Point counts of the curve (reviewed)

If an abelian variety $A$ defined over a finite field $\mathbb{F}_q$ is the Jacobian of a curve $X$, then from the data for $A$ we can produce a count of the (finite) number of points of $X$ defined over $\mathbb{F}_q$ and its extensions $\mathbb{F}_{q^2}$, $\mathbb{F}_{q^3}$, $\mathbb{F}_{q^4}$, etc.

Even if $A$ is not known to be isogenous to a Jacobian, we can still perform this computation. In that case, we refer to the virtual curve associated to $A$, since we do not know a priori if the curve whose points we are counting exists. In particular, if the point count returns a negative number, or if the point count returns a number for $\mathbb{F}_{q^n}$ smaller than for $\mathbb{F}_{q^d}$ for some factor $d \mid n$, then we know that $A$ is not isogenous to a Jacobian.