Invariants for isomorphism classes in the category $\text{bcalNT}$

The category $\text{bcalNT}$ is a category of certain commutative graded algebras over a field. It was introduced in \cite{Lobos2} as a generalization of algebras generated by Jucys-Murphy elements in the many $\mathbf{\text{End}}$ algebras of the diagrammatic Soergel category of Elias and Williamson. In the first part of this article we define certain \emph{Invariants} for the isomorphism classes in $\text{bcalNT},$ following in the same spirit of \cite{Lobos3}, where a series of \emph{Isomorphism criteria} were found. At the end, we use our invariants to provide a new lower bound for the number of isomorphism classes, improving a similar result obtained in \cite{Lobos3}.