= Solution
A <twist of an elliptic curve> $E/\mathbb Q$ is an elliptic curve that becomes isomorphic to $E$ over $\overline{\mathbb Q}$. Write
$$
E:y^2=x^3+Ax+B.
$$
For $d\in\mathbb Q^\times$, its <quadratic twist> may be written
$$
E^d:y^2=x^3+d^2Ax+d^3B.
$$
Over $\mathbb Q(\sqrt d)$, the map $(x,y)\mapsto(d^{-1}x,d^{-3/2}y)$ identifies $E^d$ with $E$.
The hypothesis $j(E)\ne0,1728$ implies
$$
\operatorname{Aut}_{\overline{\mathbb Q}}(E)=\{\pm1\}.
$$
Consequently twists are classified by
$$
H^1(G_{\mathbb Q},\{\pm1\})
\simeq\operatorname{Hom}(G_{\mathbb Q},\{\pm1\})
\simeq\mathbb Q^\times/(\mathbb Q^\times)^2,
$$
where the last identification is <Kummer theory>. In the explicit equations, $E^d\simeq_{\mathbb Q}E^e$ exactly when $d/e$ is a rational square. Every nonzero rational square class has a unique square-free integer representative, including its sign, so the twists are parametrized by the nonzero square-free integers.
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