In our case, we have all the compatible embeddings
$$\sigma_i : \mathbf{Q}(a_k) \hookrightarrow L \hookrightarrow \mathbf{C}$$
Thus the isomorphism is given by the solution of the following linear system
$$\{\sigma_i(a_k)^j\}_{i, j} \cdot v = \{\sigma_i(a_0)\}_i, \qquad v \in \mathbf{Q}^{168}$$
Distinct nodes make $\{\sigma_i(a_k)^j\}$ invertible, so the solution $v \in \mathbf{Q}^{168}$ is unique; the denominators of $v$ are bounded a priori, so enough precision pins $v$ down exactly and the isomorphism is then verified exactly.
In practice it is faster to refine the complex embeddings iteratively: their height is 4k digits, not 120k.