Given a Lattice $L$ (e.g. by its Gram-Matrix or via a basis) I would like to know whether there is an upper bound on the number of shortest vectors in $L$. Available information on $L$ includes dimension, rank as well as an upper and a lower bound on the minimum of $L$.

The question arises in the following context: The group of automorphisms of $L$ ($Aut(L)$) acts on the set of shortest vectors of $L$ and an upper bound on the cardinality of this set would yield some information on which stabilizers are to be expected.

Sphere Packings, Lattices, and Groupsby Conway and Sloane) about "kissing numbers". [The acronym "SPLAG", though risible, is not disrespectful since the authors use it themselves.] – Noam D. Elkies Sep 14 '11 at 14:32