7
$\begingroup$

(Update): Courtesy of Myerson's and Elkies' answers, we find a second cyclic quintic for $\cos\frac{2\pi}{p}$ with $p=\text{1 mod 10}$ as, $$\frac{z^5}{\beta} = 10 z^3 - 20 n^2 z^2 + 5 (3 n^4 - 25 n^2 - 625) z - 4 n^2 (n^4 - 25 n^2 - 125)$$ where $\beta=n^4 + 25 n^2 + 125.$ Its discriminant is, $$D=2^{12}\,5^{20}\,(n^4+7n^2)^2\,(n^4 + 25 n^2 + 125)^4$$ Finding a parametric cyclic quintic was one of the aims of this post.


(Original post): We have $$x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0,\quad\quad x =\sum_{k=1}^{2}\,\exp\Bigl(\tfrac{2\pi\, i\, 10^k}{11}\Bigr)$$ and so on for prime $p=10m+1$. Let $A$ be this class of quintics with $x =\sum_{k=1}^{2m}\,\exp\Bigl(\tfrac{2\pi\, i\, n^k}{p}\Bigr).$ I was trying to find a pattern to the coefficients of this infinite family, perhaps something similar to the Diophantine equation $u^2+27v^2=4N$ for the cubic case.

First, we can depress these (get rid of the $x^{n-1}$ term ) by letting $x=\frac{y-1}{5}$ to get the form, $$y^5+ay^3+by^2+cy+d=0$$ Call the depressed form of $A$ as $B$.

Questions:

  1. Is it true that for $B$, there is always an ordering of its roots such that$$\small y_1 y_2 + y_2 y_3 + y_3 y_4 + y_4 y_5 + y_5 y_1 - (y_1 y_3 + y_3 y_5 + y_5 y_2 + y_2 y_4 + y_4 y_1) = 0$$
  2. Do its coefficients $a,b,c,d$ always obey the Diophantine relations, $$a^3 + 10 b^2 - 20 a c= 2z_1^2$$ $$5 (a^2 - 4 c)^2 + 32 a b^2 = z_2^2$$ $$(a^3 + 10 b^2 - 20 a c)\,\big(5 (a^2 - 4 c)^2 + 32 a b^2\big) = 2z_1^2z_2^2 = 2(a^2 b + 20 b c - 100 a d)^2$$ for integer $z_i$?

I tested the first forty such quintics and they answer the two questions in the affirmative. But is it true for all prime $p=10m+1$?

$\endgroup$

3 Answers 3

4
$\begingroup$

I think the depressed quintic in the question is what Emma Lehmer called the reduced quintic in her paper, The quintic character of 2 and 3, Duke Math. J. Volume 18, Number 1 (1951), 11-18, MR0040338. In the proof of Theorem 4, she writes that the reduced quintic is $$F(z)=z^5-10pz^3-5pxz^2-5p\left({x^2-125w^2\over4}-p\right)z+p^2x-{x^3+625(u^2-v^2)w\over8}$$ where $16p=x^2+50u^2+50v^2+125w^2$, $xw=v^2-u^2-4uv$ determines $x$ uniquely. You might plug these into your conjectured diophantine relations, to see whether they hold.

EDIT:

A slightly different formula is given by Berndt and Evans, The determination of Gauss sums, Bull Amer Math Soc 5, no. 2 (1981) 107-129, on page 119 (formula 5.2). It goes $$ F(z)=z^5-10pz^3-5pxz^2-5p\left({x^2-125w^2\over4}-p\right)z+p^2x-p{x^3+625(u^2-v^2)w\over8} $$ the difference being the factor of $p$ in the last term. Berndt and Evans note that Lehmer inadvertently omitted this factor.

Update: (by OP)

Using the edited coefficients of $F(z)$ by Berndt and Evans, if we express it as, $$z^5+az^3+bz^2+cz+d=0$$ and taking into account $16p=x^2+50u^2+50v^2+125w^2,\,x=(v^2-u^2-4uv)/w$, then $$a^3 + 10 b^2 - 20a c =2\times125^2p^2w^2=2z_1^2$$ $$5(a^2 - 4c)^2 + 32a b^2 = 500^2 p^2(u^2 - u v - v^2)^2=z_2^2$$ $$(a^3 + 10 b^2 - 20 a c)\,\big(5 (a^2 - 4 c)^2 + 32 a b^2\big) = 2(a^2 b + 20 b c - 100 a d)^2$$ which confirms all three Diophantine relations by the OP.

$\endgroup$
4
  • $\begingroup$ I tried Lehmer's quintic on $F(z)$ and can't get it to work. Can you check if $F(z)$ has a typo, especially on the constant term? $\endgroup$ Nov 24, 2016 at 1:50
  • $\begingroup$ You may be right. I've edited to include another opinion. $\endgroup$ Nov 24, 2016 at 2:22
  • $\begingroup$ I thought so. By the way, I added an update to your answer as it was too long for a comment. I hope you don't mind. It confirms the first two Diophantine relations. And the third is confirmed by Elkies' answer. $\endgroup$ Nov 24, 2016 at 7:15
  • $\begingroup$ It's beautiful that the quadratic forms for the cubic and quintic cases are $$\alpha^2+27\beta^2=4p$$ $$\alpha^2+10\beta^2+50\gamma^2+125\delta^2=16p$$ Googling it, the one for septics is a tad more complicated. And thanks for this answer. Because of this, I found a second cyclic family of quintics rather simple in form (which has been added to the post). $\endgroup$ Nov 24, 2016 at 7:58
3
$\begingroup$

Question 1: Yes; in fact $$ \sum_{n\bmod 5} y_n y_{n+1} = \! \sum_{n\bmod 5} y_n y_{n+2} = -p/5 $$ for the "depressed" $y_n$, using an order consistent with the action of the cyclic Galois group. Likewise "for other values of 5".

This follows from the formula $|\tau|^2 = p$ for the absolute value of a nontrivial Gauss sum $\tau$, together with the observation that the quintic Gauss sums are the values at $m \neq 0$ of the discrete Fourier transform $\hat y$ of $\vec y$: $$ \tau_m = \sum_{n \bmod 5} e^{2\pi i mn/5} y_n $$ (and $\tau_0 = 0$ thanks to the "depression"). Now the $y_n$ are real, so $\bar\tau$ is the Fourier transform of the map $n \mapsto y_{-n}$; thus the autocorrelation function $$ c_j = \sum_{n \bmod 5} y_n y_{n+j} $$ is the convolution of $\vec y$ with $n \mapsto y_{-n}$, whence the Fourier transform $\hat c$ is $\tau \bar \tau = |\tau|^2$. But that's $0$ at $m=0$, and $p$ otherwise; therefore $c_0 = 4p/5$ and $c_j = -p/5$ for other $j$. In particular $c_1 = c_2$, QED.

P.S. I was using $x = y - \frac15$, not your $x = \frac{y-1}{5}$, so most formulas above will have to be scaled by suitable powers of $5$ to apply to your $y$'s.

$\endgroup$
2
  • $\begingroup$ This also confirms the third Diophantine relation. Thanks and I wish I could accept two answers. $\endgroup$ Nov 24, 2016 at 7:23
  • $\begingroup$ It's been a while. Perhaps this new question in transforming a two-parameter $5T2$ quintic into a one-parameter $5T1$ quintic might be of interest. $\endgroup$ Dec 30, 2022 at 13:22
1
$\begingroup$

CW. Here are the first 250 examples, by the method of Gauss in section VII of the Disquisitiones. I followed Galois Theory by David A. Cox, chapter 9. The quintic for a prime $p \equiv 1 \pmod 5$ is $$ x^5 + x^4 - 2 \left( \frac{p-1}{5} \right) x^3 + a x^2 + b x + c $$ with integers $a,b,c.$ Gerry has indicated that there is likely to be additional detail in Storer.

  x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1   p  11 p.root  2 exps 10^k  d  =  11^4
  x^5 + x^4 - 12 x^3 - 21 x^2 + 1 x + 5   p  31 p.root  3 exps 6^k  d  =  5^2 31^4
  x^5 + x^4 - 16 x^3 + 5 x^2 + 21 x - 9   p  41 p.root  6 exps 3^k  d  =  3^6 41^4
  x^5 + x^4 - 24 x^3 - 17 x^2 + 41 x - 13   p  61 p.root  2 exps 21^k  d  =  29^2 61^4
  x^5 + x^4 - 28 x^3 + 37 x^2 + 25 x + 1   p  71 p.root  7 exps 23^k  d  =  23^2 71^4
  x^5 + x^4 - 40 x^3 + 93 x^2 - 21 x - 17   p  101 p.root  2 exps 32^k  d  =  17^2 101^4
  x^5 + x^4 - 52 x^3 - 89 x^2 + 109 x + 193   p  131 p.root  2 exps 18^k  d  =  79^2 131^4
  x^5 + x^4 - 60 x^3 - 12 x^2 + 784 x + 128   p  151 p.root  6 exps 23^k  d  =  2^18 151^4
  x^5 + x^4 - 72 x^3 - 123 x^2 + 223 x - 49   p  181 p.root  2 exps 17^k  d  =  7^2 149^2 181^4
  x^5 + x^4 - 76 x^3 - 359 x^2 - 437 x - 155   p  191 p.root  19 exps 11^k  d  =  5^2 11^2 191^4
  x^5 + x^4 - 84 x^3 - 59 x^2 + 1661 x + 269   p  211 p.root  2 exps 26^k  d  =  31^2 67^2 211^4
  x^5 + x^4 - 96 x^3 - 212 x^2 + 1232 x + 512   p  241 p.root  7 exps 11^k  d  =  2^16 11^2 241^4
  x^5 + x^4 - 100 x^3 - 20 x^2 + 1504 x + 1024   p  251 p.root  6 exps 2^k  d  =  2^18 5^4 251^4
  x^5 + x^4 - 108 x^3 - 401 x^2 - 13 x + 845   p  271 p.root  6 exps 12^k  d  =  5^2 13^4 271^4
  x^5 + x^4 - 112 x^3 - 191 x^2 + 2257 x + 967   p  281 p.root  3 exps 6^k  d  =  193^2 281^4
  x^5 + x^4 - 124 x^3 + 535 x^2 - 413 x - 539   p  311 p.root  17 exps 11^k  d  =  7^4 13^2 311^4
  x^5 + x^4 - 132 x^3 - 887 x^2 - 1843 x - 1027   p  331 p.root  3 exps 13^k  d  =  13^2 31^2 331^4
  x^5 + x^4 - 160 x^3 + 369 x^2 + 879 x - 29   p  401 p.root  3 exps 26^k  d  =  29^2 401^4 433^2
  x^5 + x^4 - 168 x^3 + 219 x^2 + 3853 x - 3517   p  421 p.root  2 exps 32^k  d  =  223^2 239^2 421^4
  x^5 + x^4 - 172 x^3 - 724 x^2 + 1824 x + 1728   p  431 p.root  7 exps 47^k  d  =  2^20 3^4 431^4
  x^5 + x^4 - 184 x^3 - 129 x^2 + 4551 x + 5419   p  461 p.root  2 exps 13^k  d  =  163^2 461^4 491^2
  x^5 + x^4 - 196 x^3 + 59 x^2 + 2019 x + 1377   p  491 p.root  2 exps 32^k  d  =  3^4 17^2 229^2 491^4
  x^5 + x^4 - 208 x^3 - 771 x^2 + 4143 x + 2083   p  521 p.root  3 exps 24^k  d  =  61^2 521^4 577^2
  x^5 + x^4 - 216 x^3 + 1147 x^2 - 805 x - 2629   p  541 p.root  2 exps 11^k  d  =  11^2 311^2 541^4
  x^5 + x^4 - 228 x^3 + 868 x^2 + 3056 x - 7552   p  571 p.root  3 exps 2^k  d  =  2^22 31^2 571^4
  x^5 + x^4 - 240 x^3 + 1755 x^2 - 3731 x + 2399   p  601 p.root  7 exps 17^k  d  =  5^2 13^2 17^2 601^4
  x^5 + x^4 - 252 x^3 + 2095 x^2 - 5785 x + 5069   p  631 p.root  3 exps 24^k  d  =  89^2 631^4
  x^5 + x^4 - 256 x^3 - 564 x^2 + 5328 x - 5120   p  641 p.root  3 exps 21^k  d  =  2^16 5^2 61^2 641^4
  x^5 + x^4 - 264 x^3 - 185 x^2 + 16837 x + 4851   p  661 p.root  2 exps 32^k  d  =  3^16 7^2 661^4
  x^5 + x^4 - 276 x^3 - 1299 x^2 + 5329 x + 15581   p  691 p.root  3 exps 11^k  d  =  379^2 397^2 691^4
  x^5 + x^4 - 280 x^3 + 2047 x^2 - 3791 x + 1699   p  701 p.root  2 exps 23^k  d  =  17^2 19^2 23^2 701^4
  x^5 + x^4 - 300 x^3 - 2313 x^2 - 3761 x - 571   p  751 p.root  3 exps 11^k  d  =  41^2 631^2 751^4
  x^5 + x^4 - 304 x^3 + 2831 x^2 - 8925 x + 8775   p  761 p.root  6 exps 3^k  d  =  3^4 5^2 23^2 761^4
  x^5 + x^4 - 324 x^3 - 3471 x^2 - 12431 x - 13603   p  811 p.root  3 exps 12^k  d  =  7^4 47^2 811^4
  x^5 + x^4 - 328 x^3 - 1215 x^2 + 3573 x + 2179   p  821 p.root  2 exps 32^k  d  =  37^4 109^2 821^4
  x^5 + x^4 - 352 x^3 - 2361 x^2 + 4257 x + 9967   p  881 p.root  3 exps 29^k  d  =  29^2 881^4 953^2
  x^5 + x^4 - 364 x^3 - 2988 x^2 - 1392 x + 9856   p  911 p.root  17 exps 22^k  d  =  2^18 7^2 11^2 911^4
  x^5 + x^4 - 376 x^3 + 3877 x^2 - 13445 x + 15271   p  941 p.root  2 exps 12^k  d  =  191^2 941^4
  x^5 + x^4 - 388 x^3 + 1476 x^2 + 8304 x + 7168   p  971 p.root  6 exps 2^k  d  =  2^20 7^2 13^2 971^4
  x^5 + x^4 - 396 x^3 + 2101 x^2 + 8039 x - 1819   p  991 p.root  6 exps 30^k  d  =  107^2 991^4 1399^2
  x^5 + x^4 - 408 x^3 + 531 x^2 + 28539 x + 9963   p  1021 p.root  10 exps 19^k  d  =  3^14 13^2 19^2 1021^4
  x^5 + x^4 - 412 x^3 - 701 x^2 + 14467 x + 38437   p  1031 p.root  14 exps 7^k  d  =  7^2 17^2 31^2 107^2 1031^4
  x^5 + x^4 - 420 x^3 + 967 x^2 + 28789 x - 14151   p  1051 p.root  7 exps 3^k  d  =  3^14 53^2 1051^4
  x^5 + x^4 - 424 x^3 - 297 x^2 + 41031 x + 18787   p  1061 p.root  2 exps 10^k  d  =  17^2 37^2 739^2 1061^4
  x^5 + x^4 - 436 x^3 + 131 x^2 + 42453 x - 3015   p  1091 p.root  2 exps 32^k  d  =  3^4 5^2 13^2 353^2 1091^4
  x^5 + x^4 - 460 x^3 - 3545 x^2 + 4825 x + 41629   p  1151 p.root  17 exps 19^k  d  =  7^2 13^2 47^2 89^2 1151^4
  x^5 + x^4 - 468 x^3 - 1733 x^2 + 31795 x + 54201   p  1171 p.root  2 exps 3^k  d  =  3^18 7^4 1171^4
  x^5 + x^4 - 472 x^3 + 2740 x^2 + 7360 x - 45056   p  1181 p.root  7 exps 2^k  d  =  2^28 19^2 1181^4
  x^5 + x^4 - 480 x^3 - 6101 x^2 - 25711 x - 35329   p  1201 p.root  11 exps 41^k  d  =  7^2 349^2 1201^4
  x^5 + x^4 - 492 x^3 - 837 x^2 + 48787 x + 1853   p  1231 p.root  3 exps 6^k  d  =  503^2 1231^4 1669^2
  x^5 + x^4 - 516 x^3 - 2427 x^2 + 8407 x + 31217   p  1291 p.root  2 exps 11^k  d  =  11^2 19^2 1291^4 2531^2
  x^5 + x^4 - 520 x^3 - 6609 x^2 - 26811 x - 27821   p  1301 p.root  2 exps 12^k  d  =  11^4 31^2 1301^4
  x^5 + x^4 - 528 x^3 + 687 x^2 + 53305 x + 5831   p  1321 p.root  13 exps 7^k  d  =  7^2 17^4 331^2 1321^4
  x^5 + x^4 - 544 x^3 - 5825 x^2 - 14873 x + 8551   p  1361 p.root  3 exps 28^k  d  =  71^2 1097^2 1361^4
  x^5 + x^4 - 552 x^3 + 4585 x^2 - 4375 x - 8461   p  1381 p.root  2 exps 24^k  d  =  13^2 137^2 139^2 1381^4
  x^5 + x^4 - 580 x^3 - 7371 x^2 - 28161 x - 30959   p  1451 p.root  2 exps 10^k  d  =  53^2 881^2 1451^4
  x^5 + x^4 - 588 x^3 + 765 x^2 + 46413 x + 59049   p  1471 p.root  6 exps 3^k  d  =  3^18 113^2 1471^4
  x^5 + x^4 - 592 x^3 - 1007 x^2 + 61657 x - 28457   p  1481 p.root  3 exps 11^k  d  =  13^2 197^2 937^2 1481^4
  x^5 + x^4 - 604 x^3 + 5621 x^2 + 411 x - 25731   p  1511 p.root  11 exps 44^k  d  =  3^8 487^2 1511^4
  x^5 + x^4 - 612 x^3 + 8145 x^2 - 36695 x + 48749   p  1531 p.root  2 exps 18^k  d  =  11^2 19^2 41^2 1531^4
  x^5 + x^4 - 628 x^3 - 2325 x^2 + 63393 x + 27169   p  1571 p.root  2 exps 6^k  d  =  13^2 17^2 23^2 223^2 1571^4
  x^5 + x^4 - 640 x^3 + 4675 x^2 + 12475 x - 95561   p  1601 p.root  3 exps 12^k  d  =  13^2 43^2 1601^4 1613^2
  x^5 + x^4 - 648 x^3 + 843 x^2 + 20671 x - 30037   p  1621 p.root  2 exps 22^k  d  =  7^6 29^2 139^2 1621^4
  x^5 + x^4 - 688 x^3 - 2547 x^2 + 71511 x + 146323   p  1721 p.root  3 exps 6^k  d  =  1721^4 1753^2 2141^2
  x^5 + x^4 - 696 x^3 - 3273 x^2 + 69835 x + 130931   p  1741 p.root  2 exps 32^k  d  =  13^2 23^2 73^2 113^2 1741^4
  x^5 + x^4 - 720 x^3 + 1657 x^2 + 78149 x - 139081   p  1801 p.root  11 exps 22^k  d  =  23^2 97^2 1801^4 2099^2
  x^5 + x^4 - 724 x^3 + 8548 x^2 - 18704 x - 25088   p  1811 p.root  6 exps 2^k  d  =  2^20 7^2 37^2 1811^4
  x^5 + x^4 - 732 x^3 + 9741 x^2 - 39491 x + 30353   p  1831 p.root  3 exps 15^k  d  =  17^2 29^2 47^2 1831^4
  x^5 + x^4 - 744 x^3 + 3201 x^2 + 76435 x - 105625   p  1861 p.root  2 exps 7^k  d  =  5^2 7^4 17^2 443^2 1861^4
  x^5 + x^4 - 748 x^3 - 6511 x^2 + 12633 x - 3087   p  1871 p.root  14 exps 7^k  d  =  3^8 7^2 11^2 61^2 1871^4
  x^5 + x^4 - 760 x^3 + 1749 x^2 + 84009 x + 69211   p  1901 p.root  2 exps 10^k  d  =  19^2 227^2 991^2 1901^4
  x^5 + x^4 - 772 x^3 + 6411 x^2 + 2379 x - 72047   p  1931 p.root  2 exps 6^k  d  =  271^2 1931^4 3301^2
  x^5 + x^4 - 780 x^3 + 1795 x^2 + 40175 x - 150571   p  1951 p.root  3 exps 6^k  d  =  139^2 1951^4 4723^2
  x^5 + x^4 - 804 x^3 - 6596 x^2 + 14624 x + 162752   p  2011 p.root  3 exps 2^k  d  =  2^20 433^2 2011^4
  x^5 + x^4 - 832 x^3 - 1415 x^2 + 34195 x - 8621   p  2081 p.root  3 exps 15^k  d  =  37^2 163^2 521^2 2081^4
  x^5 + x^4 - 844 x^3 + 7853 x^2 - 1959 x - 38583   p  2111 p.root  7 exps 43^k  d  =  3^4 107^2 1283^2 2111^4
  x^5 + x^4 - 852 x^3 - 1449 x^2 + 78489 x - 162567   p  2131 p.root  2 exps 17^k  d  =  3^16 23^2 37^2 2131^4
  x^5 + x^4 - 856 x^3 + 4539 x^2 + 88449 x - 66173   p  2141 p.root  2 exps 12^k  d  =  13^2 269^2 349^2 2141^4
  x^5 + x^4 - 864 x^3 + 3717 x^2 + 72333 x - 263169   p  2161 p.root  23 exps 7^k  d  =  3^16 7^4 19^2 2161^4
  x^5 + x^4 - 888 x^3 - 12171 x^2 - 27647 x + 59483   p  2221 p.root  2 exps 17^k  d  =  31^2 47^2 449^2 2221^4
  x^5 + x^4 - 900 x^3 + 2071 x^2 + 105779 x - 270751   p  2251 p.root  7 exps 37^k  d  =  41^2 101^2 2239^2 2251^4
  x^5 + x^4 - 912 x^3 + 7573 x^2 + 54817 x - 114777   p  2281 p.root  7 exps 17^k  d  =  3^16 17^2 2281^4
  x^5 + x^4 - 924 x^3 + 13219 x^2 - 53911 x + 23357   p  2311 p.root  3 exps 12^k  d  =  97^2 1549^2 2311^4
  x^5 + x^4 - 936 x^3 + 4963 x^2 + 100457 x - 286453   p  2341 p.root  7 exps 22^k  d  =  2309^2 2341^4 3191^2
  x^5 + x^4 - 940 x^3 - 188 x^2 + 53584 x - 146816   p  2351 p.root  13 exps 17^k  d  =  2^22 883^2 2351^4
  x^5 + x^4 - 948 x^3 + 1233 x^2 + 115591 x + 218393   p  2371 p.root  2 exps 18^k  d  =  7^2 17^2 53^2 1489^2 2371^4
  x^5 + x^4 - 952 x^3 + 5524 x^2 + 37696 x + 38080   p  2381 p.root  3 exps 2^k  d  =  2^20 5^2 7^2 41^2 2381^4
  x^5 + x^4 - 964 x^3 + 11380 x^2 - 13328 x + 4096   p  2411 p.root  6 exps 2^k  d  =  2^18 61^2 2411^4
  x^5 + x^4 - 976 x^3 + 293 x^2 + 121347 x + 279027   p  2441 p.root  6 exps 3^k  d  =  3^4 7^2 79^2 2153^2 2441^4
  x^5 + x^4 - 1008 x^3 - 3731 x^2 + 95677 x + 400935   p  2521 p.root  17 exps 33^k  d  =  3^16 5^4 53^2 2521^4
  x^5 + x^4 - 1012 x^3 - 6783 x^2 + 106383 x + 22681   p  2531 p.root  2 exps 18^k  d  =  19^2 47^2 2531^4 6569^2
  x^5 + x^4 - 1020 x^3 + 7449 x^2 + 100489 x - 226999   p  2551 p.root  6 exps 54^k  d  =  541^2 2551^4 4787^2
  x^5 + x^4 - 1036 x^3 + 311 x^2 + 69729 x - 229779   p  2591 p.root  7 exps 14^k  d  =  3^6 11^2 17^2 157^2 2591^4
  x^5 + x^4 - 1048 x^3 - 9121 x^2 + 96327 x + 317115   p  2621 p.root  2 exps 3^k  d  =  3^8 5^2 101^2 103^2 2621^4
  x^5 + x^4 - 1068 x^3 - 3953 x^2 + 131285 x + 479621   p  2671 p.root  7 exps 30^k  d  =  2381^2 2671^4 5923^2
  x^5 + x^4 - 1084 x^3 - 17025 x^2 - 70833 x - 5219   p  2711 p.root  7 exps 13^k  d  =  199^2 2663^2 2711^4
  x^5 + x^4 - 1092 x^3 + 17260 x^2 - 69280 x + 49664   p  2731 p.root  3 exps 35^k  d  =  2^18 223^2 2731^4
  x^5 + x^4 - 1096 x^3 - 21599 x^2 - 147707 x - 323645   p  2741 p.root  2 exps 11^k  d  =  5^2 7^2 11^2 53^2 2741^4
  x^5 + x^4 - 1116 x^3 - 5247 x^2 + 107487 x - 194643   p  2791 p.root  6 exps 3^k  d  =  3^14 11^2 17^2 19^2 2791^4
  x^5 + x^4 - 1120 x^3 - 8627 x^2 + 115939 x + 42283   p  2801 p.root  3 exps 15^k  d  =  7^2 17^2 131^2 443^2 2801^4
  x^5 + x^4 - 1140 x^3 + 8325 x^2 + 90639 x + 59049   p  2851 p.root  2 exps 32^k  d  =  3^18 5^2 59^2 2851^4
  x^5 + x^4 - 1144 x^3 - 801 x^2 + 323499 x + 60547   p  2861 p.root  2 exps 32^k  d  =  541^2 1553^2 2861^4
  x^5 + x^4 - 1188 x^3 + 1545 x^2 + 159103 x - 518671   p  2971 p.root  10 exps 15^k  d  =  2221^2 2971^4 8009^2
  x^5 + x^4 - 1200 x^3 - 15245 x^2 - 3025 x - 121   p  3001 p.root  14 exps 13^k  d  =  11^4 607^2 3001^4
  x^5 + x^4 - 1204 x^3 - 843 x^2 + 163413 x - 452099   p  3011 p.root  2 exps 7^k  d  =  7^2 277^2 3011^4 9001^2
  x^5 + x^4 - 1216 x^3 + 24693 x^2 - 180903 x + 449887   p  3041 p.root  3 exps 48^k  d  =  59^2 181^2 3041^4
  x^5 + x^4 - 1224 x^3 - 22284 x^2 - 111200 x - 102400   p  3061 p.root  6 exps 57^k  d  =  2^22 5^2 47^2 3061^4
  x^5 + x^4 - 1248 x^3 + 4744 x^2 + 267632 x - 312880   p  3121 p.root  7 exps 37^k  d  =  2^30 5^2 53^2 3121^4
  x^5 + x^4 - 1272 x^3 - 2163 x^2 + 366553 x + 86171   p  3181 p.root  7 exps 10^k  d  =  37^2 139^2 991^2 3181^4
  x^5 + x^4 - 1276 x^3 + 383 x^2 + 369237 x + 112797   p  3191 p.root  11 exps 13^k  d  =  3^6 13^2 131^2 179^2 3191^4
  x^5 + x^4 - 1288 x^3 + 17780 x^2 - 30432 x - 285696   p  3221 p.root  10 exps 2^k  d  =  2^24 3^6 13^2 3221^4
  x^5 + x^4 - 1300 x^3 - 260 x^2 + 409600 x + 16384   p  3251 p.root  6 exps 2^k  d  =  2^26 379^2 3251^4
  x^5 + x^4 - 1308 x^3 - 17925 x^2 - 32867 x + 201749   p  3271 p.root  3 exps 17^k  d  =  23^4 3271^4 6271^2
  x^5 + x^4 - 1320 x^3 - 264 x^2 + 298384 x - 461296   p  3301 p.root  6 exps 2^k  d  =  2^36 11^4 3301^4
  x^5 + x^4 - 1332 x^3 + 21052 x^2 - 68512 x - 169024   p  3331 p.root  3 exps 19^k  d  =  2^20 19^2 83^2 3331^4
  x^5 + x^4 - 1344 x^3 - 24468 x^2 - 114032 x + 10496   p  3361 p.root  22 exps 13^k  d  =  2^16 7^2 13^2 59^2 3361^4
  x^5 + x^4 - 1348 x^3 - 4989 x^2 + 184569 x - 405767   p  3371 p.root  2 exps 21^k  d  =  23^2 73^2 3371^4 11411^2
  x^5 + x^4 - 1356 x^3 - 16548 x^2 + 7840 x + 22016   p  3391 p.root  3 exps 37^k  d  =  2^18 43^2 139^2 3391^4
  x^5 + x^4 - 1384 x^3 - 969 x^2 + 407955 x + 392875   p  3461 p.root  2 exps 7^k  d  =  5^2 7^2 11^2 107^2 593^2 3461^4
  x^5 + x^4 - 1396 x^3 + 14383 x^2 + 21337 x - 310823   p  3491 p.root  2 exps 10^k  d  =  59^2 151^2 769^2 3491^4
  x^5 + x^4 - 1404 x^3 - 15027 x^2 + 127351 x + 178397   p  3511 p.root  7 exps 24^k  d  =  811^2 3511^4 9479^2
  x^5 + x^4 - 1416 x^3 - 3116 x^2 + 111584 x - 253888   p  3541 p.root  7 exps 23^k  d  =  2^30 13^2 23^2 3541^4
  x^5 + x^4 - 1428 x^3 - 26711 x^2 - 120157 x + 1193   p  3571 p.root  2 exps 12^k  d  =  17^2 43^2 2083^2 3571^4
  x^5 + x^4 - 1432 x^3 + 11889 x^2 + 159111 x - 889745   p  3581 p.root  2 exps 32^k  d  =  5^2 1087^2 3581^4 4049^2
  x^5 + x^4 - 1452 x^3 - 9731 x^2 + 184571 x + 1043525   p  3631 p.root  15 exps 22^k  d  =  5^2 7^2 83^2 89^2 103^2 3631^4
  x^5 + x^4 - 1468 x^3 - 5433 x^2 + 99675 x + 55381   p  3671 p.root  13 exps 26^k  d  =  1223^2 3671^4 13901^2
  x^5 + x^4 - 1476 x^3 + 7825 x^2 + 161341 x - 934899   p  3691 p.root  2 exps 3^k  d  =  3^16 7^2 379^2 3691^4
  x^5 + x^4 - 1480 x^3 + 3405 x^2 + 438909 x - 76301   p  3701 p.root  2 exps 18^k  d  =  5^2 719^2 1753^2 3701^4
  x^5 + x^4 - 1504 x^3 - 27380 x^2 - 115568 x - 141824   p  3761 p.root  3 exps 21^k  d  =  2^16 347^2 3761^4
  x^5 + x^4 - 1528 x^3 + 5808 x^2 + 349056 x + 268288   p  3821 p.root  3 exps 2^k  d  =  2^28 41^2 47^2 3821^4
  x^5 + x^4 - 1540 x^3 - 27265 x^2 - 93225 x + 296649   p  3851 p.root  2 exps 10^k  d  =  3^4 19^2 53^2 401^2 3851^4
  x^5 + x^4 - 1552 x^3 - 6520 x^2 + 356400 x + 1131984   p  3881 p.root  13 exps 3^k  d  =  2^36 3^6 7^2 3881^4
  x^5 + x^4 - 1564 x^3 - 1095 x^2 + 240417 x + 968281   p  3911 p.root  13 exps 46^k  d  =  101^2 137^2 2069^2 3911^4
  x^5 + x^4 - 1572 x^3 + 36637 x^2 - 309637 x + 899921   p  3931 p.root  2 exps 11^k  d  =  11^2 37^2 3931^4
  x^5 + x^4 - 1600 x^3 - 20325 x^2 + 123999 x + 321199   p  4001 p.root  3 exps 12^k  d  =  5^2 661^2 2081^2 4001^4
  x^5 + x^4 - 1608 x^3 - 5951 x^2 + 144563 x + 741143   p  4021 p.root  2 exps 28^k  d  =  557^2 4021^4 15101^2
  x^5 + x^4 - 1620 x^3 - 20579 x^2 + 86659 x + 46569   p  4051 p.root  10 exps 3^k  d  =  3^18 19^2 23^2 4051^4
  x^5 + x^4 - 1636 x^3 - 7691 x^2 + 486469 x + 840277   p  4091 p.root  2 exps 17^k  d  =  7^4 53^2 59^2 179^2 4091^4
  x^5 + x^4 - 1644 x^3 - 25817 x^2 + 20851 x + 713517   p  4111 p.root  12 exps 3^k  d  =  3^14 7^2 11^2 61^2 4111^4
  x^5 + x^4 - 1680 x^3 + 3865 x^2 + 220939 x - 1110981   p  4201 p.root  11 exps 26^k  d  =  3^16 5^2 557^2 4201^4
  x^5 + x^4 - 1684 x^3 - 13812 x^2 + 104736 x + 850432   p  4211 p.root  6 exps 2^k  d  =  2^18 11^2 41^2 43^2 4211^4
  x^5 + x^4 - 1692 x^3 - 2877 x^2 + 523087 x - 537907   p  4231 p.root  3 exps 26^k  d  =  47^2 107^2 4231^4 11587^2
  x^5 + x^4 - 1696 x^3 - 41901 x^2 - 364251 x - 1068713   p  4241 p.root  3 exps 7^k  d  =  7^2 37^2 157^2 4241^4
  x^5 + x^4 - 1704 x^3 - 26759 x^2 - 58495 x + 197075   p  4261 p.root  2 exps 18^k  d  =  5^2 11^2 181^2 859^2 4261^4
  x^5 + x^4 - 1708 x^3 - 6321 x^2 + 520857 x - 131555   p  4271 p.root  7 exps 21^k  d  =  5^2 73^2 173^2 677^2 4271^4
  x^5 + x^4 - 1756 x^3 + 18091 x^2 + 84799 x - 1017779   p  4391 p.root  14 exps 7^k  d  =  7^2 13^2 41^2 293^2 4391^4
  x^5 + x^4 - 1768 x^3 + 2299 x^2 + 288037 x + 1086355   p  4421 p.root  3 exps 21^k  d  =  5^2 107^2 163^2 373^2 4421^4
  x^5 + x^4 - 1776 x^3 + 36061 x^2 - 187801 x + 212051   p  4441 p.root  21 exps 7^k  d  =  7^2 4051^2 4441^4
  x^5 + x^4 - 1780 x^3 + 39703 x^2 - 291131 x + 657073   p  4451 p.root  2 exps 19^k  d  =  199^2 2833^2 4451^4
  x^5 + x^4 - 1792 x^3 + 23839 x^2 + 25237 x - 17273   p  4481 p.root  3 exps 12^k  d  =  83^2 157^2 367^2 4481^4
  x^5 + x^4 - 1824 x^3 - 46887 x^2 - 425669 x - 1275133   p  4561 p.root  11 exps 51^k  d  =  71^2 601^2 4561^4
  x^5 + x^4 - 1836 x^3 - 45359 x^2 - 381457 x - 1043275   p  4591 p.root  11 exps 6^k  d  =  5^2 7^2 61^2 131^2 4591^4
  x^5 + x^4 - 1848 x^3 + 30129 x^2 - 51977 x + 18623   p  4621 p.root  2 exps 10^k  d  =  11^2 13^4 823^2 4621^4
  x^5 + x^4 - 1860 x^3 + 32185 x^2 - 82825 x - 645151   p  4651 p.root  3 exps 18^k  d  =  13^2 431^2 929^2 4651^4
  x^5 + x^4 - 1876 x^3 + 38091 x^2 - 188991 x - 87119   p  4691 p.root  2 exps 28^k  d  =  29^2 67^2 1949^2 4691^4
  x^5 + x^4 - 1888 x^3 - 16429 x^2 + 111189 x + 356823   p  4721 p.root  6 exps 3^k  d  =  3^8 179^2 2029^2 4721^4
  x^5 + x^4 - 1900 x^3 - 380 x^2 + 655600 x + 1166464   p  4751 p.root  19 exps 13^k  d  =  2^26 13^2 827^2 4751^4
  x^5 + x^4 - 1920 x^3 - 14787 x^2 + 227529 x - 519777   p  4801 p.root  7 exps 51^k  d  =  3^24 23^2 4801^4
  x^5 + x^4 - 1932 x^3 - 3285 x^2 + 620455 x + 1603589   p  4831 p.root  3 exps 13^k  d  =  13^2 577^2 4831^4 13469^2
  x^5 + x^4 - 1944 x^3 - 15944 x^2 + 437840 x + 2070800   p  4861 p.root  11 exps 2^k  d  =  2^28 5^2 7^2 131^2 4861^4
  x^5 + x^4 - 1948 x^3 + 7404 x^2 + 661248 x + 126208   p  4871 p.root  11 exps 23^k  d  =  2^20 23^2 1951^2 4871^4
  x^5 + x^4 - 1972 x^3 - 47732 x^2 - 353888 x - 714752   p  4931 p.root  6 exps 2^k  d  =  2^18 37^2 43^2 4931^4
  x^5 + x^4 - 1980 x^3 - 25151 x^2 + 141559 x + 1840701   p  4951 p.root  6 exps 28^k  d  =  3^16 11^4 37^2 4951^4
  x^5 + x^4 - 2004 x^3 + 38685 x^2 - 179033 x - 151879   p  5011 p.root  2 exps 13^k  d  =  7^4 13^2 5011^4 7649^2
  x^5 + x^4 - 2008 x^3 + 7632 x^2 + 506880 x + 1048576   p  5021 p.root  3 exps 2^k  d  =  2^36 307^2 5021^4
  x^5 + x^4 - 2020 x^3 - 55965 x^2 - 558681 x - 1920311   p  5051 p.root  2 exps 7^k  d  =  5^2 7^2 19^2 5051^4
  x^5 + x^4 - 2032 x^3 + 16869 x^2 + 164787 x - 1406333   p  5081 p.root  3 exps 12^k  d  =  17^2 37^2 131^2 137^2 5081^4
  x^5 + x^4 - 2040 x^3 - 15711 x^2 + 611059 x + 1680431   p  5101 p.root  6 exps 31^k  d  =  31^2 37^2 97^2 443^2 5101^4
  x^5 + x^4 - 2068 x^3 - 38679 x^2 - 138831 x + 415573   p  5171 p.root  2 exps 31^k  d  =  911^2 5171^4 10639^2
  x^5 + x^4 - 2092 x^3 - 14019 x^2 + 646719 x + 156853   p  5231 p.root  7 exps 13^k  d  =  17^2 47^2 67^2 1097^2 5231^4
  x^5 + x^4 - 2104 x^3 + 30093 x^2 + 1431 x - 483113   p  5261 p.root  2 exps 10^k  d  =  7^2 19^2 197^2 829^2 5261^4
  x^5 + x^4 - 2112 x^3 - 3591 x^2 + 1045807 x + 554723   p  5281 p.root  7 exps 34^k  d  =  5^12 13^2 71^2 5281^4
  x^5 + x^4 - 2140 x^3 + 58433 x^2 - 579021 x + 1955853   p  5351 p.root  11 exps 61^k  d  =  3^4 61^2 83^2 5351^4
  x^5 + x^4 - 2152 x^3 + 12484 x^2 + 730912 x - 1768448   p  5381 p.root  3 exps 2^k  d  =  2^20 173^2 439^2 5381^4
  x^5 + x^4 - 2172 x^3 - 3693 x^2 + 397723 x + 2132261   p  5431 p.root  3 exps 19^k  d  =  2069^2 5431^4 23057^2
  x^5 + x^4 - 2176 x^3 - 26552 x^2 + 428272 x + 1542352   p  5441 p.root  3 exps 11^k  d  =  2^30 7^2 61^2 5441^4
  x^5 + x^4 - 2188 x^3 + 52084 x^2 - 369008 x + 520960   p  5471 p.root  7 exps 11^k  d  =  2^18 5^4 197^2 5471^4
  x^5 + x^4 - 2200 x^3 - 38947 x^2 + 71469 x + 1456443   p  5501 p.root  2 exps 28^k  d  =  3^6 839^2 991^2 5501^4
  x^5 + x^4 - 2208 x^3 + 2871 x^2 + 739549 x - 2199577   p  5521 p.root  11 exps 62^k  d  =  5521^4 8761^2 17191^2
  x^5 + x^4 - 2212 x^3 + 51549 x^2 - 331683 x + 333637   p  5531 p.root  10 exps 11^k  d  =  43^2 67^2 1117^2 5531^4
  x^5 + x^4 - 2232 x^3 + 12948 x^2 + 231232 x - 1390912   p  5581 p.root  6 exps 23^k  d  =  2^20 47^2 739^2 5581^4
  x^5 + x^4 - 2236 x^3 + 671 x^2 + 1123299 x - 651483   p  5591 p.root  11 exps 22^k  d  =  3^6 5^16 7^2 5591^4
  x^5 + x^4 - 2256 x^3 - 4964 x^2 + 814064 x + 2491136   p  5641 p.root  14 exps 17^k  d  =  2^16 19^2 29^2 31^2 37^2 5641^4
  x^5 + x^4 - 2260 x^3 + 16501 x^2 + 193219 x - 872471   p  5651 p.root  2 exps 31^k  d  =  2129^2 5651^4 24049^2
  x^5 + x^4 - 2280 x^3 + 39451 x^2 + 83189 x - 1117261   p  5701 p.root  2 exps 23^k  d  =  23^2 487^2 839^2 5701^4
  x^5 + x^4 - 2284 x^3 + 26956 x^2 + 119200 x - 1640000   p  5711 p.root  19 exps 37^k  d  =  2^24 5^6 37^2 5711^4
  x^5 + x^4 - 2296 x^3 - 10793 x^2 + 758455 x + 3020251   p  5741 p.root  2 exps 11^k  d  =  13^2 17^2 71^2 79^2 131^2 5741^4
  x^5 + x^4 - 2316 x^3 + 12277 x^2 + 234605 x - 310279   p  5791 p.root  6 exps 23^k  d  =  23^2 103^2 5791^4 26701^2
  x^5 + x^4 - 2320 x^3 + 51745 x^2 - 281975 x - 254921   p  5801 p.root  3 exps 14^k  d  =  29^2 41^2 59^2 97^2 5801^4
  x^5 + x^4 - 2328 x^3 - 61004 x^2 - 492736 x - 1226752   p  5821 p.root  6 exps 79^k  d  =  2^24 71^2 5821^4
  x^5 + x^4 - 2340 x^3 + 40489 x^2 + 92399 x - 809359   p  5851 p.root  2 exps 32^k  d  =  109^2 127^2 281^2 5851^4
  x^5 + x^4 - 2344 x^3 + 27664 x^2 + 481024 x - 2321408   p  5861 p.root  3 exps 2^k  d  =  2^36 223^2 5861^4
  x^5 + x^4 - 2352 x^3 - 15761 x^2 + 263657 x + 1863767   p  5881 p.root  31 exps 19^k  d  =  1303^2 5881^4 25579^2
  x^5 + x^4 - 2392 x^3 + 19857 x^2 + 380583 x - 3135089   p  5981 p.root  3 exps 17^k  d  =  131^2 151^2 3407^2 5981^4
  x^5 + x^4 - 2404 x^3 + 22361 x^2 + 218031 x + 165969   p  6011 p.root  2 exps 19^k  d  =  3^4 19^2 463^2 479^2 6011^4
  x^5 + x^4 - 2436 x^3 + 49459 x^2 - 91901 x - 1079103   p  6091 p.root  7 exps 42^k  d  =  3^20 277^2 6091^4
  x^5 + x^4 - 2440 x^3 - 43195 x^2 - 37875 x + 1792179   p  6101 p.root  2 exps 3^k  d  =  3^8 4813^2 6101^4
  x^5 + x^4 - 2448 x^3 - 33543 x^2 + 261685 x - 163369   p  6121 p.root  7 exps 14^k  d  =  1373^2 6121^4 24169^2
  x^5 + x^4 - 2452 x^3 + 69403 x^2 - 693833 x + 2257393   p  6131 p.root  2 exps 17^k  d  =  17^4 73^2 6131^4
  x^5 + x^4 - 2460 x^3 + 5659 x^2 + 296429 x - 555019   p  6151 p.root  3 exps 12^k  d  =  11^2 2687^2 2749^2 6151^4
  x^5 + x^4 - 2484 x^3 - 1739 x^2 + 1278671 x - 1187575   p  6211 p.root  2 exps 32^k  d  =  5^18 79^2 6211^4
  x^5 + x^4 - 2488 x^3 + 9456 x^2 + 687744 x - 3571712   p  6221 p.root  3 exps 2^k  d  =  2^28 29^2 349^2 6221^4
  x^5 + x^4 - 2508 x^3 - 9281 x^2 + 1278983 x + 2108549   p  6271 p.root  11 exps 31^k  d  =  5^18 59^2 6271^4
  x^5 + x^4 - 2520 x^3 - 32009 x^2 + 248209 x - 376701   p  6301 p.root  10 exps 56^k  d  =  3^14 13^2 331^2 6301^4
  x^5 + x^4 - 2524 x^3 + 23477 x^2 + 395523 x + 920061   p  6311 p.root  7 exps 35^k  d  =  3^6 149^2 6311^4 9497^2
  x^5 + x^4 - 2544 x^3 + 30024 x^2 + 547504 x - 3454768   p  6361 p.root  19 exps 17^k  d  =  2^32 17^2 101^2 6361^4
  x^5 + x^4 - 2568 x^3 + 3339 x^2 + 456867 x - 2589165   p  6421 p.root  6 exps 11^k  d  =  3^14 5^2 11^2 103^2 6421^4
  x^5 + x^4 - 2580 x^3 - 71477 x^2 - 641281 x - 1727167   p  6451 p.root  3 exps 7^k  d  =  7^2 13^2 61^2 433^2 6451^4
  x^5 + x^4 - 2592 x^3 - 56255 x^2 - 132005 x + 1570899   p  6481 p.root  7 exps 26^k  d  =  3^14 79^2 163^2 6481^4
  x^5 + x^4 - 2596 x^3 - 51149 x^2 - 116111 x + 20281   p  6491 p.root  2 exps 10^k  d  =  17^2 701^2 1129^2 6491^4
  x^5 + x^4 - 2608 x^3 - 22693 x^2 + 834375 x - 123129   p  6521 p.root  6 exps 3^k  d  =  3^10 43^2 71^2 167^2 6521^4
  x^5 + x^4 - 2620 x^3 - 20177 x^2 + 860749 x + 4430413   p  6551 p.root  17 exps 51^k  d  =  73^2 179^2 6551^4 16141^2
  x^5 + x^4 - 2628 x^3 + 62556 x^2 - 385376 x + 682688   p  6571 p.root  3 exps 2^k  d  =  2^22 1367^2 6571^4
  x^5 + x^4 - 2632 x^3 - 11056 x^2 + 746496 x - 2396160   p  6581 p.root  14 exps 2^k  d  =  2^34 3^10 5^2 6581^4
  x^5 + x^4 - 2664 x^3 - 41831 x^2 + 236279 x + 3775847   p  6661 p.root  6 exps 18^k  d  =  5419^2 6661^4 14447^2
  x^5 + x^4 - 2676 x^3 + 14185 x^2 + 519971 x + 1946201   p  6691 p.root  2 exps 32^k  d  =  13^4 197^2 1291^2 6691^4
  x^5 + x^4 - 2680 x^3 + 73175 x^2 - 652731 x + 1583739   p  6701 p.root  2 exps 3^k  d  =  3^4 5^2 103^2 293^2 6701^4
  x^5 + x^4 - 2704 x^3 + 11629 x^2 + 542719 x - 3765533   p  6761 p.root  3 exps 7^k  d  =  7^6 17^2 47^2 251^2 6761^4
  x^5 + x^4 - 2712 x^3 - 85983 x^2 - 976247 x - 3792469   p  6781 p.root  2 exps 10^k  d  =  43^2 1451^2 6781^4
  x^5 + x^4 - 2716 x^3 + 75516 x^2 - 668832 x + 1807360   p  6791 p.root  7 exps 66^k  d  =  2^18 5^2 101^2 6791^4
  x^5 + x^4 - 2736 x^3 - 67589 x^2 - 404221 x + 519071   p  6841 p.root  22 exps 23^k  d  =  7^2 19^2 23^2 53^2 59^2 6841^4
  x^5 + x^4 - 2748 x^3 + 10444 x^2 + 1042688 x - 4390912   p  6871 p.root  3 exps 7^k  d  =  2^24 7^2 43^2 223^2 6871^4
  x^5 + x^4 - 2764 x^3 - 1935 x^2 + 1834677 x - 73679   p  6911 p.root  7 exps 37^k  d  =  19^2 487^2 6481^2 6911^4
  x^5 + x^4 - 2784 x^3 - 1949 x^2 + 1850735 x + 976175   p  6961 p.root  13 exps 23^k  d  =  5^2 23^2 79^2 6961^4 7949^2
  x^5 + x^4 - 2788 x^3 - 38201 x^2 + 345427 x - 449615   p  6971 p.root  2 exps 18^k  d  =  5^2 1279^2 6121^2 6971^4
  x^5 + x^4 - 2796 x^3 - 55089 x^2 + 783 x + 3065445   p  6991 p.root  6 exps 15^k  d  =  3^14 5^2 29^2 157^2 6991^4
  x^5 + x^4 - 2800 x^3 + 6441 x^2 + 1015089 x - 4471961   p  7001 p.root  3 exps 12^k  d  =  53^2 197^2 7001^4 26681^2
  x^5 + x^4 - 2848 x^3 - 53265 x^2 + 133533 x + 3765799   p  7121 p.root  3 exps 24^k  d  =  6911^2 7121^4 11311^2
  x^5 + x^4 - 2860 x^3 - 22025 x^2 + 544849 x + 4734589   p  7151 p.root  7 exps 28^k  d  =  5^2 61^2 499^2 521^2 7151^4
  x^5 + x^4 - 2884 x^3 - 74129 x^2 - 476849 x - 903431   p  7211 p.root  2 exps 17^k  d  =  11^2 109^2 1217^2 7211^4
  x^5 + x^4 - 2928 x^3 - 98687 x^2 - 1212709 x - 5172037   p  7321 p.root  7 exps 13^k  d  =  23^2 521^2 7321^4
  x^5 + x^4 - 2932 x^3 - 78295 x^2 - 597975 x - 991611   p  7331 p.root  2 exps 10^k  d  =  3^6 17^2 19^2 31^2 41^2 7331^4
  x^5 + x^4 - 2940 x^3 - 37343 x^2 + 204299 x + 1280189   p  7351 p.root  6 exps 13^k  d  =  11^4 13^2 233^2 269^2 7351^4
  x^5 + x^4 - 2964 x^3 + 12747 x^2 + 1078093 x + 3011081   p  7411 p.root  2 exps 26^k  d  =  7^6 149^2 5077^2 7411^4
  x^5 + x^4 - 2980 x^3 + 81365 x^2 - 636375 x + 1494549   p  7451 p.root  2 exps 32^k  d  =  3^10 7^4 109^2 7451^4
  x^5 + x^4 - 2992 x^3 - 5087 x^2 + 2017117 x - 273017   p  7481 p.root  6 exps 15^k  d  =  17^2 31^2 241^2 1069^2 7481^4
  x^5 + x^4 - 3016 x^3 + 61233 x^2 + 49107 x - 3271433   p  7541 p.root  2 exps 11^k  d  =  7309^2 7541^4 9091^2
  x^5 + x^4 - 3024 x^3 - 2117 x^2 + 2273381 x + 560927   p  7561 p.root  13 exps 57^k  d  =  7^2 653^2 1931^2 7561^4
  x^5 + x^4 - 3036 x^3 + 31275 x^2 + 447201 x + 1100061   p  7591 p.root  6 exps 44^k  d  =  3^16 23^2 61^2 7591^4
  x^5 + x^4 - 3048 x^3 + 95415 x^2 - 1045967 x + 3677819   p  7621 p.root  2 exps 19^k  d  =  71^2 2221^2 7621^4
  x^5 + x^4 - 3072 x^3 - 5223 x^2 + 2083333 x - 628849   p  7681 p.root  17 exps 13^k  d  =  61^2 149^2 7681^4 20107^2
  x^5 + x^4 - 3076 x^3 + 62451 x^2 - 171417 x - 1464455   p  7691 p.root  2 exps 6^k  d  =  5^4 863^2 1889^2 7691^4
  x^5 + x^4 - 3096 x^3 + 31893 x^2 + 542737 x + 1451627   p  7741 p.root  7 exps 6^k  d  =  1009^2 7741^4 38959^2
  x^5 + x^4 - 3136 x^3 - 108833 x^2 - 1372865 x - 6001529   p  7841 p.root  12 exps 6^k  d  =  919^2 7841^4
  x^5 + x^4 - 3160 x^3 - 24335 x^2 + 412369 x + 1819819   p  7901 p.root  2 exps 12^k  d  =  5^4 19^2 31^2 71^2 149^2 7901^4
  x^5 + x^4 - 3180 x^3 + 70923 x^2 - 58901 x - 2697427   p  7951 p.root  6 exps 48^k  d  =  4219^2 7951^4 11633^2
  x^5 + x^4 - 3204 x^3 - 50309 x^2 + 178421 x + 380129   p  8011 p.root  14 exps 21^k  d  =  2131^2 8011^4 36083^2
  x^5 + x^4 - 3232 x^3 - 5495 x^2 + 2460115 x + 602659   p  8081 p.root  3 exps 22^k  d  =  3121^2 8081^4 16759^2
  x^5 + x^4 - 3240 x^3 + 72261 x^2 - 131301 x - 110241   p  8101 p.root  6 exps 10^k  d  =  3^18 853^2 8101^4
  x^5 + x^4 - 3244 x^3 + 111283 x^2 - 1402619 x + 6141997   p  8111 p.root  11 exps 19^k  d  =  71^2 449^2 8111^4

======================================

$\endgroup$
3
  • 1
    $\begingroup$ Thanks. Courtesy of the answers by Myerson and Elkies, the depressed forms of these satisfy the three Diophantine relations in the post. And I edited the post to include a second simple cyclic family which, incidentally, contains $p=151$. $\endgroup$ Nov 24, 2016 at 7:51
  • 1
    $\begingroup$ I checked Storer's book, he doesn't do the quintic. $\endgroup$ Nov 24, 2016 at 11:46
  • $\begingroup$ @GerryMyerson:The corrected Lehmer quintic does satisfy all three Diophantine relations. (I made a small typo when I was checking it earlier.) $\endgroup$ Nov 24, 2016 at 13:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.