Bounding Selmer Groups of Superelliptic Jacobians via Class Groups

Let $K$ be a number field containing a primitive $p$-th root of unity $ζ_p$. Let $f(x)\in K[x]$ be a monic integral polynomial, and let $f_0$ denote its radical. Let $C/K$ be the superelliptic curve defined by $y^p=f(x)$, and let $J$ be its Jacobian variety. The variety $J$ admits multiplication by