Prescribed ideal classes and elliptic points in Ennola cubic fields

Source abstract

For every prime $\ell\geq5$ and integer $n\geq2$, we construct an Ennola cubic field with a specified prime above $\ell$ of exact ideal-class order $2^n$. The same squareclass obstructions give four independent rational points on associated elliptic curves. A Pell equation describes all parameters satisfying the required square and congruence conditions. A genus-one base change gives five independent sections whose generated subgroup is saturated at two. It also yields infinitely many geometrically nonisomorphic rational fibres of rank at least five.