Rational Points of Projective Curves: X^3+Y^3+Z^3 = nXYZ (n \in [-100..-1])

[2004.12.05]X^3+Y^3+Z^3=nXYZ(n \in [-100..-1])の有理点

• x^3+y^3+1=-4xyのグラフ(jpeg)
• y^2=x^3+16416*x+1772496のグラフ(jpeg)
• x^3+y^3+z^3=nxyの有理点を計算するプログラム(pari/gp)

gp> forstep(n=-1,-100,-1,print("E_{",n,"}_{tors}=",elltors(ec(n),1)))
E_{-1}_{tors}=[6, [6], [[327, 6048]]]
E_{-2}_{tors}=[3, [3], [[48, 756]]]
E_{-3}_{tors}=[3, [3], [[27, 972]]]
E_{-4}_{tors}=[3, [3], [[12, 1404]]]
E_{-5}_{tors}=[3, [3], [[3, 2052]]]
E_{-6}_{tors}=[3, [3], [[0, 2916]]]
E_{-7}_{tors}=[3, [3], [[3, 3996]]]
E_{-8}_{tors}=[3, [3], [[12, 5292]]]
E_{-9}_{tors}=[3, [3], [[27, 6804]]]
E_{-10}_{tors}=[3, [3], [[48, 8532]]]
E_{-11}_{tors}=[3, [3], [[75, 10476]]]
E_{-12}_{tors}=[3, [3], [[108, 12636]]]
E_{-13}_{tors}=[3, [3], [[147, 15012]]]
E_{-14}_{tors}=[3, [3], [[192, 17604]]]
E_{-15}_{tors}=[3, [3], [[243, 20412]]]
E_{-16}_{tors}=[3, [3], [[300, 23436]]]
E_{-17}_{tors}=[3, [3], [[363, 26676]]]
E_{-18}_{tors}=[3, [3], [[432, 30132]]]
E_{-19}_{tors}=[3, [3], [[507, 33804]]]
E_{-20}_{tors}=[3, [3], [[588, 37692]]]
E_{-21}_{tors}=[3, [3], [[675, 41796]]]
E_{-22}_{tors}=[3, [3], [[768, 46116]]]
E_{-23}_{tors}=[3, [3], [[867, 50652]]]
E_{-24}_{tors}=[3, [3], [[972, 55404]]]
E_{-25}_{tors}=[3, [3], [[1083, 60372]]]
E_{-26}_{tors}=[3, [3], [[1200, 65556]]]
E_{-27}_{tors}=[3, [3], [[1323, 70956]]]
E_{-28}_{tors}=[3, [3], [[1452, 76572]]]
E_{-29}_{tors}=[3, [3], [[1587, 82404]]]
E_{-30}_{tors}=[3, [3], [[1728, 88452]]]
E_{-31}_{tors}=[3, [3], [[1875, 94716]]]
E_{-32}_{tors}=[3, [3], [[2028, 101196]]]
E_{-33}_{tors}=[3, [3], [[2187, 107892]]]
E_{-34}_{tors}=[3, [3], [[2352, 114804]]]
E_{-35}_{tors}=[3, [3], [[2523, 121932]]]
E_{-36}_{tors}=[3, [3], [[2700, 129276]]]
E_{-37}_{tors}=[3, [3], [[2883, 136836]]]
E_{-38}_{tors}=[3, [3], [[3072, 144612]]]
E_{-39}_{tors}=[3, [3], [[3267, 152604]]]
E_{-40}_{tors}=[3, [3], [[3468, 160812]]]
E_{-41}_{tors}=[3, [3], [[3675, 169236]]]
E_{-42}_{tors}=[3, [3], [[3888, 177876]]]
E_{-43}_{tors}=[3, [3], [[4107, 186732]]]
E_{-44}_{tors}=[3, [3], [[4332, 195804]]]
E_{-45}_{tors}=[3, [3], [[4563, 205092]]]
E_{-46}_{tors}=[3, [3], [[4800, 214596]]]
E_{-47}_{tors}=[3, [3], [[5043, 224316]]]
E_{-48}_{tors}=[3, [3], [[5292, 234252]]]
E_{-49}_{tors}=[3, [3], [[5547, 244404]]]
E_{-50}_{tors}=[3, [3], [[5808, 254772]]]
E_{-51}_{tors}=[3, [3], [[6075, 265356]]]
E_{-52}_{tors}=[3, [3], [[6348, 276156]]]
E_{-53}_{tors}=[3, [3], [[6627, 287172]]]
E_{-54}_{tors}=[3, [3], [[6912, 298404]]]
E_{-55}_{tors}=[3, [3], [[7203, 309852]]]
E_{-56}_{tors}=[3, [3], [[7500, 321516]]]
E_{-57}_{tors}=[3, [3], [[7803, 333396]]]
E_{-58}_{tors}=[3, [3], [[8112, 345492]]]
E_{-59}_{tors}=[3, [3], [[8427, 357804]]]
E_{-60}_{tors}=[3, [3], [[8748, 370332]]]
E_{-61}_{tors}=[3, [3], [[9075, 383076]]]
E_{-62}_{tors}=[3, [3], [[9408, 396036]]]
E_{-63}_{tors}=[3, [3], [[9747, 409212]]]
E_{-64}_{tors}=[3, [3], [[10092, 422604]]]
E_{-65}_{tors}=[3, [3], [[10443, 436212]]]
E_{-66}_{tors}=[3, [3], [[10800, 450036]]]
E_{-67}_{tors}=[3, [3], [[11163, 464076]]]
E_{-68}_{tors}=[3, [3], [[11532, 478332]]]
E_{-69}_{tors}=[3, [3], [[11907, 492804]]]
E_{-70}_{tors}=[3, [3], [[12288, 507492]]]
E_{-71}_{tors}=[3, [3], [[12675, 522396]]]
E_{-72}_{tors}=[3, [3], [[13068, 537516]]]
E_{-73}_{tors}=[3, [3], [[13467, 552852]]]
E_{-74}_{tors}=[3, [3], [[13872, 568404]]]
E_{-75}_{tors}=[3, [3], [[14283, 584172]]]
E_{-76}_{tors}=[3, [3], [[14700, 600156]]]
E_{-77}_{tors}=[3, [3], [[15123, 616356]]]
E_{-78}_{tors}=[3, [3], [[15552, 632772]]]
E_{-79}_{tors}=[3, [3], [[15987, 649404]]]
E_{-80}_{tors}=[3, [3], [[16428, 666252]]]
E_{-81}_{tors}=[3, [3], [[16875, 683316]]]
E_{-82}_{tors}=[3, [3], [[17328, 700596]]]
E_{-83}_{tors}=[3, [3], [[17787, 718092]]]
E_{-84}_{tors}=[3, [3], [[18252, 735804]]]
E_{-85}_{tors}=[3, [3], [[18723, 753732]]]
E_{-86}_{tors}=[3, [3], [[19200, 771876]]]
E_{-87}_{tors}=[3, [3], [[19683, 790236]]]
E_{-88}_{tors}=[3, [3], [[20172, 808812]]]
E_{-89}_{tors}=[3, [3], [[20667, 827604]]]
E_{-90}_{tors}=[3, [3], [[21168, 846612]]]
E_{-91}_{tors}=[3, [3], [[21675, 865836]]]
E_{-92}_{tors}=[3, [3], [[22188, 885276]]]
E_{-93}_{tors}=[3, [3], [[22707, 904932]]]
E_{-94}_{tors}=[3, [3], [[23232, 924804]]]
E_{-95}_{tors}=[3, [3], [[23763, 944892]]]
E_{-96}_{tors}=[3, [3], [[24300, 965196]]]
E_{-97}_{tors}=[3, [3], [[24843, 985716]]]
E_{-98}_{tors}=[3, [3], [[25392, 1006452]]]
E_{-99}_{tors}=[3, [3], [[25947, 1027404]]]
E_{-100}_{tors}=[3, [3], [[26508, 1048572]]]
time = 2mn, 58,252 ms.

bash-2.05a$ mwrank3
Program mwrank: uses 2-descent (via 2-isogeny if possible) to
determine the rank of an elliptic curve E over Q, and list a
set of points which generate E(Q) modulo 2E(Q).
and finally search for further points on the curve.
For more details see the file mwrank.doc.
For details of algorithms see the author's book.

Please acknowledge use of this program in published work, 
and send problems to John.Cremona@nottingham.ac.uk.

Version compiled on Feb 11 2003 at 17:40:15 by GCC 3.2.1
using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floating point)
Using LiDIA multiprecision floating point with 15 decimal places.
Enter curve: [0, 0, 0, 16416, 1772496]

Curve [0,0,0,16416,1772496] :   Working with minimal curve [0,1,1,13,42]
        [u,r,s,t] = [6,12,0,108]
No points of order 2
Basic pair: I=-608, J=-65648
disc=-5208682752
2-adic index bound = 2
By Lemma 5.1(a), 2-adic index = 1
2-adic index = 1
One (I,J) pair
Looking for quartics with I = -608, J = -65648
Looking for Type 3 quartics:
Trying positive a from 1 up to 12 (square a first...)
(1,0,-158,1092,-2131)   --nontrivial...(x:y:z) = (1 : 1 : 0)
Point = [26 : 136 : 1]
        height = 0.353081695469717
Rank of B=im(eps) increases to 1
(1,0,-74,364,-507)      --trivial
(1,0,-2,52,-51) --trivial
(1,-1,13,5,-66) --trivial
Trying positive a from 1 up to 12 (...then non-square a)
Trying negative a from -1 down to -10
(-3,-4,22,56,49)        --trivial
Finished looking for Type 3 quartics.
Mordell rank contribution from B=im(eps) = 1
Selmer  rank contribution from B=im(eps) = 1
Sha     rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer  rank contribution from A=ker(eps) = 0
Sha     rank contribution from A=ker(eps) = 0
Rank = 1
Points generating E(Q)/2E(Q):
Point [948 : 29484 : 1], height = 0.353081695469717
After descent, rank of points found is 1
Transferring points back to original curve [0,0,0,16416,1772496]

Generator 1 is [948 : 29484 : 1]; height 0.353081695469717

The rank has been determined unconditionally.
The basis given is for a subgroup of full rank of the Mordell-Weil group
 (modulo torsion), possibly of index greater than 1.
Regulator (of this subgroup) = 0.353081695469717

 (3 seconds)
Enter curve: [0,0,0,0,0]

bash-2.05a$ 

gp>  read("de15.gp")
time = 10 ms.
gp>  rpE([948, 29484],-4,10)
[948, 29484]
[220, 4004]
[57, 1701]
[-3956/121, 1458548/1331]
[-2055804/26569, 855655164/4330747]
[-96767/1764, -62290943/74088]
[2499086820/117787609, -1865839213677348/1278348920477]
[282774610732/1942781929, -230498118954737452/85631999084533]
[20019951125049/38933209225, -92880196223625290907/7682106178230875]
[172821720838216252/22376901758929, -71855179206070433760879292/105852210749178196967]
time = 15 ms.

gp>  rpCC([948, 29484],-4,10)
[2, -1, 1]
[7, -4, 9]
[31, -95, 134]
[-1544, -5551, 3663]
[-502111, -589174, 264223]
[-225749735, -101167849, 118880244]
[-179042138354, 10807191487, 193411019969]
[-293255735344481, 339980855190512, 613059833705121]
[-2124787285654006081, 4755227249429348801, 3370973297269526290]
[-85451648778129165330452, 111952689700086312034601, 20590283114024812857639]
time = 15 ms.

[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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