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 -13/35 -8/35 0/1 1/5 3/14 2/7 3/10 2/5 3/7 1/2 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/0 -4/5 -1/7 -11/14 -2/15 -7/9 -9/70 -10/13 -13/105 -3/4 -4/35 -8/11 -11/105 -5/7 -1/10 -7/10 -2/21 -9/13 -13/140 -2/3 -3/35 -9/14 -2/25 -7/11 -11/140 -12/19 -19/245 -5/8 -8/105 -3/5 -1/14 -4/7 -1/15 -5/9 -9/140 -11/20 -4/63 -6/11 -11/175 -1/2 -2/35 -5/11 -11/210 -4/9 -9/175 -3/7 -1/20 -2/5 -1/21 -5/13 -13/280 -3/8 -8/175 -13/35 -1/22 -10/27 -27/595 -7/19 -19/420 -4/11 -11/245 -5/14 -2/45 -6/17 -17/385 -1/3 -3/70 -4/13 -13/315 -7/23 -23/560 -10/33 -33/805 -3/10 -2/49 -2/7 -1/25 -3/11 -11/280 -1/4 -4/105 -3/13 -13/350 -8/35 -1/27 -5/22 -22/595 -2/9 -9/245 -3/14 -2/55 -1/5 -1/28 -1/6 -6/175 0/1 -1/35 1/5 -1/42 3/14 -2/85 2/9 -9/385 3/13 -13/560 1/4 -4/175 3/11 -11/490 2/7 -1/45 3/10 -2/91 4/13 -13/595 1/3 -3/140 5/14 -2/95 4/11 -11/525 7/19 -19/910 3/8 -8/385 2/5 -1/49 3/7 -1/50 4/9 -9/455 9/20 -4/203 5/11 -11/560 1/2 -2/105 6/11 -11/595 5/9 -9/490 4/7 -1/55 3/5 -1/56 8/13 -13/735 5/8 -8/455 22/35 -1/57 17/27 -27/1540 12/19 -19/1085 7/11 -11/630 9/14 -2/115 11/17 -17/980 2/3 -3/175 9/13 -13/770 16/23 -23/1365 23/33 -33/1960 7/10 -2/119 5/7 -1/60 8/11 -11/665 3/4 -4/245 10/13 -13/805 27/35 -1/62 17/22 -22/1365 7/9 -9/560 11/14 -2/125 4/5 -1/63 5/6 -6/385 1/1 -1/70 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,-70,-57) (-1/1,-4/5) -> (-2/5,-5/13) Hyperbolic Matrix(111,88,140,111) (-4/5,-11/14) -> (11/14,4/5) Hyperbolic Matrix(179,140,280,219) (-11/14,-7/9) -> (7/11,9/14) Hyperbolic Matrix(181,140,-490,-379) (-7/9,-10/13) -> (-10/27,-7/19) Hyperbolic Matrix(37,28,70,53) (-10/13,-3/4) -> (1/2,6/11) Hyperbolic Matrix(87,64,140,103) (-3/4,-8/11) -> (8/13,5/8) Hyperbolic Matrix(61,44,140,101) (-8/11,-5/7) -> (3/7,4/9) Hyperbolic Matrix(99,70,140,99) (-5/7,-7/10) -> (7/10,5/7) Hyperbolic Matrix(193,134,-350,-243) (-7/10,-9/13) -> (-5/9,-11/20) Hyperbolic Matrix(177,122,280,193) (-9/13,-2/3) -> (12/19,7/11) Hyperbolic Matrix(99,64,-280,-181) (-2/3,-9/14) -> (-5/14,-6/17) Hyperbolic Matrix(219,140,280,179) (-9/14,-7/11) -> (7/9,11/14) Hyperbolic Matrix(193,122,280,177) (-7/11,-12/19) -> (2/3,9/13) Hyperbolic Matrix(111,70,-490,-309) (-12/19,-5/8) -> (-5/22,-2/9) Hyperbolic Matrix(13,8,-70,-43) (-5/8,-3/5) -> (-1/5,-1/6) Hyperbolic Matrix(41,24,70,41) (-3/5,-4/7) -> (4/7,3/5) Hyperbolic Matrix(39,22,140,79) (-4/7,-5/9) -> (3/11,2/7) Hyperbolic Matrix(233,128,-770,-423) (-11/20,-6/11) -> (-10/33,-3/10) Hyperbolic Matrix(53,28,70,37) (-6/11,-1/2) -> (3/4,10/13) Hyperbolic Matrix(17,8,70,33) (-1/2,-5/11) -> (3/13,1/4) Hyperbolic Matrix(107,48,-350,-157) (-5/11,-4/9) -> (-4/13,-7/23) Hyperbolic Matrix(101,44,140,61) (-4/9,-3/7) -> (5/7,8/11) Hyperbolic Matrix(29,12,70,29) (-3/7,-2/5) -> (2/5,3/7) Hyperbolic Matrix(37,14,140,53) (-5/13,-3/8) -> (1/4,3/11) Hyperbolic Matrix(811,302,1050,391) (-3/8,-13/35) -> (27/35,17/22) Hyperbolic Matrix(1079,400,1400,519) (-13/35,-10/27) -> (10/13,27/35) Hyperbolic Matrix(87,32,280,103) (-7/19,-4/11) -> (4/13,1/3) Hyperbolic Matrix(61,22,280,101) (-4/11,-5/14) -> (3/14,2/9) Hyperbolic Matrix(149,52,-490,-171) (-6/17,-1/3) -> (-7/23,-10/33) Hyperbolic Matrix(103,32,280,87) (-1/3,-4/13) -> (4/11,7/19) Hyperbolic Matrix(41,12,140,41) (-3/10,-2/7) -> (2/7,3/10) Hyperbolic Matrix(79,22,140,39) (-2/7,-3/11) -> (5/9,4/7) Hyperbolic Matrix(59,16,70,19) (-3/11,-1/4) -> (5/6,1/1) Hyperbolic Matrix(33,8,70,17) (-1/4,-3/13) -> (5/11,1/2) Hyperbolic Matrix(881,202,1400,321) (-3/13,-8/35) -> (22/35,17/27) Hyperbolic Matrix(659,150,1050,239) (-8/35,-5/22) -> (5/8,22/35) Hyperbolic Matrix(101,22,280,61) (-2/9,-3/14) -> (5/14,4/11) Hyperbolic Matrix(29,6,140,29) (-3/14,-1/5) -> (1/5,3/14) Hyperbolic Matrix(51,8,70,11) (-1/6,0/1) -> (8/11,3/4) Hyperbolic Matrix(43,-8,70,-13) (0/1,1/5) -> (3/5,8/13) Hyperbolic Matrix(309,-70,490,-111) (2/9,3/13) -> (17/27,12/19) Hyperbolic Matrix(157,-48,350,-107) (3/10,4/13) -> (4/9,9/20) Hyperbolic Matrix(181,-64,280,-99) (1/3,5/14) -> (9/14,11/17) Hyperbolic Matrix(379,-140,490,-181) (7/19,3/8) -> (17/22,7/9) Hyperbolic Matrix(57,-22,70,-27) (3/8,2/5) -> (4/5,5/6) Hyperbolic Matrix(537,-242,770,-347) (9/20,5/11) -> (23/33,7/10) Hyperbolic Matrix(243,-134,350,-193) (6/11,5/9) -> (9/13,16/23) Hyperbolic Matrix(341,-222,490,-319) (11/17,2/3) -> (16/23,23/33) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-70,1) Matrix(27,22,-70,-57) -> Matrix(13,2,-280,-43) Matrix(111,88,140,111) -> Matrix(29,4,-1820,-251) Matrix(179,140,280,219) -> Matrix(61,8,-3500,-459) Matrix(181,140,-490,-379) -> Matrix(111,14,-2450,-309) Matrix(37,28,70,53) -> Matrix(17,2,-910,-107) Matrix(87,64,140,103) -> Matrix(37,4,-2100,-227) Matrix(61,44,140,101) -> Matrix(39,4,-1960,-201) Matrix(99,70,140,99) -> Matrix(41,4,-2450,-239) Matrix(193,134,-350,-243) -> Matrix(107,10,-1680,-157) Matrix(177,122,280,193) -> Matrix(87,8,-4970,-457) Matrix(99,64,-280,-181) -> Matrix(99,8,-2240,-181) Matrix(219,140,280,179) -> Matrix(101,8,-6300,-499) Matrix(193,122,280,177) -> Matrix(103,8,-6090,-473) Matrix(111,70,-490,-309) -> Matrix(181,14,-4900,-379) Matrix(13,8,-70,-43) -> Matrix(27,2,-770,-57) Matrix(41,24,70,41) -> Matrix(29,2,-1610,-111) Matrix(39,22,140,79) -> Matrix(61,4,-2730,-179) Matrix(233,128,-770,-423) -> Matrix(347,22,-8470,-537) Matrix(53,28,70,37) -> Matrix(33,2,-2030,-123) Matrix(17,8,70,33) -> Matrix(37,2,-1610,-87) Matrix(107,48,-350,-157) -> Matrix(193,10,-4690,-243) Matrix(101,44,140,61) -> Matrix(79,4,-4760,-241) Matrix(29,12,70,29) -> Matrix(41,2,-2030,-99) Matrix(37,14,140,53) -> Matrix(87,4,-3850,-177) Matrix(811,302,1050,391) -> Matrix(659,30,-40880,-1861) Matrix(1079,400,1400,519) -> Matrix(881,40,-54600,-2479) Matrix(87,32,280,103) -> Matrix(177,8,-8120,-367) Matrix(61,22,280,101) -> Matrix(179,8,-7630,-341) Matrix(149,52,-490,-171) -> Matrix(319,14,-7770,-341) Matrix(103,32,280,87) -> Matrix(193,8,-9240,-383) Matrix(41,12,140,41) -> Matrix(99,4,-4480,-181) Matrix(79,22,140,39) -> Matrix(101,4,-5530,-219) Matrix(59,16,70,19) -> Matrix(51,2,-3290,-129) Matrix(33,8,70,17) -> Matrix(53,2,-2730,-103) Matrix(881,202,1400,321) -> Matrix(1079,40,-61530,-2281) Matrix(659,150,1050,239) -> Matrix(811,30,-46200,-1709) Matrix(101,22,280,61) -> Matrix(219,8,-10430,-381) Matrix(29,6,140,29) -> Matrix(111,4,-4690,-169) Matrix(51,8,70,11) -> Matrix(59,2,-3570,-121) Matrix(43,-8,70,-13) -> Matrix(83,2,-4690,-113) Matrix(309,-70,490,-111) -> Matrix(601,14,-34300,-799) Matrix(157,-48,350,-107) -> Matrix(457,10,-23170,-507) Matrix(181,-64,280,-99) -> Matrix(379,8,-21840,-461) Matrix(379,-140,490,-181) -> Matrix(671,14,-41650,-869) Matrix(57,-22,70,-27) -> Matrix(97,2,-6160,-127) Matrix(537,-242,770,-347) -> Matrix(1117,22,-66360,-1307) Matrix(243,-134,350,-193) -> Matrix(543,10,-32200,-593) Matrix(341,-222,490,-319) -> Matrix(809,14,-48020,-831) 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: ((1,4,16,27,35,38,21,40,39,33,26,8,7,25,43,45,30,46,44,34,47,32,11,3,10,31,41,18,37,36,42,23,17,5,2)(6,22,28,29,9,14,13)(12,24,15,20,19); (1,2,8,22,24,45,33,32,28,44,38,15,4,14,37,47,35,41,48,39,42,43,21,6,5,20,36,30,29,25,31,34,19,9,3)(7,12,11,16,40,18,17)(10,23,13,27,26); (2,6,23,39,16,15,22,21,44,46,36,14,19,5,18,35,13,4,3,12,34,28,8,27,11,33,48,41,25,17,10,9,30,24,7)(20,38,47,31,26,45,42)(29,32,37,40,43)) ----------------------------------------------------------------------- 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 0/1 2/7 2/5 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/35 1/4 -4/175 2/7 -1/45 1/3 -3/140 3/8 -8/385 2/5 -1/49 3/7 -1/50 1/2 -2/105 4/7 -1/55 3/5 -1/56 2/3 -3/175 5/7 -1/60 8/11 -11/665 3/4 -4/245 4/5 -1/63 5/6 -6/385 1/1 -1/70 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(13,-3,35,-8) (0/1,1/4) -> (1/3,3/8) Hyperbolic Matrix(76,-21,105,-29) (1/4,2/7) -> (5/7,8/11) Hyperbolic Matrix(19,-6,35,-11) (2/7,1/3) -> (1/2,4/7) Hyperbolic Matrix(57,-22,70,-27) (3/8,2/5) -> (4/5,5/6) Hyperbolic Matrix(41,-17,70,-29) (2/5,3/7) -> (4/7,3/5) Hyperbolic Matrix(24,-11,35,-16) (3/7,1/2) -> (2/3,5/7) Hyperbolic Matrix(27,-17,35,-22) (3/5,2/3) -> (3/4,4/5) Hyperbolic Matrix(59,-43,70,-51) (8/11,3/4) -> (5/6,1/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,-35,1) Matrix(13,-3,35,-8) -> Matrix(43,1,-2065,-48) Matrix(76,-21,105,-29) -> Matrix(134,3,-8085,-181) Matrix(19,-6,35,-11) -> Matrix(46,1,-2485,-54) Matrix(57,-22,70,-27) -> Matrix(97,2,-6160,-127) Matrix(41,-17,70,-29) -> Matrix(99,2,-5495,-111) Matrix(24,-11,35,-16) -> Matrix(51,1,-3010,-59) Matrix(27,-17,35,-22) -> Matrix(57,1,-3535,-62) Matrix(59,-43,70,-51) -> Matrix(121,2,-7805,-129) 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 0/1 -1/35 1 35 1/4 -4/175 1 35 2/7 -1/45 7 5 1/3 -3/140 1 35 3/8 -8/385 1 35 2/5 -1/49 5 7 3/7 -1/50 7 5 1/2 -2/105 1 35 1/0 0/1 35 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(13,-3,35,-8) (0/1,1/4) -> (1/3,3/8) Hyperbolic Matrix(53,-14,140,-37) (1/4,3/11) -> (3/8,5/13) Glide Reflection Matrix(29,-8,105,-29) (4/15,2/7) -> (4/15,2/7) Reflection Matrix(16,-5,35,-11) (2/7,1/3) -> (3/7,1/2) Glide Reflection Matrix(41,-16,105,-41) (8/21,2/5) -> (8/21,2/5) Reflection Matrix(29,-12,70,-29) (2/5,3/7) -> (2/5,3/7) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,70,1) (0/1,1/0) -> (-1/35,0/1) Matrix(13,-3,35,-8) -> Matrix(43,1,-2065,-48) Matrix(53,-14,140,-37) -> Matrix(177,4,-8540,-193) Matrix(29,-8,105,-29) -> Matrix(134,3,-5985,-134) (4/15,2/7) -> (-3/133,-1/45) Matrix(16,-5,35,-11) -> Matrix(46,1,-2345,-51) Matrix(41,-16,105,-41) -> Matrix(146,3,-7105,-146) (8/21,2/5) -> (-3/145,-1/49) Matrix(29,-12,70,-29) -> Matrix(99,2,-4900,-99) (2/5,3/7) -> (-1/49,-1/50) Matrix(-1,1,0,1) -> Matrix(-1,0,105,1) (1/2,1/0) -> (-2/105,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.