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.
