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Keywords:
elliptic curve; torsion subgroup; rank
elliptic curve; torsion subgroup; rank
Summary:
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 $m$.
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