Can there be two different elliptic curve $E\_{1}$ and $E\_{2}$ and two different rational points $P\_{1}$ and $P\_{2}$ such that $P\_{1}, P\_{2} \in E\_{1}$ and $P\_{1}, P\_{2} \in E\_{2}$ but $P\_{1} + P\_{2}$ is a different point for $E\_{1}$ and $E\_{2}$. If so, is it easy for find an example?


Or given two different rational points $P\_{1}$ and $P\_{2}$ is there a unique elliptic curve $E$ such that $P\_{1}, P\_{2} \in E$


Thank you



Can two different elliptic curves have rational points in common
Yes. Let $E\_1$ be the curve defined by the equation $y^2=x(x-1)(x-2)$ and $E\_2$ be the curve $y^2=x(x-1)(x-3)$. Let $P\_1=(0,0)$ and $P\_2=(1,0)$. These points are certainly on both curves.


On $E\_1$, $P\_1+P\_2=(2,0)$ but on $E\_2$ it's $(3,0)$. (Assuming that the point at infinity is the identity.)


In fact, there is a unique elliptic curve passing through every $9$ points in general position.


