The computation is based of the Honda-Tate theorem which states that isogeny classes of abelian varieties over finite fields are completely determined by the characteristic polynomial of their Frobenius automorphism acting on the first $\ell$-adic cohomology group. For a given dimension $g$ and base field of size $q$, a complete list of all Weil polynomials that do occur can be enumerated using a technique developed by Kedlaya [MR:2459990, arXiv:math/0608104]. In 2016, Dupuy, Kedlaya, Roe, and Vincent improved upon Kedlaya's original code to generate these tables and the data they contain [arXiv:2003.05380].
See also the article of Kedlaya and Sutherland [MR:3540942, arXiv:1511.06945], where these techniques are used to compute Weil polynomials for K3 surfaces.
The determination of which isogeny classes contain Jacobians includes the constraint that if $X$ is a curve over $\F_q$, then the number of degree-$n$ places of $X$ is nonnegative for all positive integers $n$; together with additional constraints from: Howe and Lauter, New methods for bounding the number of points on curves over finite fields [MR:2987661, arXiv:1202.6308, 10.4171/119-1/12]. This incorporates prior results from the following papers.
- Howe and Lauter, Improved upper bounds for the number of points on curves over finite fields [MR:2038778]
- Howe, Maisner, Nart, and Ritzenthaler, Principally polarizable isogeny classes of abelian surfaces over finite fields [MR:2367179, 10.4310/MRL.2008.v15.n1.a11]
- Howe, Nart, and Ritzenthaler, Jacobians in isogeny classes of abelian surfaces over finite fields [MR:2514865]
- Korchmáros and Torres, On the genus of a maximal curve [MR:1923698]
- Serre, Rational points on curves over finite fields, URL (see also manYPoints database)
- Stohr and Voloch, Weierstrass points and curves over finite field [MR:0812443, 10.1112/plms/s3-52.1.1]
- Zaytsev, An improvement of the Hasse-Weil-Serre bound for curves over some finite fields [MR:3249822, 10.1016/j.ffa.2014.06.003]
- Zaytsev, Optimal curves of low genus over finite fields [MR:3426586, 10.1016/j.ffa.2015.09.008]
The list of curves with Jacobian in a given isogeny class were computed by:
- Andrew Sutherland (dimensions $2$ and $3$ except as otherwise indicated);
- Everett Howe (hyperelliptics in dimensions $3$, $4$, and $5$ over $\F_q$ with $q$ odd; [arXiv:2401.15255]);
- Xavier Xarles (dimension $4$ over $\F_2$; [arXiv:2007.07822]);
- Kiran Kedlaya (hyperelliptics in dimensions $3$ and $4$ over $\F_q$ with $q>2$ even, using the method of Xarles);
- Jonas Bergström, Carel Faber, and Sam Payne (dimension $4$ over $\F_3$; [arXiv:2206.07759]);
- Dusan Dragutinović (dimension $5$ over $\F_2$; [arXiv:2202.07809]);
- Steve Huang, Kiran Kedlaya and Jun Bo Lau (dimension $6$ over $\F_2$; [arXiv:2402.00716]).
Cyclic isogeny classes and lists of noncyclic primes primes were computed by Haonan Gu, using results of Alejandro J. Giangreco-Maidana [arXiv:1806.10842].