Congruences between Ramanujan's tau function and elliptic curves, and Mazur--Tate elements at additive primes

We show that if $E\mathrm{/}\mathbb{Q}$ is an elliptic curve with a rational $p$-torsion for $p=2$ or $3$, then there is a congruence relation between Ramanujan’s tau function and $E$ modulo $p$. We make use of such congruences to compute the Iwasawa invariants of $2$-adic and $3$-adic Mazur–Tate elements attached to Ramanujan’s tau function. We also investigate numerically the Iwasawa invariants of the Mazur–Tate elements attached to an elliptic curve with additive reduction at a fixed prime number.