  Let $p\in\{3,5,7\}$ and let $\Delta$ denote the weight twelve modular form
arising from Ramanujan's tau function. We show that $\Delta$ is congruent to an
Eisenstein series $E_{k,\chi, \psi}$ modulo $p$ for explicit choices of $k$ and
Dirichlet characters $\chi$ and $\psi$. We then prove formulae describing the
Iwasawa invariants of the Mazur--Tate elements attached to $\Delta$, confirming
numerical data gathered by the authors in a previous work.
