Quaternary universal forms over \(\mathbb Q(\sqrt{13})\) (Q1039634)

Let \(F=\mathbb{Q}(\sqrt{m})\) be a real quadratic field with the ring of integers \(\mathcal{O}\). A totally positive definite integral \(n\)-ary quadratic form \(f=f(x_1,\dots,x_n)\) with coefficients in \(\mathcal{O}\) is called \textit{universal} if it represents all totally positive integers in \(\mathcal{O}\). The author proves that there are only two quaternary universal forms (up to equivalence) over \(\mathbb{Q}(\sqrt{13})\): \[ x_1^2+x_2^2+\frac{5+\sqrt{13}}{2}x_3^2+\frac{5-\sqrt{13}}{2}x_4^2+2x_3x_4 \] and \[ x_1^2+2x_2^2+\frac{5+\sqrt{13}}{2}x_3^2+\frac{5-\sqrt{13}}{2}x_4^2+2x_2x_3+2x_2x_4. \]