Twists of 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.

First assume K is a field of characteristic different from 2. Let E be an elliptic curve over K of the form:

${\displaystyle y^{2}=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,}$

Given ${\displaystyle d\in K\setminus K^{2}}$ and ${\displaystyle d\neq 0}$, the quadratic twist of ${\displaystyle E}$ is the curve ${\displaystyle E^{d}}$, defined by the equation:

${\displaystyle dy^{2}=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,}$

or equivalently

${\displaystyle y^{2}=x^{3}+da_{2}x^{2}+d^{2}a_{4}x+d^{3}a_{6}.\,}$

The two elliptic curves ${\displaystyle E}$ and ${\displaystyle E^{d}}$ are not isomorphic over ${\displaystyle K}$, but over the field extension ${\displaystyle K({\sqrt {d}})}$.

Now assume K is of characteristic 2. Let E be an elliptic curve over K of the form:

${\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,}$

Given ${\displaystyle d\in K}$ such that ${\displaystyle X^{2}+X+d}$ is an irreducible polynomial over K, the quadratic twist of E is the curve Ed, defined by the equation:

${\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+(a_{2}+da_{1}^{2})x^{2}+a_{4}x+a_{6}+da_{3}^{2}.\,}$

The two elliptic curves ${\displaystyle E}$ and ${\displaystyle E^{d}}$ are not isomorphic over ${\displaystyle K}$, but over the field extension ${\displaystyle K[X]/(X^{2}+X+d)}$.

If ${\displaystyle K}$ is a finite field with ${\displaystyle q}$ elements, then for all ${\displaystyle x}$ there exist a ${\displaystyle y}$ such that the point ${\displaystyle (x,y)}$ belongs to either ${\displaystyle E}$ or ${\displaystyle E^{d}}$. In fact, there are always exactly two such ${\displaystyle y}$ unless the point belongs to both curves (which can happen if the characteristic is not ${\displaystyle 2}$).

As a consequence,

${\displaystyle |E(K)|+|E^{d}(K)|=2q+2}$ or equivalently ${\displaystyle t_{E^{d}}=-t_{E}}$

where ${\displaystyle t_{E}}$ 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 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 ${\displaystyle K}$ 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.

References

• P. Stevenhagen (2008). Elliptic Curves (PDF). Universiteit Leiden.
• C. L. Stewart and J. Top (1995). On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms. Journal of the American Mathematical Society, Vol. 8, No. 4 (Oct., 1995), pp. 943–973. JSTOR 2152834.