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

A twist of an elliptic curve $E/\mathbb{Q}$ is an elliptic curve that becomes isomorphic to $E$ over $\overline{\mathbb{Q}}$. WriteFor $d\in {\mathbb{Q}}^{\times }$, its quadratic twist may be writtenOver $\mathbb{Q}(\sqrt{d})$, the map $(x,y)\mapsto (d^{-1}x,d^{-3/2}y)$ identifies $E^d$ with $E$.

The hypothesis $j(E)\ne 0,1728$ impliesConsequently twists are classified bywhere 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.