Lambda-invariants of Mazur--Tate elements attached to Ramanujan's tau function and congruences with Eisenstein series

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.