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High Quality Content by WIKIPEDIA articles! In mathematics an elliptic curve E over a field K has its 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 a 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.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 obtain 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.