The problem of the characterization of primes of the form $p=x^2+ny^2$ is solved (e.g. in the book of Cox), and there are also a few examples for special numbers $n$. My question is:
Does anyone know a source where one can find more examples of how the characterization looks like in spaciel cases (e.g. for all $n$ up to, say, $20$)?
For $n <= 25 $ you are covered by idoneal numbers (one class per genus) together with two articles, Hudson+Williams 1991 and then Liu + Williams. I put both at http://zakuski.utsa.edu/~jagy/inhom.html
Added: you get all $n \leq 34$ with a few values in the tables of Henri Cohen there. But 26 and 29 are in a rather different manner of reporting, some serious work to extract the appropriate monic cubic univariate polynomials from the given sextics. The relevant phrase is field compositum, not something I know how to do. Hmmm, unless those field tables Noam once mentioned have it.
$$ x^2 + y^2: \; p=2 \; \mbox{OR} \; (-1|p)= 1 $$ $$ x^2 + 2 y^2: \; p=2 \; \mbox{OR} \; (-2|p)= 1 $$ $$ x^2 + 3 y^2: \; p=3 \; \mbox{OR} \; (-3|p)= 1 $$ $$ x^2 + 4 y^2: \; \; (-1|p)= 1 $$ $$ x^2 + 5 y^2: \; p=5 \; \mbox{OR} \; \; (-5|p)= 1 \; \mbox{AND} \; (p|5)= 1 $$ $$ x^2 + 6 y^2: \; (-6|p)= 1 \; \mbox{AND} \; (p|3)= 1 $$ $$ x^2 + 7 y^2: \; p=7 \; \mbox{OR} \; \; \; (-7|p)= 1 $$ $$ x^2 + 8 y^2: \; \; p \equiv 1 \pmod 8 $$ $$ x^2 + 9 y^2: \; \; p \equiv 1 \pmod {12} $$ $$ x^2 + 10 y^2: \; (-10|p)= 1 \; \mbox{AND} \; (p|5)= 1 $$ $$ x^2 + 11 y^2: \; p=11 \; \mbox{OR} \; \; (-11|p)= 1 \; \mbox{AND} \; z^3 + z^2 - z + 1 \; \mbox{has three distinct roots} \; \pmod p $$ $$ x^2 + 12 y^2: \; \; p \equiv 1 \pmod {12} $$ $$ x^2 + 13 y^2: \; p=13 \; \mbox{OR} \;\; (-13|p)= 1 \; \mbox{AND} \; (-1|p)= 1 $$ $$ x^2 + 14 y^2: \; (-14|p)= 1 \; \mbox{AND} \; z^4 + 2 z^2 - 7 \; \mbox{has four distinct roots} \; \pmod p $$ $$ x^2 + 15 y^2: \; (-15|p)= 1 \; \mbox{AND} \; (p|3)= 1 $$ $$ x^2 + 16 y^2: \; \; p \equiv 1 \pmod 8 $$ $$ x^2 + 17 y^2: \; p=17 \; \mbox{OR} \; (-17|p)= 1 \; \mbox{AND} \; z^4 - 2 z^2 + 17 \; \mbox{has four distinct roots} \; \pmod p $$ $$ x^2 + 18 y^2: \; (-2|p)= 1 \; \mbox{AND} \; (p|3)= 1 $$ $$ x^2 + 19 y^2: \; p=19 \; \mbox{OR} \; \; (-19|p)= 1 \; \mbox{AND} \; z^3 - 2 z + 2 \; \mbox{has three distinct roots} \; \pmod p $$ $$ x^2 + 20 y^2: (-5|p)= 1 \; \mbox{AND} \; z^4 - 2 z^2 + 5 \; \mbox{has four distinct roots} \; \pmod p $$ $$ x^2 + 21 y^2: \; (p|7)= 1 \; \mbox{AND} \; (p|3)= 1 \; \mbox{AND} \; (-1|p)= 1 $$ $$ x^2 + 22 y^2: \; (-22|p)= 1 \; \mbox{AND} \; (p|11)= 1 $$ $$ x^2 + 23 y^2: \; p=23 \; \mbox{OR} \; \; (-23|p)= 1 \; \mbox{AND} \; z^3 - z + 1 \; \mbox{has three distinct roots} \; \pmod p $$ $$ x^2 + 24 y^2: \; \; p \equiv 1 \pmod {24} $$ $$ x^2 + 25 y^2: \; (-1|p)= 1 \; \mbox{AND} \; (p|5)= 1 $$ $$ x^2 + 26 y^2: \; \; (-26|p)= 1 \; \mbox{AND} \; \; (p|13)= 1 \; \mbox{AND} \; z^3 - z - 2 \; \mbox{has three distinct roots} \; \pmod p $$ $$ x^2 + 27 y^2: \; \; (-3|p)= 1 \; \mbox{AND} \; z^3 - 2 \; \mbox{has three distinct roots} \; \pmod p $$ $$ x^2 + 28 y^2: \; (-7|p)= 1 \; \mbox{AND} \; p \equiv 1 \pmod 4 $$ $$ x^2 + 29 y^2: \; p=29 \; \mbox{OR} \; \; (-29|p)= 1 \; \mbox{AND} \; \; p \equiv 1 \pmod 4 \; \mbox{AND} \; z^3 - z^2 - 2 \; \mbox{has three distinct roots} \; \pmod p $$