Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins

Question

(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.

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(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 $p=10m+1$?

Answer 1

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.

Answer 2

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.

Answer 3

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

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