  Let $\lambda$ and $\kappa$ be cardinal numbers such that $\kappa$ is infinite
and either $2\leq \lambda\leq \kappa$, or $\lambda=2^\kappa$. We prove that
there exists a lattice $L$ with exactly $\lambda$ many congruences, $2^\kappa$
many ideals, but only $\kappa$ many filters. Furthermore, if $\lambda\geq 2$ is
an integer of the form $2^m\cdot 3^n$, then we can choose $L$ to be a modular
lattice generating one of the minimal modular nondistributive congruence
varieties described by Ralph Freese in 1976, and this $L$ is even relatively
complemented for $\lambda=2$. Related to some earlier results of George
Gr\"atzer and the first author, we also prove that if $P$ is a bounded ordered
set (in other words, a bounded poset) with at least two elements, $G$ is a
group, and $\kappa$ is an infinite cardinal such that $\kappa\geq |P|$ and
$\kappa\geq |G|$, then there exists a lattice $L$ of cardinality $\kappa$ such
that (i) the principal congruences of $L$ form an ordered set isomorphic to
$P$, (ii) the automorphism group of $L$ is isomorphic to $G$, (iii) $L$ has
$2^\kappa$ many ideals, but (iv) $L$ has only $\kappa$ many filters.
