Geodesic
On a family of elliptic curves of rank at least 2
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 681-693
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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 $m \not \equiv 0 \pmod 3$, $m \not \equiv 0 \pmod 4$ and $m \equiv 2 \pmod {64}$ with neither $p$ nor $q$ dividing $m$.
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 $m \not \equiv 0 \pmod 3$, $m \not \equiv 0 \pmod 4$ and $m \equiv 2 \pmod {64}$ with neither $p$ nor $q$ dividing $m$.
DOI : 10.21136/CMJ.2022.0106-21
Classification : 11G05, 14G05
Keywords: elliptic curve; torsion subgroup; rank 
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Chakraborty, Kalyan; Sharma, Richa. On a family of elliptic curves of rank at least 2. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 681-693. doi: 10.21136/CMJ.2022.0106-21

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