  For an infinite family of monogenic trinomials $P(X) = X^3\pm 3rbX-b$ in
$\mathbb{Z}\lbrack X\rbrack$, arithmetical invariants of the cubic number field
$L = \mathbb{Q}(\theta)$, generated by a zero $\theta$ of $P(X)$, and of its
Galois closure $N = L(\sqrt{d(L)})$ are determined. The conductor $f$ of the
cyclic cubic relative extension $N/K$, where $K = \mathbb{Q}(\sqrt{d(L)})$
denotes the unique quadratic subfield of $N$, is proved to be of the form
$3^eb$ with $e\in\lbrace 1,2\rbrace$, which admits statements concerning
primitive ambiguous principal ideals, lattice minima, and independent units in
$L$. The number $m$ of non-isomorphic cubic fields $L_1,\ldots,L_m$ sharing a
common discriminant $d(L_i) = d(L)$ with $L$ is determined.
