  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 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.
