On a family of elliptic curves of rank at least 2

Chakraborty, Kalyan, and Sharma, Richa. "On a family of elliptic curves of rank at least 2." Czechoslovak Mathematical Journal 72.3 (2022): 681-693. <http://eudml.org/doc/298384>.

@article{Chakraborty2022,
abstract = {Let $C_\{m\} \colon y^\{2\} = x^\{3\} - m^\{2\}x +p^\{2\}q^\{2\}$ be a family of elliptic curves over $\mathbb \{Q\}$, where $m$ is a positive integer and $p$, $q$ are distinct odd primes. We study the torsion part and the rank of $C_m(\mathbb \{Q\})$. More specifically, we prove that the torsion subgroup of $C_\{m\}(\mathbb \{Q\})$ is trivial and the $\mathbb \{Q\}$-rank of this family is at least 2, whenever $m \lnot \equiv 0 \hspace\{4.44443pt\}(\@mod \; 3)$, $m \lnot \equiv 0 \hspace\{4.44443pt\}(\@mod \; 4)$ and $m \equiv 2 \hspace\{4.44443pt\}(\@mod \; 64)$ with neither $p$ nor $q$ dividing $m$.},
author = {Chakraborty, Kalyan, Sharma, Richa},
journal = {Czechoslovak Mathematical Journal},
keywords = {elliptic curve; torsion subgroup; rank},
language = {eng},
number = {3},
pages = {681-693},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a family of elliptic curves of rank at least 2},
url = {http://eudml.org/doc/298384},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Chakraborty, Kalyan
AU - Sharma, Richa
TI - On a family of elliptic curves of rank at least 2
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 3
SP - 681
EP - 693
AB - Let $C_{m} \colon y^{2} = x^{3} - m^{2}x +p^{2}q^{2}$ be a family of elliptic curves over $\mathbb {Q}$, where $m$ is a positive integer and $p$, $q$ are distinct odd primes. We study the torsion part and the rank of $C_m(\mathbb {Q})$. More specifically, we prove that the torsion subgroup of $C_{m}(\mathbb {Q})$ is trivial and the $\mathbb {Q}$-rank of this family is at least 2, whenever $m \lnot \equiv 0 \hspace{4.44443pt}(\@mod \; 3)$, $m \lnot \equiv 0 \hspace{4.44443pt}(\@mod \; 4)$ and $m \equiv 2 \hspace{4.44443pt}(\@mod \; 64)$ with neither $p$ nor $q$ dividing $m$.
LA - eng
KW - elliptic curve; torsion subgroup; rank
UR - http://eudml.org/doc/298384
ER -