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The integral $t$-motivic cohomology and the class module of a (rigidanalytically trivial) Anderson $t$-motive were introduced by the first authorin [Gaz22b]. This paper is devoted to their determination in the particularcase of tensor powers of the Carlitz $t$-motive, namely, the function fieldcounterpart $\underline{A}(n)$ of Tate twists $\mathbb{Z}(n)$. We find out thatthese modules are in relation with fundamental objects of function fieldarithmetic: integral $t$-motivic cohomology governs linear relations amongCarlitz polylogarithms, its torsion is expressed in terms of the denominator ofBernoulli-Carlitz numbers and the Fitting ideal of class modules is a specialzeta value. We also express the regulator of $\underline{A}(n)$ for positive$n$ in terms of generalized Carlitz polylogarithms; after establishing theiralgebraic relations using difference Galois theory together with theAnderson-Brownawell-Papanikolas criterion, we prove that the regulator is anisomorphism if, and only if, $n$ is prime to the characteristic.

Carlitz twists: their motivic cohomology, regulators, zeta values and polylogarithms