Let $y^2 = x^3 + ax + b$ be an elliptic curve defined over $\mathbb{Z}$. If $b=a^2$, find a point of infinite order on $\mathcal{E}(\mathbb{Q})$.
The previous part of the question implies that I should use Nagell-Lutz, so am looking for a point $(x, y)$ on the curve such that $x$ or $y$ isn't an integer, or $y \neq 0$ and $y^2 \nmid a^3(4+27a)$ but am having no luck!
Any help greatly appreciated!
Let $a>0$ be an integer, and let $E: y^2=x^3+ax+a^2$. There is an obvious point $P=(x_0,y_0)$ on this curve with integer coordinates. The problem is that $x_0$, and $y_0$ are integers, and $y_0^2$ does divide $a^3(4+27a)$, so we cannot immediately conclude that $P$ is of infinite order.
Hint: Calculate $2P$ to show that $P$ has infinite order.