If $$E : y^2 = x^3 + ax^2 + bx + c$$ is an elliptic curve defined over the Rational Field, there exists at least $2$ points in $E$ with integer coordinates? Any theorem state this or something in the same way?

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    $\begingroup$ No, there can be none (e.g. $y^2=x^3+6$) and there can be exactly one (e.g. $y^2=x^3+27$). $\endgroup$ – Wojowu Mar 3 '18 at 21:51

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