On the rank of elliptic curves in elementary cubic extensions (Q2337583)

Summary: We give a method for explicitly constructing an elementary cubic extension \(L\) over which an elliptic curve \(E_D\colon y^2 + Dy = x^3\) \((D \in \mathbb{Q}^*)\) has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of \(E_D\) and a generic polynomial for cyclic cubic extensions. In our method, the extension degree \([L : \mathbb{Q}]\) often becomes small.