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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.

azerbajdzan
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    For any $ad-bc=\pm 1$ and any torsion points $P_1,P_2,P_3$ with $P_3\ne O$ you can replace $T_2,Q_1,Q_2$ by $P_3,aQ_1+bQ_2+P_1,cQ_1+dQ_2+P_2$ and every generators are of this form. – reuns Oct 30 '19 at 16:37
  • @reuns That's an answer. Why are you answering in the comments? – Arthur Oct 30 '19 at 16:44
  • Well the answer is that $GL_2(\Bbb{Z}) = { \det(M) = \pm 1}$ – reuns Oct 30 '19 at 16:46
  • @reuns: Thank you for your answer. Is it a hard problem to express given random point on elliptic curve as a linear combination of given generators or not? I mean to find coefficients $a$, $b$, $c$ for random point. I guess it is easy for rank 1 curves but I can not imagine how to do it for higher ranks. – azerbajdzan Nov 04 '19 at 22:25

2 Answers2

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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 $A$.

Slup
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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 $Q_1,Q_2$.

Arthur
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