The setup is a lattice $L$, a finitely generated free abelian group with symmetric positive definite integral bilinear form $\langle \ ,\ \rangle:L\otimes_{\mathbb{Z}} L\rightarrow \mathbb{Z}$, for which we have a canonical spanning set $e_x$, where $\langle e_x,e_y \rangle$ is explicitly given. There is a lot of redundancy between these $e_x$ however, and it is not necessarily possible to find a subset of them that is a basis for $L$.
Is there some way to still compute the discriminant/volume of $(L, \langle\ , \ \rangle)$ from this data, without using a basis? By discriminant/volume, we mean, pick a basis $v_i$ of $L$, and look at $\det[\langle v_i,v_j\rangle]_{ij}$.
An example of something similar to this would be computing the trace of a linear map $\phi$ by finding $\phi$ invariant subspaces which span your space, taking the associated complex of vector spaces, and then taking an alternating sum of the traces on these potentially simpler pieces.
An example of a lattice where no two proper subsets of our generators generate is given by $(6,10,15)=\mathbb{Z}$.