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 K {\displaystyle K} is a field of characteristic different from 2. Let E {\displaystyle E} be an elliptic curve over K {\displaystyle K} of the form:
y 2 = x 3 + a 2 x 2 + a 4 x + a 6 . {\displaystyle y^{2}=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,}
Given d ≠ 0 {\displaystyle d\neq 0} not a square in K {\displaystyle K} , the quadratic twist of E {\displaystyle E} is the curve E d {\displaystyle E^{d}} , defined by the equation:
d y 2 = x 3 + a 2 x 2 + a 4 x + a 6 . {\displaystyle 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 . {\displaystyle y^{2}=x^{3}+da_{2}x^{2}+d^{2}a_{4}x+d^{3}a_{6}.\,}
The two elliptic curves E {\displaystyle E} and E d {\displaystyle E^{d}} are not isomorphic over K {\displaystyle K} , but rather over the field extension K ( d ) {\displaystyle K({\sqrt {d}})} . Qualitatively speaking, the arithmetic of a curve and its quadratic twist can look very different in the field K {\displaystyle K} , 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 K {\displaystyle K} is of characteristic 2. Let E {\displaystyle E} be an elliptic curve over K {\displaystyle K} of the form:
y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 . {\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,}
Given d ∈ K {\displaystyle d\in K} such that X 2 + X + d {\displaystyle X^{2}+X+d} is an irreducible polynomial over K {\displaystyle K} , the quadratic twist of E {\displaystyle E} is the curve E d {\displaystyle E^{d}} , defined by the equation:
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