Unit Indices of Shanks Orders

Abstract

For an integer $t\geq-1$, let $θ_t$ be the largest real root of $g_t(X)=X^3-tX^2-(t+3)X-1$, and put $R_t=\mathbb{Z}[θ_t]\subseteq\mathcal{O}_t=\mathcal{O}_{\mathbb{Q}(θ_t)}$, $N_t=[\mathcal{O}_t:R_t]$, and $\varepsilon_t=[\mathcal{O}_t^\times:R_t^\times]$. We prove that if $N_t$ is squarefree, then $\varepsilon_t=3$ for $t=3$, $\varepsilon_t=7$ for $t=5$, and $\varepsilon_t=1$ otherwise. We also prove that if $N_t=27$, then $\varepsilon_t=13$ for $t=12$ and $\varepsilon_t=1$ otherwise. The squarefree proof determines the local conductor at every prime divisor of $N_t$, proves that $\varepsilon_t\mid N_t$, and combines an argument ruling out cubic powers with regulator and congruence bounds. When $N_t$ is prime, we also determine the Picard kernel, the ideal class monoid, and every fiber in the corresponding classification of integral matrices up to conjugacy. An appendix uses computer assistance to determine the rational points needed for the case of divisibility by $13$ and proves, using Magma's saturation-enabled genus-two Chabauty routine, that $13\mid\varepsilon_t$ exactly for $t=12,66$