Complex numbers modulo integers Is there a "nice" way to think about the quotient group $\mathbb{C} / \mathbb{Z}$?
Bonus points for $\mathbb{C}/2\mathbb{Z}$ (or even $\mathbb{C}/n\mathbb{Z}$ for $n$ an integer) and how it relates to $\mathbb{C} / \mathbb{Z}$.
By "nice" I mean something like:


*

*$\mathbb{R}/\mathbb{Z}$ is isomorphic to the circle group via the exponential map $\theta \mapsto e^{i\theta}$, and

*$\mathbb{C}/\Lambda$ is a complex torus for $\Lambda$ an integer lattice (an integer lattice is a discrete subgroup of the form $\alpha\mathbb{Z} + \beta\mathbb{Z}$ where $\alpha,\beta$ are linearly independent over $\mathbb{R}$.)


Intuitively, it seems like it should be something like a circle or elliptic curve.

$\mathbb C/\mathbb Z$ is isomorphic to $\mathbb C/ n\mathbb Z$ for any non-zero complex number $n$.
You can show that $\mathbb C/\mathbb Z\cong (\mathbb C\setminus\{0\},\times)$ via the homomorphism $z \to e^{2\pi iz}$.
You can also think of this as $S^1\times\mathbb R$, which is topologically a cylinder.
