A Cute Diophantine Problem

problem.md

Find all non-trivial $A$, $B$, $C$, ..., $G$ $\in$ ℤ such that

$A^2 = B^3 + C^3$

$B^2 = C^3 + D^3$

$C^2 = D^3 + E^3$

$D^2 = E^3 + F^3$

$E^2 = F^3 + G^3$

$F^2 = G^3 + A^3$

if any such solutions exist.

What is a minimal number field (if one exists) over which a non-trivial solution exists?