If I consider the ideal $(2)\mathcal{O}_{\mathbb{Q}(\zeta_3)}$ in $\mathcal{O}_{\mathbb{Q}(\zeta_3)}$ sage claims it factors as $$ (-2 \zeta_3^2 - 1)^2 $$ Why is this true? If I multiply the ideal out, I get \begin{align} (-2 \zeta_3 - 1)^2 & = 4 \zeta_3 + 4 \zeta_3^2 + 1 \\ & = 4 \zeta_3(1 + \zeta_3) + 1 \end{align} I don't think this is $3$. For reference, here is the sage code I entered:
sage: k = CyclotomicField(3)
sage: k.factor(3)
(Fractional ideal (-2*zeta3 - 1))^2