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A note on the Selmer group of the elliptic curve \(y^ 2=x^ 3+Dx\). (English) Zbl 1040.11042

Summary: We present an explicit formula for the Selmer rank of the elliptic curve \(y^2=x^3+Dx\). Using this formula, we give some results analogous to B. Iskra’s theorem [ibid. 72, 168–169 (1996; Zbl 0877.11014)].

MSC:

11G05 Elliptic curves over global fields

Citations:

Zbl 0877.11014
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References:

[1] Aoki, N.: On the 2-Selmer groups of elliptic curves arising from the congruent number problem. Comment. Math. Univ. St. Paul., 48 , 77-101 (1999). · Zbl 0934.11030
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[5] Iskra, B.: Non-congruent numbers with arbitrarily many prime factors congruent to 3 modulo 8. Proc. Japan Acad., 72A , 168-169 (1996). · Zbl 0877.11014 · doi:10.3792/pjaa.72.168
[6] Goto, T.: Calculation of Selmer groups of elliptic curves with rational 2-torsions and \(\theta\)-congruent number problem. Comment. Math. Univ. St. Paul (to appear). · Zbl 1079.11029
[7] Heath-Brown, D. R.: The size of Selmer groups for the congruent number problem. II. Invent. Math., 118 , 331-370 (1994). · Zbl 0815.11032 · doi:10.1007/BF01231536
[8] Silverman, J. H.: The Arithmetic of Elliptic Curves. Graduate Texts in Math., vol. 106, Springer, New York (1986). · Zbl 0585.14026
[9] Yoshida, S.: On the equation \(y^ 2=x^ 3+pqx\). Comment. Math. Univ. St. Paul., 49 , 23-42 (2000). · Zbl 0983.11033
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