Diophantine equation generated by the subfield of a circular field

Two forms $f(x,y,z)$ and $g(x,y,z)$ of degree $3$ were constructed, with their values being the norms of numbers in the subfields of degree $3$ of the circular fields $K_{13}$ and $K_{19}$, respectively. Using the decomposition law in a circular field, Diophantine equations $f(x,y,z)=a$ and $g(x,y,z)=b$, where $a,b\in\mathbb{Z},\ a\ne0,\ b\ne 0$ were solved. The assertions that, based on the canonical decomposition of the numbers $a$ и $b$ into prime factors, make it possible to determine whether the equations $f(x,y,z)=a$ and $g(x,y,z)=b$ have solutions were proved.