PlanetMath: Mordell curve

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Mordell curve

(Definition)

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

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

Examples:

Let $E_1/\mathbb{Q}\colon y^2=x^3+2$ , this is a Mordell curve with Mordell-Weil group $E_1(\mathbb{Q})\simeq \mathbb{Z}$ and generated by .
Let $E_2/\mathbb{Q}\colon y^2=x^3+109858299531561$ , then $E_2(\mathbb{Q})\simeq \mathbb{Z}/3\mathbb{Z}\bigoplus {\mathbb{Z}}^5$ . See generators here.
In general, a Mordell curve of the form has torsion group isomorphic to $\mathbb{Z}/3\mathbb{Z}$ generated by .
Let $E_3/\mathbb{Q}\colon y^2=x^3+496837487681$ then this is a Mordell curve with $E_3(\mathbb{Q})\simeq {\mathbb{Z}}^8$ . See generators here.
Here you can find a list of the minimal-known positive and negative k for Mordell curves of given rank, and the Mordell curves with maximum rank known (see BS-D conjecture).