Weber modular curves and modular isogenies

We study the modular curves defined by Weber functions, and associated modular polynomials, action of ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$, and parametrizations of elliptic curves with a view to the study of the isogeny graphs that they determine, particularly for supersingular elliptic curves. In addition to applications to efficient isogeny computation in cryptographic applications, we present an application to explicit Galois representations.