Classification of integral lattices with large class number

A detailed exposition of Kneser's neighbour method for quadratic lattices over totally real number fields, and of the sub-procedures needed for its implementation, is given. Using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices, various results about genera of $\ell$-elementary lattices, for small prime level $\ell,$ are obtained. For instance, the class number of $12$-dimensional $7$-elementary even lattices of determinant $7^6$ is $395$; no extremal lattice in the sense of Quebbemann exists. The implementation incorporates as essential parts previous programs of W. Plesken and B. Souvignier.