A twist of rank 4 of an elliptic curve of conductor 15
Let \(E\) be a given elliptic curve, defined over \(\mathbb Q\), let \(d\) be a rational integer without quadratic factor, denote by \(E_d\) the twist of \(E\) by \(\sqrt{d}\). A conjecture due to \textit{T. Honda} [Jap. J. Math. 30, 84-101 (1960; Zbl 0109.39602)] predicts that, as \(d\) varies, the rank of the \(\mathbb Z\)-module \(E_d(\mathbb Q)\) is bounded by a constant which depends only on \(E\). In order to justify this conjecture the following problem arises. Find explicitly some example of an elliptic curve \(E\) and some integer \(d\) such that the difference between the rank of \(E_d\) and the rank of \(E\) is as big as possible. Moreover it is reasonable to require that \(E\) has the smallest possible conductor among all its twists. In this article, applying techniques which were in part already used by \textit{C. L. Stewart} and \textit{J. Top} [J. Am. Math. Soc. 8, 943-973 (1995; Zbl 0857.11026)], the authors sketch the proof that the elliptic curve \(E\) denoted 15A1 by \textit{J. E. Cremona} [Algorithms for modular elliptic curves, Cambridge University Press (1997; Zbl 0872.14041)], which has rank \(0\), has a twist \(E_d\) whose rank is \(4\). A numerical verification is needed, which is possible only after some (interesting) reductions. This example is suggested by the Birch-Swinnerton-Dyer conjecture.
scientific article; zbMATH DE number 1192517