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Let 
$K$ be an algebraic number field of degree 
$d\geqslant 3$ , 
$\unicode[STIX]{x1D70E}_{1},\unicode[STIX]{x1D70E}_{2},\ldots ,\unicode[STIX]{x1D70E}_{d}$ the embeddings of 
$K$ into 
$\mathbb{C}$ , 
$\unicode[STIX]{x1D6FC}$ a non-zero element in 
$K$ , 
$a_{0}\in \mathbb{Z}$ , 
$a_{0}>0$ and Let 
$$\begin{eqnarray}F_{0}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC})Y).\end{eqnarray}$$
$\unicode[STIX]{x1D710}$ 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 
$\unicode[STIX]{x1D710}^{a}$ ( 
$a\in \mathbb{Z}$ ) of 
$\unicode[STIX]{x1D710}$ by setting Given 
$$\begin{eqnarray}F_{a}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})Y).\end{eqnarray}$$
$m>0$ , our main result is an effective upper bound for the size of solutions 
$(x,y,a)\in \mathbb{Z}^{3}$ of the Diophantine inequalities for which 
$$\begin{eqnarray}0<|F_{a}(x,y)|\leqslant m\end{eqnarray}$$
$xy\not =0$ and 
$\mathbb{Q}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})=K$ . Our estimate is explicit in terms of its dependence on 
$m$ , the regulator of 
$K$ and the heights of 
$F_{0}$ and of 
$\unicode[STIX]{x1D710}$ ; it also involves an effectively computable constant depending only on 
$d$ .
