(Twist)
Let \(X\) be a variety over a field \(k\). A twist of \(X\) is a variety \(T\) over \(k\) such that \(T \times_k \Spec K \cong X \times_k \Spec K\) as \(K\)-varieties for some field extension \(K/k\).
(Quadratic twists)
Let \(k\) be a field of characteristic not \(2\), \(f \in k[x]\) a monic cubic without repeated roots, \(E \colon y^2 = f(x)\), and \(d \in k^\times\). The quadratic twist \(E_d \colon d y^2 = f(x)\) is a twist of \(E\): over \(K = k(\sqrt{d})\), the substitution \(y \mapsto \sqrt{d}\, y\) is an isomorphism \(E_d \times_k K \to E \times_k K\).
Let \(E\) and \(d\) be as above. If \(d \in (k^\times)^2\), then \(E_d \cong E\) over \(k\). If \(j(E) \neq 0, 1728\) and \(d \notin (k^\times)^2\), then \(E_d \not\cong E\) over \(k\).
The hypothesis on \(j\) is needed. For \(E \colon y^2 = x^3 - x\), with \(j(E) = 1728\), and \(d = -1\), the twist \(E_{-1} \colon -y^2 = x^3 - x\) is isomorphic to \(E\) over every field \(k\) of characteristic not \(2\), by \((x, y) \mapsto (-x, y)\), whether or not \(-1\) is a square in \(k\).