Conditions under which the Lattice Generated by a Subset of Lattice Vectors is Equivalent to the Original Lattice

Question

In general, the sublattice $L_2$ generated by subset of vectors of $L_1$ need not have the same rank as $L_1$. Even if it does, it may be a proper sublattice of $L_1$. However, if the rank and determinant of $L_2$ matches those of $L_1$, then is it possible to conclude that $L_2 = L_1$? If not, what are the minimal necessary conditions on the subset? Clearly, if the subset includes a basis of $L_1$, it's sufficient, but I'm looking for conditions which are easier to test.

Answer 1

sigh $\;$ Answered my own question again. If the rank and determinants are equal, then the index of $L_2$ in $L_1$ is 1, i.e. they're the same lattice, perhaps with a different choice of basis. This page has more details.