Groups play an important role in mathematics. The LMFDB has data on two types of groups: Galois groups and lattices.

Galois groups appear in various places. Naturally, they appear for global and local number fields, but also as Galois groups of local factors of L-functons. Strictly speaking, these are transitive subgroups of $S_n$ up to conjugation, but due to their application here, we refer to them as simply Galois groups.

Sato-Tate groups provide a generalization of Galois groups to motives. In particular, the Sato-Tate group of a curve determines the asymptotic distribution of the number of points on the curve modulo $p$ for different primes $p$, just as the Galois group of a number field determines the asymptotic distribution of the degrees of the irreducible factors modulo $p$ of a defining polynomial for the field.

Lattices appear in multiple places in number theory, from the ring of integers of a number field to the Mordell-Weil group (modulo torsion) of an elliptic curve over $\Q$, or more generally, of an abelian variety over a number field.

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