Models of some genus one curves with applications to descent (Q557570)

The classical method for finding the rank of an elliptic curve \(E\) defined over the rationals is performing an \(n\)-descent for \(n=2\) or \(n = 3\). One important step in performing such descents is computing the Selmer group, whose elements are certain curves of genus \(1\) that become isomorphic to \(E\) over the algebraic closure of \(\mathbb Q\). For \(n = 2\), these curves of genus \(1\) have the form \(y^2 = f(x)\) for some quartic polynomial \(f\), and for \(n=3\) they are certain cubic curves (at least if we adjoin the cube roots of unity to \(\mathbb Q\)). In this article, the author derives families of genus \(1\) curves for special cases when \(n = 3\) and \(n = 5\), based on her previous work concerning the period-index obstruction [J. Number Theory 95, No. 2, 329--339 (2002); erratum ibid. 109, No. 2, 390 (2004; Zbl 1033.11029)]