This question is not very precise; I hope it is suitable for the site.


I have come to a situation where I have to study rational points on an elliptic curve defined over $\mathbb{Q}$. I don't know much about the curve, let alone its equation. I already have one rational point, which sits on a bounded real connected component. What I want to avoid is that this is the only rational point (other than the marked point).


I am not sure what to use about my curve that will help me get there, so I turn the questio the other way round:



> 
> What is known about elliptic curves $E$ over $\mathbb{Q}$ such that $E(\mathbb{Q}) \cong \mathbb{Z}/2 \mathbb{Z}$?
> 
> 
> 



Elliptic curves with Mordell-Weil group Z/2Z over Q
Mazur's theorem ensures that there are exactly 15 possible cases for the torsion part of the Mordell-Weil group of an elliptic curve: the cyclic groups $\mathbb{Z}\_n$ (with $1\leq n\leq 10$ or $n=12$) and the groups $\mathbb{Z}\_2\times\mathbb{Z}\_n$ for $n=2,4,6,8$.


In his paper [Universal Bounds on The Torsion of Elliptic Curves, Proc. London. Math. Soc.(1976) 33, 193-237](http://plms.oxfordjournals.org/content/s3-33/2/193.extract) , Daniel Sion Kubert (who was a student of Mazur) presents in table 3 (page 217) a list of parametrizations for the different possible cases. 


In particular, curves with a $\mathbb{Z}\_2$ torsion are parametrized by the following family:
$$
\mathbb{Z}\_2\ \text{torsion}:\quad y^2=x(x^2+a x+ b), \quad b^2(a^2-4b)\neq 0.
$$


The example given in Francesco's answer is a special case with $a=0$. 
As another example, the case with torsion $\mathbb{Z}\_2\times \mathbb{Z}\_2$ is parametrized by the Legendre family:


$$
\mathbb{Z}\_2\times\mathbb{Z}\_2 \ \text{torsion}:\quad y^2=x(x+r)(x+s), \quad r\neq 0 \neq s \neq r.
$$


A slight generalization of the Hesse family parametrizes the curves with torsion $\mathbb{Z}\_3$:


$$
\mathbb{Z}\_3 \ \text{torsion}:\quad y^2+a\_1 x y +a\_3 y =x^3, \quad a\_3^3( a\_1^3-27 a\_3)\neq 0.
$$
For the other groups you might have to use Tate's normal form 
$$
E(b,c): \quad y^2+(1-c)x y - b y =x^3- b x^2
$$
and the condition for a given torsion is expressed as an algebraic condition on $b$ and $c$.


For example for $\mathbb{Z}\_4$, we have $c=0$ and $b^4(1+16b)\neq 0$, which gives: 
$$
\mathbb{Z}\_4 \ \text{torsion}:\quad E(b,c=0): \quad y^2+x y - b y =x^3- b x^2, \quad b^4(1+16b)\neq 0.
$$


For a review, you can read chapter 4 of the book of Husemoller . A friendly short review is also available in section 2 of [this string theory paper](http://arxiv.org/PS_cache/hep-th/pdf/9805/9805206v2.pdf)  by Aspinwall and Morrison ( they don't present all the 15 cases but for those they analyze, they express everything in Weierstrass form).


