Beilinson's determinant formula for the adjoint motive of an elliptic curve

Let $A/F$ be an elliptic curve and let $M=\operatorname{Ad}(h^1(A)_{\mathbb Q})$. Let $r$ be the Beilinson regulator, $r_1$ and $r_2$ the numbers of real and complex places of $F$, and let $H_\sigma$ denote the generators of $H^1_{\mathcal D}(M,\mathbb{R}(1))$ defined from the complex periods. Then

Beilinson's conjecture. One has

$$\dim_{\mathbb Q}r\left(H^1_{\mathcal M}(M,\mathbb{Q}(1))\right)=r_2,$$

and

$$\bigwedge^{r_2}r\left(H^1_{\mathcal M}(M,\mathbb{Q}(1))\right)=L(1,M)(2\pi i)^{-r_1-r_2}\prod_{\sigma\in\Sigma_F^{\mathbb R}}\Omega_{\sigma,1}^{-1}\Omega_{\sigma,2}^{-1}\prod_{\sigma\in\Sigma_F^{\mathbb C}}\operatorname{Im}(\Omega_{\sigma,1}\overline{\Omega_{\sigma,2}})^{-2}\left(\bigwedge_{\sigma\in\Sigma_F^{\mathbb C}}H_\sigma\right)\mathbb{Q}.$$

This conjecture predicts the determinant of the motivic regulator in terms of the special value $L(1,M)$ and the period basis of Deligne cohomology. It is presented as a consequence of Beilinson's conjectures and remains conjectural.