In the Sensual Quadratic Form page 44, J.H. Conway is discussing isospectral lattices of dimension 4 when he states the following:
''... we find all solutions of the equation(s) $x^2+7y^2+13z^2+19w^2=48$ ... to find possible vectors of norm 4 ... in $L^+(1,7,13,19)$ and $L^-(1,7,13,19)$.''
Here $L^+$ and $L^-$ are the isospectral lattices in question. I know that 'all solutions' in this case refers to integer solutions, but I wonder how one goes about finding these. The statement is written in such a way that it would seem easy to do, yet I have no idea where to start.
It suffices to find all solutions with $x$, $y$, $z$ and $w$ nonnegative. Because the quadratic form is positive definite, we see that $$y^2\leq\frac{48}{7},\qquad z^2\leq\frac{48}{13},\qquad w^2\leq\frac{48}{19},$$ and so $y\leq2$, $z\leq1$, $w\leq1$, and for each choice of $(y,z,w)$ there is at most one (nonnegative) value of $x$ that satisfies the equation. So this leaves just $12$ cases to check, which isn't too much work to do by hand even.