Say I have two unimodular lattices $A$ and $B$, represented by their Gram matrices.
Question: Is there an algorithm to decide whether $A$ and $B$ are isometric, i.e. whether there exists a matrix $S \in SL(n,\mathbb Z)$ such that $B=SAS^T$?
Background: I'm developing an algorithm to test whether two toric manifolds have the same $\mathbb Z$-cohomology graded ring. If the manifold is (complex)-even dimensional, the intersection form of the middle dimension homology is an integral lattice, and manifolds with the same cohomology ring have isometric intersection forms.