##
**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 \(\text{sha} (4105033)= 635^2\).

Reviewer: Roelof J.Stroeker (Rotterdam)

### MSC:

11G05 | Elliptic curves over global fields |

14H52 | Elliptic curves |

11D25 | Cubic and quartic Diophantine equations |

### Software:

ecdata### References:

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