Unit Indices of Shanks Orders

Abstract

For an integer $t\geq-1$, let $\theta_t$ be the largest real root of $g_t(X)=X^3-tX^2-(t+3)X-1$, and set $R_t=\mathbb{Z}[\theta_t]\subseteq\mathcal{O}_t=\mathcal{O}_{\mathbb{Q}(\theta_t)}$, $N_t=[\mathcal{O}_t:R_t]$, and $\varepsilon_t=[\mathcal{O}_t^\times:R_t^\times]$. We determine $\varepsilon_t$ when $N_t$ is squarefree, $27$, or $343$: the only nontrivial indices in these cases are $\varepsilon_3=3$, $\varepsilon_5=7$, $\varepsilon_{12}=13$, and $\varepsilon_{54}=19$. Local conductor calculations give $\varepsilon_t\mid N_t$ for squarefree $N_t$. For arbitrary $N_t$, a regulator comparison shows that $t\geq2N_t$ implies $\varepsilon_t=1$. When $N_t=p^3$ with $p\neq3$ a rational prime, this bound and the index criterion leave at most four possible parameters with $\varepsilon_t>1$ for each fixed $p$. For arbitrary additive index, we also prove that $13\mid\varepsilon_t$ if and only if $t=12$ or $66$, with unit index $13$ in both cases. The proof determines the rational solutions of a plane quartic equation by an explicit genus-two descent and a local Chabauty argument, proving Louboutin's Conjecture 19 on its integral solutions. When $N_t$ is a rational prime, we determine the Picard kernel, the cardinality and fibers of the ideal class monoid over $\operatorname{Pic}(\mathcal{O}_t)$, and the corresponding integral matrix-conjugacy classes.