Gross's conjectural formula for the Tate–Shafarevich group of a Gross curve

Let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$, and let $A=A(q)$ be the Gross curve. Let $F=\mathbb{Q}(j)$, let $H=FK$ be the Hilbert class field of $K$, and write $h$ for the class number of $K$. Put

$$\epsilon_q(c):=\left(\frac{c}{q}\right).$$

Gross's conjectural formula. The order of the Tate–Shafarevich group of the Gross curve over $H$ is

$$\#(\text{\cyrfont X}(E/H)) = \frac{q^{3h/2}L(E/H,1)}{2^{3h-4}\pi^h\displaystyle\prod_{0<c<q}\Gamma\left(\frac{c}{q}\right)^{\epsilon_q(c)}}.$$

The formula is presented as conjectural and is intended to give the order predicted by the Birch–Swinnerton-Dyer conjecture. The authors report numerical calculations for primes $q\equiv 7\pmod 8$ up to $q=4831$, with nonzero values of $L(E/H,1)$ and agreement with related computations over $F$; the general formula remains conditional on the Birch–Swinnerton-Dyer conjecture.