The Beilinson–Bloch conjecture for the μ6-invariant Jacobian motive

Let $C=C_{a,b}$ be a bielliptic Picard curve over a number field $k$, let $J$ be its Jacobian, and let $E^\Delta$ be the elliptic curve appearing in the associated decomposition. Write ${\rm CH}^2_{(1)}(J)^{\mu_6}$ for the $\mu_6$-invariant part of the indicated graded Chow group, and let $\mathfrak h^1(E^\Delta)$ be the first Chow-motive component of $E^\Delta$.

Beilinson–Bloch conjecture. The Beilinson–Bloch conjecture for the motive $\mathfrak h^3(J)^{\mu_6}$ predicts

$$\operatorname{rk}{\rm CH}^2_{(1)}(J)^{\mu_6}=\operatorname*{ord}_{s=1}L(\mathfrak h^1(E^\Delta),s).$$

By the weak Birch and Swinnerton-Dyer conjecture, the right-hand side equals $\operatorname{rk}E^\Delta(k)$, so this prediction is equivalent to the weak Birch and Swinnerton-Dyer conjecture for $E^\Delta$.