Counting elliptic curves with a rational \(N\)-isogeny for small \(N\) (Q6556227)

Let \(E\) be an elliptic curve over \(\mathbb{Q}\). An isogeny \(\phi : E \to E'\) between two elliptic curves is said to be cyclic of degree \(N\) if \(\mathrm{Ker}(\phi)(\mathbb{Q}) \cong \mathbb{Z}/N\mathbb{Z}\), and it is said to be rational if \(\mathrm{Ker}(\phi)\) is stable under the action of the absolute Galois group \(G_{\mathbb{Q}}\). In this paper, the authors count the number of rational elliptic curves of bounded naive height that have a rational cyclic isogeny of degree \(N\), for \(N \in \{2, 3, 4, 5, 6, 8, 9, 12, 16, 18\}\).