# Mordell curve

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.
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