Twists of elliptic 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.
Applications of twists include cryptography, the solution of Diophantine equations, and when generalized to hyperelliptic curves, the study of the Sato–Tate conjecture.

Quadratic twist

First assume is a field of characteristic different from 2. Let be an elliptic curve over of the form:
Given not a square in, the quadratic twist of is the curve, defined by the equation:
or equivalently
The two elliptic curves and are not isomorphic over, but rather over the field extension. Qualitatively speaking, the arithmetic of a curve and its quadratic twist can look very different in the field, while the complex analysis of the curves is the same; and so a family of curves related by twisting becomes a useful setting in which to study the arithmetic properties of elliptic curves.
Twists can also be defined when the base field is of characteristic 2. Let be an elliptic curve over of the form:
Given such that is an irreducible polynomial over, the quadratic twist of is the curve, defined by the equation:
The two elliptic curves and are not isomorphic over, but over the field extension.

Quadratic twist over finite fields

If is a finite field with elements, then for all there exist a such that the point belongs to or . In fact, if is on just one of the curves, there is exactly one other on that same curve.
As a consequence, or equivalently, where is the trace of the Frobenius endomorphism of the curve.

Quartic twist

It is possible to "twist" elliptic curves with j-invariant equal to 1728 by quartic characters; twisting a curve by a quartic twist, one obtains precisely four curves: one is isomorphic to, one is its quadratic twist, and only the other two are really new. Also in this case, twisted curves are isomorphic over the field extension given by the twist degree.

Cubic twist

Analogously to the quartic twist case, an elliptic curve over with j-invariant equal to zero can be twisted by cubic characters. The curves obtained are isomorphic to the starting curve over the field extension given by the twist degree.

Generalization

Twists can be defined for other smooth projective curves as well. Let be a field and be curve over that field, i.e., a projective variety of dimension 1 over that is irreducible and geometrically connected. Then a twist of is another smooth projective curve for which there exists a -isomorphism between and, where the field is the algebraic closure of.