  Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by
the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed
positive integer. We compute several interesting arithmetic invariants of
$J_m$: its decomposition up to isogeny into simple abelian varieties, the
minimal field $\mathbb{Q}(\operatorname{End}(J_m))$ over which its
endomorphisms are defined, its connected monodromy field
$\mathbb{Q}(\varepsilon_{J_m})$, and its Sato-Tate group. Currently, there is
no general algorithm that computes these last two invariants. Furthermore, for
large enough values of $m$, the abelian varieties $J_m$ provide non-trivial
examples of high-dimensional phenomena, such as degeneracy and the
non-triviality of the extension
$\mathbb{Q}(\varepsilon_{J_m})/\mathbb{Q}(\operatorname{End}(J_m))$. We also
describe the Sato-Tate group of an abelian variety, generalizing existing
results that apply only to non-degenerate varieties, and prove an extension of
a well-known formula of Gross-Koblitz that relates values of the classical and
$p$-adic gamma functions at rational arguments.
