简单的高秩曲线

wayne

无意间 留意到一篇论文, http://dx.doi.org/10.1155/S0161171200002210
有一族曲线, $y^2=x^3-t^2x+1$,看似简单, 却有大量的高秩.
$1<t<1000$的统计情况如下, 秩为7的竟然有9个,分别是$ t = 347,443,614,757,778,784,857,877,888$

1. 1 1
   
2. 2 2
   
3. 3 220
   
4. 4 413
   
5. 5 272
   
6. 6 82
   
7. 7 9
   

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1. (20252716.12:27:03)> E=ellinit([-347^2,1])
   
2. %29 = [0, 0, 0, -120409, 1, 0, -240818, 4, -14498327281, 5779632, -864, 111726661732987024, 12066479467162645248/6982916358311689, Vecsmall([1]), [Vecsmall([128, 1])], [0, 0, 0, 0, 0, 0, 0, 0]]
   
3. (20254316.23:43:03)> ellrank(E,1)
   
4. %30 = [7, 7, 0, [[-347, 1], [-345, 691], [-181, 3983], [-163, 3911], [-105, 3389], [-77, 2969], [469, 6833]]]
   
5. (20254316.23:43:11)> E=ellinit([-888^2,1])
   
6. %31 = [0, 0, 0, -788544, 1, 0, -1577088, 4, -621801639936, 37850112, -864, 31380348951148363344, 125521395804593455104/72639696646176767, Vecsmall([1]), [Vecsmall([128, 1])], [0, 0, 0, 0, 0, 0, 0, 0]]
   
7. (20254316.23:43:29)> ellrank(E,1)
   
8. %32 = [7, 7, 0, [[-886, 1773], [-820, 9759], [-705, 14336], [-228, 12959], [-76, 7713], [-1, 888], [890, 1779]]]

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nyy

秩是生成元个数，还是元素个数？