Rational Points on Elliptic Curves $y^{2}=x^{3}+a^{3}$ in ${\bf F}_p$ where $p\equiv 1\pmod6$ is Prime

THE NUMBER OF POINTS ON ELLIPTIC CURVES y2= x3+ Ax AND y2= x3+ B3MOD 24

“…In [11], three algorithms to find the number of points on an elliptic curve over a finite field are given. Also in [3], [4] the number of rational points on Frey elliptic curves E : y 2 = x 3 − n 2 x and E : y 2 = x 3 + a 3 are found. The purpose of this paper is to give a straightforward proof of the number of points mod 24 on elliptic curves over finite fields.…”

“…The purpose of this paper is to give a straightforward proof of the number of points mod 24 on elliptic curves over finite fields. One found the number of points on E : y 2 = x 3 + Ax over F p ( [2], [3], [6], [8], [10]). In 2003, H. Park, D. Kim and H. Lee, calculated the number of points on elliptic curves E 0 A : y 2 = x 3 + Ax over F p mod 8 ( [5], [8]).…”

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

“…Here we shall deal with Bachet elliptic curves y 2 = x 3 + a 3 modulo p. Let N p,a denote the number of rational points on this curve. Some results on these curves have been given in [1], and [2].…”

“…We show that this group is isomorphic to a direct product of two cyclic groups C n and C nm , i.e. 2 Bachet Elliptic curves having a group of the form C n × C nm Let E be the curve in (1) . Then its twist is defined as the curve y 2 ≡ x 3 + g 3 a 3 , where g is an element of Q ′ p , the set of quadratic non-residues modulo p. As usual, Q p denotes the set of quadratic residues modulo p. Here note that if…”

“…16 Let p ≡ 1 (mod 6) be a prime. Let E be the curve given by(1) .Then a) a ∈ Q p iff E (F p ) has 2 or 8 elements of order 3. b) a ∈ Q ′ p iff E (F p ) has no elements of order 3.Proof. It is clear from Corollary 8 and Theorem 13.…”