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

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