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

We work with detail the Drinfeld module over the ring $$A=F_2[x,y]\mathrm{/}(y^2+y=x^3+x+1)\mathrm{.}$$ 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 ${\mathrm{\ell }}_k$ 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.