in collaboration with han, padurariu, and park, 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 $c_5\cdot h^{1/6} (\log h)^2$ 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 $\mathscr{x}_0(5)$ and the generalized fermat equation $x^2 + y^2 = z^4$ with $\mathbb{g}_m$ action of weights $(4, 4, 2)$.

pretalk: i will explain how to count isomorphism classes of elliptic curves over the rationals. on the way, i will introduce some basic stacky notions: torsors, quotient stacks, weighted projective stacks, and canonical rings.

[pre-talk at 3:00pm]