Twists of curves

In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is isomorphic to E over an algebraic closure of K. In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny. Some curves have higher order twists such as cubic and quartic twists. The curve and its twists have the same j-invariant.

First assume K is a field of characteristic different from 2. Let E be an elliptic curve over K of the form:

Given ${\displaystyle d\in K\setminus K^{2}}$ and ${\displaystyle d\neq 0}$, the quadratic twist of ${\displaystyle E}$ is the curve ${\displaystyle E^{d}}$, defined by the equation:

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