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 ].
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 , 10.1023/A1017553432375 ]
- 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 $3$) and Xavier Xarles (dimension $4$ over $\F_2$; [arXiv:2007.07822 ]).