Tabulation of Cubic Function Fields with Imaginary and Unusual Hessian

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We give a general method for tabulating all cubic function fields over whose discriminant D has odd degree, or even degree such that the leading coefficient of − 3D is a non-square in , up to a given bound on \(|D| = q^{\deg(D)}\). The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields. We present numerical data for cubic function fields over and over with \(\deg(D) \leq 7\) and \(\deg(D)\) odd in both cases.