The discriminant norm conjecture for cubic roots

Let $u\in S_d$, write $\nu=u^{1/3}$, and let $c=c(u)$ be the product of the distinct rational primes dividing $u$ but not $d$. Let $c'$ be the odd part of $c$. Let $F=\mathbb{Q}(\sqrt{-3d})$, let $M_c$ be the corresponding ring class field, and let $D(F(\nu)/F)$ denote the relative discriminant. Discriminant norm conjecture. If $\nu\in M_c$, then

$$N\left(D(F(\nu)/F)c^{-4}\right)= \begin{cases} 1,&3\nmid d,\\ 3^6,&3\mid d, \end{cases}$$

and if $\nu\notin M_c$, then

$$N\left(D(F(\nu)/F)c^{-4}\right)= \begin{cases} 3^8,&3\nmid d,\\ 3^{18},&3\mid d. \end{cases}$$

When $d\equiv3\pmod 4$, each occurrence of $c^{-4}$ is instead replaced by $c'^{-4}$. The conjecture predicts precise discriminant norms according to whether the cubic root lies in the relevant ring class field, complementing the paper's class-field criteria and numerical evidence.