On explicit units in a family of cubic fields and the arithmetic of curves arising from powers of these units

For an integer b≠0,1 let Kb be the cubic field generated by the unique real root of x³-3bx-b³, let η₀ be its fundamental unit, and let ηb be the explicit unit studied in the companion paper (arXiv:2609.37389). Write ηb=η₀ⁿ⁽ᵇ⁾. Deciding whether $n(b)$ is divisible by q amounts to finding the integral points of a plane curve Cq of degree $2q$. We prove that for every integer q≥1 the curve Cq is irreducible with g(Cq)=q-12, the genus of a smooth plane curve of degree q rather than of degree $2q$. The proof is not a genus computation but a structural one: Cq is the inverse image, under the q-power map of a two-dimensional torus, of the curve traced by the minimal polynomial of ηb as b varies, and the formula falls out of two applications of Riemann-Hurwitz in the resulting tower. As a consequence the problem splits into exactly three regimes: $q=2$ rational, $q=3$ elliptic, and q≥4 of general type. In particular only finitely many b have q n(b) for each q≥4. A lower bound for linear forms in two logarithms gives $n(b)<6082138$ for every b, so n(b)∈\1,2\ for all but finitely many b, though not effectively. We add two computations: $n(b)$ is determined outright for |b|≤10³⁰, and the rational points of C₄ are determined by quadratic Chabauty, giving 4 n(b) b∈\-75,-3,5,96\. We state precisely which parts of the latter rest on certified arithmetic and which do not. Finally, we show that Kb Kb' with b≠ b' holds for ,b'\=\-1,5\, and otherwise only if one of the two parameters, c say, satisfies |c|>10³⁰ and $n(c)$ is divisible by a prime q≥5. There are only finitely many such pairs.