Elliptic curves with square-free $\Delta$

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.