  This article explores the connection between radical isogenies and modular
curves. Radical isogenies are formulas designed for the computation of chains
of isogenies of fixed small degree $N$, introduced by Castryck, Decru, and
Vercauteren at Asiacrypt 2020. One significant advantage of radical isogeny
formulas over other formulas with a similar purpose is that they eliminate the
need to generate a point of order $N$ that generates the kernel of the isogeny.
While radical isogeny formulas were originally developed using elliptic curves
in Tate normal form, Onuki and Moriya have proposed radical isogeny formulas of
degrees $3$ and $4$ on Montgomery curves and attempted to obtain a simpler form
of radical isogenies using enhanced elliptic and modular curves. In this
article, we translate the original setup of radical isogenies in Tate normal
form into the language of modular curves. Additionally, we solve an open
problem introduced by Onuki and Moriya regarding radical isogeny formulas on
$X_0(N).$
