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 3/28 -13/2 1/9 -6/1 2/17 -11/2 1/9 -16/3 4/35 -5/1 1/8 -9/2 1/9 -13/3 1/8 -4/1 0/1 -11/3 1/8 -7/2 1/7 -10/3 2/17 -13/4 1/8 -3/1 1/8 -14/5 6/43 -39/14 1/7 -25/9 3/20 -11/4 1/7 -8/3 4/29 -13/5 1/7 -5/2 1/7 -12/5 8/53 -7/3 5/32 -9/4 5/31 -11/5 1/6 -13/6 1/6 -2/1 2/11 -13/7 1/5 -11/6 1/5 -20/11 8/39 -9/5 5/24 -25/14 13/61 -16/9 4/19 -7/4 5/23 -26/15 2/9 -19/11 13/58 -12/7 8/35 -29/17 7/30 -17/10 3/13 -5/3 1/4 -13/8 1/4 -8/5 4/15 -19/12 3/11 -11/7 1/4 -25/16 3/13 -39/25 1/4 -14/9 6/23 -3/2 1/3 -13/9 1/3 -10/7 2/5 -37/26 1/3 -27/19 5/14 -17/12 3/7 -7/5 1/4 -25/18 3/7 -43/31 1/0 -18/13 2/7 -11/8 1/3 -26/19 2/5 -15/11 1/2 -4/3 0/1 -13/10 1/3 -9/7 1/2 -32/25 4/11 -23/18 3/7 -14/11 2/3 -5/4 1/3 -26/21 2/5 -21/17 5/12 -16/13 4/9 -27/22 7/15 -11/9 1/2 -17/14 1/1 -23/19 1/0 -52/43 0/1 -29/24 1/5 -6/5 2/5 -13/11 1/2 -7/6 3/5 -8/7 4/7 -17/15 9/14 -26/23 2/3 -9/8 5/7 -1/1 1/0 0/1 0/1 1/1 1/22 7/6 3/61 13/11 1/20 6/5 2/39 11/9 1/20 16/13 4/79 5/4 1/19 9/7 1/20 13/10 1/19 4/3 0/1 11/8 1/19 7/5 1/18 10/7 2/39 13/9 1/19 3/2 1/19 14/9 6/109 39/25 1/18 25/16 3/53 11/7 1/18 8/5 4/73 13/8 1/18 5/3 1/18 12/7 8/141 7/4 5/87 9/5 5/86 11/6 1/17 13/7 1/17 2/1 2/33 13/6 1/16 11/5 1/16 20/9 8/127 9/4 5/79 25/11 13/204 16/7 4/63 7/3 5/78 26/11 2/31 19/8 13/201 12/5 8/123 29/12 7/107 17/7 3/46 5/2 1/15 13/5 1/15 8/3 4/59 19/7 3/44 11/4 1/15 25/9 3/46 39/14 1/15 14/5 6/89 3/1 1/14 13/4 1/14 10/3 2/27 37/11 1/14 27/8 5/69 17/5 3/40 7/2 1/15 25/7 3/40 43/12 1/11 18/5 2/29 11/3 1/14 26/7 2/27 15/4 1/13 4/1 0/1 13/3 1/14 9/2 1/13 32/7 4/55 23/5 3/40 14/3 2/25 5/1 1/14 26/5 2/27 21/4 5/67 16/3 4/53 27/5 7/92 11/2 1/13 17/3 1/12 23/4 1/11 52/9 0/1 29/5 1/16 6/1 2/27 13/2 1/13 7/1 3/38 8/1 4/51 17/2 9/113 26/3 2/25 9/1 5/62 1/0 1/11 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(19,-2,48,-5) Matrix(27,182,4,27) -> Matrix(55,-6,706,-77) Matrix(25,156,4,25) -> Matrix(35,-4,464,-53) Matrix(51,286,-28,-157) -> Matrix(55,-6,266,-29) Matrix(131,702,-92,-493) -> Matrix(17,-2,60,-7) Matrix(79,416,-64,-337) -> Matrix(69,-8,164,-19) Matrix(27,130,-16,-77) -> Matrix(15,-2,68,-9) Matrix(53,234,12,53) -> Matrix(17,-2,230,-27) Matrix(25,104,6,25) -> Matrix(1,0,6,1) Matrix(27,104,-20,-77) -> Matrix(1,0,-6,1) Matrix(79,286,-50,-181) -> Matrix(17,-2,60,-7) Matrix(53,182,-30,-103) -> Matrix(19,-2,86,-9) Matrix(79,260,24,79) -> Matrix(33,-4,454,-55) Matrix(25,78,8,25) -> Matrix(15,-2,218,-29) Matrix(157,442,-92,-259) -> Matrix(73,-10,314,-43) Matrix(391,1092,140,391) -> Matrix(85,-12,1268,-179) Matrix(701,1950,252,701) -> Matrix(41,-6,622,-91) Matrix(441,1222,-310,-859) -> Matrix(15,-2,38,-5) Matrix(105,286,-76,-207) -> Matrix(15,-2,38,-5) Matrix(79,208,30,79) -> Matrix(57,-8,848,-119) Matrix(51,130,20,51) -> Matrix(41,-6,622,-91) Matrix(53,130,-42,-103) -> Matrix(13,-2,46,-7) Matrix(131,312,-76,-181) -> Matrix(105,-16,466,-71) Matrix(79,182,-56,-129) -> Matrix(13,-2,20,-3) Matrix(129,286,-106,-235) -> Matrix(37,-6,68,-11) Matrix(131,286,60,131) -> Matrix(109,-18,1738,-287) Matrix(25,52,12,25) -> Matrix(23,-4,374,-65) Matrix(27,52,14,27) -> Matrix(21,-4,352,-67) Matrix(155,286,84,155) -> Matrix(89,-18,1518,-307) Matrix(259,468,-202,-365) -> Matrix(19,-4,62,-13) Matrix(727,1300,-524,-937) -> Matrix(19,-4,24,-5) Matrix(467,832,-380,-677) -> Matrix(37,-8,88,-19) Matrix(389,676,164,285) -> Matrix(163,-36,2522,-557) Matrix(391,676,166,287) -> Matrix(161,-36,2500,-559) Matrix(519,884,-428,-729) -> Matrix(17,-4,30,-7) Matrix(79,130,48,79) -> Matrix(25,-6,446,-107) Matrix(129,208,80,129) -> Matrix(31,-8,562,-145) Matrix(131,208,-114,-181) -> Matrix(1,0,-2,1) Matrix(365,572,-298,-467) -> Matrix(15,-4,34,-9) Matrix(1249,1950,800,1249) -> Matrix(25,-6,446,-107) Matrix(701,1092,450,701) -> Matrix(47,-12,850,-217) Matrix(235,364,-184,-285) -> Matrix(15,-4,34,-9) Matrix(53,78,36,53) -> Matrix(7,-2,130,-37) Matrix(181,260,126,181) -> Matrix(11,-4,212,-77) Matrix(183,260,-164,-233) -> Matrix(11,-4,14,-5) Matrix(469,650,-412,-571) -> Matrix(9,-2,14,-3) Matrix(493,676,132,181) -> Matrix(11,-4,146,-53) Matrix(495,676,134,183) -> Matrix(9,-4,124,-55) Matrix(79,104,60,79) -> Matrix(1,0,16,1) Matrix(181,234,140,181) -> Matrix(5,-2,98,-39) Matrix(833,1066,-690,-883) -> Matrix(5,-2,18,-7) Matrix(545,676,104,129) -> Matrix(31,-12,416,-161) Matrix(547,676,106,131) -> Matrix(29,-12,394,-163) Matrix(2235,2704,386,467) -> Matrix(1,0,16,1) Matrix(2237,2704,388,469) -> Matrix(1,0,6,1) Matrix(131,156,110,131) -> Matrix(9,-4,178,-79) Matrix(155,182,132,155) -> Matrix(11,-6,222,-121) Matrix(597,676,68,77) -> Matrix(43,-28,536,-349) Matrix(599,676,70,79) -> Matrix(41,-28,514,-351) Matrix(1,0,2,1) -> Matrix(1,0,22,1) Matrix(157,-182,44,-51) -> Matrix(41,-2,554,-27) Matrix(235,-286,106,-129) -> Matrix(121,-6,1916,-95) Matrix(571,-702,170,-209) -> Matrix(39,-2,566,-29) Matrix(337,-416,64,-79) -> Matrix(157,-8,2100,-107) Matrix(103,-130,42,-53) -> Matrix(37,-2,574,-31) Matrix(77,-104,20,-27) -> Matrix(1,0,-6,1) Matrix(207,-286,76,-105) -> Matrix(39,-2,566,-29) Matrix(129,-182,56,-79) -> Matrix(41,-2,636,-31) Matrix(285,-442,118,-183) -> Matrix(183,-10,2800,-153) Matrix(781,-1222,232,-363) -> Matrix(37,-2,500,-27) Matrix(181,-286,50,-79) -> Matrix(37,-2,500,-27) Matrix(77,-130,16,-27) -> Matrix(35,-2,508,-29) Matrix(181,-312,76,-131) -> Matrix(281,-16,4338,-247) Matrix(103,-182,30,-53) -> Matrix(35,-2,438,-25) Matrix(157,-286,28,-51) -> Matrix(103,-6,1322,-77) Matrix(209,-468,46,-103) -> Matrix(63,-4,898,-57) Matrix(573,-1300,160,-363) -> Matrix(63,-4,772,-49) Matrix(365,-832,68,-155) -> Matrix(125,-8,1672,-107) Matrix(365,-884,64,-155) -> Matrix(61,-4,778,-51) Matrix(77,-208,10,-27) -> Matrix(1,0,-2,1) Matrix(207,-572,38,-105) -> Matrix(59,-4,782,-53) Matrix(129,-364,28,-79) -> Matrix(59,-4,782,-53) Matrix(77,-260,8,-27) -> Matrix(55,-4,674,-49) Matrix(181,-650,22,-79) -> Matrix(31,-2,388,-25) 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): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 42 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, lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z. ----------------------------------------------------------------------- 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 1/22 7/6 3/61 13/11 1/20 6/5 2/39 11/9 1/20 16/13 4/79 5/4 1/19 9/7 1/20 13/10 1/19 4/3 0/1 11/8 1/19 7/5 1/18 10/7 2/39 13/9 1/19 3/2 1/19 14/9 6/109 39/25 1/18 25/16 3/53 11/7 1/18 8/5 4/73 13/8 1/18 5/3 1/18 12/7 8/141 7/4 5/87 9/5 5/86 11/6 1/17 13/7 1/17 2/1 2/33 13/6 1/16 11/5 1/16 20/9 8/127 9/4 5/79 25/11 13/204 16/7 4/63 7/3 5/78 26/11 2/31 19/8 13/201 12/5 8/123 29/12 7/107 17/7 3/46 5/2 1/15 13/5 1/15 8/3 4/59 19/7 3/44 11/4 1/15 25/9 3/46 39/14 1/15 14/5 6/89 3/1 1/14 13/4 1/14 10/3 2/27 37/11 1/14 27/8 5/69 17/5 3/40 7/2 1/15 25/7 3/40 43/12 1/11 18/5 2/29 11/3 1/14 26/7 2/27 15/4 1/13 4/1 0/1 13/3 1/14 9/2 1/13 32/7 4/55 23/5 3/40 14/3 2/25 5/1 1/14 26/5 2/27 21/4 5/67 16/3 4/53 27/5 7/92 11/2 1/13 17/3 1/12 23/4 1/11 52/9 0/1 29/5 1/16 6/1 2/27 13/2 1/13 7/1 3/38 8/1 4/51 17/2 9/113 26/3 2/25 9/1 5/62 1/0 1/11 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(1,0,11,1) Matrix(157,-182,44,-51) -> Matrix(41,-2,554,-27) Matrix(155,-182,23,-27) -> Matrix(121,-6,1553,-77) Matrix(131,-156,21,-25) -> Matrix(79,-4,1047,-53) Matrix(235,-286,106,-129) -> Matrix(121,-6,1916,-95) Matrix(571,-702,170,-209) -> Matrix(39,-2,566,-29) Matrix(337,-416,64,-79) -> Matrix(157,-8,2100,-107) Matrix(103,-130,42,-53) -> Matrix(37,-2,574,-31) Matrix(181,-234,41,-53) -> Matrix(39,-2,527,-27) Matrix(79,-104,19,-25) -> Matrix(1,0,-5,1) Matrix(77,-104,20,-27) -> Matrix(1,0,-6,1) Matrix(207,-286,76,-105) -> Matrix(39,-2,566,-29) Matrix(129,-182,56,-79) -> Matrix(41,-2,636,-31) Matrix(181,-260,55,-79) -> Matrix(77,-4,1059,-55) Matrix(53,-78,17,-25) -> Matrix(37,-2,537,-29) Matrix(285,-442,118,-183) -> Matrix(183,-10,2800,-153) Matrix(701,-1092,251,-391) -> Matrix(217,-12,3237,-179) Matrix(1249,-1950,449,-701) -> Matrix(107,-6,1623,-91) Matrix(781,-1222,232,-363) -> Matrix(37,-2,500,-27) Matrix(181,-286,50,-79) -> Matrix(37,-2,500,-27) Matrix(129,-208,49,-79) -> Matrix(145,-8,2157,-119) Matrix(79,-130,31,-51) -> Matrix(107,-6,1623,-91) Matrix(77,-130,16,-27) -> Matrix(35,-2,508,-29) Matrix(181,-312,76,-131) -> Matrix(281,-16,4338,-247) Matrix(103,-182,30,-53) -> Matrix(35,-2,438,-25) Matrix(157,-286,28,-51) -> Matrix(103,-6,1322,-77) Matrix(155,-286,71,-131) -> Matrix(307,-18,4895,-287) Matrix(27,-52,13,-25) -> Matrix(67,-4,1089,-65) Matrix(209,-468,46,-103) -> Matrix(63,-4,898,-57) Matrix(573,-1300,160,-363) -> Matrix(63,-4,772,-49) Matrix(365,-832,68,-155) -> Matrix(125,-8,1672,-107) Matrix(287,-676,121,-285) -> Matrix(559,-36,8649,-557) Matrix(365,-884,64,-155) -> Matrix(61,-4,778,-51) Matrix(77,-208,10,-27) -> Matrix(1,0,-2,1) Matrix(207,-572,38,-105) -> Matrix(59,-4,782,-53) Matrix(129,-364,28,-79) -> Matrix(59,-4,782,-53) Matrix(77,-260,8,-27) -> Matrix(55,-4,674,-49) Matrix(181,-650,22,-79) -> Matrix(31,-2,388,-25) Matrix(183,-676,49,-181) -> Matrix(55,-4,729,-53) Matrix(233,-1066,40,-183) -> Matrix(27,-2,392,-29) Matrix(131,-676,25,-129) -> Matrix(163,-12,2187,-161) Matrix(469,-2704,81,-467) -> Matrix(1,0,5,1) Matrix(79,-676,9,-77) -> Matrix(351,-28,4375,-349) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 42 ----------------------------------------------------------------------- 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 11 1 2/1 2/33 1 13 13/6 1/16 11 1 11/5 1/16 1 13 9/4 5/79 1 13 7/3 5/78 1 13 26/11 2/31 9 1 12/5 8/123 1 13 5/2 1/15 1 13 13/5 1/15 7 1 8/3 4/59 1 13 11/4 1/15 1 13 25/9 3/46 1 13 39/14 1/15 9 1 14/5 6/89 1 13 3/1 1/14 1 13 13/4 1/14 3 1 10/3 2/27 1 13 27/8 5/69 1 13 17/5 3/40 1 13 7/2 1/15 1 13 18/5 2/29 1 13 11/3 1/14 1 13 26/7 2/27 1 1 4/1 0/1 1 13 13/3 1/14 1 1 9/2 1/13 1 13 23/5 3/40 1 13 14/3 2/25 1 13 5/1 1/14 1 13 26/5 2/27 3 1 16/3 4/53 1 13 27/5 7/92 1 13 11/2 1/13 1 13 17/3 1/12 1 13 23/4 1/11 1 13 52/9 0/1 5 1 6/1 2/27 1 13 13/2 1/13 5 1 7/1 3/38 1 13 8/1 4/51 1 13 26/3 2/25 7 1 9/1 5/62 1 13 1/0 1/11 1 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(1,0,1,-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(1,0,22,-1) (0/1,1/0) -> (0/1,1/11) Matrix(1,0,1,-1) -> Matrix(1,0,33,-1) (0/1,2/1) -> (0/1,2/33) Matrix(25,-52,12,-25) -> Matrix(65,-4,1056,-65) (2/1,13/6) -> (2/33,1/16) Matrix(131,-286,60,-131) -> Matrix(287,-18,4576,-287) (13/6,11/5) -> (1/16,9/143) Matrix(129,-286,23,-51) -> Matrix(95,-6,1219,-77) Matrix(79,-182,23,-53) -> Matrix(31,-2,387,-25) Matrix(155,-364,66,-155) -> Matrix(311,-20,4836,-311) (7/3,26/11) -> (5/78,2/31) Matrix(131,-312,55,-131) -> Matrix(247,-16,3813,-247) (26/11,12/5) -> (2/31,8/123) Matrix(53,-130,11,-27) -> Matrix(31,-2,449,-29) Matrix(51,-130,20,-51) -> Matrix(91,-6,1380,-91) (5/2,13/5) -> (3/46,1/15) Matrix(79,-208,30,-79) -> Matrix(119,-8,1770,-119) (13/5,8/3) -> (1/15,4/59) Matrix(105,-286,29,-79) -> Matrix(29,-2,391,-27) Matrix(207,-572,38,-105) -> Matrix(59,-4,782,-53) Matrix(701,-1950,252,-701) -> Matrix(91,-6,1380,-91) (25/9,39/14) -> (3/46,1/15) Matrix(391,-1092,140,-391) -> Matrix(179,-12,2670,-179) (39/14,14/5) -> (1/15,6/89) Matrix(129,-364,28,-79) -> Matrix(59,-4,782,-53) Matrix(25,-78,8,-25) -> Matrix(29,-2,420,-29) (3/1,13/4) -> (1/15,1/14) Matrix(79,-260,24,-79) -> Matrix(55,-4,756,-55) (13/4,10/3) -> (1/14,2/27) Matrix(209,-702,39,-131) -> Matrix(29,-2,391,-27) Matrix(77,-260,8,-27) -> Matrix(55,-4,674,-49) Matrix(51,-182,7,-25) -> Matrix(27,-2,337,-25) Matrix(155,-572,42,-155) -> Matrix(55,-4,756,-55) (11/3,26/7) -> (1/14,2/27) Matrix(27,-104,7,-27) -> Matrix(1,0,27,-1) (26/7,4/1) -> (0/1,2/27) Matrix(25,-104,6,-25) -> Matrix(1,0,28,-1) (4/1,13/3) -> (0/1,1/14) Matrix(53,-234,12,-53) -> Matrix(27,-2,364,-27) (13/3,9/2) -> (1/14,1/13) Matrix(131,-598,23,-105) -> Matrix(27,-2,337,-25) Matrix(51,-260,10,-51) -> Matrix(55,-4,756,-55) (5/1,26/5) -> (1/14,2/27) Matrix(79,-416,15,-79) -> Matrix(107,-8,1431,-107) (26/5,16/3) -> (2/27,4/53) Matrix(415,-2392,72,-415) -> Matrix(1,0,22,-1) (23/4,52/9) -> (0/1,1/11) Matrix(53,-312,9,-53) -> Matrix(1,0,27,-1) (52/9,6/1) -> (0/1,2/27) Matrix(25,-156,4,-25) -> Matrix(53,-4,702,-53) (6/1,13/2) -> (2/27,1/13) Matrix(27,-182,4,-27) -> Matrix(77,-6,988,-77) (13/2,7/1) -> (1/13,3/38) Matrix(25,-208,3,-25) -> Matrix(101,-8,1275,-101) (8/1,26/3) -> (4/51,2/25) Matrix(53,-468,6,-53) -> Matrix(249,-20,3100,-249) (26/3,9/1) -> (2/25,5/62) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.