The Group Structure of Bachet Elliptic Curves over Finite Fields Fₚ

Original · EN

Bachet elliptic curves are the curves y²=x³+a³ 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 (mod6) and p≡1 (mod6), respectively. A result of Washington is restated in a more specific way saying that if E(Fₚ)Zₙ×Zₙ, then p≡7 (mod12) and p=n²∓n+1.