Unit Indices of Shanks Orders

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 set $R_t=\mathbb{Z}[θ_t]\subseteq\mathcal{O}_t=\mathcal{O}_{\mathbb{Q}(θ_t)}$, $N_t=[\mathcal{O}_t:R_t]$, and $\varepsilon_t=[\mathcal{O}_t^\times:R_t^\times]$. We determine $\varepsilon_t$ when $N_t$ is