Intrinsic Subgroups and the $\mathrm{\ell }$-adic Galois image

Let $X$ be a geometrically irreducible smooth projective curve over a field $k$. Yamazaki et al. define a biadditive symmetric pairing $⟨-,-⟩$ on the torsion subgroup of the Picard group $\mathrm{P}\mathrm{i}\mathrm{c}(X)$ with values in $k^{\times }\otimes \mathbb{Q}\mathrm{/}\mathbb{Z}$. The intrinsic subgroup $\mathrm{P}\mathrm{i}\mathrm{c}(X{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}^{\mathrm{i}\mathrm{s}}$ is the kernel of this pairing. When $X$ is an elliptic curve $E$, we can identify $E\simeq {\mathrm{P}\mathrm{i}\mathrm{c}}^0(E)$. We classify $E(k{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}^{\mathrm{i}\mathrm{s}}$ in purely algebraic terms for many elliptic curves over an arbitrary field $k$. We give a generalization of the analytic methods of Yamazaki et al. from $\mathbb{Q}$ to an arbitrary field $k\subset \mathbb{C}$. Lastly, for $k=\mathbb{Q}$, we describe an algorithm to explicitly compute $E(\mathbb{Q}{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}^{\mathrm{i}\mathrm{s}}$.