Rank of the family of elliptic curves $y^2 = x^3- 5px$
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| Format: | Preprint |
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2025
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| _version_ | 1866908756523089920 |
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| author | Ghosh, Arkabrata |
| author_facet | Ghosh, Arkabrata |
| contents | This article considers the family of elliptic curves given by $E_{p}: y^2=x^3-5px$ and certain conditions on an odd prime $p$. More specifically, we have shown that if $p \equiv 7, 23 \pmod {40}$, then the rank of $E_{p}$ is zero for both $ \mathbb{Q} $ and $ \mathbb{Q}(i) $. Furthermore, if the prime $ p $ is of the form $ 40k_1 + 3 $ or $ 40k_2 + 27$, where $k_1, k_2 \in \mathbb{Z}$ such that $(5k_1+1)$ or $(5k_2 +4)$ are perfect squares, then the given family of elliptic curves has rank one over $\mathbb{Q}$ and rank two over $\mathbb{Q}(i)$. Moreover, if the prime $ p $ is of the form $ 40k_3 + 11 $ or $ 40k_4 + 19$ where $k_3 ~\text{and}~ k_4 \in \mathbb{Z}$ such that $(160k_3+49)$ or $(160k_4 + 81) $ are perfect squares, then the given family of elliptic curves has rank at least one over $\mathbb{Q}$ and rank at least two over $\mathbb{Q}(i)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_09169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rank of the family of elliptic curves $y^2 = x^3- 5px$ Ghosh, Arkabrata Number Theory 11D25, 11G05, 14G05 This article considers the family of elliptic curves given by $E_{p}: y^2=x^3-5px$ and certain conditions on an odd prime $p$. More specifically, we have shown that if $p \equiv 7, 23 \pmod {40}$, then the rank of $E_{p}$ is zero for both $ \mathbb{Q} $ and $ \mathbb{Q}(i) $. Furthermore, if the prime $ p $ is of the form $ 40k_1 + 3 $ or $ 40k_2 + 27$, where $k_1, k_2 \in \mathbb{Z}$ such that $(5k_1+1)$ or $(5k_2 +4)$ are perfect squares, then the given family of elliptic curves has rank one over $\mathbb{Q}$ and rank two over $\mathbb{Q}(i)$. Moreover, if the prime $ p $ is of the form $ 40k_3 + 11 $ or $ 40k_4 + 19$ where $k_3 ~\text{and}~ k_4 \in \mathbb{Z}$ such that $(160k_3+49)$ or $(160k_4 + 81) $ are perfect squares, then the given family of elliptic curves has rank at least one over $\mathbb{Q}$ and rank at least two over $\mathbb{Q}(i)$. |
| title | Rank of the family of elliptic curves $y^2 = x^3- 5px$ |
| topic | Number Theory 11D25, 11G05, 14G05 |
| url | https://arxiv.org/abs/2509.09169 |