Computing the Galois action

$$Q : a_0 x^2 + a_1 x y + \cdots + a_9 w^2 = 0 \subset \mathbf{P}^3, \quad [L := \mathbf{Q}(\{a_i\}_i):\mathbf{Q}] = 168$$

$Q \cap X$ decomposes into a pair of quartics over $K$ a quadratic extension of $L$.

Goal

Compute $K$ and $\operatorname{Gal}(K/\mathbf{Q})$ acting on $\Lambda_Q$.

The direct computation of $\operatorname{Gal}(K/\mathbf{Q})$ looks hopeless.

We guess that $K = F(\sqrt[14]{u})$ where $[F : \mathbf{Q}] = 24$ and $\operatorname{Gal}(F/\mathbf{Q}) = C_3 \times \operatorname{PGL}(2,7)$.
Note, $\#\operatorname{Gal}(F/\mathbf{Q})$ is 14 times smaller than $\#\operatorname{Aut} \operatorname{Pic} \overline{X}$.

There is a new paper about computing Galois groups of this kind of polynomial [Elsenhans-Steel].

  1. Can we compute $\operatorname{Gal}(K/\mathbf{Q})$?
  2. $\operatorname{Gal}(K/\mathbf{Q}) \overset{?}{=} \operatorname{Aut} \Lambda$?
  3. $H^1(\operatorname{Gal}(\bar k/k), \operatorname{Pic} \overline{X}) = ?$