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Can the base ring of a polyhedron be restricted?

I have a polyhedron P with rational vertices

sage: A = matrix([[1,0,0],[1,0,2],[1,1,1],[1,3/2,0]])
sage: A
[  1   0   0]
[  1   0   2]
[  1   1   1]
[  1 3/2   0]
sage: P = Polyhedron(A)
sage: P
A 2-dimensional polyhedron in QQ^3 defined as the convex hull

If we scale P by a factor of two, then we get a lattice polytope.

sage: (2*P).is_lattice_polytope()
True

However, the base ring of P is still QQ

sage: (2*P).parent()
Polyhedra in QQ^3

Since 2*P is a lattice polytope it seems like it should be possible to restrict the base ring of 2*P to ZZ. Is this possible?

of 4 vertices

Can the base ring of a polyhedron be restricted?

I have a polyhedron P with rational vertices

sage: A = matrix([[1,0,0],[1,0,2],[1,1,1],[1,3/2,0]])
sage: A
[  1   0   0]
[  1   0   2]
[  1   1   1]
[  1 3/2   0]
sage: P = Polyhedron(A)
sage: P
A 2-dimensional polyhedron in QQ^3 defined as the convex hull
hull  of 4 vertices

If we scale P by a factor of two, then we get a lattice polytope.

sage: (2*P).is_lattice_polytope()
True

However, the base ring of P is still QQ

sage: (2*P).parent()
Polyhedra in QQ^3

Since 2*P is a lattice polytope it seems like it should be possible to restrict the base ring of 2*P to ZZ. Is this possible?

~~of 4 vertices~~