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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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.

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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