The number of cubic extensions of F_q(T) of discriminant q^M is C_1 q^M - C_2(M) q^(5M/6) + O(q^((2/3+epsilon)M)), with explicit constants and an unconditional omega-result for refined counts.
Improved error estimates for the Davenport–Heilbronn theorems,
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On the counting function of cubic function fields
The number of cubic extensions of F_q(T) of discriminant q^M is C_1 q^M - C_2(M) q^(5M/6) + O(q^((2/3+epsilon)M)), with explicit constants and an unconditional omega-result for refined counts.