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 $\mathrm{\Delta }$ denote the weight twelve modular form arising from Ramanujan’s tau function. We show that $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$, confirming numerical data gathered by the authors in a previous work.