you don't seem to be mentioning Gauss composition. You have a genus of forms, equivalent to $\langle 1,8,27 \rangle,$ then $\langle 3,8,29 \rangle,$ then $\langle 9,8,3 \rangle.$ These are convenient for Dirichlet's description of composition.

There is a cancellation for principal forms: if $x^2 + 8xy + 27 y^2$ represents both $p$ and $np,$ it also represents $n.$

You mention 15, $$ \left( 3x^2 + 8xy + 9 y^2 \right) \left( 9z^2 + 8zw + 3 w^2 \right) = \color{magenta}{ u^2 + 8 uv + 27 v^2,} $$ where $u = 3xw + 9 yz +8yw$ and $v=xz-yw.$

Lots more...A prime $p \neq 2,11$ with Legendre symbol $(-44|p) = 1$ is represented by $x^2 + 8xy + 27 y^2$ if and only if the polynomial $z^3 + z^2 - z + 1$ factors into three distinct linear factors $\pmod p.$ Cubic because class number $h(-44) = 3$

Oh, Dirichlet composition is available everywhere, I copied from D. A. Cox, Primes of the Form $x^2 + n y^2$

there is always more

you don't seem to be mentioning Gauss composition. You have a genus of forms, equivalent to $\langle 1,8,27 \rangle,$ then $\langle 3,8,9 \rangle,$ then $\langle 9,8,3 \rangle.$ These are convenient for Dirichlet's description of composition.

There is a cancellation for principal forms: if $x^2 + 8xy + 27 y^2$ represents both $p$ and $np,$ it also represents $n.$

You mention 15, $$ \left( 3x^2 + 8xy + 9 y^2 \right) \left( 9z^2 + 8zw + 3 w^2 \right) = \color{magenta}{ u^2 + 8 uv + 27 v^2,} $$ where $u = 3xw + 9 yz +8yw$ and $v=xz-yw.$

Lots more...A prime $p \neq 2,11$ with Legendre symbol $(-44|p) = 1$ is represented by $x^2 + 8xy + 27 y^2$ if and only if the polynomial $z^3 + z^2 - z + 1$ factors into three distinct linear factors $\pmod p.$ Cubic because class number $h(-44) = 3$

Oh, Dirichlet composition is available everywhere, I copied from D. A. Cox, Primes of the Form $x^2 + n y^2$

there is always more