Rank of eliptic curve \(y^2=x^3+px\) with \(p\equiv 7\pmod{16}\) (Q2837911)
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scientific article; zbMATH DE number 6184902
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Rank of eliptic curve \(y^2=x^3+px\) with \(p\equiv 7\pmod{16}\) |
scientific article; zbMATH DE number 6184902 |
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8 July 2013
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rank of an elliptic curve
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prime number
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Rank of eliptic curve \(y^2=x^3+px\) with \(p\equiv 7\pmod{16}\) (English)
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The authors use the classical formula \(2^r=\frac{|\alpha(\Gamma)||\overline{\alpha}(\overline{\Gamma})|}{4}\) and simple \(p\)-adic approximation to compute the rank \(r\) of elliptic curves \(E_p\,: y^2=x^3+px\) (\(p\) a prime). They show that the rank is bounded by 2 and that it is equal to 0 if \(p\equiv 7 \pmod{16}\) (these results were already mentioned in Chapter III, Section 6, Example 4 of [\textit{J. H. Silverman} and \textit{J. T. Tate}, Rational points on elliptic curves. New York: Springer-Verlag (1992; Zbl 0752.14034)]).
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