  The so-called class-invariant homomorphism $\psi$ measures the Galois module
structure of torsors--under a finite flat group scheme $G$--which lie in the
image of a coboundary map associated to an isogeny between (N\'eron models of)
abelian varieties with kernel $G$. When the varieties are elliptic curves with
semi-stable reduction and the order of $G$ is coprime to 6, is is known that
the homomorphism $\psi$ vanishes on torsion points. In this paper, using Weil
restrictions of elliptic curves, we give the construction, for any prime number
$p>2$, of an abelian variety $A$ of dimension $p$ endowed with an isogeny (with
kernel $\mu_p$) whose coboundary map is surjective. In the case when $A$ has
rank zero and the $p$-part of the Picard group of the base is non-trivial, we
obtain examples where $\psi$ does not vanishes on torsion points.
