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.
Contents
Quadratic twist
First assume K is a field of characteristic different from 2. Let E be an elliptic curve over K of the form:
Given
or equivalently
The two elliptic curves
Now assume K is of characteristic 2. Let E be an elliptic curve over K of the form:
Given
The two elliptic curves
Quadratic twist over finite fields
If
As a consequence,
where
Quartic twist
It is possible to "twist" elliptic curves with j-invariant equal to 1728 by quartic characters; twisting a curve E by a quartic twist, one obtains precisely four curves: one is isomorphic to E, 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
Examples
- Twisted Hessian curves
- Twisted Edwards curve
- Twisted tripling-oriented Doche–Icart–Kohel curve