# Show that quadratic form is anisotropic over rational numbers

How to determine if quadratic form $$q(x,y,z)=-6x^2+ \frac{7}{2}y^2-\frac{25}{7}z^2$$ is anisotropic over $$\mathbb{Q}$$? This quadratic form is a diagonalisation of another quadratic form. Is there any criteria or a theorem that could give me the answer to the question? Any help is welcome.

Just check that it's anisotropic over some $$\Bbb Q_p$$ (it is clearly isotropic over $$\Bbb R$$). The only $$p$$ that could possibly work are $$p\in\{2,3,5,7\}$$. (The Hasse-Minkowski theorem proves that if $$q$$ is isotropic locally then it is isotropic over $$\Bbb Q$$.)
Incidentally, $$q$$ is equivalent to $$-6x^2+14y^2-7z^2$$ over $$\Bbb Q$$. I think that's anisotropic over $$\Bbb Q_3$$.