Abstract: In this work, we first give in what fields Fp, the cubic
root of unity lies in F*p, in Qp and in K*p where Qp and K*p denote
the sets of quadratic and non-zero cubic residues modulo p. Then we
use these to obtain some results on the classification of the Bachet
elliptic curves y2 ≡ x3 +a3 modulo p, for p ≡ 1 (mod 6) is prime.