  Ramification invariants are necessary, but not in general sufficient, to
determine the Galois module structure of ideals in local number field
extensions. This insufficiency is associated with elementary abelian
extensions, where one can define a refined ramification filtration -- one with
more ramification breaks [JNTB 17 (2005)]. The first refined break number comes
from the usual ramification filtration and is therefore necessary. Here we
study the second refined break number.
