) Matrix(38,-91,5,-12) -> Matrix(17,-1,220,-13) Matrix(274,-663,81,-196) -> Matrix(19,-1,284,-15) Matrix(51,-130,20,-51) -> Matrix(65,-4,1056,-65) (5/2,13/5) -> (2/33,1/16) Matrix(14,-39,5,-14) -> Matrix(15,-1,224,-15) (13/5,3/1) -> (1/16,1/14) Matrix(25,-78,8,-25) -> Matrix(31,-2,480,-31) (3/1,13/4) -> (1/16,1/15) Matrix(79,-260,24,-79) -> Matrix(89,-6,1320,-89) (13/4,10/3) -> (1/15,3/44) Matrix(155,-572,42,-155) -> Matrix(1,0,24,-1) (11/3,26/7) -> (0/1,1/12) Matrix(209,-780,56,-209) -> Matrix(1,0,36,-1) (26/7,15/4) -> (0/1,1/18) Matrix(25,-104,6,-25) -> Matrix(29,-2,420,-29) (4/1,13/3) -> (1/15,1/14) Matrix(14,-65,3,-14) -> Matrix(13,-1,168,-13) (13/3,5/1) -> (1/14,1/12) Matrix(25,-156,4,-25) -> Matrix(27,-2,364,-27) (6/1,13/2) -> (1/14,1/13) Matrix(27,-182,4,-27) -> Matrix(25,-2,312,-25) (13/2,7/1) -> (1/13,1/12) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.