Conductor distributions of elliptic curves

Abstract

We determine the distribution of the conductors $N$ of rational elliptic curves when ordered by naive height $H$, in the form of an explicit density function for the ratios $N/H$. Our work is essentially an effective version of the Brumer--McGuinness--Watkins heuristic. Applying our results to the problem of enumerating elliptic curves by conductor gives the strongest bounds yet for the number of elliptic curves which have conductor much smaller than their height for ranges up to $H\ll N^{1.2165}$.