  The main point of the paper is to take the explicit motivic Chabauty-Kim
method developed in papers of Dan-Cohen--Wewers and Dan-Cohen and the author
and make it work for non-rational curves. In particular, we calculate the
abstract form of an element of the Chabauty-Kim ideal for
$\mathbb{Z}[1/\ell]$-points on a punctured elliptic curve, and lay some
groundwork for certain kinds of higher genus curves. For this purpose, we
develop an "explicit Tannakian Chabauty-Kim method" using
$\mathbb{Q}_{p}$-Tannakian categories of Galois representations in place of
$\mathbb{Q}$-linear motives. In future work, we intend to use this method to
explicitly apply the Chabauty-Kim method to a curve of positive genus in a
situation where Quadratic Chabauty does not apply.
