  We study the arithmetic of the twist of the Fermat quartic defined by $X^4 +
Y^4 + Z^4 = 0$ which has no $\mathbb{Q}$-rational point. We calculate the
Mordell--Weil group of the Jacobian variety explicilty. We show that the degree
$0$ part of the Picard group is a free $\mathbb{Z}/2\mathbb{Z}$-module of rank
$2$, whereas the Mordell--Weil group is a free $\mathbb{Z}/2\mathbb{Z}$-module
of rank $3$. Thus the relative Brauer group is non-trivial. We also show that
this quartic violates the local-global property for linear determinantal
representations.
