= Modulo-eight obstruction to a two-isogeny descent class
{title2=$N^2=3U^4+5U^2V^2-2V^4\quad\text{has no primitive solution}$}
A proposed square class in a <two-isogeny descent> can be excluded by testing its quartic modulo $8$. For $N^2=3U^4+5U^2V^2-2V^4$ with $\gcd(U,V)=1$, both variables odd give residue $6$; odd-even gives $3$ or $7$; even-odd gives $6$ or $2$. None is a square modulo $8$, excluding class $3$ and, by the subgroup property of the descent image, its entire coset.
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