# 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:

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

Examples:

1. 1.

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 $(-1,1)$.

2. 2.

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 http://math.bu.edu/people/alozano/Torsion.htmlgenerators here.

3. 3.

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

4. 4.

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 http://math.bu.edu/people/alozano/Mordell.htmlgenerators here.

5. 5.

http://www.maths.nott.ac.uk/personal/pmxtow/mordellc.htmHere 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).

Title Mordell curve MordellCurve 2013-03-22 13:49:57 2013-03-22 13:49:57 alozano (2414) alozano (2414) 5 alozano (2414) Definition msc 14H52 EllipticCurve BirchAndSwinnertonDyerConjecture ArithmeticOfEllipticCurves Mordell curve