Twists of elliptic curves (awaiting review)

A twist of an elliptic curve $E$ defined over a field $K$ is another elliptic curve $E'$, also defined over $K$, which is isomorphic to $E$ over the algebraic closure of $K$.

Two elliptic curves are twists if an only if they have the same $j$-invariant.

For elliptic curves $E$ with $j(E)\not=0, 1728$, the only twists of $E$ are its quadratic twists $E^{(d)}$. Provided that the characteristic of $K$ is not $2$, the nontrivial quadratic twists of $E$ are in bijection with the nontrivial elements $d$ of $K^*/(K^*)^2$, and $E^{(d)}$ is isomorphic to $E$ over the quadratic extension $K(\sqrt{d})$.

Over fields of characteristic not $2$ or $3$, elliptic curves with $j$-invariant $1728$ also admit quartic twists, parametrised by $K^*/(K^*)^4$, and elliptic curves with $j$-invariant $0$ also admit sextic twists, parametrised by $K^*/(K^*)^6$. Elliptic curves $E$ over fields $K$ of characteristic $2$ and $3$ with $j(E)=0=1728$ have nonabelian automorphism groups, and their twists are more complicated to describe, being in all cases parametrised by $H^1(\Gal(\overline{K}/K), \Aut(E))$.

Elliptic curve twists are a special case of twists of abelian varieties.