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 Mordell curve with MordellWeil 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 ${E}_{2}(\mathbb{Q})\simeq \mathbb{Z}/3\mathbb{Z}\oplus {\mathbb{Z}}^{5}$. 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 ${E}_{3}/\mathbb{Q}:{y}^{2}={x}^{3}+496837487681$ 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 minimalknown positive and negative k for Mordell curves of given rank, and the Mordell curves with maximum rank known (see BSD conjecture).
Title  Mordell curve 

Canonical name  MordellCurve 
Date of creation  20130322 13:49:57 
Last modified on  20130322 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 