$$\det(1 - t\operatorname{Frob}_q \mid H^2_{\mathrm{et}}(X_p^{\mathrm{al}}, \mathbf{Q}_\ell)) = t^{22}P_2(1/t).$$
The Hasse-Weil zeta function $Z_{X_p}(t)$ can be written as
$$Z_{X_p}(t) := \exp\left( \sum_{m=1}^{\infty} \frac{\# X_p(\mathbf{F}_{q^m})}{m} t^m \right) = \frac{1}{(1-t)\, t^{22}P_2(1/t)\, (1-q^2 t)}.$$
One may deduce $P_2$ from the point counts $\# X_p(\mathbf{F}_{q^m})$ for $m \leq b_2/2 + 1 = 12$.