TITLE: Primes of the form $ 2 x^2 + 2 xy + 3 y^2 $
QUESTION [5 upvotes]: Why is every prime $3,7 (\bmod 20)$ of the form
$$ 2 x^2 + 2 xy + 3 y^2 $$

I do not think that form is the norm of an abelian ring ?
How to prove this ?

REPLY [2 votes]: These are the norms of the non-principal ideals of $\Bbb Z[\sqrt{-5}]$.
The ideals of prime norm in this ring have norms $2$, $5$ and $p$
with $p\equiv1,3,7,9\pmod{20}$. The principal ideals cannot have norms
$\equiv3,7$ and the non-principal ones cannot have norms $\equiv1,9$.