  Discriminant ideals of noncommutative algebras $A$, which are module finite
over a central sublagebra $C$, are key invariants that carry important
information about $A$, such as the sum of the squares of the dimensions of its
irreducible modules with a given central character. There has been substantial
research on the computation of discriminants, but very little is known about
the computation of discriminant ideals. In this paper we carry out a detailed
investigation of the lowest discriminant ideals of Cayley-Hamilton Hopf
algebras in the sense of De Concini, Reshetikhin, Rosso and Procesi, whose
identity fiber algebras are basic. The lowest discriminant ideals are the most
complicated ones, because they capture the most degenerate behaviour of the
fibers in the exact opposite spectrum of the picture from the Azumaya locus. We
provide a description of the zero sets of the lowest discriminant ideals of
Cayley-Hamilton Hopf algebras in terms of maximally stable modules of Hopf
algebras, irreducible modules that are stable under tensoring with the maximal
possible number of irreducible modules with trivial central character. In
important situations, this is shown to be governed by the actions of the
winding automorphism groups. The results are illustrated with applications to
the group algebras of central extensions of abelian groups, big quantum Borel
subalgebras at roots of unity and quantum coordinate rings at roots of unity.
