Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, these primes go into sequences... Note that, within a few hours, another guy had run the tables much higher with a one-line Maple command. Some days it does not pay to get up.

I thought of one I really do not understand. Discriminant $205$ has four classes of forms, $$ \langle 1, 13, -9 \rangle, \; \langle -1, 13, 9 \rangle, \; \langle 3, 13, -3 \rangle, \; \langle -3, 13, 3 \rangle. $$

The third and fourth are opposites so in the same genus, although distinct. The first two are in the principal genus, but they are not opposites, one is $-1$ times the other; in particular, they get diffeent positive primes, although both do residues $\pmod 5$ and $\pmod {41}.$ For $\langle 1, 13, -9 \rangle$ we get $$ 1,5,59,131,139,241,269,271,359,409, \ldots, $$ while for $\langle -1, 13, 9 \rangle$ we get $$ 31,41,61,251,349,379,389,401,419,431, \ldots. $$

For positive forms, low class number, there are polynomials, such as in Cox's book, such that primes represented by the principal form are those for which the polynomial factors a certain way. For a prime $p \equiv 1 \pmod 3,$ Gauss showed that $2$ is a cubic residue if an only if $p = u^2 + 27 v^2.$ Jacobi showed that $3$ is a cubic residue if an only if $p = u^2 + uv + 61 v^2.$ All I found in Henri Cohen's tables was the fact that $\mathbb Q(\sqrt {205})$ has class number $2$ and $L_K = \mathbb Q(\sqrt 5),$ appendix 12C on pages 533 and 534. See related information at IT'S A LINK.

Let's see, $34$ is the smallest number where it is a surprise that there is no solution to $x^2 - 34 y^2 = -1.$ The smallest such odd number is $205,$ as there is no solution to $x^2 - 205 y^2 = -1.$ For prime $p \equiv 1 \pmod 4,$ there is always a solution to $x^2 - p y^2 = -1.$ Proof in Mordell's book. Anyway, this is why $\langle 1, 13, -9 \rangle, \; \langle -1, 13, 9 \rangle$ are distinct classes.

So, that is the question, can I distinguish the represented (positive) primes by factoring some polynomial mod these primes?