Prescribed ideal classes and elliptic points in Ennola cubic fields

For every prime $\mathrm{\ell }\ge 5$ and integer $n\ge 2$, we construct an Ennola cubic field with a specified prime above $\mathrm{\ell }$ 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.