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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 125 / 1 / b / Solution

A twist of an elliptic curve $E/K$ is an elliptic curve over $K$ that becomes isomorphic to $E$ over an algebraic closure of $K$. Twists are classified by the Galois-cohomology set $H^1(G_K,\operatorname{Aut}(E_{\overline K}))$.


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