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Computation of the $ 2$-rank of pure cubic fields

Authors: H. Eisenbeis, G. Frey and B. Ommerborn
Journal: Math. Comp. 32 (1978), 559-569
MSC: Primary 12A30; Secondary 12A50
MathSciNet review: 0480416
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Abstract: For $ k \in {\mathbf{Z}}\backslash \{ 0\} $ there is a close connection between a certain subgroup of the Selmer group of the elliptic curve given by: $ {y^2} = {x^3} + k$, and the group of elements of order 2 of the class group $ {\text{Cl}}(k)$ of $ {\mathbf{Q}}(\sqrt[3]{k})$ denoted by $ {\text{Cl}_2}(k)$ (cf. [4]). In the following paper we give some consequences of this fact, that make the computation of $ {\text{Cl}_2}(k)$ considerably easier. For $ k < 10\,000$ we compute $ {\text{Cl}_2}(k)$ by methods developed in [2], and by using [1] we get the structure of the 2-primary part of $ {\text{Cl}}(k)$ with the exception of 39 cases.

References [Enhancements On Off] (What's this?)

  • [1] P. BARRUCAND, H. C. WILLIAMS & L. BANIUK, "A computational technique for determining the class number of a pure cubic field," Math. Comp., v. 30, 1976, pp. 312-323. MR 0392913 (52:13726)
  • [2] B. J. BIRCH & H. P. F. SWINNERTON-DYER, "Notes on elliptic curves. I," J. Reine Angew. Math., v. 212, 1963, pp. 7-25. MR 0146143 (26:3669)
  • [3] H. EISENBEIS & B. OMMERBORN, Die Berechnung der 2-Klassenzahl rein kubischer Körper mit Hilfe der Selmergruppen gewisser elliptischer Kurven, Diplomarbeit, Saarbrücken, 1977.
  • [4] G. FREY, "Die Klassengruppen quadratischer und kubischer Zahlkörper und die Selmergruppen gewisser elliptischer Kurven," Manuscripta Math., v. 16, 1975, pp. 333-362. MR 0379504 (52:409)
  • [5] E. LUTZ, "Sur l'équation $ {y^2} = {x^3} - AX - B$ dans les corps $ \mathfrak{p}$-adiques," J. Reine Angew. Math., v. 177, 1937, pp. 237-247.
  • [6] A. NÉRON, "Modèles minimaux des variétés abéliennes sur les corps locaux et globaux," Publ. Math. Inst. Hautes Études Sci., v. 21, 1964, 128 pp. MR 0179172 (31:3423)
  • [7] J. TATE, WC Groups Over $ \mathfrak{P}$-Adic Fields, Séminaire Bourbaki, Vol. 10, No. 156, 1957, 13 pp.

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Keywords: Pure cubic fields, elements of order 2 of the class group, Selmer group of elliptic curves, computation of 2-coverings of elliptic curves
Article copyright: © Copyright 1978 American Mathematical Society

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