Quartic covering in a two-isogeny descent
= Quartic covering in a two-isogeny descent
{title2=$C_d$}
For a square class $d$ dividing $b$, the associated quartic covering is
$$
C_d:\qquad w^2=du^4+au^2v^2+\frac bdv^4.
$$
It has a nontrivial rational point exactly when $d$ occurs in the square-class image attached to $E:y^2=x(x^2+ax+b)$.