Twists of curves

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In 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.

Quadratic twist[edit]

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

y^2 = x^3 + a_2 x^2 +a_4 x + a_6. \,

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

dy^2 = x^3 + a_2 x^2 + a_4 x + a_6. \,

or equivalently

y^2 = x^3 + d a_2 x^2 + d^2 a_4 x + d^3 a_6. \,

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

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

y^2 + a_1 x y +a_3 y = x^3 + a_2 x^2 +a_4 x + a_6. \,

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

y^2 + a_1 x y +a_3 y = x^3 + (a_2 + d a_1^2) x^2 +a_4 x + a_6 + d a_3^2. \,

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

Quadratic twist over finite fields[edit]

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

As a consequence,

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

where t_E is the trace of the Frobenius endomorphism of the curve.

Quartic twist[edit]

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[edit]

Analogously to the quartic twist case, an elliptic curve over 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.

Examples[edit]

1.Twisted Hessian curves

2.Twisted Edwards curve

3.Twisted tripling-oriented Doche–Icart–Kohel curve

References[edit]