A quintic of genus 1 that contradicts the Hasse principle (Q1769636)

The author gives an explicit plane quintic \(C\) with five nodes that has rational points over all completions of \({\mathbb Q}\), but has no rational points over \({\mathbb Q}\). To show the curve has no \({\mathbb Q}\)-rational points, the author makes use of the algebraic number field \({\mathbb Q}(\zeta +\bar\zeta)\), where \(\zeta\) is a primitive eleventh root of unity. This number field was also used by D.~F.~Coray in an appendix to his 1974 Cambridge dissertation to construct a smooth plane quintic that violates the Hasse principle. The author shows that the curve \(C\) gives rise to an element of order 5 of the Tate-Shafarevich group Ш\((E/{\mathbb Q})\), where \(E\) denotes the Jacobian of \(C\). An element of order 5 of a Tate-Shafarevich group has also been constructed, using different methods, by \textit{T.~A.~ Fisher} [J. Eur. Math. Soc. (JEMS) 3, No.~2, 169--201 (2001; Zbl 1007.11031)].