$$Q : a_0 x^2 + a_1 x y + \cdots + a_9 w^2 = 0 \subset \mathbf{P}^3, \quad [L := \mathbf{Q}(\{a_i\}_i):\mathbf{Q}] = 168$$
$Q \cap X$ decomposes into a pair of quartics over $K$ a quadratic extension of $L$.
Compute $K$ and $\operatorname{Gal}(K/\mathbf{Q})$ acting on $\Lambda_Q$.
Via the identification with the original classes we have $\frac{1}{2 \pi i} \left( \int_C \omega \right)_{\omega \in F^1} \in K^{21}$.
These can be reconstructed in the same fashion as we reconstructed $a_i$.
Unclear how to certify this step! What are the denominators of $\frac{1}{2 \pi i} \int_C \omega$?
For $Q$ smooth, $K = L(\sqrt{\operatorname{disc} Q})$ [Costa-Sertöz].