Modulo-eight obstruction to a two-isogeny descent class ($N^2=3U^4+5U^2{V}^2-2V^4\text{has no primitive solution}$)

A proposed square class in a two-isogeny descent can be excluded by testing its quartic modulo $8$. For $N^2=3U^4+5U^2{V}^2-2V^4$ 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.