On the Family of Elliptic Curves $y^2=x^3-5pqx$
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915533276839936 |
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| author | Ghosh, Arkabrata |
| author_facet | Ghosh, Arkabrata |
| contents | This article considers the family of elliptic curves given by $E_{pq}: y^2=x^3-5pqx$ and certain conditions on odd primed $p$ and $q$. More specifically, we have proved that if $p \equiv 33 \pmod {40}$ and $ q \equiv 7 \pmod {40}$, then the rank of $E_{pq}$ is zero over both $ \mathbb{Q} $ and $ \mathbb{Q}(i) $. Furthermore, if the primes $ p $ and $q$ are of the form $ 40k + 33 $ and $ 40l + 27$, where $k,l \in \mathbb{Z}$ such that $(25k+ 5l +21)$ is a perfect square, then the given family of elliptic curves has rank one over $\mathbb{Q}$ and rank two over $\mathbb{Q}(i)$. Finally, we have shown that torsion of $E_{pq}$ over $\mathbb{Q}$ is isomorphic to $ \mathbb{Z}/ 2\mathbb{Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21655 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Family of Elliptic Curves $y^2=x^3-5pqx$ Ghosh, Arkabrata Number Theory 11D25, 11G05, 14G05 This article considers the family of elliptic curves given by $E_{pq}: y^2=x^3-5pqx$ and certain conditions on odd primed $p$ and $q$. More specifically, we have proved that if $p \equiv 33 \pmod {40}$ and $ q \equiv 7 \pmod {40}$, then the rank of $E_{pq}$ is zero over both $ \mathbb{Q} $ and $ \mathbb{Q}(i) $. Furthermore, if the primes $ p $ and $q$ are of the form $ 40k + 33 $ and $ 40l + 27$, where $k,l \in \mathbb{Z}$ such that $(25k+ 5l +21)$ is a perfect square, then the given family of elliptic curves has rank one over $\mathbb{Q}$ and rank two over $\mathbb{Q}(i)$. Finally, we have shown that torsion of $E_{pq}$ over $\mathbb{Q}$ is isomorphic to $ \mathbb{Z}/ 2\mathbb{Z}$. |
| title | On the Family of Elliptic Curves $y^2=x^3-5pqx$ |
| topic | Number Theory 11D25, 11G05, 14G05 |
| url | https://arxiv.org/abs/2505.21655 |