Rational Points of Projective Curves: X^3+Y^3+Z^3 = nXYZ (n \in [101..200])

[2006.11.18]X^3+Y^3+Z^3=nXYZ(n \in [101..200])の有理点

• x^3+y^3+z^3=nxyzの有理点を計算するプログラム(pari/gp)

gp>  read("de15.gp")
time = 3 ms.
gp>  for(n=101,200,print("E_",n,"_{tors}=",elltors(ec(n),1)))
E_101_{tors}=[3, [3], [[34347, 1135404]]]
E_102_{tors}=[3, [3], [[34992, 1157652]]]
E_103_{tors}=[3, [3], [[35643, 1180116]]]
E_104_{tors}=[3, [3], [[36300, 1202796]]]
E_105_{tors}=[3, [3], [[36963, 1225692]]]
E_106_{tors}=[3, [3], [[37632, 1248804]]]
E_107_{tors}=[3, [3], [[38307, 1272132]]]
E_108_{tors}=[3, [3], [[38988, 1295676]]]
E_109_{tors}=[3, [3], [[39675, 1319436]]]
E_110_{tors}=[3, [3], [[40368, 1343412]]]
E_111_{tors}=[3, [3], [[41067, 1367604]]]
E_112_{tors}=[3, [3], [[41772, 1392012]]]
E_113_{tors}=[3, [3], [[42483, 1416636]]]
E_114_{tors}=[3, [3], [[43200, 1441476]]]
E_115_{tors}=[3, [3], [[43923, 1466532]]]
E_116_{tors}=[3, [3], [[44652, 1491804]]]
E_117_{tors}=[3, [3], [[45387, 1517292]]]
E_118_{tors}=[3, [3], [[46128, 1542996]]]
E_119_{tors}=[3, [3], [[46875, 1568916]]]
E_120_{tors}=[3, [3], [[47628, 1595052]]]
E_121_{tors}=[3, [3], [[48387, 1621404]]]
E_122_{tors}=[3, [3], [[49152, 1647972]]]
E_123_{tors}=[3, [3], [[49923, 1674756]]]
E_124_{tors}=[3, [3], [[50700, 1701756]]]
E_125_{tors}=[3, [3], [[51483, 1728972]]]
E_126_{tors}=[3, [3], [[52272, 1756404]]]
E_127_{tors}=[3, [3], [[53067, 1784052]]]
E_128_{tors}=[3, [3], [[53868, 1811916]]]
E_129_{tors}=[3, [3], [[54675, 1839996]]]
E_130_{tors}=[3, [3], [[55488, 1868292]]]
E_131_{tors}=[3, [3], [[56307, 1896804]]]
E_132_{tors}=[3, [3], [[57132, 1925532]]]
E_133_{tors}=[3, [3], [[57963, 1954476]]]
E_134_{tors}=[3, [3], [[58800, 1983636]]]
E_135_{tors}=[3, [3], [[59643, 2013012]]]
E_136_{tors}=[3, [3], [[60492, 2042604]]]
E_137_{tors}=[3, [3], [[61347, 2072412]]]
E_138_{tors}=[3, [3], [[62208, 2102436]]]
E_139_{tors}=[3, [3], [[63075, 2132676]]]
E_140_{tors}=[3, [3], [[63948, 2163132]]]
E_141_{tors}=[3, [3], [[64827, 2193804]]]
E_142_{tors}=[3, [3], [[65712, 2224692]]]
E_143_{tors}=[3, [3], [[66603, 2255796]]]
E_144_{tors}=[3, [3], [[67500, 2287116]]]
E_145_{tors}=[3, [3], [[68403, 2318652]]]
E_146_{tors}=[3, [3], [[69312, 2350404]]]
E_147_{tors}=[3, [3], [[70227, 2382372]]]
E_148_{tors}=[3, [3], [[71148, 2414556]]]
E_149_{tors}=[3, [3], [[72075, 2446956]]]
E_150_{tors}=[3, [3], [[73008, 2479572]]]
E_151_{tors}=[3, [3], [[73947, 2512404]]]
E_152_{tors}=[3, [3], [[74892, 2545452]]]
E_153_{tors}=[3, [3], [[75843, 2578716]]]
E_154_{tors}=[3, [3], [[76800, 2612196]]]
E_155_{tors}=[3, [3], [[77763, 2645892]]]
E_156_{tors}=[3, [3], [[78732, 2679804]]]
E_157_{tors}=[3, [3], [[79707, 2713932]]]
E_158_{tors}=[3, [3], [[80688, 2748276]]]
E_159_{tors}=[3, [3], [[81675, 2782836]]]
E_160_{tors}=[3, [3], [[82668, 2817612]]]
E_161_{tors}=[3, [3], [[83667, 2852604]]]
E_162_{tors}=[3, [3], [[84672, 2887812]]]
E_163_{tors}=[3, [3], [[85683, 2923236]]]
E_164_{tors}=[3, [3], [[86700, 2958876]]]
E_165_{tors}=[3, [3], [[87723, 2994732]]]
E_166_{tors}=[3, [3], [[88752, 3030804]]]
E_167_{tors}=[3, [3], [[89787, 3067092]]]
E_168_{tors}=[3, [3], [[90828, 3103596]]]
E_169_{tors}=[3, [3], [[91875, 3140316]]]
E_170_{tors}=[3, [3], [[92928, 3177252]]]
E_171_{tors}=[3, [3], [[93987, 3214404]]]
E_172_{tors}=[3, [3], [[95052, 3251772]]]
E_173_{tors}=[3, [3], [[96123, 3289356]]]
E_174_{tors}=[3, [3], [[97200, 3327156]]]
E_175_{tors}=[3, [3], [[98283, 3365172]]]
E_176_{tors}=[3, [3], [[99372, 3403404]]]
E_177_{tors}=[3, [3], [[100467, 3441852]]]
E_178_{tors}=[3, [3], [[101568, 3480516]]]
E_179_{tors}=[3, [3], [[102675, 3519396]]]
E_180_{tors}=[3, [3], [[103788, 3558492]]]
E_181_{tors}=[3, [3], [[104907, 3597804]]]
E_182_{tors}=[3, [3], [[106032, 3637332]]]
E_183_{tors}=[3, [3], [[107163, 3677076]]]
E_184_{tors}=[3, [3], [[108300, 3717036]]]
E_185_{tors}=[3, [3], [[109443, 3757212]]]
E_186_{tors}=[3, [3], [[110592, 3797604]]]
E_187_{tors}=[3, [3], [[111747, 3838212]]]
E_188_{tors}=[3, [3], [[112908, 3879036]]]
E_189_{tors}=[3, [3], [[114075, 3920076]]]
E_190_{tors}=[3, [3], [[115248, 3961332]]]
E_191_{tors}=[3, [3], [[116427, 4002804]]]
E_192_{tors}=[3, [3], [[117612, 4044492]]]
E_193_{tors}=[3, [3], [[118803, 4086396]]]
E_194_{tors}=[3, [3], [[120000, 4128516]]]
E_195_{tors}=[3, [3], [[121203, 4170852]]]
E_196_{tors}=[3, [3], [[122412, 4213404]]]
E_197_{tors}=[3, [3], [[123627, 4256172]]]
E_198_{tors}=[3, [3], [[124848, 4299156]]]
E_199_{tors}=[3, [3], [[126075, 4342356]]]
E_200_{tors}=[3, [3], [[127308, 4385772]]]
time = 5mn, 43,197 ms.

• X^3+Y^3+Z^3=nXYZ(n \in [0..100])の有理点

[参考文献]

• [1]Joseph H. Silverman, John Tate(著), 足立 恒雄, 木田 雅成, 小松 啓一, 田谷 久雄(訳), "楕円曲線論入門", シュプリンガー・フェアラーク東京, 1995, ISBN4-431-70683-6, {3900円}.
  
• [2]Joseph H. Silverman, "The Arithmetic of Elliptic Curves", GTM 106, Springer-Verlag New York Inc., 1986, ISBN0-387-96203-4.
  
• [3]Erik Dofs, "Solutions of x^3+y^3+z^3=nxyz", 1995, ACTA Mathematica LXXIII.3
  

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