  For a totally real field $F$, a finite extension $\mathbf{F}$ of
$\mathbf{F}_p$ and a Galois character $\chi: G_F \to \mathbf{F}^{\times}$
unramified away from a finite set of places $\Sigma \supset \{\mathfrak{p} \mid
p\}$ consider the Bloch-Kato Selmer group $H:=H^1_{\Sigma}(F, \chi^{-1})$. In
an earlier paper of the authors it was proved that the number $d$ of
isomorphism classes of (non-semisimple, reducible) residual representations
$\overline{\rho}$ giving rise to lines in $H$ which are modular by some
$\rho_f$ (also unramified outside $\Sigma$) satisfies $d \geq n:=
\dim_{\mathbf{F}} H$. This was proved under the assumption that the order of a
congruence module is greater than or equal to that of a divisible Selmer group.
We show here that if in addition the relevant local Eisenstein ideal $J$ is
non-principal, then $d >n$. When $F=\mathbf{Q}$ we prove the desired bounds on
the congruence module and the Selmer group. We also formulate a congruence
condition implying the non-principality of $J$ that can be checked in practice,
allowing us to furnish an example where $d>n$.
