curves with rational subgroups of order three

In this article we present a characterization of elliptic curves defined over a finite field F q which possess a rational subgroup of order three. There are two posible cases depending on the rationality of the points in these groups. We show that for finite fields F q , q ≡ −1 mod 3, all ellip-tic curves with a point of order 3, they have another rational subgroup whose points are not defined over the finite field. If q ≡ −1 mod 3, this is no true; but there exits a one to one correspondence between curves with points of order 3 and curves with rational subgroups whose points are not rational.