When is a complex elliptic curve the product of two real algebraic curves? (Q810608)

Recall that a complex projective variety \(V\subseteq {\mathbb{P}}^ n({\mathbb{C}})\) can be, in the obvious way, considered as a real algebraic variety and, as such, will be denoted by \(V_{{\mathbb{R}}}\). Moreover, it can be shown that \(V_{{\mathbb{R}}}\) is an affine real algebraic variety, by which we mean a locally ringed space isomorphic to an algebraic subset of \({\mathbb{R}}^ n\), for some n, equipped with the sheaf of \({\mathbb{R}}\)-valued regular functions. - The paper contains the full answer to the following problem: Given a complex elliptic curve E, decide when the underlying real algebraic surface \(E_{{\mathbb{R}}}\) is biregularly isomorphic to the product of two real algebraic curves.