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): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 12 Genus: 19 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 0/1 1/1 3/2 24/11 7/3 5/2 27/8 11/3 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 1/70 -4/9 1/63 -7/16 9/560 -3/7 4/245 -5/12 1/60 -2/5 3/175 -7/18 11/630 -5/13 8/455 -3/8 1/56 -1/3 2/105 -2/7 1/49 -3/11 8/385 -4/15 11/525 -1/4 3/140 -2/9 1/45 -1/5 4/175 -2/11 9/385 -1/6 1/42 0/1 1/35 1/4 1/28 3/11 2/55 5/18 23/630 2/7 9/245 3/10 13/350 1/3 4/105 4/11 3/77 3/8 11/280 2/5 1/25 5/12 17/420 3/7 2/49 4/9 13/315 5/11 16/385 1/2 3/70 5/9 2/45 9/16 5/112 4/7 11/245 11/19 6/133 7/12 19/420 3/5 8/175 2/3 1/21 5/7 12/245 3/4 1/20 4/5 9/175 9/11 4/77 5/6 11/210 6/7 13/245 1/1 2/35 7/6 13/210 13/11 24/385 6/5 11/175 5/4 9/140 4/3 1/15 11/8 19/280 7/5 12/175 3/2 1/14 8/5 13/175 13/8 3/40 5/3 8/105 22/13 1/13 17/10 27/350 12/7 19/245 7/4 11/140 9/5 2/25 11/6 17/210 2/1 3/35 13/6 19/210 24/11 1/11 11/5 16/175 9/4 13/140 16/7 23/245 23/10 33/350 7/3 2/21 19/8 27/280 12/5 17/175 5/2 1/10 13/5 18/175 21/8 29/280 29/11 8/77 8/3 11/105 11/4 3/28 3/1 4/35 13/4 17/140 23/7 6/49 10/3 13/105 27/8 1/8 44/13 57/455 17/5 22/175 7/2 9/70 18/5 23/175 47/13 12/91 29/8 37/280 11/3 2/15 15/4 19/140 4/1 1/7 5/1 6/35 6/1 1/5 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(35,16,94,43) (-1/2,-4/9) -> (4/11,3/8) Hyperbolic Matrix(199,88,52,23) (-4/9,-7/16) -> (15/4,4/1) Hyperbolic Matrix(65,28,58,25) (-7/16,-3/7) -> (1/1,7/6) Hyperbolic Matrix(151,64,92,39) (-3/7,-5/12) -> (13/8,5/3) Hyperbolic Matrix(169,70,70,29) (-5/12,-2/5) -> (12/5,5/2) Hyperbolic Matrix(359,140,100,39) (-2/5,-7/18) -> (7/2,18/5) Hyperbolic Matrix(315,122,142,55) (-7/18,-5/13) -> (11/5,9/4) Hyperbolic Matrix(167,64,60,23) (-5/13,-3/8) -> (11/4,3/1) Hyperbolic Matrix(65,24,46,17) (-3/8,-1/3) -> (7/5,3/2) Hyperbolic Matrix(41,12,58,17) (-1/3,-2/7) -> (2/3,5/7) Hyperbolic Matrix(51,14,142,39) (-2/7,-3/11) -> (1/3,4/11) Hyperbolic Matrix(119,32,264,71) (-3/11,-4/15) -> (4/9,5/11) Hyperbolic Matrix(83,22,298,79) (-4/15,-1/4) -> (5/18,2/7) Hyperbolic Matrix(53,12,128,29) (-1/4,-2/9) -> (2/5,5/12) Hyperbolic Matrix(75,16,14,3) (-2/9,-1/5) -> (5/1,6/1) Hyperbolic Matrix(41,8,46,9) (-1/5,-2/11) -> (6/7,1/1) Hyperbolic Matrix(123,22,218,39) (-2/11,-1/6) -> (9/16,4/7) Hyperbolic Matrix(59,8,22,3) (-1/6,0/1) -> (8/3,11/4) Hyperbolic Matrix(35,-8,22,-5) (0/1,1/4) -> (3/2,8/5) Hyperbolic Matrix(119,-32,212,-57) (1/4,3/11) -> (5/9,9/16) Hyperbolic Matrix(517,-142,142,-39) (3/11,5/18) -> (29/8,11/3) Hyperbolic Matrix(239,-70,140,-41) (2/7,3/10) -> (17/10,12/7) Hyperbolic Matrix(193,-60,74,-23) (3/10,1/3) -> (13/5,21/8) Hyperbolic Matrix(87,-34,64,-25) (3/8,2/5) -> (4/3,11/8) Hyperbolic Matrix(181,-76,312,-131) (5/12,3/7) -> (11/19,7/12) Hyperbolic Matrix(109,-48,134,-59) (3/7,4/9) -> (4/5,9/11) Hyperbolic Matrix(103,-48,88,-41) (5/11,1/2) -> (7/6,13/11) Hyperbolic Matrix(117,-64,64,-35) (1/2,5/9) -> (9/5,11/6) Hyperbolic Matrix(355,-204,134,-77) (4/7,11/19) -> (29/11,8/3) Hyperbolic Matrix(239,-140,70,-41) (7/12,3/5) -> (17/5,7/2) Hyperbolic Matrix(35,-22,8,-5) (3/5,2/3) -> (4/1,5/1) Hyperbolic Matrix(87,-64,34,-25) (5/7,3/4) -> (5/2,13/5) Hyperbolic Matrix(97,-76,60,-47) (3/4,4/5) -> (8/5,13/8) Hyperbolic Matrix(295,-242,128,-105) (9/11,5/6) -> (23/10,7/3) Hyperbolic Matrix(103,-88,48,-41) (5/6,6/7) -> (2/1,13/6) Hyperbolic Matrix(569,-674,168,-199) (13/11,6/5) -> (44/13,17/5) Hyperbolic Matrix(109,-134,48,-59) (6/5,5/4) -> (9/4,16/7) Hyperbolic Matrix(27,-34,4,-5) (5/4,4/3) -> (6/1,1/0) Hyperbolic Matrix(89,-124,28,-39) (11/8,7/5) -> (3/1,13/4) Hyperbolic Matrix(215,-362,98,-165) (5/3,22/13) -> (24/11,11/5) Hyperbolic Matrix(409,-694,188,-319) (22/13,17/10) -> (13/6,24/11) Hyperbolic Matrix(181,-312,76,-131) (12/7,7/4) -> (19/8,12/5) Hyperbolic Matrix(119,-212,32,-57) (7/4,9/5) -> (11/3,15/4) Hyperbolic Matrix(119,-222,52,-97) (11/6,2/1) -> (16/7,23/10) Hyperbolic Matrix(223,-528,68,-161) (7/3,19/8) -> (13/4,23/7) Hyperbolic Matrix(695,-1828,192,-505) (21/8,29/11) -> (47/13,29/8) Hyperbolic Matrix(267,-884,74,-245) (23/7,10/3) -> (18/5,47/13) Hyperbolic Matrix(433,-1458,128,-431) (10/3,27/8) -> (27/8,44/13) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,70,1) Matrix(35,16,94,43) -> Matrix(129,-2,3290,-51) Matrix(199,88,52,23) -> Matrix(251,-4,1820,-29) Matrix(65,28,58,25) -> Matrix(123,-2,2030,-33) Matrix(151,64,92,39) -> Matrix(243,-4,3220,-53) Matrix(169,70,70,29) -> Matrix(239,-4,2450,-41) Matrix(359,140,100,39) -> Matrix(459,-8,3500,-61) Matrix(315,122,142,55) -> Matrix(457,-8,4970,-87) Matrix(167,64,60,23) -> Matrix(227,-4,2100,-37) Matrix(65,24,46,17) -> Matrix(111,-2,1610,-29) Matrix(41,12,58,17) -> Matrix(99,-2,2030,-41) Matrix(51,14,142,39) -> Matrix(193,-4,4970,-103) Matrix(119,32,264,71) -> Matrix(383,-8,9240,-193) Matrix(83,22,298,79) -> Matrix(381,-8,10430,-219) Matrix(53,12,128,29) -> Matrix(181,-4,4480,-99) Matrix(75,16,14,3) -> Matrix(89,-2,490,-11) Matrix(41,8,46,9) -> Matrix(87,-2,1610,-37) Matrix(123,22,218,39) -> Matrix(341,-8,7630,-179) Matrix(59,8,22,3) -> Matrix(81,-2,770,-19) Matrix(35,-8,22,-5) -> Matrix(57,-2,770,-27) Matrix(119,-32,212,-57) -> Matrix(331,-12,7420,-269) Matrix(517,-142,142,-39) -> Matrix(659,-24,4970,-181) Matrix(239,-70,140,-41) -> Matrix(379,-14,4900,-181) Matrix(193,-60,74,-23) -> Matrix(267,-10,2590,-97) Matrix(87,-34,64,-25) -> Matrix(151,-6,2240,-89) Matrix(181,-76,312,-131) -> Matrix(493,-20,10920,-443) Matrix(109,-48,134,-59) -> Matrix(243,-10,4690,-193) Matrix(103,-48,88,-41) -> Matrix(191,-8,3080,-129) Matrix(117,-64,64,-35) -> Matrix(181,-8,2240,-99) Matrix(355,-204,134,-77) -> Matrix(489,-22,4690,-211) Matrix(239,-140,70,-41) -> Matrix(309,-14,2450,-111) Matrix(35,-22,8,-5) -> Matrix(43,-2,280,-13) Matrix(87,-64,34,-25) -> Matrix(121,-6,1190,-59) Matrix(97,-76,60,-47) -> Matrix(157,-8,2100,-107) Matrix(295,-242,128,-105) -> Matrix(423,-22,4480,-233) Matrix(103,-88,48,-41) -> Matrix(151,-8,1680,-89) Matrix(569,-674,168,-199) -> Matrix(737,-46,5880,-367) Matrix(109,-134,48,-59) -> Matrix(157,-10,1680,-107) Matrix(27,-34,4,-5) -> Matrix(31,-2,140,-9) Matrix(89,-124,28,-39) -> Matrix(117,-8,980,-67) Matrix(215,-362,98,-165) -> Matrix(313,-24,3430,-263) Matrix(409,-694,188,-319) -> Matrix(597,-46,6580,-507) Matrix(181,-312,76,-131) -> Matrix(257,-20,2660,-207) Matrix(119,-212,32,-57) -> Matrix(151,-12,1120,-89) Matrix(119,-222,52,-97) -> Matrix(171,-14,1820,-149) Matrix(223,-528,68,-161) -> Matrix(291,-28,2380,-229) Matrix(695,-1828,192,-505) -> Matrix(887,-92,6720,-697) Matrix(267,-884,74,-245) -> Matrix(341,-42,2590,-319) Matrix(433,-1458,128,-431) -> Matrix(561,-70,4480,-559) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 12 Genus: 19 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 48 Degree of the the map Y: 48 Permutation triple for Y: ((1,6,14,17,41,44,45,22,38,34,27,8,26,30,46,32,31,47,39,35,37,13,4,3,12,36,40,43,19,42,33,11,23,7,2)(5,18,25,28,29,10,9)(15,21,20,24,16); (1,4,16,39,28,27,14,13,38,48,36,26,11,3,10,32,15,8,7,18,33,22,6,21,25,44,47,31,19,9,24,23,40,17,5)(12,34,46,42,20,41,35)(29,37,43,45,30); (1,2,8,28,21,32,34,13,29,39,44,20,6,5,19,37,41,40,48,38,33,46,45,25,7,24,42,31,10,30,36,35,16,9,3)(4,14,22,43,23,26,15)(11,18,17,27,12)) ----------------------------------------------------------------------- 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, lambda1, lambda2, lambda1+lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z+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): 48 Minimal number of generators: 9 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: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 5/2 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 0/1 1/35 1/3 4/105 2/5 1/25 1/2 3/70 3/5 8/175 2/3 1/21 3/4 1/20 1/1 2/35 4/3 1/15 3/2 1/14 2/1 3/35 5/2 1/10 8/3 11/105 3/1 4/35 4/1 1/7 5/1 6/35 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(0,-1,1,2) (-1/1,1/0) -> (-1/1,0/1) Parabolic Matrix(10,-3,17,-5) (0/1,1/3) -> (1/2,3/5) Hyperbolic Matrix(55,-21,21,-8) (1/3,2/5) -> (5/2,8/3) Hyperbolic Matrix(13,-6,11,-5) (2/5,1/2) -> (1/1,4/3) Hyperbolic Matrix(35,-22,8,-5) (3/5,2/3) -> (4/1,5/1) Hyperbolic Matrix(24,-17,17,-12) (2/3,3/4) -> (4/3,3/2) Hyperbolic Matrix(13,-11,6,-5) (3/4,1/1) -> (2/1,5/2) Hyperbolic Matrix(10,-17,3,-5) (3/2,2/1) -> (3/1,4/1) Hyperbolic Matrix(16,-43,3,-8) (8/3,3/1) -> (5/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(0,-1,1,2) -> Matrix(1,0,35,1) Matrix(10,-3,17,-5) -> Matrix(27,-1,595,-22) Matrix(55,-21,21,-8) -> Matrix(76,-3,735,-29) Matrix(13,-6,11,-5) -> Matrix(24,-1,385,-16) Matrix(35,-22,8,-5) -> Matrix(43,-2,280,-13) Matrix(24,-17,17,-12) -> Matrix(41,-2,595,-29) Matrix(13,-11,6,-5) -> Matrix(19,-1,210,-11) Matrix(10,-17,3,-5) -> Matrix(13,-1,105,-8) Matrix(16,-43,3,-8) -> Matrix(19,-2,105,-11) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 48 Minimal number of generators: 9 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: 3 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 -1/1 0/1 35 1 1/1 2/35 1 35 4/3 1/15 7 5 3/2 1/14 5 7 2/1 3/35 1 35 5/2 1/10 7 5 3/1 4/35 1 35 4/1 1/7 5 7 1/0 1/0 1 35 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(11,-13,5,-6) (1/1,4/3) -> (2/1,5/2) Glide Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(10,-17,3,-5) (3/2,2/1) -> (3/1,4/1) Hyperbolic Matrix(21,-55,8,-21) (5/2,11/4) -> (5/2,11/4) Reflection Matrix(8,-23,1,-3) (8/3,3/1) -> (5/1,1/0) Glide Reflection Matrix(5,-24,1,-5) (4/1,6/1) -> (4/1,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(0,1,1,0) -> Matrix(1,0,35,-1) (-1/1,1/1) -> (0/1,2/35) Matrix(11,-13,5,-6) -> Matrix(16,-1,175,-11) Matrix(17,-24,12,-17) -> Matrix(29,-2,420,-29) (4/3,3/2) -> (1/15,1/14) Matrix(10,-17,3,-5) -> Matrix(13,-1,105,-8) Matrix(21,-55,8,-21) -> Matrix(29,-3,280,-29) (5/2,11/4) -> (1/10,3/28) Matrix(8,-23,1,-3) -> Matrix(9,-1,35,-4) Matrix(5,-24,1,-5) -> Matrix(6,-1,35,-6) (4/1,6/1) -> (1/7,1/5) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.