We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \mathbb{P}^2$ be a plane quintic curve describing the locus of degenerate conics of $f$, and let $\pi: \widetilde{C} \to C$ be the etale double cover parametrizing the lines on these degenerate conics. Let $H$ be the hyperplane section on $C$, and consider $M = \pi^*H$. I need to work out the dimension $h^0(M)$. If $k$ had characteristic not $2$ then we know $h^0(M) = 4$, since $h^0(M) = h^0(\pi_*M) = h^0(H) + h^0(H + \eta)$ where $\eta$ is some line bundle satisfying $\eta^2 \cong \mathcal{O}_C$, and we know $h^0(H) = 3$, and $h^0(H + \eta) = 1$ as in Beauville's paper on singularities of the Theta divisor.

However this approach does not work in characteristic $2$. Here we simply get a short exact sequence $$0 \to \mathcal{O}_C \to \pi_*\mathcal{O}_{\widetilde{C}} \to \mathcal{O}_C \to 0$$ because in characteristic $2$ we deal with Artin-Schreier extensions. After twisting by $H$ and taking long exact sequence, this bounds $3 \leq h^0(M) \leq 6$ for us. It would suffice in my case to know that this quantity is even, so essentially I want to ensure that even in characteristic $2$, the theta divisor is $\textbf{even}$.

I know that $det(\pi_*M) = 2H$ has $h^0(2H) = 11$ since $C$ is genus $6$ and $2H$ is the canonical divisor, but I am not sure if this is of any use. I also tried to manually compute the global sections from cocycle data but that seems like a nightmare.