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,[1] the solution of Diophantine equations,[2][3] and when generalized to hyperelliptic curves, the study of the Sato–Tate conjecture.
is a field of characteristic different from 2.
be an elliptic curve over
, the quadratic twist of
, defined by the equation: or equivalently The two elliptic curves
, 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.
[5] Twists can also be defined when the base field
be an elliptic curve over
is an irreducible polynomial over
, defined by the equation: The two elliptic curves
, but over the field extension
is a finite field with
on that same curve (which can happen if the characteristic is not
is the trace of the Frobenius endomorphism of the curve.
It is possible to "twist" elliptic curves with j-invariant equal to 1728 by quartic characters;[6] twisting a curve
by a quartic twist, one obtains precisely four curves: one is isomorphic to
Also in this case, twisted curves are isomorphic over the field extension given by the twist degree.
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.
Twists can be defined for other smooth projective curves as well.
be curve over that field, i.e., a projective variety of dimension 1 over
that is irreducible and geometrically connected.
is another smooth projective curve for which there exists a
is the algebraic closure of