 Quadratic twist of an elliptic curve

For an elliptic curve $E:y^2=f(x)$ over a field of characteristic other than two and $d\in K^{\times }$, the curve $E^{(d)}:d{y}^2=f(x)$ is its quadratic twist by $d$. It becomes isomorphic to $E$ after adjoining $\sqrt{d}$; over a finite field its Frobenius trace is multiplied by the quadratic character of $d$.