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On the equation \(y^2 = x^3 - pqx\). (English) Zbl 1423.11106

Summary: We consider certain quartic twists of an elliptic curve. We establish the rank of these curves under the Birch and Swinnerton-Dyer conjecture and obtain bounds on the size of Shafarevich-Tate group of these curves. We also establish a reduction between the problem of factoring integers of a certain form and the problem of computing rational points on these twists.

MSC:

11G05 Elliptic curves over global fields
11D25 Cubic and quartic Diophantine equations
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture

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References:

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