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): 240 Minimal number of generators: 41 Number of equivalence classes of cusps: 8 Genus: 17 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 2/1 19/7 4/1 19/4 38/7 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -7/1 7/228 -6/1 3/95 -5/1 5/152 -14/3 7/209 -9/2 9/266 -13/3 13/380 -4/1 2/57 -7/2 7/190 -3/1 3/76 -8/3 4/95 -5/2 5/114 -7/3 7/152 -9/4 9/190 -20/9 10/209 -11/5 11/228 -2/1 1/19 -7/4 7/114 -19/11 1/16 -12/7 6/95 -5/3 5/76 -8/5 4/57 -19/12 1/14 -30/19 15/209 -11/7 11/152 -3/2 3/38 -10/7 5/57 -7/5 7/76 -4/3 2/19 -13/10 13/114 -9/7 9/76 -14/11 7/57 -19/15 1/8 -5/4 5/38 -16/13 8/57 -27/22 27/190 -38/31 1/7 -11/9 11/76 -6/5 3/19 -7/6 7/38 -1/1 1/0 0/1 0/1 1/1 1/76 7/6 7/494 6/5 3/209 5/4 5/342 14/11 7/475 9/7 9/608 13/10 13/874 4/3 2/133 7/5 7/456 3/2 3/190 8/5 4/247 5/3 5/304 7/4 7/418 9/5 9/532 20/11 10/589 11/6 11/646 2/1 1/57 7/3 7/380 19/8 1/54 12/5 6/323 5/2 5/266 8/3 4/209 19/7 1/52 30/11 15/779 11/4 11/570 3/1 3/152 10/3 5/247 7/2 7/342 4/1 2/95 13/3 13/608 9/2 9/418 14/3 7/323 19/4 1/46 5/1 5/228 16/3 8/361 27/5 27/1216 38/7 1/45 11/2 11/494 6/1 3/133 7/1 7/304 1/0 1/38 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(9,76,2,17) (-7/1,1/0) -> (13/3,9/2) Hyperbolic Matrix(23,152,18,119) (-7/1,-6/1) -> (14/11,9/7) Hyperbolic Matrix(13,76,-6,-35) (-6/1,-5/1) -> (-11/5,-2/1) Hyperbolic Matrix(63,304,-40,-193) (-5/1,-14/3) -> (-30/19,-11/7) Hyperbolic Matrix(33,152,28,129) (-14/3,-9/2) -> (7/6,6/5) Hyperbolic Matrix(17,76,2,9) (-9/2,-13/3) -> (7/1,1/0) Hyperbolic Matrix(71,304,-32,-137) (-13/3,-4/1) -> (-20/9,-11/5) Hyperbolic Matrix(21,76,8,29) (-4/1,-7/2) -> (5/2,8/3) Hyperbolic Matrix(23,76,-10,-33) (-7/2,-3/1) -> (-7/3,-9/4) Hyperbolic Matrix(27,76,-16,-45) (-3/1,-8/3) -> (-12/7,-5/3) Hyperbolic Matrix(29,76,8,21) (-8/3,-5/2) -> (7/2,4/1) Hyperbolic Matrix(31,76,-20,-49) (-5/2,-7/3) -> (-11/7,-3/2) Hyperbolic Matrix(307,684,-250,-557) (-9/4,-20/9) -> (-16/13,-27/22) Hyperbolic Matrix(43,76,-30,-53) (-2/1,-7/4) -> (-3/2,-10/7) Hyperbolic Matrix(131,228,-104,-181) (-7/4,-19/11) -> (-19/15,-5/4) Hyperbolic Matrix(221,380,82,141) (-19/11,-12/7) -> (8/3,19/7) Hyperbolic Matrix(47,76,34,55) (-5/3,-8/5) -> (4/3,7/5) Hyperbolic Matrix(239,380,100,159) (-8/5,-19/12) -> (19/8,12/5) Hyperbolic Matrix(529,836,112,177) (-19/12,-30/19) -> (14/3,19/4) Hyperbolic Matrix(107,152,-88,-125) (-10/7,-7/5) -> (-11/9,-6/5) Hyperbolic Matrix(55,76,34,47) (-7/5,-4/3) -> (8/5,5/3) Hyperbolic Matrix(175,228,-142,-185) (-4/3,-13/10) -> (-5/4,-16/13) Hyperbolic Matrix(59,76,52,67) (-13/10,-9/7) -> (1/1,7/6) Hyperbolic Matrix(119,152,18,23) (-9/7,-14/11) -> (6/1,7/1) Hyperbolic Matrix(659,836,242,307) (-14/11,-19/15) -> (19/7,30/11) Hyperbolic Matrix(1177,1444,216,265) (-27/22,-38/31) -> (38/7,11/2) Hyperbolic Matrix(1179,1444,218,267) (-38/31,-11/9) -> (27/5,38/7) Hyperbolic Matrix(129,152,28,33) (-6/5,-7/6) -> (9/2,14/3) Hyperbolic Matrix(67,76,52,59) (-7/6,-1/1) -> (9/7,13/10) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(63,-76,34,-41) (6/5,5/4) -> (11/6,2/1) Hyperbolic Matrix(241,-304,88,-111) (5/4,14/11) -> (30/11,11/4) Hyperbolic Matrix(233,-304,128,-167) (13/10,4/3) -> (20/11,11/6) Hyperbolic Matrix(53,-76,30,-43) (7/5,3/2) -> (7/4,9/5) Hyperbolic Matrix(49,-76,20,-31) (3/2,8/5) -> (12/5,5/2) Hyperbolic Matrix(45,-76,16,-27) (5/3,7/4) -> (11/4,3/1) Hyperbolic Matrix(377,-684,70,-127) (9/5,20/11) -> (16/3,27/5) Hyperbolic Matrix(33,-76,10,-23) (2/1,7/3) -> (3/1,10/3) Hyperbolic Matrix(97,-228,20,-47) (7/3,19/8) -> (19/4,5/1) Hyperbolic Matrix(45,-152,8,-27) (10/3,7/2) -> (11/2,6/1) Hyperbolic Matrix(53,-228,10,-43) (4/1,13/3) -> (5/1,16/3) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(9,76,2,17) -> Matrix(67,-2,3116,-93) Matrix(23,152,18,119) -> Matrix(129,-4,8740,-271) Matrix(13,76,-6,-35) -> Matrix(63,-2,1292,-41) Matrix(63,304,-40,-193) -> Matrix(241,-8,3344,-111) Matrix(33,152,28,129) -> Matrix(119,-4,8360,-281) Matrix(17,76,2,9) -> Matrix(59,-2,2508,-85) Matrix(71,304,-32,-137) -> Matrix(233,-8,4864,-167) Matrix(21,76,8,29) -> Matrix(55,-2,2888,-105) Matrix(23,76,-10,-33) -> Matrix(53,-2,1140,-43) Matrix(27,76,-16,-45) -> Matrix(49,-2,760,-31) Matrix(29,76,8,21) -> Matrix(47,-2,2280,-97) Matrix(31,76,-20,-49) -> Matrix(45,-2,608,-27) Matrix(307,684,-250,-557) -> Matrix(377,-18,2660,-127) Matrix(43,76,-30,-53) -> Matrix(33,-2,380,-23) Matrix(131,228,-104,-181) -> Matrix(97,-6,760,-47) Matrix(221,380,82,141) -> Matrix(159,-10,8284,-521) Matrix(47,76,34,55) -> Matrix(29,-2,1900,-131) Matrix(239,380,100,159) -> Matrix(141,-10,7600,-539) Matrix(529,836,112,177) -> Matrix(307,-22,14136,-1013) Matrix(107,152,-88,-125) -> Matrix(45,-4,304,-27) Matrix(55,76,34,47) -> Matrix(21,-2,1292,-123) Matrix(175,228,-142,-185) -> Matrix(53,-6,380,-43) Matrix(59,76,52,67) -> Matrix(17,-2,1216,-143) Matrix(119,152,18,23) -> Matrix(33,-4,1444,-175) Matrix(659,836,242,307) -> Matrix(177,-22,9196,-1143) Matrix(1177,1444,216,265) -> Matrix(267,-38,12008,-1709) Matrix(1179,1444,218,267) -> Matrix(265,-38,11932,-1711) Matrix(129,152,28,33) -> Matrix(23,-4,1064,-185) Matrix(67,76,52,59) -> Matrix(9,-2,608,-135) Matrix(1,0,2,1) -> Matrix(1,0,76,1) Matrix(63,-76,34,-41) -> Matrix(139,-2,8132,-117) Matrix(241,-304,88,-111) -> Matrix(545,-8,28272,-415) Matrix(233,-304,128,-167) -> Matrix(537,-8,31616,-471) Matrix(53,-76,30,-43) -> Matrix(129,-2,7676,-119) Matrix(49,-76,20,-31) -> Matrix(125,-2,6688,-107) Matrix(45,-76,16,-27) -> Matrix(121,-2,6232,-103) Matrix(377,-684,70,-127) -> Matrix(1061,-18,47804,-811) Matrix(33,-76,10,-23) -> Matrix(109,-2,5396,-99) Matrix(97,-228,20,-47) -> Matrix(325,-6,14896,-275) Matrix(45,-152,8,-27) -> Matrix(197,-4,8816,-179) Matrix(53,-228,10,-43) -> Matrix(281,-6,12692,-271) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 240 Minimal number of generators: 41 Number of equivalence classes of cusps: 8 Genus: 17 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 40 Degree of the the map Y: 40 Permutation triple for Y: ((2,6,19,17,27,37,34,31,13,4,3,12,26,25,36,33,24,22,7)(5,18,35,38,23,8,11,29,21,10,9,28,39,32,16,15,20,30,14); (1,4,16,34,18,25,38,24,23,37,20,22,8,7,21,6,15,31,39,40,35,19,14,13,11,3,10,12,30,27,9,26,28,33,32,17,5,2)(29,36); (1,2,8,24,28,27,32,31,16,33,29,13,15,4,14,12,11,22,38,40,39,26,10,7,20,6,5,19,21,36,18,17,35,34,23,25,9,3)(30,37)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1, lambda2, lambda1+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): 60 Minimal number of generators: 11 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: 4 Genus: 4 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 19/7 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/76 5/4 5/342 4/3 2/133 7/5 7/456 3/2 3/190 8/5 4/247 5/3 5/304 7/4 7/418 2/1 1/57 7/3 7/380 19/8 1/54 12/5 6/323 5/2 5/266 8/3 4/209 19/7 1/52 11/4 11/570 3/1 3/152 7/2 7/342 4/1 2/95 5/1 5/228 1/0 1/38 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(31,-38,9,-11) (1/1,5/4) -> (3/1,7/2) Hyperbolic Matrix(29,-38,13,-17) (5/4,4/3) -> (2/1,7/3) Hyperbolic Matrix(55,-76,21,-29) (4/3,7/5) -> (5/2,8/3) Hyperbolic Matrix(27,-38,5,-7) (7/5,3/2) -> (5/1,1/0) Hyperbolic Matrix(49,-76,20,-31) (3/2,8/5) -> (12/5,5/2) Hyperbolic Matrix(47,-76,13,-21) (8/5,5/3) -> (7/2,4/1) Hyperbolic Matrix(45,-76,16,-27) (5/3,7/4) -> (11/4,3/1) Hyperbolic Matrix(21,-38,5,-9) (7/4,2/1) -> (4/1,5/1) Hyperbolic Matrix(145,-342,53,-125) (7/3,19/8) -> (19/7,11/4) Hyperbolic Matrix(159,-380,59,-141) (19/8,12/5) -> (8/3,19/7) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,38,1) Matrix(31,-38,9,-11) -> Matrix(69,-1,3382,-49) Matrix(29,-38,13,-17) -> Matrix(67,-1,3686,-55) Matrix(55,-76,21,-29) -> Matrix(131,-2,6878,-105) Matrix(27,-38,5,-7) -> Matrix(65,-1,2926,-45) Matrix(49,-76,20,-31) -> Matrix(125,-2,6688,-107) Matrix(47,-76,13,-21) -> Matrix(123,-2,5966,-97) Matrix(45,-76,16,-27) -> Matrix(121,-2,6232,-103) Matrix(21,-38,5,-9) -> Matrix(59,-1,2774,-47) Matrix(145,-342,53,-125) -> Matrix(487,-9,25270,-467) Matrix(159,-380,59,-141) -> Matrix(539,-10,28082,-521) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 60 Minimal number of generators: 11 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: 4 Genus: 4 Degree of H/liftables -> H/(image of liftables): 1 ----------------------------------------------------------------------- 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 38 1 2/1 1/57 2 19 7/3 7/380 1 38 5/2 5/266 1 38 8/3 4/209 2 19 19/7 1/52 19 2 11/4 11/570 1 38 3/1 3/152 1 38 7/2 7/342 1 38 4/1 2/95 2 19 5/1 5/228 1 38 1/0 1/38 1 38 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(17,-38,4,-9) (2/1,7/3) -> (4/1,5/1) Glide Reflection Matrix(31,-76,11,-27) (7/3,5/2) -> (11/4,3/1) Glide Reflection Matrix(29,-76,8,-21) (5/2,8/3) -> (7/2,4/1) Glide Reflection Matrix(113,-304,42,-113) (8/3,19/7) -> (8/3,19/7) Reflection Matrix(153,-418,56,-153) (19/7,11/4) -> (19/7,11/4) Reflection Matrix(11,-38,2,-7) (3/1,7/2) -> (5/1,1/0) Glide Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,76,-1) (0/1,1/0) -> (0/1,1/38) Matrix(1,0,1,-1) -> Matrix(1,0,114,-1) (0/1,2/1) -> (0/1,1/57) Matrix(17,-38,4,-9) -> Matrix(55,-1,2584,-47) Matrix(31,-76,11,-27) -> Matrix(107,-2,5510,-103) Matrix(29,-76,8,-21) -> Matrix(105,-2,5092,-97) Matrix(113,-304,42,-113) -> Matrix(417,-8,21736,-417) (8/3,19/7) -> (4/209,1/52) Matrix(153,-418,56,-153) -> Matrix(571,-11,29640,-571) (19/7,11/4) -> (1/52,11/570) Matrix(11,-38,2,-7) -> Matrix(49,-1,2204,-45) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.