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

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.