Volume of Smallest Rational Tetrahedron

Theorem
The only rational tetrahedron whose edge lengths are less than $157$ has:


 * edges of length $117$, $80$, $53$, $52$, $51$, $84$
 * faces of area $1800$, $1890$, $2016$, $1170$
 * volume of $18 \, 144$.