On the trigonometric roots of a cubic

Question

We have, $$x^3+x^2-2x-1=0,\quad\quad x =\sum_{k=1}^{2}\,\exp\Bigl(\tfrac{2\pi\, i\, (6^k)}{7}\Bigr)\\ x^3+x^2-4x+1=0,\quad\quad x =\sum_{k=1}^{4}\,\exp\Bigl(\tfrac{2\pi\, i\, (5^k)}{13}\Bigr)\\ x^3+x^2-6x-7=0,\quad\quad x =\sum_{k=1}^{6}\,\exp\Bigl(\tfrac{2\pi\, i\, (8^k)}{19}\Bigr)$$ and so on. It seems the general formula is, given a prime $p=6m+1$ and the unique solution $u,v$ to, $$u^2+27v^2=4p$$ then,

$$x^3+x^2-2mx+\frac{1-3p\pm p u}{27}=0\tag1$$

for the appropriate sign $\pm$ and where $x =\sum_{k=1}^{2m}\,\exp\Bigl(\tfrac{2\pi\, i\, (\color{blue}c^k)}{p}\Bigr)$ and appropriate integer $\color{blue}c$. For example, for $p=7$, we have $$(-1)^2+27\times1^2=4\times7$$ and using $u=-1$ on formula $(1)$, we recover the first example.

Q: Is it possible to find a formula for the sign $\pm$ and/or $\color{blue}c\,$ as a function of $p$?

$\color{blue}{Update}$: It seems the answer to the part about the sign is yes. While checking the OEIS for the sequence of solutions $u=1, 5, 7, 4, 11, 8,\dots$, it turns out there is signed version using the Kronecker symbol(?). Thus,

$$x^3+x^2-2mx+\frac{1-3p- p\,u'}{27}=0\tag2$$

where $u'=1, -5, 7, 4, -11, -8,\dots$ and given by A123489.

Answer 1

If $p = 3n+1$ and $u,v$ satisfy $u^2+27v^2 = 4p$ with $u \equiv 1 \pmod 3$

then taking this relation modulo $27$ we get $p \equiv 7u^2$, and then $1-3p-pu \equiv 1-21u^2-7u^3 \equiv 7(1-u)^3 - 12(1-u)^2 + 9(1-u) \equiv 0$ because $1-u \equiv 0 \pmod 3$.

Then if we call $w = \frac 1 {27} (1-3p-pu)$, $w$ is an integer.
If $p$ is odd then $w = u = v \pmod 2$ and $n$ is even.

The cubic $x^3+x^2-n+w$ has discriminant $D = n^2+4n^3-4w-27w^2-18nw$.

$27D = 3(p-1)^2+4(p-1)^3-4(1-3p-pu)-(1-3p-pu)^2-6(p-1)(1-3p-pu)$ which after simplification gives $27D = p^2(4p-u^2) = 27p^2v^2$ and so $D = p^2v^2$

Since this is a square, the Galois group is cyclic and after computing the interpolating polynomials (or by factoring the cubic and solving the remaining quadratic), we get that if $x$ is a root then the other two are

$\frac 1v(x^2 + x(1+((n+9w)/p-v)/2) - (n+((n^2-3w)/p+v)/2))$ and its conjugate obtained by switching the sign of $v$.

Except the division by $v$ at the front, the divisions inside those coefficients all result in integers because of parity considerations, and the equations $27w \equiv -3n \equiv 1 \pmod p$.

Therefore the lattice $\Bbb Z[x_1] = \langle 1,x_1,x_1^2 \rangle$ has index $|v|$ in the lattice $\Bbb Z[x_1,x_2,x_3] = \langle 1,x_1,x_2 \rangle = \langle x_1, x_2, x_3\rangle$, and so the discriminant of this lattice is $p^2$. Since there is no unramified extension of $\Bbb Q$, it can't get any smaller and so $\Bbb Z[x_1,x_2,x_3]$ is its ring of integers and we have an integral normal basis.

Now this extension must be the class group of conductor $p$ corresponding to the unique subgroup $H$ of index $3$ in $G = (\Bbb Z/p \Bbb Z)^*$, which is trigonometric by the Kronecker Weber theorem.
Then we have another integral normal basis for the ring of integers, $\{s_K = \sum_{k \in K} \zeta_p^k \mid K \in G/H \}$

Looking at how the Galois group can act on those elements shows that we must have $x_1 = \pm s_K$ for some $K$. Finally, since $x_1+x_2+x_3 = -1 = \sum s_K$, we have $x_1 = s_K$ for some $K \in G/H$

Answer 2

CW the cubics for $p < 1000,$ method of Gauss near the end of the Disquisitiones, with a leisurely modern discussion in Galois Theory by David A. Cox. Worked examples, for the primes up to 100, in REUSCHLE.

Found a table in Cohen, his "d" tends to equal $p^2,$ where mine is sometimes $4p^2$

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

jagy@phobeusjunior:~$ ./cubic_cyclic_gauss_loop | grep exps
  x^3 + x^2   - 2 x - 1   p  7 p.root  3 exps 6^k  d = 7^2
  x^3 + x^2   - 4 x + 1   p  13 p.root  2 exps 5^k  d = 13^2
  x^3 + x^2   - 6 x - 7   p  19 p.root  2 exps 8^k  d = 19^2
  x^3 + x^2   - 10 x - 8   p  31 p.root  3 exps 15^k  d = 2^2 * 31^2
  x^3 + x^2   - 12 x + 11   p  37 p.root  2 exps 8^k  d = 37^2
  x^3 + x^2   - 14 x + 8   p  43 p.root  3 exps 2^k  d = 2^2 * 43^2
  x^3 + x^2   - 20 x - 9   p  61 p.root  2 exps 8^k  d = 3^2 * 61^2
  x^3 + x^2   - 22 x + 5   p  67 p.root  2 exps 3^k  d = 3^2 * 67^2
  x^3 + x^2   - 24 x - 27   p  73 p.root  5 exps 7^k  d = 3^2 * 73^2
  x^3 + x^2   - 26 x + 41   p  79 p.root  3 exps 12^k  d = 79^2
  x^3 + x^2   - 32 x - 79   p  97 p.root  5 exps 19^k  d = 97^2
  x^3 + x^2   - 34 x - 61   p  103 p.root  5 exps 3^k  d = 3^2 * 103^2
  x^3 + x^2   - 36 x - 4   p  109 p.root  6 exps 2^k  d = 2^4 * 109^2
  x^3 + x^2   - 42 x + 80   p  127 p.root  3 exps 5^k  d = 2^2 * 127^2
  x^3 + x^2   - 46 x + 103   p  139 p.root  2 exps 8^k  d = 139^2
  x^3 + x^2   - 50 x - 123   p  151 p.root  6 exps 3^k  d = 3^2 * 151^2
  x^3 + x^2   - 52 x + 64   p  157 p.root  5 exps 2^k  d = 2^4 * 157^2
  x^3 + x^2   - 54 x - 169   p  163 p.root  2 exps 5^k  d = 163^2
  x^3 + x^2   - 60 x - 67   p  181 p.root  2 exps 6^k  d = 5^2 * 181^2
  x^3 + x^2   - 64 x + 143   p  193 p.root  5 exps 11^k  d = 3^2 * 193^2
  x^3 + x^2   - 66 x + 59   p  199 p.root  3 exps 12^k  d = 5^2 * 199^2
  x^3 + x^2   - 70 x - 125   p  211 p.root  2 exps 8^k  d = 5^2 * 211^2
  x^3 + x^2   - 74 x - 256   p  223 p.root  3 exps 13^k  d = 2^2 * 223^2
  x^3 + x^2   - 76 x - 212   p  229 p.root  6 exps 2^k  d = 2^4 * 229^2
  x^3 + x^2   - 80 x + 125   p  241 p.root  7 exps 17^k  d = 5^2 * 241^2
  x^3 + x^2   - 90 x + 261   p  271 p.root  6 exps 24^k  d = 3^2 * 271^2
  x^3 + x^2   - 92 x + 236   p  277 p.root  5 exps 2^k  d = 2^4 * 277^2
  x^3 + x^2   - 94 x + 304   p  283 p.root  3 exps 2^k  d = 2^2 * 283^2
  x^3 + x^2   - 102 x - 216   p  307 p.root  5 exps 2^k  d = 2^2 * 3^2 * 307^2
  x^3 + x^2   - 104 x + 371   p  313 p.root  10 exps 7^k  d = 313^2
  x^3 + x^2   - 110 x - 49   p  331 p.root  3 exps 7^k  d = 7^2 * 331^2
  x^3 + x^2   - 112 x + 25   p  337 p.root  10 exps 5^k  d = 7^2 * 337^2
  x^3 + x^2   - 116 x - 517   p  349 p.root  2 exps 6^k  d = 349^2
  x^3 + x^2   - 122 x + 435   p  367 p.root  6 exps 3^k  d = 3^2 * 367^2
  x^3 + x^2   - 124 x - 221   p  373 p.root  2 exps 8^k  d = 7^2 * 373^2
  x^3 + x^2   - 126 x + 365   p  379 p.root  2 exps 8^k  d = 5^2 * 379^2
  x^3 + x^2   - 132 x - 544   p  397 p.root  5 exps 15^k  d = 2^4 * 397^2
  x^3 + x^2   - 136 x - 515   p  409 p.root  21 exps 11^k  d = 5^2 * 409^2
  x^3 + x^2   - 140 x - 343   p  421 p.root  2 exps 8^k  d = 7^2 * 421^2
  x^3 + x^2   - 144 x - 16   p  433 p.root  5 exps 42^k  d = 2^6 * 433^2
  x^3 + x^2   - 146 x - 504   p  439 p.root  15 exps 3^k  d = 2^2 * 3^2 * 439^2
  x^3 + x^2   - 152 x - 220   p  457 p.root  13 exps 5^k  d = 2^6 * 457^2
  x^3 + x^2   - 154 x + 343   p  463 p.root  3 exps 7^k  d = 7^2 * 463^2
  x^3 + x^2   - 162 x - 505   p  487 p.root  3 exps 7^k  d = 7^2 * 487^2
  x^3 + x^2   - 166 x + 536   p  499 p.root  7 exps 2^k  d = 2^2 * 3^2 * 499^2
  x^3 + x^2   - 174 x - 891   p  523 p.root  2 exps 8^k  d = 3^2 * 523^2
  x^3 + x^2   - 180 x + 521   p  541 p.root  2 exps 8^k  d = 7^2 * 541^2
  x^3 + x^2   - 182 x - 81   p  547 p.root  2 exps 8^k  d = 3^4 * 547^2
  x^3 + x^2   - 190 x - 719   p  571 p.root  3 exps 7^k  d = 7^2 * 571^2
  x^3 + x^2   - 192 x + 171   p  577 p.root  5 exps 14^k  d = 3^4 * 577^2
  x^3 + x^2   - 200 x + 512   p  601 p.root  7 exps 31^k  d = 2^6 * 601^2
  x^3 + x^2   - 202 x - 1169   p  607 p.root  3 exps 6^k  d = 607^2
  x^3 + x^2   - 204 x + 999   p  613 p.root  2 exps 8^k  d = 3^2 * 613^2
  x^3 + x^2   - 206 x + 321   p  619 p.root  2 exps 3^k  d = 3^4 * 619^2
  x^3 + x^2   - 210 x - 1075   p  631 p.root  3 exps 27^k  d = 5^2 * 631^2
  x^3 + x^2   - 214 x - 1024   p  643 p.root  11 exps 2^k  d = 2^2 * 3^2 * 643^2
  x^3 + x^2   - 220 x - 1273   p  661 p.root  2 exps 8^k  d = 3^2 * 661^2
  x^3 + x^2   - 224 x - 997   p  673 p.root  5 exps 10^k  d = 7^2 * 673^2
  x^3 + x^2   - 230 x + 128   p  691 p.root  3 exps 2^k  d = 2^2 * 5^2 * 691^2
  x^3 + x^2   - 236 x + 1313   p  709 p.root  2 exps 8^k  d = 709^2
  x^3 + x^2   - 242 x + 1104   p  727 p.root  5 exps 3^k  d = 2^2 * 3^2 * 727^2
  x^3 + x^2   - 244 x + 1276   p  733 p.root  6 exps 2^k  d = 2^4 * 733^2
  x^3 + x^2   - 246 x - 520   p  739 p.root  3 exps 2^k  d = 2^2 * 5^2 * 739^2
  x^3 + x^2   - 250 x + 1057   p  751 p.root  3 exps 6^k  d = 7^2 * 751^2
  x^3 + x^2   - 252 x + 729   p  757 p.root  2 exps 8^k  d = 3^4 * 757^2
  x^3 + x^2   - 256 x - 1481   p  769 p.root  11 exps 7^k  d = 5^2 * 769^2
  x^3 + x^2   - 262 x - 991   p  787 p.root  2 exps 3^k  d = 3^4 * 787^2
  x^3 + x^2   - 270 x + 1592   p  811 p.root  3 exps 2^k  d = 2^2 * 811^2
  x^3 + x^2   - 274 x + 61   p  823 p.root  3 exps 5^k  d = 11^2 * 823^2
  x^3 + x^2   - 276 x - 307   p  829 p.root  2 exps 7^k  d = 11^2 * 829^2
  x^3 + x^2   - 284 x + 1011   p  853 p.root  2 exps 5^k  d = 3^4 * 853^2
  x^3 + x^2   - 286 x - 509   p  859 p.root  2 exps 8^k  d = 11^2 * 859^2
  x^3 + x^2   - 292 x + 1819   p  877 p.root  2 exps 6^k  d = 877^2
  x^3 + x^2   - 294 x + 1439   p  883 p.root  2 exps 8^k  d = 7^2 * 883^2
  x^3 + x^2   - 302 x - 739   p  907 p.root  2 exps 8^k  d = 11^2 * 907^2
  x^3 + x^2   - 306 x - 1872   p  919 p.root  7 exps 3^k  d = 2^2 * 3^2 * 919^2
  x^3 + x^2   - 312 x - 2221   p  937 p.root  5 exps 15^k  d = 937^2
  x^3 + x^2   - 322 x + 1361   p  967 p.root  5 exps 3^k  d = 3^4 * 967^2
  x^3 + x^2   - 330 x - 2349   p  991 p.root  6 exps 3^k  d = 3^2 * 991^2
  x^3 + x^2   - 332 x - 480   p  997 p.root  7 exps 2^k  d = 2^4 * 3^2 * 997^2
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=