Mordell curveMordellCurve2003-08-07 15:36:032004-03-14 12:17:11DefinitionMordell curveMordellrank% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{cor}{Corollary}A \emph{Mordell curve} is an elliptic curve $E/K$, for some field
$K$, which admits a model by a Weierstrass equation of the form:
$$y^2=x^3+k,\quad k\in K$$
{\bf Examples}:
\begin{enumerate}
\item 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 $(-1,1)$.
\item 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
\PMlinkexternal{generators}{http://math.bu.edu/people/alozano/Torsion.html}
here.
\item 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
$(0,n)$.
\item 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
\PMlinkexternal{generators}{http://math.bu.edu/people/alozano/Mordell.html}
here.
\item
\PMlinkexternal{Here}{http://www.maths.nott.ac.uk/personal/pmxtow/mordellc.htm}
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).
\end{enumerate}