Mordell curve

A Mordell curve is an elliptic curve $E/K$, for some field$K$, which admits a model by a Weierstrass equation of the form:

$$y^2=x^3+k,k\in K$$

Examples:

1. Let $E_1/\mathbb{Q}:y^2=x^3+2$, this is a Mordellcurve with Mordell-Weil group $E_1(\mathbb{Q})\simeq \mathbb{Z}$and generated by $(-1,1)$.

2. Let $E_2/\mathbb{Q}:y^2=x^3+109858299531561$, then$E_2(\mathbb{Q})\simeq \mathbb{Z}/3\mathbb{Z}\oplus {\mathbb{Z}}^5$. Seehttp://math.bu.edu/people/alozano/Torsion.htmlgeneratorshere.

3. In general, a Mordell curve of the form $y^2=x^3+n^2$ hastorsion group isomorphic to $\mathbb{Z}/3\mathbb{Z}$ generated by$(0,n)$.

4. Let $E_3/\mathbb{Q}:y^2=x^3+496837487681$ then this isa Mordell curve with $E_3(\mathbb{Q})\simeq {\mathbb{Z}}^8$. Seehttp://math.bu.edu/people/alozano/Mordell.htmlgeneratorshere.

5. http://www.maths.nott.ac.uk/personal/pmxtow/mordellc.htmHereyou can find a list of the minimal-known positive and negative kfor Mordell curves of given rank, and the Mordell curves withmaximum rank known (see BS-D conjecture).