Labels for isogeny classes of abelian varieties over finite fields (reviewed)

The LMFDB uses a systematic system to label isogeny classes defined over finite fields. The label format is g.q.iso, where

• $g$ is the dimension of the abelian varieties contained in the isogeny class,

• $q$ is the cardinality of the field over which the abelian varieties and the isogenies are defined, and

• iso specifies the isogeny class.

The label iso is obtained in the following manner: If the Weil $q$-polynomial of the isogeny class is $$1 + a_1 x + a_2 x^2 + \cdots +a_gx^g+ qa_{g-1}x^{g+1} \cdots + q^{g-1}a_1 x^{2g-1} + q^g x^{2g},$$ the label contains the integer coefficients $a_1, \ldots a_g$, encoded in base 26 with the symbols a, b, c... z, where a = 0, and separated by underscores. Negative numbers are distinguished from positive numbers by a leading a. For example, ae_j_ap denotes the polynomial $$1 -4x + 9x^2 - 15x^3 +9qx^4 - 4q^2 x^5 + q^3 x^6,$$ where $q$ is the cardinality of the field, because e = 4, j = 9 and p = 15.