Elliptic curves of type \(y^2 = x^3 - 3pqx\) having ranks zero and one.(English)Zbl 1532.11074
Summary: The group of rational points on an elliptic curve over \(\mathbb{Q}\) is always a finitely generated Abelian group, hence isomorphic to \(\mathbb{Z}^r \times G\) with \(G\) a finite Abelian group. Here, \(r\) is the rank of the elliptic curve. In this paper, we determine sufficient conditions that need to be set on the prime numbers \(p\) and \(q\) so that the elliptic curve \(E: y^2 = x^3 - 3pqx\) over \(\mathbb{Q}\) would possess a rank zero or one. Specifically, we verify that if distinct primes \(p\) and \(q\) satisfy the congruence \(p \equiv q \equiv 5\pmod{24}\), then \(E\) has rank zero. Furthermore, if \(p \equiv 5\pmod{12}\) is considered instead of a modulus of 24, then \(E\) has rank zero or one. Lastly, for primes of the form \(p = 24k + 17\) and \(q = 24\ell + 5\), where \(9k + 3\ell + 7\) is a perfect square, we show that \(E\) has rank one.
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