On Lattice Barycentric Tetrahedra

A lattice tetrahedron T ⊂ R3 is a tetrahedron whose four vertices are all in the lattice Z3. Lattice tetrahedra are preserved by those affine linear maps of the form ~v 7→ A~v +~b, such that A is an element of GL(3,Z) and ~b is an element of the lattice Z3. Such affine linear maps are called unimodular maps. We say that a lattice tetrahedron whose barycentre is its only non-vertex lattice point is lattice barycentric. The notation T (a,b,c) describes that lattice tetrahedron with vertices {0,e1,e2,ae1 +be2 +ce3}. Our result is then that all such lattice barycentric tetrahedra T (a,b,c) are unimodularly equivalent to T (3,3,4) or T (7,11,20).