This is something I made up years ago when Kap got interested in the "tribonacci" numbers. A few weeks ago, I put this at MSE but no takers. Oh, years ago I asked some guy at MSRI about this, he muttered something about norm forms and laughed at me.
Given integers $a,b,c,$ and cubic form $$ f(a,b,c) = a^3 + b^3 + c^3 + a^2 b - a b^2 + 3 a^2 c - a c^2 + b^2 c - b c^2 - 4 a b c $$ $$ f(a,b,c) = \det \left( \begin{array}{ccc} a & b & c \\\ c & a + c & b + c \\\ b + c & b + 2 c & a + b + 2 c \end{array} \right) . $$ what primes $p$ can be integrally represented as $$ p = f(a,b,c)? $$
(A): I think it is all primes $(p| 11) = -1 ,$ and all $p = u^2 + 11 v^2$ in integers, but not any $q = 3 u^2 + 2 u v + 4 v^2.$ Note that, if $-p$ is represented, so is $p.$
(B): I also suspect that if prime $q = 3 u^2 + 2 u v + 4 v^2$ and $f(a,b,c) \equiv 0 \pmod q,$ then all three $a,b,c \equiv 0 \pmod q,$ and $f(a,b,c) \equiv 0 \pmod {q^3}.$
Note that if $f$ integrally represents both $m,n$ then it represents $mn.$ That is because $f(a,b,c) = \det(aI + b X + c X^2),$ where $$ X = \left( \begin{array}{ccc} 0 & 1 & 0 \\\ 0 & 0 & 1 \\\ 1 & 1 & 1 \end{array} \right) $$ Then $X^3 = X^2 + X + I$ and $X^4 = 2 X^2 + 2 X + I.$
If all suspicions are correct, we can correctly describe all numbers integrally represented by this polynomial: positive or negative are unimportant, most prime factors are unimportant, all that matters is that every exponent of a prime factor $q = 3 u^2 + 2 u v + 4 v^2$ must be divisible by 3.
I should have done this last time: most of the class field part has already been done, by Hudson and Williams (1991), Theorem 1 and Table 1 on page 134. You get my version of the polynomial by negating their variable $x.$
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p a b c
2 0 1 1
7 0 -11 6
11 0 -3 2
13 0 -1 2
17 -1 0 2
19 1 2 4
29 0 -7 4
41 0 3 2
43 0 4 -1
47 0 5 -2
53 0 1 4
61 0 46 -25
73 2 -36 19
79 0 3 4
83 0 24 -13
101 -1 12 -6
103 0 15 -8
107 1 -9 5
109 1 2 6
127 1 -2 4
131 1 7 -3
139 1 -6 4
149 -1 4 2
151 0 -20 11
163 0 5 2
167 -1 1 5
173 0 6 -1
193 1 -52 28
197 0 9 -4
199 -1 5 1
211 0 -12 7
227 -2 0 5
233 0 -16 9
239 0 -6 5
241 0 -4 5
257 0 -1 6
263 2 4 9
269 -1 0 6
271 2 8 -3
277 1 -7 5
281 0 2 7
283 -1 2 6
293 -1 -8 6
307 2 -1 6
311 0 5 6
337 -2 5 2
347 1 7 5
349 0 19 -10
359 -1 9 -3
373 2 5 10
397 1 -1 7
401 0 -68 37
409 3 -77 41
419 0 -7 6
421 0 7 2
431 1 -14 8
439 0 8 -1
457 0 1 8
461 0 -2 7
479 1 -8 6
491 0 7 4
499 0 13 -6
503 -1 -36 20
523 0 9 -2
541 2 -12 7
547 1 -11 7
557 -1 25 -13
563 -2 -11 8
569 0 8 1
571 1 -3 7
587 0 -29 16
593 3 -25 13
599 -1 0 8
601 0 7 6
607 0 11 -4
613 0 4 9
617 2 -1 8
659 0 8 3
673 0 -6 7
677 0 -17 10
683 -1 4 8
701 2 13 -6
733 1 10 -2
739 -1 14 -6
743 -2 1 8
757 0 81 -44
761 -1 8 2
769 0 -25 14
773 -1 7 5
787 2 5 12
809 -1 -10 8
811 -4 0 7
821 -1 3 9
827 2 10 7
853 0 -11 8
857 -2 3 8
863 0 9 2
877 -2 -15 10
883 0 -14 9
887 2 -3 8
907 0 -5 8
911 0 8 7
919 0 -2 9
929 1 7 11
937 3 8 14
941 3 -1 9
953 -1 6 8
967 1 13 -5
991 1 -35 19
997 -3 7 3
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