In the present article, the author considers elliptic curves over \(\mathbb{Q}\) given by \(w^3=P(u)\), where \(P(u)\) is a monic irreducible cubic polynomial over \(\mathbb{Q}\), with the distinguished point on the projective curve given by the unique rational point at infinity in the above model. First, a bijection between the rational points on the curve and the set \({\mathcal{W}}(\xi)=\{q\xi+r | \text{Norm}_{K\big/\mathbb{Q}}(q\xi+r) = 1,q,r\in \mathbb{Q}\}\) is established, where \(\xi\in K\) is a root of \(P(u)\) in a cubic number field. The rest of the article is devoted to a case by case study of these elliptic curves. They are all either isomorphic to pure cubic twists of the Fermat cubic, and that case is not considered further in this paper, or to a curve with Weierstrass model \(y^2 = x^3-B^2(B+3)\), \(B\in \mathbb{Q}^\times\). For the curves in the latter family, all but two have rank at least 1 and a point of infinite order is exhibited on each of them.
For the entire collection see [Zbl 1131.11004].