Let $K$ be the field $\mathbb{Q}(\sqrt[3]{2})$.
I want to construct an explicit morphism from $K$ to the fraction field of
$\mathbb{Q}[X,Y,Z]/(X^3 + 2Y^3 + 4Z^3 - 6XYZ)$
but this doesn't seem to be that easy. Can someone help me?
Of course I just have to send $\sqrt[3]{2}$ to some explicit polynomial/rational function in $X$, $Y$ and $Z$...?
EDIT. Should I start thinking that it is impossible?