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.

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.

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.

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.

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.

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
Canonical name	MordellCurve
Date of creation	2013-03-22 13:49:57
Last modified on	2013-03-22 13:49:57
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	5
Author	alozano (2414)
Entry type	Definition
Classification	msc 14H52
Related topic	EllipticCurve
Related topic	BirchAndSwinnertonDyerConjecture
Related topic	ArithmeticOfEllipticCurves
Defines	Mordell curve