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arXiv · 2401.00215

On families of elliptic curves $E_{p,q}:y^2=x^3-pqx$ that intersect the same line $L_{a,b}:y=\frac{a}{b}x$ of rational slope

Eldar Sultanow·Anja Jeschke·Amir Darwish Tfiha·Madjid Tehrani·William J Buchanan
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Abstract

Let $p$ and $q$ be two distinct odd primes, $p<q$ and $E_{p,q}:y^2=x^3-pqx$ be an elliptic curve. Fix a line $L_{a.b}:y=\frac{a}{b}x$ where $a\in \mathbb{Z},b\in \mathbb{N}$ and $(a,b)=1$. We study sufficient conditions that $p$ and $q$ must satisfy so that there are infinitely many elliptic curves $E_{p,q}$ that intersect $L_{a,b}$.

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Eldar Sultanow, Anja Jeschke, Amir Darwish Tfiha, Madjid Tehrani, William J Buchanan. 2023-12-30. On families of elliptic curves $E_{p,q}:y^2=x^3-pqx$ that intersect the same line $L_{a,b}:y=\frac{a}{b}x$ of rational slope. https://arxiv.org/abs/2401.00215

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