We present a method for tabulating all cubic function fields over$\mathbb{F}_q(t)$ whose discriminant $D$ has either odd degree or even degreeand the leading coefficient of $-3D$ is a non-square in $\mathbb{F}_{q}^*$, upto a given bound $B$ on the degree of $D$. Our method is based on ageneralization of Belabas' method for tabulating cubic number fields. The maintheoretical ingredient is a generalization of a theorem of Davenport andHeilbronn to cubic function fields, along with a reduction theory for binarycubic forms that provides an efficient way to compute equivalence classes ofbinary cubic forms. The algorithm requires $O(B^4 q^B)$ field operations as $B\rightarrow \infty$. The algorithm, examples and numerical data for$q=5,7,11,13$ are included.
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机译：我们提出了一种方法，用于制表所有超过\\ mathbb {F} _q（t）$的三次函数字段的表，该函数的判别式$ D $具有奇数或偶数的度，并且$ -3D $的前导系数在$ \中是非平方的mathbb {F} _ {q} ^ * $，直到$ D $的给定边界$ B $。我们的方法基于Belabas方法的一般化，用于制表立方数字段。主要的理论成分是Davenport和Heilbronn定理对三次函数域的推广，以及针对二元三次形式的归约理论，它提供了一种有效的方法来计算二元三次形式的等价类。该算法要求$ O（B ^ 4 q ^ B）$字段操作为$ B \ rightarrow \ infty $。包括$ q = 5,7,11,13 $的算法，示例和数值数据。
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