= Solution
\b[True; there are exactly five.] On a simple <module>, the central element $Y$ acts as zero: its <image> is a <submodule>, and surjectivity would contradict $Y^6=0$. The <nilpotent central elements annihilate simple modules> argument shows that the action factors through
$$
\mathbb C[X,Y]/(X^5-1,Y)\cong\mathbb C[X]/(X^5-1)\cong\mathbb C^5
$$
by the <Chinese remainder theorem>. Therefore the simple <modules> are one-dimensional with
$$
\boxed{X\mapsto\zeta_5^j,\qquad Y\mapsto0,\qquad j=0,\ldots,4.}
$$
This also rules out hypothetical infinite-dimensional simple <modules>, rather than counting only finite-dimensional ones.
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