Are generators of elliptic curves unique?

Question

Example of an elliptic curve of rank 2 with torsion $Z/3Z$: $$y^2 + x*y + y = x^3 - 75*x + 242$$ Torsion points $T_1, T_2,T_3$ are: ${(0 : 1 : 0), (4 : -6 : 1), (4 : 1 : 1)}$

Generator points $Q_1,Q_2$ (according Sage) are:${(-10 : 8 : 1), (-3 : 22 : 1)}$

As far as I know every rational point on this curve can be expressed as a linear combination of these points.

I mean any rationals point is expressible this way: $$P=a*T_2+b*Q_1+c*Q_2$$ where $-1\leq a\leq 1$ and $b,c\in \mathbb{Z}$.

My question is if there are any other two points except $Q_1$ and $Q_2$ that can be used to produce all rational points on the given elliptic curve.

Answer 1

Group of $\mathbb{Q}$-rational points on your elliptic curve $E$ is abstractly isomorphic with $A = \mathbb{Z}\oplus \mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z}$. Your question really concerns minimal generating sets of $A$ and has nothing to do with elliptic curves in this formulation. There are many minimal generating sets of $\mathbb{Z}\oplus \mathbb{Z}$ and hence many minimal generating sets of $A$.

Answer 2

The group of rational points is isomorphic to $\Bbb Z_3\times\Bbb Z\times\Bbb Z$, with $Q_1$ represented by $(0,1,0)$ and $Q_2$ represented by $(0,0,1)$. You may just as well use, for instance $(0,1,1)$ and $(0,1,2)$ in place of the $Q_1,Q_2$.