Twists of elliptic curves
Introduction 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, the solution of Diophantine equations, and when generalized to hyperelliptic curves, the study of the Sato–Tate conjecture.
Quadratic twist First assume  is a field of characteristic different from 2.Let  be an elliptic curve over  of the form:: Given  not a square in , the '''quadratic twist''' of  is the curve , defined by the equation:: or equivalently: The two elliptic curves  and  are not isomorphic over , 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.Twists can also be defined when the base field  is of characteristic 2.Let  be an elliptic curve over  of the form:: Given  such that  is an irreducible polynomial over , the '''quadratic twist''' of  is the curve , 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.
Quartic twist It is possible to "twist" elliptic curves with j-invariant equal to 1728 by quartic characters; twisting a curve  by a '''quartic twist''', one obtains precisely four curves: one is isomorphic to , 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  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.
Generalization Twists can be defined for other smooth projective curves as well. Let  be a field and  be curve over that field, i.e., a projective variety of dimension 1 over  that is irreducible and geometrically connected. Then a twist  of  is another smooth projective curve for which there exists a -isomorphism between  and , where the field  is the algebraic closure of .
Examples * Twisted Hessian curves* Twisted Edwards curve* Twisted tripling-oriented Doche–Icart–Kohel curve
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