Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 12A30, 12A50

Retrieve articles in all journals with MSC: 12A30, 12A50


Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1978-0480416-4
PII: S 0025-5718(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