More precisely, let $X$ be a one-sheeted hyperboloid and $Y$ a sphere in $\mathbb{CP}^3$, both defined over $\mathbb{Q}$. Can every divisor class on $X$ be defined over some real extension of $\mathbb{Q}$, and can every divisor class on $Y$ be defined over some imaginary quadratic extension of $\mathbb{Q}$?
I am trying to understand the phenomenon in which varieties can be isomorphic over certain fields, but not over smaller ones. In this case, it's clear that $X$ and $Y$ are isomorphic as varieties over $\mathbb{C}$, as for example the map $[x:y:z]\mapsto [x:y:iz]$ is an isomorphism from $X$ to $Y$.
Now if I can show that the divisor classes of $X$ and $Y$ have the properties described above, then this should suffice to show that they are not isomorphic as varieties over $\mathbb{Q}$.
A quick secondary question: is there a name for the smallest extension of the field of definition of a variety such that every divisor class is defined over that extension? Perhaps the "splitting field of a variety" or something like this?