[[Lattice subgroup#Classical lattice]]
# Covolume of a classical lattice

Let $L \leq_{\mathbb{Z}} \mathbb{R}^n$ be a [[Lattice subgroup#Classical lattice|complete classical lattice]].
The **covolume** of $L$ is the [[Measure space|measure]] of the [[Fundamental domain]], #m/def/geo
or equivalently
$$
\begin{align*}
\opn{covol} (L) = \opn{vol} ( \mathbb{R}^n / L)
\end{align*}
$$
where $\opn{vol}$ is an induced measure.

## Properties

1. Suppose $L' \leq_{\mathbb{Z}} L$ is a complete sublattice.
  Then $\opn{covol}(L') = \abs{L / L'} \opn{covol}(L)$. ^P1

> [!missing]- Proof
> #missing/proof

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