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

[2006.12.02]X^3+Y^3+Z^3=nXYZ(n \in [-200..-101])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], [[27075, 1069956]]]
E_{-102}_{tors}=[3, [3], [[27648, 1091556]]]
E_{-103}_{tors}=[3, [3], [[28227, 1113372]]]
E_{-104}_{tors}=[3, [3], [[28812, 1135404]]]
E_{-105}_{tors}=[3, [3], [[29403, 1157652]]]
E_{-106}_{tors}=[3, [3], [[30000, 1180116]]]
E_{-107}_{tors}=[3, [3], [[30603, 1202796]]]
E_{-108}_{tors}=[3, [3], [[31212, 1225692]]]
E_{-109}_{tors}=[3, [3], [[31827, 1248804]]]
E_{-110}_{tors}=[3, [3], [[32448, 1272132]]]
E_{-111}_{tors}=[3, [3], [[33075, 1295676]]]
E_{-112}_{tors}=[3, [3], [[33708, 1319436]]]
E_{-113}_{tors}=[3, [3], [[34347, 1343412]]]
E_{-114}_{tors}=[3, [3], [[34992, 1367604]]]
E_{-115}_{tors}=[3, [3], [[35643, 1392012]]]
E_{-116}_{tors}=[3, [3], [[36300, 1416636]]]
E_{-117}_{tors}=[3, [3], [[36963, 1441476]]]
E_{-118}_{tors}=[3, [3], [[37632, 1466532]]]
E_{-119}_{tors}=[3, [3], [[38307, 1491804]]]
E_{-120}_{tors}=[3, [3], [[38988, 1517292]]]
E_{-121}_{tors}=[3, [3], [[39675, 1542996]]]
E_{-122}_{tors}=[3, [3], [[40368, 1568916]]]
E_{-123}_{tors}=[3, [3], [[41067, 1595052]]]
E_{-124}_{tors}=[3, [3], [[41772, 1621404]]]
E_{-125}_{tors}=[3, [3], [[42483, 1647972]]]
E_{-126}_{tors}=[3, [3], [[43200, 1674756]]]
E_{-127}_{tors}=[3, [3], [[43923, 1701756]]]
E_{-128}_{tors}=[3, [3], [[44652, 1728972]]]
E_{-129}_{tors}=[3, [3], [[45387, 1756404]]]
E_{-130}_{tors}=[3, [3], [[46128, 1784052]]]
E_{-131}_{tors}=[3, [3], [[46875, 1811916]]]
E_{-132}_{tors}=[3, [3], [[47628, 1839996]]]
E_{-133}_{tors}=[3, [3], [[48387, 1868292]]]
E_{-134}_{tors}=[3, [3], [[49152, 1896804]]]
E_{-135}_{tors}=[3, [3], [[49923, 1925532]]]
E_{-136}_{tors}=[3, [3], [[50700, 1954476]]]
E_{-137}_{tors}=[3, [3], [[51483, 1983636]]]
E_{-138}_{tors}=[3, [3], [[52272, 2013012]]]
E_{-139}_{tors}=[3, [3], [[53067, 2042604]]]
E_{-140}_{tors}=[3, [3], [[53868, 2072412]]]
E_{-141}_{tors}=[3, [3], [[54675, 2102436]]]
E_{-142}_{tors}=[3, [3], [[55488, 2132676]]]
E_{-143}_{tors}=[3, [3], [[56307, 2163132]]]
E_{-144}_{tors}=[3, [3], [[57132, 2193804]]]
E_{-145}_{tors}=[3, [3], [[57963, 2224692]]]
E_{-146}_{tors}=[3, [3], [[58800, 2255796]]]
E_{-147}_{tors}=[3, [3], [[59643, 2287116]]]
E_{-148}_{tors}=[3, [3], [[60492, 2318652]]]
E_{-149}_{tors}=[3, [3], [[61347, 2350404]]]
E_{-150}_{tors}=[3, [3], [[62208, 2382372]]]
E_{-151}_{tors}=[3, [3], [[63075, 2414556]]]
E_{-152}_{tors}=[3, [3], [[63948, 2446956]]]
E_{-153}_{tors}=[3, [3], [[64827, 2479572]]]
E_{-154}_{tors}=[3, [3], [[65712, 2512404]]]
E_{-155}_{tors}=[3, [3], [[66603, 2545452]]]
E_{-156}_{tors}=[3, [3], [[67500, 2578716]]]
E_{-157}_{tors}=[3, [3], [[68403, 2612196]]]
E_{-158}_{tors}=[3, [3], [[69312, 2645892]]]
E_{-159}_{tors}=[3, [3], [[70227, 2679804]]]
E_{-160}_{tors}=[3, [3], [[71148, 2713932]]]
E_{-161}_{tors}=[3, [3], [[72075, 2748276]]]
E_{-162}_{tors}=[3, [3], [[73008, 2782836]]]
E_{-163}_{tors}=[3, [3], [[73947, 2817612]]]
E_{-164}_{tors}=[3, [3], [[74892, 2852604]]]
E_{-165}_{tors}=[3, [3], [[75843, 2887812]]]
E_{-166}_{tors}=[3, [3], [[76800, 2923236]]]
E_{-167}_{tors}=[3, [3], [[77763, 2958876]]]
E_{-168}_{tors}=[3, [3], [[78732, 2994732]]]
E_{-169}_{tors}=[3, [3], [[79707, 3030804]]]
E_{-170}_{tors}=[3, [3], [[80688, 3067092]]]
E_{-171}_{tors}=[3, [3], [[81675, 3103596]]]
E_{-172}_{tors}=[3, [3], [[82668, 3140316]]]
E_{-173}_{tors}=[3, [3], [[83667, 3177252]]]
E_{-174}_{tors}=[3, [3], [[84672, 3214404]]]
E_{-175}_{tors}=[3, [3], [[85683, 3251772]]]
E_{-176}_{tors}=[3, [3], [[86700, 3289356]]]
E_{-177}_{tors}=[3, [3], [[87723, 3327156]]]
E_{-178}_{tors}=[3, [3], [[88752, 3365172]]]
E_{-179}_{tors}=[3, [3], [[89787, 3403404]]]
E_{-180}_{tors}=[3, [3], [[90828, 3441852]]]
E_{-181}_{tors}=[3, [3], [[91875, 3480516]]]
E_{-182}_{tors}=[3, [3], [[92928, 3519396]]]
E_{-183}_{tors}=[3, [3], [[93987, 3558492]]]
E_{-184}_{tors}=[3, [3], [[95052, 3597804]]]
E_{-185}_{tors}=[3, [3], [[96123, 3637332]]]
E_{-186}_{tors}=[3, [3], [[97200, 3677076]]]
E_{-187}_{tors}=[3, [3], [[98283, 3717036]]]
E_{-188}_{tors}=[3, [3], [[99372, 3757212]]]
E_{-189}_{tors}=[3, [3], [[100467, 3797604]]]
E_{-190}_{tors}=[3, [3], [[101568, 3838212]]]
E_{-191}_{tors}=[3, [3], [[102675, 3879036]]]
E_{-192}_{tors}=[3, [3], [[103788, 3920076]]]
E_{-193}_{tors}=[3, [3], [[104907, 3961332]]]
E_{-194}_{tors}=[3, [3], [[106032, 4002804]]]
E_{-195}_{tors}=[3, [3], [[107163, 4044492]]]
E_{-196}_{tors}=[3, [3], [[108300, 4086396]]]
E_{-197}_{tors}=[3, [3], [[109443, 4128516]]]
E_{-198}_{tors}=[3, [3], [[110592, 4170852]]]
E_{-199}_{tors}=[3, [3], [[111747, 4213404]]]
E_{-200}_{tors}=[3, [3], [[112908, 4256172]]]
time = 5mn, 52,164 ms.

• X^3+Y^3+Z^3=nXYZ(n \in [-100..-1])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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