Counting 5-isogenies of elliptic curves over $\mathbb{Q}$

We show that the number of $5$-isogenies of elliptic curves defined over $\mathbb{Q}$ with naive height bounded by $H>0$ is asymptotic to $C_5\cdot H^{1\mathrm{/}6}(\log H{)}^2$ for some explicitly computable constant $C_5>0$. This settles the asymptotic count of rational points on the genus zero modular curves $X_0(m)$. We leverage an explicit $\mathbb{Q}$-isomorphism between the stack ${\mathcal{X}}_0(5)$ and the generalized Fermat equation $x^2+y^2=z^4$ with ${\mathbb{G}}_m$-action of weights $(4,4,2)$.