Let $\mathbb{C}^n / \Lambda$ be a complex torus of complex dimension $n$, with the lattice $\Lambda$ satisfying all the nice properties required for it to embed in some $\mathbb{CP}^N$. What are some explicit examples of such tori, by giving an actual family of polynomials cutting it out, for $n > 1$?
I know for the "easy" case $n=1$, where we just have elliptic curves, we get the standard cubic $y^2 = x^3 + ax + b$. (Side question: this describes a curve in $\mathbb{C}^2$ and is not a homogeneous polynomial, so it doesn't immediately give an embedding in $\mathbb{CP}^2$, where I believe it usually lies in. Does this cubic induce a natural, single homogeneous polynomial whose solution set in $\mathbb{CP}^2$ is the embedded elliptic curve?)
What possible polynomials, and how many, would we get when describing the embedding of a complex torus of dimension $2$, let's say?