Non-trivial Ш in the Jacobian of an infinite family of curves of genus 2 (Q1032661)

Let \(C\) be the curve of genus two over \(\mathbb Q\) given by \(C:y^{2}=2qx(-x^{2}+2x+1)(x^{2}+4x+2)\), where \(q\) is a prime congruent to 13 modulo 24, and let \(J\) denote the Jacobian variety of \(C\). The main result of this article shows that the 2-part of the Tate-Shafarevich group of \(J\) is non-trivial. For the proof the authors employ the method of descent via Richelot isogeny utilized in [\textit{E. V. Flynn}, Acta Arith. 66, No. 1, 23--43 (1994; Zbl 0835.14009)], and provide us with complete detail of their computation.