Mordell–Weil group

Examples

Constructing explicit examples of the Mordell–Weil group of an abelian variety is a non-trivial process which is not always guaranteed to be successful, so we instead specialize to the case of a specific elliptic curve ${\displaystyle E/\mathbb{Q}}$. Let ${\displaystyle E}$ be defined by the Weierstrass equation

${\displaystyle y^2=x(x-6)(x+6)}$

over the rational numbers. It has discriminant ${\displaystyle {\mathrm{\Delta }}_E=2^{12}\cdot 3^6}$ (and this polynomial can be used to define a global model ${\displaystyle \mathcal{E}/\mathbb{Z}}$). It can be found

${\displaystyle E(\mathbb{Q})\cong \mathbb{Z}/2\times \mathbb{Z}/2\times \mathbb{Z}}$

through the following procedure. First, we find some obvious torsion points by plugging in some numbers, which are

${\displaystyle \mathrm{\infty },(0,0),(6,0),(-6,0)}$

In addition, after trying some smaller pairs of integers, we find ${\displaystyle (-3,9)}$ is a point which is not obviously torsion. One useful result for finding the torsion part of ${\displaystyle E(\mathbb{Q})}$ is that the torsion of prime to ${\displaystyle p}$, for ${\displaystyle E}$ having good reduction to ${\displaystyle p}$, denoted ${\displaystyle E(\mathbb{Q}{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s},p}}$ injects into ${\displaystyle E({\mathbb{F}}_p)}$, so

${\displaystyle E(\mathbb{Q}{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s},p}\hookrightarrow E({\mathbb{F}}_p)}$

We check at two primes ${\displaystyle p=5,7}$ and calculate the cardinality of the sets

${\displaystyle \begin{array}{cc}\mathrm{\#}E({\mathbb{F}}_5)& =8=2^3\\ \mathrm{\#}E({\mathbb{F}}_7)& =12=2^2\cdot 3\end{array}}$

note that because both primes only contain a factor of ${\displaystyle 2^2}$, we have found all the torsion points. In addition, we know the point ${\displaystyle (-3,9)}$ has infinite order because otherwise there would be a prime factor shared by both cardinalities, so the rank is at least ${\displaystyle 1}$. Now, computing the rank is a more arduous process consisting of calculating the group ${\displaystyle E(\mathbb{Q})/2E(\mathbb{Q})\cong (\mathbb{Z}/2{)}^{r+2}}$ where ${\displaystyle r=\mathrm{rank}(E(\mathbb{Q}))}$ using some long exact sequences from homological algebra and the Kummer map.