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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 |
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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. |
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