THE GROUP STRUCTURE OF BACHET ELLIPTIC CURVES OVER FINITE FIELDS F p

AbstractBachet elliptic curves are the curves y 2 = x 3 +a 3 and in this work thegroup structure E(F p ) of these curves over finite fields F p is considered. Itis shown that there are two possible structures E(F p ) ∼= C p+1 or E(F p ) ∼=C n ×C nm , for m,n∈ N,according to p≡ 5 (mod6) and p≡ 1 (mod6),respectively. A result of Washington is restated in a more specific waysaying that if E(F p ) ∼= Z n ×Z n ,then p≡ 7 (mod12) and p= n 2 ∓n+1. 1 Introduction 12 Let p be a prime. We shall consider the elliptic curvesE : y 2 ≡ x 3 +a 3 (modp) (1)where a is an element of F ∗p = F − {0}. Let us denote the group of the pointson E by E (F p ).If Fis a field, then an elliptic curve over Fhas, after a change of variables,a formy 2 = x 3 +Ax +Bwhere A and B ∈ Fwith 4A 3 +27B 2 6= 0 in F. Here D = −164A 3 +27B 2 iscalled the discriminant of the curve. Elliptic curves are studied over finite andinfinite fields. Here we take Fto be a finite prime field F p with characteristicp > 3. Then A,B ∈ F p and the set of points (x,y) ∈ F