Session p. 215

p[ρ_, prec_] := 
      Block[{$MinPrecision = prec}, 
        p /. FindRoot[
            ArcCot[ρ] == ((p π)/2) + 
                Arg[EllipticK[(E^(2 I p π))]], {p, (10^(-8)), (10^(-3))}, AccuracyGoal -> prec, 
            WorkingPrecision -> prec]]
\(\({Precision == #, \(p\_*\) == 
              p[10, #]} &\) /@ {MachinePrecision, 17} // 
      TableForm\)