In this article, we consider the quadratic twists of the Mordell curve \(𝐸\) : 𝑦2=𝑥3−1. For a square-free integer k, the quadratic twist of \(𝐸\) is given by \(𝐸\)𝑘 : 𝑦2=𝑥3−𝑘3. We prove that there exist infinitely many k for which the rank of \(𝐸\)𝑘 is 0, by modifying existing techniques. Moreover, using simple tools, we produce precise values of k for which the rank of \(𝐸\)𝑘 is 0. We also construct an infinite family of curves {\(𝐸\)𝑘} such that the rank of each \(𝐸\)𝑘 is positive. It was conjectured by Silverman that there are infinitely many primes p for which \(𝐸\)𝑝(\(\mathbb{Q}\)) has a positive rank as well as infinitely many primes q for which \(𝐸\)𝑞(\(\mathbb{Q}\)) has rank 0. We show, assuming the Parity Conjecture that Silverman’s conjecture is true for this family of quadratic twists.
In this article, we consider the quadratic twists of the Mordell curve \(𝐸\) : 𝑦2=𝑥3−1. For a square-free integer k, the quadratic twist of \(𝐸\) is given by \(𝐸\)𝑘 : 𝑦2=𝑥3−𝑘3. We prove that there exist infinitely many k for which the rank of \(𝐸\)𝑘 is 0, by modifying existing...
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In this article, we consider the quadratic twists of the Mordell curve \(𝐸\) : 𝑦2=𝑥3−1. For a square-free integer k, the quadratic twist of \(𝐸\) is given by \(𝐸\)𝑘 : 𝑦2=𝑥3−𝑘3. We prove that there exist infinitely many k for which the rank of \(𝐸\)𝑘 is 0, by modifying existing techniques. Moreover, using simple tools, we produce precise values of k for which the rank of \(𝐸\)𝑘 is 0. We also construct an infinite family of curves {\(𝐸\)𝑘} such that the rank of each \(𝐸\)𝑘 is positive. It was conjectured by Silverman that there are infinitely many primes p for which \(𝐸\)𝑝(\(\mathbb{Q}\)) has a positive rank as well as infinitely many primes q for which \(𝐸\)𝑞(\(\mathbb{Q}\)) has rank 0. We show, assuming the Parity Conjecture that Silverman’s conjecture is true for this family of quadratic twists.
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