Summary: Let \(K\) be an algebraic number field of degree \(d\geqslant 3\), \(\sigma_{1},\sigma_{2},\ldots ,\sigma_{d}\) the embeddings of \(K\) into \(\mathbb{C}\), \(\alpha\) a non-zero element in \(K\), \(a_{0}\in \mathbb{Z}\), \(a_{0}>0\) and \[ F_{0}(X,Y)=a_{0}\prod_{i=1}^{d}(X-\sigma_{i}(\alpha)Y). \] Let \(\upsilon\) be a unit in \(K\). For \(a\in \mathbb{Z}\), we twist the binary form \(F_{0}(X,Y)\in \mathbb{Z}[X,Y]\) by the powers \(\upsilon^{a}\) (\(a\in \mathbb{Z}\)) of \(\upsilon\) by setting \[ F_{a}(X,Y)=a_{0}\prod_{i=1}^{d}(X-\sigma_{i}(\alpha\upsilon^{a})Y). \] Given \(m>0\), our main result is an effective upper bound for the size of solutions \((x,y,a)\in \mathbb{Z}^{3}\) of the Diophantine inequalities \[ 0<|F_{a}(x,y)|\leqslant m \] for which \(xy\not =0\) and \(\mathbb{Q}(\alpha\upsilon^{a})=K\). Our estimate is explicit in terms of its dependence on \(m\), the regulator of \(K\) and the heights of \(F_{0}\) and of \(\upsilon\); it also involves an effectively computable constant depending only on \(d\).