On rational functions with monodromy group $M_{11}$

TL;DR

This paper constructs new polynomials over the rationals with Galois group M_{11} by exploring families of covers of the projective line, resulting in number fields with small discriminants or limited ramification.

Contribution

It introduces new polynomials with Galois group M_{11} derived from families of covers, expanding the known examples and properties of such polynomials.

Findings

Polynomials with Galois group M_{11} over Q(t) are constructed.

Some polynomials produce number fields with minimal discriminants.

Certain polynomials have few ramified primes upon specialization.

Abstract

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.