  Under the Riemann Hypothesis for Dirichlet L-functions, we improve on the
error term in a smoothed version of an estimate for the density of elliptic
curves with square-free $\Delta=D/16$, where D is the discriminant, by T.D.
Browning and the author. To achieve this improvement, we elaborate on our
methods for counting weighted solutions of inhomogeneous cubic congruences to
power-ful moduli. The novelty lies in going a step further in the explicit
evaluation of complete exponential sums and saving a factor by averaging over
the moduli.
