  We prove the conjectural relations between Mahler measures and $L$-values of
elliptic curves of conductors 20 and 24. We also present new hypergeometric
expressions for $L$-values of CM elliptic curves of conductors 27 and 36.
Furthermore, we prove a new functional equation for the Mahler measure of the
polynomial family $(1+X)(1+Y)(X+Y)-\alpha XY$, $\alpha\in\mathbb R$.
