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On some elliptic curves with large sha. (English) Zbl 0983.11032
The class of elliptic curves considered in this paper is given by $C(n): y^2=x^3+nx,$ where $$n\in \mathbb{Z}$$. Let $$p$$ be a prime satisfying $$p\equiv 1\pmod 8$$. The author is mainly concerned with the case $$C(p^3)$$, and also with $$C(p)$$, but only because of its connection with the former. These curves have Mordell-Weil rank 0 or 2, provided the Birch and Swinnerton-Dyer conjecture holds (this conjecture is assumed to be true throughout the paper). What the author really is interested in is the order of the Tate-Shafarevich group of $$C(p^3)$$, $$\text{sha}(p)$$ for short. It turns out that, regularly, the curves $$C(p^3)$$ have large sha and this value is relatively often an integer square. For each $$k\leq 105$$ the author calculates the primes $$p< 10^5$$ for which $$\text{sha}(p)= k^2$$ (all values of $$k$$ occur). The largest sha found by the author is $$\text{sha} (4105033)= 635^2$$.
##### MSC:
 11G05 Elliptic curves over global fields 14H52 Elliptic curves 11D25 Cubic and quartic Diophantine equations
##### Keywords:
elliptic curves; Tate-Shafarevich group
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##### References:
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