Explicit Descent via 4-Isogeny on an Elliptic Curve

Abstract

We work out the complete descent via 4-isogeny for a family of rational elliptic curves with a rational point of order 4; such a family is of the form $y^2+xy+ay=x^3+a{x}^2$ where $\sqrt{-a}\in {\mathbb{Q}}^{\times }$. In the process we exhibit the 4-isogeny and the isogenous curve, explicitly present the principal homogeneous spaces, and discuss examples by computing the rank.

20 pages