Elliptic curve over $\mathbb{Q}$ with two distinct non-torsion points $P$ and $Q$ such that $nP \neq mQ$ for all $n,m \in \mathbb{Z}-\{0\}$.

Question

We know an elliptic curve over $\mathbb{Q}$ can have at most finite number of torsion points. The torsion group for $E$, $E_{tors}$ is either $\mathbb{Z}/n\mathbb{Z}$ with $1 \leq n \leq 10$ or $n = 12,$ or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ for $2 \leq n \leq 4.$ I am looking for example of an elliptic curve $E$ with two non-torsion points $P$ and $Q$ such that $nP \neq mQ$ for $m,n \in \mathbb{Z}-\{0\}.$ In other words we must have $<P> \cap <Q> = \{O\},$ where $O$ is the point at infinity.

Answer 1

See here for a general discussion on elliptic curves of high rank.

If you want the rank to be exactly 2 you can consult Cremona's tables. Here is a discussion on Math Stackexchange with links included.