Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$

We describe an algorithm that, given a positive integer $N$, computes a Gr\"obner basis of the ideal of polynomial relations among Dedekind $\eta$-functions of level $N$, i.e., among the elements of $\{\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)\}$ where $1=\delta_1<\delta_2\dots<\delta_n=N$ are the positive divisors of $N$. More precisely, we find a finite generating set (which is also a Gr\"obner basis of the ideal $\ker\phi$ where \begin{gather*} \phi:Q[E_1,\ldots,E_n] \to Q[\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)], \quad E_k\mapsto \eta(\delta_k\tau), \quad k=1,\ldots,n. \end{gather*} [NOTE: Accepted for publication in the Journal of Symbolic Computation]