In this note we determine all birational isomorphism types of real elliptic curves and show that it is the same as the orbit space of smooth cubic real curves in real projective space under linear projective equivalence. There are two families, each depending polynomially on a real parameter in a open subinterval of R. We further show that the complexification of a real elliptic curve has exactly two real forms. Thus the real elliptic curves come in pairs which are isomorphic over C. Finally, the map taking a real elliptic curve to its j-invariant maps the two families onto the real line in C, intersecting only at the value 1728, the special curve with 4 automorphisms and two topologically distinct real forms.
Broughton, Sean A., "The Birational Isomorphism Types of Smooth Real Elliptic Curves" (2004). Mathematical Sciences Technical Reports (MSTR). 44.