For cubic extensions of F_q(t), the discriminant-norm count is c1 q^{2N} minus c2^{N mod 3} q^{5N/3}, plus an error O(N^4 q^{3N/2}), with all constants explicit.
On the Davenport-Heilbronn theorems and second order terms
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The geometry of secondary terms in arithmetic statistics
For cubic extensions of F_q(t), the discriminant-norm count is c1 q^{2N} minus c2^{N mod 3} q^{5N/3}, plus an error O(N^4 q^{3N/2}), with all constants explicit.