  Building on the positive solution of Pillay's conjecture we present a notion
of "intrinsic" reduction for elliptic curves over a real closed field K. We
compare such notion with the traditional algebro-geometric reduction and
produce a classification of the group of K-points of an elliptic curve E with
three "real" roots according to the way E reduces (algebro-geometrically) and
the geometric complexity of the "intrinsically" reduced curve.
