Isolations of the sum of two squares from its proper subforms

Abstract: For a (positive definite and integral) quadratic form $f$, a quadratic form is said to be {\it an isolation of $f$ from its proper subforms} if it represents all proper subforms of $f$, but not $f$ itself. It was proved that the minimal rank of isolations of the square quadratic form $x^2$ is three, and there are exactly $15$ ternary diagonal isolations of $x^2$. Recently, it was proved that any quaternary quadratic form cannot be an isolation of the sum of two squares $I_2=x^2+y^2$, and there are quinary isolations of $I_2$. In this article, we prove that there are at most $231$ quinary isolations of $I_2$, which are listed in Table $1$. Moreover, we prove that $14$ quinary quadratic forms with dagger mark in Table $1$ are isolations of $I_2$.