Consider the group scheme $\mathcal T$ over $\mathbf P^1$ given locally (variable $t$) by the equation

$x^2 - f(t)y^2 = 1$

where $f(t)$ is a polynomial of degree $r$ with distinct roots (assume that $r$ is even and the field is algebraically closed). What is the group of torsors for $\mathcal T$, i.e. $\mathrm H^1(\mathbf P^1, \mathcal T)$?

Using the machinery of spectral sequences I was able to deduce that my group is an extension of the Jacobian $J(C)$, where $C$ is the double cover that splits $\mathcal T$:

$1 \to (\mathbb Z/2\mathbb Z)^{r-1} \to \mathrm H^1(\mathbf P^1, \mathcal T) \to J(C) \to 1$

Unfortunately, I cannot confirm whether my answer is correct. It is difficult to believe that such example hasn't been computed anywhere in the literature, but I couldn't find any reference. Would be happy if someone could give a hint or a reference.