An elliptic curve twist of an elliptic curve 
over a field 
is an elliptic
curve over 
that becomes isomorphic to 
over an algebraic closure
of 
.
Thus twists may fail to be isomorphic over the base field even though they have the same j-invariant.
When the field characteristic is not 2, a common example is a quadratic twist, which becomes
isomorphic to the original curve over a quadratic
extension of 
.
