  We compute new polynomials with Galois group $M_{11}$ over $\mathbb{Q}(t)$.
These polynomials stem from various families of covers of
$\mathbb{P}^1\mathbb{C}$ ramified over at least 4 points. Each of these
families has features that make a detailed study interesting. Some of the
polynomials lead, via specialization, to number fields with very small
discriminant or few ramified primes.
