Computation of the $2$-rank of pure cubic fields
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- by H. Eisenbeis, G. Frey and B. Ommerborn PDF
- Math. Comp. 32 (1978), 559-569 Request permission
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
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Additional Information
- © Copyright 1978 American Mathematical Society
- 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