This is a very specific question. This equation came up when attempting to compute certain subgroups of groups of Lie type. The polynomial $x^6-2x^3+8$ splits over $\mathbb{F}_q$ if and only if a particular subgroup ($\mathrm{SL}_2(8).3$) embeds in a particular group of Lie type ($E_6(q)$).
More specifically, let $q$ be a prime power congruent to $1,2,4\bmod 7$, and not a power of $2$ or $3$. We ask for solutions to the equation $x^6-2x^3+8=0$ over $\mathbb{F}_q$. If $q\equiv 2\bmod 3$ then there is always a solution, but the polynomial does not split. Thus we may in addition assume that $q\equiv 1\bmod 3$. This polynomial has a solution if and only if $1+\sqrt{-7}$ is a cube in $\mathbb{F}_q$.
If it is of help, notice that $1+\sqrt{-7}$ is the sum $2(\zeta+\zeta^2+\zeta^4)$, where $\zeta$ is a primitive $7$th root of unity.
Ideally, one would like a statement of the form 'This equation has a solution if and only if $q$ is congruent to one of […] modulo $n$'.
Best Answer
I have not looked at prime powers, just primes. The four forms are indefinite, so we must require that we are using the positive primes represented by one of the forms, except for $7$ itself. Note that the third form integrally represents $-89,$ but we ignore that. I give them in "reduced" form, which means $\langle a,b,c \rangle$ with $$ ac < 0 \; \; \; \; \mbox{AND} \; \; \; \; b > |a+c| $$
$$ x^2 + 108 xy - 108 y^2 $$ $$ 4x^2 + 108 xy - 27 y^2 $$ $$ 7x^2 + 98 xy - 89 y^2 $$ $$ 16 x^2 + 96 xy - 45 y^2 $$
these make a subgroup of all primitive forms of this discriminant.
I asked for the first prime above 100,000 from each of the four forms, here is factoring using gp-pari
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