On the computation of the class numbers of some cubic fields (Q580426)

Let \(f(x)=x^ 3+12Ax-12\) with \(A>0\). Class numbers are calculated for the cubic fields generated by the unique real root of the equations \(f(x)=0\), where A takes the values \(1\leq A\leq 36.\) If A is of the form \(A=9a^ 2\), these fields are related to the diophantine equation \[ x^ 3\quad +\quad y^ 3\quad +\quad z^ 3 = 3\quad. \] For a in the range \(1\leq a\leq 17\), the class numbers of these fields are also estimated (for \(a=1, 2, 3, 4\) the exact values are given).