Counting elliptic curves with a rational $N$-isogeny for small $N$

Abstract

We count the number of rational elliptic curves of bounded naive height that have a rational $N$-isogeny, for $N\in \{2,3,4,5,6,8,9,12,16,18\}$. For some $N$, this is done by generalizing a method of Harron and Snowden. For the remaining cases, we use the framework of Ellenberg, Satriano and Zureick-Brown, in which the naive height of an elliptic curve is the height of the corresponding point on a moduli stack.