On the Family of Elliptic Curves y²=x³-5pqx

2025/05/27 by Ghosh, Arkabrata
#11D25 #11G05 #14G05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.21655

Abstract

This article considers the family of elliptic curves given by E_pq: y²=x³-5pqx and certain conditions on odd primed p and q. More specifically, we have proved that if p ≡ 33 \pmod 40 and q ≡ 7 \pmod 40, then the rank of E_pq is zero over both ℚ and ℚ(i) . Furthermore, if the primes p and q are of the form 40k + 33 and 40l + 27, where k,l ∈ ℤ such that (25k+ 5l +21) is a perfect square, then the given family of elliptic curves has rank one over ℚ and rank two over ℚ(i). Finally, we have shown that torsion of E_pq over ℚ is isomorphic to ℤ/ 2ℤ.

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