On the family of elliptic curves $y^2=x^3-m^2x + (pqr)^2$
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909827621453824 |
|---|---|
| author | Ghosh, Arkabrata |
| author_facet | Ghosh, Arkabrata |
| contents | In this article, we consider a family of elliptic curves defined by $E_{m}: y^2= x^3 -m^2 x + (pqr)^2 $ where $m $ is a positive integer and $p, q, ~\text{and}~ r$ are distinct odd primes and study the torsion as well the rank of $E_{m}(\mathbb{Q})$. More specifically, we proved that if $m \not \equiv 0 \pmod{3}, m \not \equiv 0 \pmod{4} ~\text{and}~ m \equiv 2 \pmod {2^{k}}$ where $k \geq 5$, then the torsion subgroup of $E_{m}(\mathbb{Q})$ is trivial and lower bound of the $\mathbb{Q}$ rank of this family of elliptic curves is $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01213 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the family of elliptic curves $y^2=x^3-m^2x + (pqr)^2$ Ghosh, Arkabrata Number Theory 11G05, 14G05 In this article, we consider a family of elliptic curves defined by $E_{m}: y^2= x^3 -m^2 x + (pqr)^2 $ where $m $ is a positive integer and $p, q, ~\text{and}~ r$ are distinct odd primes and study the torsion as well the rank of $E_{m}(\mathbb{Q})$. More specifically, we proved that if $m \not \equiv 0 \pmod{3}, m \not \equiv 0 \pmod{4} ~\text{and}~ m \equiv 2 \pmod {2^{k}}$ where $k \geq 5$, then the torsion subgroup of $E_{m}(\mathbb{Q})$ is trivial and lower bound of the $\mathbb{Q}$ rank of this family of elliptic curves is $2$. |
| title | On the family of elliptic curves $y^2=x^3-m^2x + (pqr)^2$ |
| topic | Number Theory 11G05, 14G05 |
| url | https://arxiv.org/abs/2403.01213 |