  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 four
examples 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 the
exponential and logarithmic functions and relate them with the product $D_k$ of
all 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, we
obtain a formula relating both invariants. We prove also using elementary
methods a theorem due to Thakur that relate two different combinatorial symbols
important in the analysis of solitons.
