The arithmetic of a twist of the Fermat quartic

We study the arithmetic of the twist of the Fermat quartic defined by $X^4 + Y^4 + Z^4 = 0$whichhasno$$\mathbb{Q}$$-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 $\mathbb{Z}\mathrm{/}2\mathbb{Z}$-module of rank $2$, whereas the Mordell–Weil group is a free $\mathbb{Z}\mathrm{/}2\mathbb{Z}$-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.