On a family of cubics over imaginary quadratic fields</o:p>

Summary

We consider the relative Thue equations \[X^3 - t X^2 Y - (t+1) X Y^2 -Y^3=\mu,\] where the parameter <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>t$, the root of unity $\mu$ and the solutions $X$ and $Y$ are integers in the same imaginary quadratic number field. We use Baker's method to find all solutions for $|t|> 2.88 \cdot 10^{33}$.