  Bachet elliptic curves are the curves y^2=x^3+a^3 and in this work the group
structure E(F_{p}) of these curves over finite fields F_{p} is considered. It
is shown that there are two possible structures E(F_{p}){\cong}C_{p+1} or
E(F_{p}){\cong}C_{n}{\times}C_{nm}, for m,n{\in}{\mathbb{N}}, according to
p{\equiv}5 (mod6) and p{\equiv}1 (mod6), respectively. A result of Washington
is restated in a more specific way saying that if
E(F_{p}){\cong}Z_{n}{\times}Z_{n}, then p{\equiv}7 (mod12) and p=n^2{\mp}n+1.
