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We work with detail the Drinfeld module over the ring$$A=F_2[x,y]/(y^2+y=x^3+x+1).$$ The example in question is one of the fourexamples that come from quadratic imaginary fields with class number $h = 1$and rank one.  We develop specific formulas for the coefficients $d_k$ and $\ell_k$ of theexponential and logarithmic functions and relate them with the product $D_k$ ofall monic elements of $A$ of degree $k$. On the Carlitz module, $D_k$ and $d_k$coincide, but this is not true in general Drinfeld modules. On this example, weobtain a formula relating both invariants. We prove also using elementarymethods a theorem due to Thakur that relate two different combinatorial symbolsimportant in the analysis of solitons.

A description of a Drinfeld module with class number $h=1$ and rank $1$