Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface $$ \mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6, $$ where $a_i(t) \in \mathbb{F}_{q}[t]$. Is there a way to calculate the Picard $\mathbb{F}_q$-number of $\mathcal{E}$ (i.e., rank of the Picard $\mathbb{F}_q$-group $\mathrm{Pic}(\mathcal{E})$) if I don't know the Mordell-Weil rank of $\mathcal{E}$?
I am mainly interested in the cases when $\mathcal{E}$ is a rational or K3 surface and $\mathrm{char}(\mathbb{F}_q) > 3$. By abuse of notation I also denote by $\mathcal{E}$ the corresponding Kodaira-Neron model.
