Czechoslovak Mathematical Journal, Vol. 72, No. 3, pp. 681-693, 2022
On a family of elliptic curves of rank at least 2
Kalyan Chakraborty, Richa Sharma
Received March 22, 2021. Published online June 6, 2022.
Abstract: Let $C_m \colon y^2 = x^3 - m^2x +p^2q^2$ 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 $m \not\equiv0 \pmod3$, $m \not\equiv0 \pmod4$ and $m \equiv2 \pmod{64}$ with neither $p$ nor $q$ dividing $m$.
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Affiliations: Kalyan Chakraborty, Richa Sharma (corresponding author), Kerala School of Mathematics, Kunnamangalam PO, Kozhikode-673571, Kerala, India, e-mail: kalychak@ksom.res.in, richa@ksom.res.in
