Counterexamples to the Hasse principle among the twists of the Klein quartic (Q6585092)

The Klein quartic is isomorphic to the modular curve \(X(7)\) over the cyclotomic field \(\mathbb Q(\zeta_7)\), and it is given by \(x^3y + y^3z + z^3x =0\). It is the curve with the smallest genus that attains the Hurwitz upper bound \(84(g-1)\) on the number of automorphisms for a curve of a genus \(g\) in characteristic \(0\). The authors prove that there are twists of the Klein quartic that are counterexamples to the Hasse principle. Such twists are constructed from the existence of an irreducible polynomial \(f(x)=x^3+bx+c\in \mathbb Z[x]\) such that \(2\nmid c\), \(3\nmid b\), \(7\mid b\), \(7|| c\), \(p\mid b\) and \(p|| c\) for all primes \(p\ne 2,3,7\) dividing \(4b^3+27c^2\), and the splitting field of \(f(x)\) is not isomorphic to \(\mathbb Q(\zeta_7)\) or to the field numbered 6.0.214375.1. by The \(L\)-function and Modular Forms Database.