 Two-isogeny descent

Here descent is an arithmetic calculation with square classes and an isogeny of elliptic curves, rather than the gluing of objects along a covering.

For $E:y^2=x^3+a{x}^2+bx$ and its two-isogenous curve $E^{\prime }:y^2=x^3-2a{x}^2+(a^2-4b)x$, square-class maps on $E(\mathbb{Q})$ and $E^{\prime }(\mathbb{Q})$ determine the Mordell-Weil rank.