  It is well-known that every elliptic curve over the rationals admits a
parametrization by means of modular functions. In this short note, we show that
only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by
modular units. This answers a question raised by Zudilin in a recent work on
Mahler measures. Further, we give the list of all elliptic curves $E$ of
conductor up to $1000$ parametrized by modular units supported in the rational
torsion subgroup of $E$. Finally, we raise several open questions.

Number Theory