Twists of curves
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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.
Given not a square in , the quadratic twist of is the curve , defined by the equation:
The two elliptic curves and are not isomorphic over , but rather over the field extension .
Now assume K is of characteristic 2. Let E be an elliptic curve over K of the form:
Given such that is an irreducible polynomial over K, the quadratic twist of E is the curve Ed, 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 either or . In fact, if is on just one of the curves, there is exactly one other on that same curve (which can happen if the characteristic is not ).
As a consequence, or equivalently
where 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; 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.
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.
- P. Stevenhagen (2008). Elliptic Curves (PDF). Universiteit Leiden.
- F. Gouvea, B.Mazur (1991). The square-free sieve and the rank of elliptic curves (PDF). Journal of American Mathematical Society, Vol 4, Num 1.