Expired by this question Show determinant of matrix is non-zero I am moved to ask:

Given integers $a,b,c,$ and cubic form $$ f(a,b,c) = a^3+2 b^3-6 a b c+4 c^3 = \left|\begin{bmatrix} a & 2c & 2b\\b & a & 2c\\ c & b & a\end{bmatrix}\right|, $$ what primes $p$ can be integrally represented as $$ p = f(a,b,c)? $$

I think it is $3,$ all primes $p \equiv 2 \pmod 3,$ and all $p = u^2 + 27 v^2$ in integers, but not any $q = 4 u^2 + 2 u v + 7 v^2.$ I checked for $p < 10000.$

Note that, if $-p$ is represented, so is $p.$

Although it does not finish things, 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 $$ X = \begin{bmatrix} 0 & 0 & 2\\1 & 0 & 0\\ 0 & 1 & 0\end{bmatrix}. $$ Then $X^3 = 2 I$ and $X^4 = 2 X.$

I once asked a guy at MSRI about pretty much the same problem, only instead of the important polynomial being $\lambda^3 - 2$ it was $\lambda^3 - \lambda^2 - \lambda - 1.$ The phrase norm forms came up, and he laughed at me.

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```
p a b c
2 0 1 0
3 -1 0 1
5 1 0 1
11 -1 1 1
17 -1 -1 2
23 1 1 2
29 -3 10 -6
31 -1 26 -20
41 1 -2 2
43 1 -1 2
47 -1 4 -2
53 1 -4 3
59 1 3 -2
71 -1 2 2
83 3 1 3
89 1 2 3
101 3 -7 4
107 -1 0 3
109 1 -12 9
113 1 4 2
127 -1 16 -12
131 3 3 -1
137 -3 1 3
149 1 4 -1
157 -1 5 -2
167 -3 3 2
173 -3 7 -3
179 1 -31 24
191 -1 -2 4
197 5 2 -1
223 1 5 2
227 3 -2 3
229 -1 -1 4
233 1 5 -3
239 1 3 4
251 -1 -4 5
257 1 0 4
263 3 4 -3
269 -1 9 -6
277 1 5 -1
281 -1 1 4
283 -1 8 -5
293 1 -9 7
307 -1 4 3
311 3 3 5
317 -3 5 1
347 3 -12 8
353 3 -1 4
359 -5 23 -15
383 -5 28 -19
389 -3 2 4
397 1 7 -5
401 1 -5 5
419 3 6 5
431 1 -7 6
433 -1 -5 6
439 3 1 5
443 3 -4 4
449 1 8 -6
457 1 2 5
461 5 4 -2
467 -1 -1 5
479 -1 4 4
491 3 18 -16
499 -1 0 5
503 5 3 6
509 1 4 5
521 5 5 -1
557 -1 89 -70
563 3 6 -1
569 -1 7 -2
587 3 4 6
593 -7 2 5
599 1 7 -4
601 1 -22 17
617 -5 -59 50
641 3 23 -20
643 -1 3 5
647 -1 14 -10
653 1 16 -13
659 3 -10 7
677 -1 -11 10
683 -5 5 3
691 3 -2 5
701 -1 -3 6
719 5 5 -4
727 3 -5 5
733 3 9 -8
739 1 7 -2
743 3 -3 5
761 1 -14 11
773 5 -1 5
797 -3 3 5
809 1 2 6
811 1 3 6
821 3 7 6
827 -1 11 -7
839 -3 5 4
857 -9 5 4
863 -1 0 6
881 1 -4 6
887 7 3 7
911 -1 -5 7
919 -1 7 3
929 9 2 -2
941 9 3 -1
947 -3 1 6
953 -7 26 -16
971 -1 8 -1
977 1 7 5
983 3 7 -3
997 3 -11 8
```

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