Elliptic curves having the same 6-torsion of a given elliptic curve
Let \(K\) be a field of characteristic different from \(2\) and \(3\), and let \(E\) be an elliptic curve defined over \(K\). Let \(E[6]\) be its group of \(6\)-torsion points. A question raised by Mazur is: Is there a finite or an infinite number of elliptic curves \(F\), defined over \(K\), such that \(F[6]\) is isomorphic to \(E[6]\) as a \(\text{Gal}(\overline{K}/K)\)-module? This moduli-problem is parametrized by the elliptic curve \(\widetilde{E}\) of equation \(y^2 = x^3 + \Delta_E\), whereas \(\Delta_E\) is the discriminant of a Weierstrass equation of \(E\). Hence the answer to the problem stated depends on the \(K\)-rank of \(\widetilde{E}\).
scientific article; zbMATH DE number 1389293
scientific article