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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
DOI: https://doi.org/10.1090/S0025-5718-1978-0480416-4
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.


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  • [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)
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  • [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.
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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1978-0480416-4
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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