The group structure of Bachet elliptic curves over finite fields $F_p$

Abstract

Bachet elliptic curves are the curves y 2 = x 3 + a 3 and in this work the group structure E(Fp) of these curves over finite fields Fp is considered. It is shown that there are two possible structures E(Fp) ∼ = Cp+1 or E(Fp) ∼ = Cn × Cnm, for m, n ∈ N, according to p ≡ 5 (mod 6) and p ≡ 1 (mod 6) , respectively. A result of Washington is restated in a more specific way saying that if E(Fp) ∼ = Zn × Zn, then p ≡ 7 (mod 12) and p = n 2 ∓ n + 1.