Malaysian Journal of Mathematical Sciences, March 2023, Vol. 17, No. 1
Elliptic Curves of Type $y^2=x^3-3pqx$ Having Ranks Zero and One
Mina, R. J. S and Bacani, J. B.
Corresponding Email: jbbacani@up.edu.ph
Received date: 8 February 2022
Accepted date: 24 November 2022
Abstract:
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.
Keywords: elliptic curve; rank of elliptic curve; torsor
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