# Gauss periods are minimal polynomials for totally real cyclic fields of
prime degree

Jason A.C. Gallas  Max-Planck Institut PKS, Nöthnitzer Str. 38, 01187 Dresden,
Germany<EMAIL_ADDRESS>Dept. of Appl. Math., Complexity Sciences Center,
9225 Collins Avenue Suite 1208, Surfside, FL 33154, USA  Instituto de Altos
Estudos da Paraíba, Rua Silvino Lopes 419-2502
58039-190 João Pessoa, Brazil

###### Abstract.

We report extensive computational evidence that Gauss period equations are
minimal discriminant polynomials for primitive elements representing Abelian
(cyclic) polynomials of prime degrees $p$. By computing 200 period equations
up to $p=97$, we significantly extend tables in the compendious number fields
database of Klüners and Malle. Up to $p=7$, period equations reproduce known
results proved to have minimum discriminant. For $11\leq p\leq 23$, period
equations coincide with 53 known but unproved cases of minimum discriminant in
the database, and fill a gap of 19 missing cases. For $29\leq p\leq 97$, we
report 128 not previously known cases, 16 of them conjectured to be minimum
discriminant polynomials of Galois group $pT1$. The significant advantage of
period equations is that they all may be obtained analytically using a
procedure that works for fields of arbitrary degrees, and which are extremely
hard to detect by systematic numerical search.

##  1\. Introduction

The determination of number field representations by a minimal polynomial of
primitive elements of interest keeps drawing attention as may be seen from
Refs. [1]–[15], and many additional references given therein. Such polynomials
are notoriously difficult to determine, particularly for degrees higher than
10, say. For the 9 primes $p\leq 23$, they are available in the compendious
and user-friendly tables of Klüners and Malle, containing more than $5\times
10^{6}$ polynomials, and obtained by extensive systematic computer search
[16]–[19],

The aim of this paper is to report the computation of number field
representations of minimal polynomial for the 25 primes $p<100$. In contrast
with previous works, such polynomials are obtained here totally analytically.
As shown below, this was possible after realizing that minimal polynomials
coincide with the period equations described in §343 of the Disquisitiones
Arithmeticæ [20]: Omnes radices $\Omega$ in certas classes (periodos)
distribuuntur.

As an extra bonus, period equations allow minimal polynomials to be computed
analytically for arbitrarily large primes $p$, well beyond what is possible to
probe with systematic numerical search, due to the very large number and
magnitudes of the polynomial coefficients involved. Some representative large
coefficients are given in Table 4 below. Here we focus on totally real Abelian
(cyclic) fields because of their relevance as equations of motion for certain
classes of dynamical systems, namely in the so-called partition generating
limit of discrete-time quadratic and quartic dynamics [21, 22].

##  2\. Analytical expressions for period equations and field discriminants

Consider the set $\Omega$ of the $p-1$ complex roots of the equation
$x^{p}-1=0$. The key to obtain Gaussian period equations is to distribute the
$p-1$ roots in $\Omega$ into certain sums called periods [20, 23]. This is
done by choosing two positive rational integers $e$ and $f$ such that
$ef=p-1$, and partitioning the roots in $\Omega$ into $e$ disjoint classes,
thereby forming $e$ periods $\eta_{i}$, each one consisting of a sum of $f$
roots.

Table 1. Field discriminants for families of totally real cyclic fields of prime degree $p$ and Galois group $pT1$, computed with Eq. (3). Non-highlighted discriminants coincide with values in the tables of Klüners and Malle [16], while highlighted discriminants are missing in their tables. Minimal polynomials representing the 19 new fields are given in Table 2. Labels $\ell$ refer to the order in which polynomials are generated when $f$ is varied in $ef=p-1$ along rows. The $\ell=1$ column lists proven and unproven minimum field discriminants.  $p$ | $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$ | $\ell=5$ | $\ell=6$ | $\ell=7$ | $\ell=8$
---|---|---|---|---|---|---|---|---
2 | $5$ | $13$ | $17$ | $29$ | $37$ | $41$ | $53$ | $61$
3 | $7^{2}$ | $13^{2}$ | $19^{2}$ | $31^{2}$ | $37^{2}$ | $43^{2}$ | $61^{2}$ | $67^{2}$
5 | $11^{4}$ | $31^{4}$ | $41^{4}$ | $61^{4}$ | $71^{4}$ | $101^{4}$ | $131^{4}$ | $151^{4}$
7 | $29^{6}$ | $43^{6}$ | $71^{6}$ | $113^{6}$ | $127^{6}$ | $197^{6}$ | $211^{6}$ | $239^{6}$
11 | $23^{10}$ | $67^{10}$ | $89^{10}$ | $199^{10}$ | $331^{10}$ | $353^{10}$ | $397^{10}$ | $419^{10}$
13 | $53^{12}$ | $79^{12}$ | $131^{12}$ | $157^{12}$ | $313^{12}$ | $443^{12}$ | $521^{12}$ | $547^{12}$
17 | $103^{16}$ | $137^{16}$ | $239^{16}$ | $307^{16}$ | $409^{16}$ | $443^{16}$ | $613^{16}$ | $647^{16}$
19 | $191^{18}$ | $229^{18}$ | $419^{18}$ | $457^{18}$ | $571^{18}$ | $647^{18}$ | $761^{18}$ | $1103^{18}$
23 | $47^{22}$ | $139^{22}$ | $277^{22}$ | $461^{22}$ | $599^{22}$ | $691^{22}$ | $829^{22}$ | $967^{22}$

Let $r$ be any complex root in $\Omega$, say $r=e^{2\pi i/p}$, and $g$ a
primitive root of $p$. Then, the periods $\eta_{i}$ are given by the sums [20,
23, 24]

(1) $\eta_{i}=\sum_{k=0}^{f-1}r^{g^{ke+i}},\qquad i=0,1,\cdots,e-1,$

namely, $e$ sums of $f$ distinct complex roots suitably selected from $\Omega$

$\displaystyle\eta_{0}$ $\displaystyle=$ $\displaystyle r$ $\displaystyle+$
$\displaystyle r^{g^{e}}$ $\displaystyle+$ $\displaystyle r^{g^{2e}}$
$\displaystyle+$ $\displaystyle\ \cdots\ $ $\displaystyle+$ $\displaystyle
r^{g^{(f-1)e}},$ $\displaystyle\eta_{1}$ $\displaystyle=$ $\displaystyle
r^{g}$ $\displaystyle+$ $\displaystyle r^{g^{e+1}}$ $\displaystyle+$
$\displaystyle r^{g^{2e+1}}$ $\displaystyle+$ $\displaystyle\ \cdots\ $
$\displaystyle+$ $\displaystyle\ r^{g^{(f-1)e+1}},$ $\displaystyle\;\;\vdots$
$\displaystyle\eta_{e-1}$ $\displaystyle=$ $\displaystyle\ r^{g^{e-1}}$
$\displaystyle+$ $\displaystyle\ r^{g^{2me1}}$ $\displaystyle+$
$\displaystyle\ r^{g^{3e-1}}$ $\displaystyle+$ $\displaystyle\ \cdots\ $
$\displaystyle+$ $\displaystyle\ r^{g^{fe-1}}.$

For a fixed $e$, the period equation $\psi_{e}(x)$ of degree $e$ is defined as
[20, 23, 24]

(2)
$\psi_{e}(x)=\prod_{k=0}^{e-1}(x-\eta_{k})=x^{e}+x^{e-1}+\alpha_{2}x^{e-2}+\cdots+\alpha_{e},$

where the $\alpha_{\ell}$ turn out to be integers. The corresponding field
discriminant $\Delta_{e}$ of $\psi_{e}(x)$ is [25]

(3) $\Delta_{e}=\begin{cases}-p^{e-1},&\hbox{if }(e-1)\ \hbox{mod }4=1{\ \ \
\rm and\ \ \ }f{\ \rm mod\ }2=1,\\\ \phantom{-}p^{e-1},&\hbox{if\ \
otherwise}.\end{cases}$

Equivalently, Eq. (3) may be also written as

(4) $\Delta_{e}=\begin{cases}(-1)^{n_{P}}p^{e-1},&\hbox{if}\ (e-1)\ \hbox{mod
}4=1,\\\ p^{e-1},&\hbox{if\ \ otherwise},\end{cases}$

where $n_{P}$ is the number of pairs of complex roots of $\psi_{e}(x)$.

Table 2. Polynomials of Galois group $pT1$ derived analytically which fill gaps in tables of Klüners and Malle [16]. For reference, the highlighted polynomials in each group may be compared with analogous ones in the tables Klüners and Malle. $\Delta_{e}$ is the field discriminant.  $p,\Delta_{e}$ |  Polynomial coefficients ordered by decreasing powers of $x$
---|---
$11,89^{10}$ |  $1,1,-40,-19,482,84,-2185,102,3152,-781,57,-1$
$11,199^{10}$ | $1,1,-90,-115,2349,943,-26327,21284,102168,-217794,148930,-30647$
$11,331^{10}$ | $1,1,-150,-402,6577,28617,-62124,-475464,-343344,1913488,4015168,2287616$
$11,353^{10}$ | $1,1,-160,-525,6066,26034,-48369,-265374,-42966,405001,63189,-170569$
$11,397^{10}$ | $1,1,-180,-13,11655,-12159,-316973,720142,2670510,-10551746,10752776,-3098903$
$11,419^{10}$ | $1,1,-190,-547,10985,51221,-141765,-1224028,-2399676,-1263744,873500,785489$
$13,79^{12}$ |  $1,1,-36,-77,365,1193,-617,-5541,-4414,4575,6321,-411,-2196,-293$
$13,131^{12}$ | $1,1,-60,-27,1199,33,-9610,3352,33548,-20328,-47723,34869,21271,-15667$
$13,157^{12}$ | $1,1,-72,-129,1672,3386,-16810,-32367,81708,121902,-196272,-127412,217458,-61399$
$13,313^{12}$ | $1,1,-144,-161,6530,9620,-109398,-196143,512628,917970,-650724,-1134730,253950,409375$
$13,443^{12}$ | $1,1,-204,181,10752,-9116,-208418,161679,1686466,-1207646,-4904338,3051848,896956,-144209$
$13,521^{12}$ | $1,1,-240,293,19153,-45777,-616830,1795569,7791196,-23224049,-29107980,68466088,31673025,-4516075$
$13,547^{12}$ | $1,1,-252,-1123,15626,107844,-204415,-3094114,-4853400,22393129,91453411,116380476,47088126,1165671$
$17,239^{16}$ |  $1,1,-112,-47,3976,4314,-64388,-136247,422013,1631073,411840,-5840196,-11894369,-10635750$,
|  $-4739804,-938485,-54850,-619$
$17,307^{16}$ |  $1,1,-144,-241,6894,14938,-127923,-323969,847982,2194186,-2617873,-6091397,3745755,7069429,-1600190$,
|  $-3100257,-220118,208777$
$17,409^{16}$ |  $1,1,-192,-273,14752,28028,-571107,-1411675,11275657,36814399,-91832077,-461179352,-109192148$,
|  $\phantom{-}1929139488,3679722325,2767754010,828153361,45886883$
$17,443^{16}$ |  $1,1,-208,17,15287,-13881,-487578,703261,6754359,-10540902,-41136753,57683825,92010954,-95287840$,
|  $-17501435,25026156,-563260,-1246103$
$17,613^{16}$ | $1,1,-288,-265,26034,40228,-875968,-2022008,8464009,27681440,-8855367,-101412811,-87313302,38624139,$
|  $\phantom{-}67164168,7149746,-7878215,664471$
$17,647^{16}$ |  $1,1,-304,-1117,25631,126439,-773932,-4360454,10731832,64676368,-79260104,-441919082,345306489$,
|  -$1259087517,-718017711,-1025767171,183044979,202031659$
$19,571^{18}$ |  $1,1,-270,-351,28987,41181,-1648168,-2453428,55186847,85340779,-1133336624,-1826548777,14258200659$,
|
-$24108274876,-104945874488,-187074906809,398834944226,749021546949,-549551190705,-1072348621073$
$19,647^{18}$ |  $1,1,-306,79,29370,-130,-1361758,-704483,34401066,34840236,-488507308,-678792838,3902105097,6509889712$,
|
$-16762277726,-31790175776,33454527221,71781895812,-18736713884,-50781641759$
$19,761^{18}$ |  $1,1,-360,173,45376,-58762,-2558302,4227138,70534890,-120121397,-973700212,1501612590,6678374954$,
|
$-8595059019,-21259099080,21796436285,27241052007,-18814754704,-12659238391,3483379661$
$19,1103^{18}$ |  $1,1,-522,-504,101486,128414,-9620928,-15777284,481094177,987900729,-12621266990,-31913538892,160258304904$,
|
-$502132433072,-773726987936,-3469327268928,-293071915904,7722407112704,6083919470592,-276663107584$

The explicit calculation of period equations presents no more difficulty than
the inevitable length. Analytically, the method works fine for low-degree
equations. For instance, it is a standard textbook example to solve
$x^{17}-1=0$ by radicals. But the method collapses in a tangle of calculations
as soon as it is applied to higher degrees, leading to no convenient formulas
to solve polynomials by radicals. Fortunately, the method is not difficult to
implement in computer algebra systems. Using the equations above, we wrote a
procedure in MAPLE 18 (X86 64 Linux), which was used to produce the
polynomials and results summarized in our tables below.

##  3\. Results

Table 1 summarizes field discriminants calculated for the $\psi_{e}(x)$
polynomials for $p\leq 23$. The non-highlighted discriminants arise from
polynomials that coincide with the ones included in the Tables of Klüners and
Malle [16]. The 19 highlighted discriminants are from polynomials missing in
their tables. In Table 1, the labels $\ell$ refer to the order in which
polynomials are generated when $f$ is varied in $ef=p-1$ along rows.

Table 2 reports polynomial coefficients for the 19 highlighted cases in Table
1. Coefficients are arranged in order of decreasing powers of $x$, for
instance, the topmost polynomial for $p=11$ is

${x}^{11}+{x}^{10}-40\,{x}^{9}-19\,{x}^{8}+482\,{x}^{7}+84\,{x}^{6}-2185\,{x}^{5}+102\,{x}^{4}+3152\,{x}^{3}-781\,{x}^{2}+57\,x-1.$

Table 3 summarizes discriminants computed for 128 new Gaussian period
equations $\psi_{e}(x)$. Note the relatively large magnitudes of the
discriminants, e.g. $43651^{96}$ is a number of $446$ digits. Of particular
interest are the discriminants listed in the $\ell=1$ column which, based on
what happens in all previous cases, are conjectured to be minimum field
discriminant polynomials. These polynomials are given explicitly in Table 4.
The very large magnitude of their coefficients should make them rather
difficult, if not impossible, to determine by systematic computer searches.
Note that polynomials for additional primes $p$ are not difficult to be
determined, thanks to the analytical expression for $\psi_{e}(x)$.

In Tables 2 and 4, the $\ell=1$ period equations are also exceptional in that
they are monogenic polynomials, meaning that their ordinary discriminant
$\Delta_{\psi}$ of $\psi_{e}(x)$ coincides with their field discriminant
$\Delta_{e}$, a rather rare occurrence [26] that misled Dedekind for some time
[27]. For other values of $\ell$ in Tables 1 and 3 one has

$\Delta_{\psi}=k^{2}\;\Delta_{e},\qquad k^{2}\neq 1.$

Table 3. Field discriminants for 128 new $\psi_{e}(x)$ period equations which significantly extend the known results in Table 1. Labels $\ell$ refer to the order in which polynomials are generated when $f$ is varied in $ef=p-1$ along rows. Discriminants on the $\ell=1$ column are conjectured to be minimum field discriminants. The corresponding polynomials are given explicitly in Table 4. $p$ | $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$ | $\ell=5$ | $\ell=6$ | $\ell=7$ | $\ell=8$
---|---|---|---|---|---|---|---|---
29 | $59^{28}$ | $233^{28}$ | $349^{28}$ | $523^{28}$ | $929^{28}$ | $1103^{28}$ | $1277^{28}$ | $1451^{28}$
31 | $311^{30}$ | $373^{30}$ | $683^{30}$ | $1117^{30}$ | $1303^{30}$ | $1427^{30}$ | $1489^{30}$ | $1613^{30}$
37 | $149^{36}$ | $223^{36}$ | $593^{36}$ | $1259^{36}$ | $1481^{36}$ | $1777^{36}$ | $1999^{36}$ | $2221^{36}$
41 | $83^{40}$ | $739^{40}$ | $821^{36}$ | $1231^{40}$ | $1559^{40}$ | $1723^{40}$ | $2297^{40}$ | $2543^{40}$
43 | $173^{42}$ | $431^{42}$ | $947^{42}$ | $1033^{42}$ | $1291^{42}$ | $1549^{42}$ | $1721^{42}$ | $1979^{42}$
47 | $283^{46}$ | $659^{46}$ | $941^{46}$ | $1129^{46}$ | $1223^{46}$ | $1693^{46}$ | $1787^{46}$ | $2069^{46}$
53 | $107^{52}$ | $743^{52}$ | $1061^{52}$ | $1697^{52}$ | $2333^{52}$ | $2969^{56}$ | $3181^{52}$ | $3499^{52}$
59 | $709^{58}$ | $827^{58}$ | $1063^{58}$ | $1181^{58}$ | $1889^{58}$ | $2243^{58}$ | $2833^{58}$ | $3187^{58}$
61 | $367^{60}$ | $977^{60}$ | $1709^{60}$ | $1831^{60}$ | $2441^{60}$ | $3539^{60}$ | $4027^{60}$ | $4271^{60}$
67 | $269^{66}$ | $1609^{66}$ | $1877^{66}$ | $2011^{66}$ | $3083^{66}$ | $3217^{66}$ | $4021^{66}$ | $4289^{66}$
71 | $569^{70}$ | $1279^{70}$ | $2131^{70}$ | $2273^{70}$ | $2557^{70}$ | $2699^{70}$ | $4261^{70}$ | $5113^{70}$
73 | $293^{72}$ | $1607^{72}$ | $1753^{72}$ | $3359^{72}$ | $3797^{72}$ | $3943^{72}$ | $4673^{72}$ | $5987^{72}$
79 | $317^{78}$ | $2213^{78}$ | $2687^{78}$ | $3319^{78}$ | $3793^{78}$ | $5531^{78}$ | $6163^{78}$ | $6637^{78}$
83 | $167^{82}$ | $1163^{82}$ | $1993^{82}$ | $2657^{82}$ | $4483^{82}$ | $5147^{82}$ | $5479^{82}$ | $6143^{82}$
89 | $179^{88}$ | $4273^{88}$ | $5519^{88}$ | $6053^{88}$ | $7477^{88}$ | $8011^{88}$ | $9257^{88}$ | $9781^{88}$
97 | $389^{96}$ | $3881^{96}$ | $10477^{96}$ | $11059^{96}$ | $23087^{96}$ | $25997^{96}$ | $42293^{96}$ | $43651^{96}$

##  4\. Conclusions

The main message of this paper is the identification of minimal polynomials
for totally real cyclic fields as being Gaussian period equations. This fact
allowed the easy analytical determination of period equations and field
discriminants for the 25 cyclic fields of prime degree $p\leq 100$. Such
period equation were found to coincide with all known minimum discriminant
polynomials for totally real cyclic fields and to significantly extend the
minimal polynomials by 128 new cases. Up to $p=7$, the 4 period equations
coincide with known polynomials proven to have minimum discriminants. For
$11\leq p\leq 23$, the 5 equations coincide with tabulated polynomials having
known but still unproven minimum discriminant. The remaining 16 equations, for
$29\leq p\leq 97$, we reported 128 not previously known minimal polynomials,
16 of them conjectured to also have minimum discriminants The findings
reported here suggest that polynomials representing totally real number fields
with minimum discriminants are given by Gaussian period equations which may be
determined to very high degrees with no major difficulty. Although we limited
our tables to just $\ell=8$ cases, we have investigated much higher $\ell$
values with no difficulty, always corroborating the results described above.

The ability of the Gaussian period equations to reproduce minimal polynomials
is almost uncanny, and we hope this work to set the agenda for the study of
their general properties and new applications. The identification of Gaussian
period equations as representatives of minimal discriminant fields was also
found to be valid for fields of arbitrary degrees, and for fields of
signatures other than totally real. These results will be presented elsewhere.

## Acknowledgments

This work was partially supported by the Max-Planck Institut PKS, Dresden,
Germany, and CNPq, Brazil, Grant No. PQ-305305/2020-4.

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##  Appendix A Table 4: Gaussian period equations extending Table 1

Table 4. Analytically derived coefficients of Gaussian period equations which significantly extend the known cases shown in Table 1 of totally real cyclic fields of prime degree and Galois group $pT1$. These polynomials are conjectured to have minimum field discriminants $\Delta_{e}$. Numbers in parenthesis give the number of digits of $\Delta_{e}$.  $p,\Delta_{e}$ |  Polynomial coefficients ordered by decreasing powers of $x$
---|---
$29,59^{28}$ |  $1,1,-28,-27,351,325,-2600,-2300,12650,10626,-42504,-33649,100947,74613,-170544,-116280,203490,125970$,
(50) |  $-167960,-92378,92378,43758,-31824,-12376,6188,1820,-560,-105,15,1$
$31,311^{30}$ |  $1,1,-150,-117,9434,4958,-328880,-78328,7098810,-507925,-100297691,39217806,950942678,-677567200,-6056859921$,
(75) |  $\phantom{-}6293114670,25303767350,-35315191500,-65145555320,122001770825,87706314807$,
|
$-253609179978,-19776440614,299693882286,-98200795275,-175359843704,113820457794,30260338943$,
|  $-38729335872,5811762482,1056847214,26458109$
$37,149^{36}$ |  $1,1,-72,-67,2278,1964,-41849,-33262,497142,362207,-4027127,-2672215,22871279,13719683,-92273266,-49606380$,
(79) |  $\phantom{-}265291071,126402144,-540999671,-224635365,773506030,271832880,-761221384,-214848879,502240682,103923082$,
|
$-213804070,-27920639,55105551,3603129,-7708785,-235974,497852,22640,-12819,-1020,28,1$
$41,83^{40}$ |  $1,1,-40,-39,741,703,-8436,-7770,66045,58905,-376992,-324632,1623160,1344904,-5379616,-4272048,13884156$,
(77) |  $\phantom{-}10518300,-28048800,-20160075,44352165,30045015,-54627300,-34597290,51895935,30421755,-37442160,-20058300$,
|
$\phantom{-}20058300,9657700,-7726160,-3268760,2042975,735471,-346104,-100947,33649,7315,-1540,-210,21,1$
$43,173^{42}$ |  $1,1,-84,-79,3160,2786,-70521,-58076,1042773,798942,-10810701,-7671648,81126975,53056499,-448758019,-268953093$,
(94) |  $\phantom{-}1846875875,1007873658,-5671315301,-2797394685,12962901258,5730420663,-21895326590,-8590544711,27001558938$,
|
$\phantom{-}9297003415,-23896748020,-7121607714,14834334408,3755010451,-6269987218,-1310830451,1732796449,287529242$,
|  $-294221159,-37041429,27505221,2599666,-1134285,-90620,12893,370,-54,1$
$47,283^{46}$ |  $1,1,-138,-315,8338,29804,-276833,-1433626,5033859,41190458,-30657314,-748097961,-742659788,8506344013,21519259357$,
(113) |  $-52948548811,-268879641855,30332528938,1920252236103,2430736233424,-7367030656288,-21401598866455,5373046913681$,
|
$\phantom{-}89240242581627,85735098102709,-174332690558567,-418760640969237,-24016008538438,845143941649693,850093833789498$,
|
$-563502754038610,-1652337279635119,-688139958495907,1132046847057208,1399925903803443,129366579867529$,
|
$-739733924950208,-464611724348759,61382685657965,169596960618209,46703998783537,-17972540150505,-11555848466611$,
|  $-591153994800,757301830397,134933083169,-1756074840,-132954859$
$53,107^{52}$ |  $1,1,-52,-51,1275,1225,-19600,-18424,211876,194580,-1712304,-1533939,10737573,9366819,-53524680,-45379620,215553195$,
(106) |  $\phantom{-}177232627,-708930508,-563921995,1917334783,1471442973,-4280561376,-3159461968,7898654920,5586853480$,
|
$-12033222880,-8122425444,15084504396,9669554100,-15471286560,-9364199760,12875774670,7307872110,-8597496600$,
|
$-4537567650,4537567650,2203961430,-1855967520,-818809200,573166440,225792840,-129024480,-44352165,20160075$,
|  $\phantom{-}5852925,-2035800,-475020,118755,20475,-3276,-351,27,1$
$59,709^{58}$ |  $1,1,-348,-773,54710,181658,-5108670,-22380037,313864889,1735112733,-13244902303,-92489323874,384874485628$,
(166) |  $\phantom{-}3563558672105,-7224424298771,-102284588313436,59507208676899,2226843428814170,1091206962010061$,
|
$-37110651573131926,-50769946526736009,473955610044577175,1034887463549316633,-4599282007018108614$,
|
$-14157628456571325637,33064370721903906102,142351088507771542259,-164489546968394181788,-1088242867211629932856$,
|
$\phantom{-}434724808533168556934,6421863971239504423218,821416241786189985129,-29458267038177484817142$,
|
$-15268230232813063311459,105361097493218843884743,86357419790172643990359,-294180307521278014201331$,
|
$-310904964780280912527690,641819365276550155193989,795098801640263156488019,-1096213896340607733888156$,
|
$-1490164806719968998511703,1470729980576933657823174,2055743062031065158143660,-1554777323587753672707889$,
|
$-2062294848329571542288479,1289894730680748454358450,1461538845678796120527548,-820373077133361026883291$,
|
$-693872641742794094654444,378259269617360523640282,199906595981055748193122,-114544083882472863320184$,
|
$-27852340198822864548711,19330914962494272607266,388755699390176021351,-1296740570702255596842$,
|  $\phantom{-}197731971687386122768,-8442821363163035954,-4584804110356861$
$61,367^{60}$ |  $1,1,-180,-413,14645,52673,-688601,-3528285,19601342,147066683,-294411763,-4094132898,-515253399,77730183943$,
(154) |  $\phantom{-}136127326190,-976171853534,-3440181895427,6820700430038,49495243759604,5003873089663,-454773118963937$,
|
$-708332454647506,2503079468369386,8524707843825128,-4353506873916376,-55965866407852931,-50586504885325349$,
|
$\phantom{-}208588686855790630,488766704388771586,-248616968576483612,-2111939395471861259,-1621839831538649260$,
|
$\phantom{-}4656502567948889185,9658116194858556310,-1497012549402345042,-23738135773030440428,-20637249171326524602$,
|
$\phantom{-}24569280463351126387,57740462789299170679,14579305016676768566,-65771381500668227611,-72094504453297506055$,
|
$\phantom{-}13570829828275968614,78797457647783506537,45464576927398898131,-26397733136489030972,-45803590606881784425$,
|
$-13479501565046232768,13463013755243374867,12223486303700829006,1531995617799024854,-2479700985326385279$,
|
$-1203853517042779781,45547818199089944,173724631575165523,33343190173223859,-8176066599449700$,
|  $-3216273109307699,787254613563,91700724228752,3245424268762,-934868803397$
$67,269^{66}$ |  $1,1,-132,-127,8128,7514,-310489,-275412,8255097,7015522,-162469429,-132074732,2457934103,1908054027,-29302580527$,
(161) |  $-21682477578,280016358684,197116015784,-2170586468374,-1450574505983,13762204743256,8711498507598,-71769908578626$,
|
$-42926578989207,308917467707281,174121538109300,-1099328139842573,-582252857213765,3235005088514751$,
|
$\phantom{-}1605012951198723,-7862532891652511,-3641973093600359,15743247597494598,6784769007620029,-25870357696878572$,
|
$-10338065796135117,34707160257704653,12822873516191615,-37759031607934407,-12874525720258242,33032386678911041$,
|
$\phantom{-}10396393258198832,-22995232909296191,-6702394382020432,12575711941082825,3418837407991456,-5318199438079755$,
|
$-1363375197739315,1705561117819514,417633543753729,-404848315593940,-95674661443562,68987247039926$,
|
$\phantom{-}15749316188706,-8123019166449,-1758233811824,632045599194,122538815988,-31180078280,-4726423766$,
|  $\phantom{-}955488656,84545195,-16930746,-385257,133369,-3611,-84,1$
$71,569^{70}$ |  $1,1,-280,-259,36362,31280,-2918219,-2341868,162684328,121979735,-6712586533,-4705153431,213256136347,139745967919$,
(193) |  $-5356937166534,-3281290140692,108406810097566,62068478295763,-1791341741101037,-959090929705315,24407831840018355$,
|
$\phantom{-}12233821563475166,-276147939713855185,-129846912904598319,2606654089690226864,1153616129990347133$,
|
$-20587950147002420775,-8616230414933743750,136232462096831704169,54248999168315159885,-755063095484716789902$,
|
$-288262405489471572571,3499387584477151458385,1291958342775348433576,-13519262273839572481549$,
|
$-4871234681829364379697,43337921624309469265580,15374832177489501594722,-114567325220911193557243$,
|
$-40314399744765933113883,247818220129501437242168,86899209178511442518249,-434439541957702412239512$,
|
$-151897369076330346742203,610281064922154821463337,211710160779065690463654,-678145211065889672618174$,
|
$-230622710469268836012522,587757024818899676327921,191925680146871604212121,-391624630310001080455072$,
|
$-119033636843278373618263,197766785967009160429706,53612258357612675910509,-74608688043563231273283$,
|
$-17058617253176590894815,20687510333056339496635,3713992963764087642008,-4130593027985267121538$,
|
$-531046062111838509185,578162928492769417198,46961262145834098724,-54781357679840005659,-2287539667564813302$,
|
$\phantom{-}3352222063105017682,40620243008078108,-123514842854460529,1010652258526922,2420447510811051,-55851319642213$,
|  $-18452351744696,786556882087$
Table 5. Continuation of Table 4. $p,\Delta_{K}$ |  Polynomial coefficients for decreasing powers of $x$
---|---
$73,293^{72}$ |  $1,1,-144,-139,9730,9056,-410441,-367906,12132108,10460357,-267391523,-221468185,4565619563,3627532367$,
(178) |  $-61950350404,-47147580804,679947392088,494890787591,-6113510799041,-4248263015489,45441140233604$,
|
$\phantom{-}30093580771161,-281045946680083,-177038034694018,1452868601357593,868715820105921,-6295670418447183$,
|
$-3565186787476080,22902199637195106,12253312487223193,-69962703250113206,-35272462117714863$,
|
$\phantom{-}179338461559168548,84955105196536588,-385048845118205565,-170849729564878739,690518720387088459$,
|
$\phantom{-}285992431892776900,-1030365887765938986,-396815828472409411,1273033339340609215,453948542708929508$,
|
$-1294487651212092317,-425372577783968644,1075460998363558124,323938213588417248,-723673836702904009$,
|
$-198620283444369134,390347789427434985,96972004303780760,-166728320860948405,-37205870049815405$,
|
$\phantom{-}55579282507755157,11040773363511690,-14210001829233769,-2484259728799330,2727333323515561$,
|
$\phantom{-}413148457058309,-382313978331429,-49086744961361,37718946870926,3978758518003,-2483523023306$,
|
$-206824721197,100506678700,6386688064,-2177670239,-106831868,20248339,685305,-67318,-1667,55,1$
$79,317^{78}$ |  $1,1,-156,-151,11476,10742,-529833,-479120,17237859,15043356,-420695673,-353920938,8005510587,6484821551$,
(196) |  $-121897544509,-94959796018,1512506583433,1131648317743,-15496341553928,-11120185043808,132376432838106$,
|
$\phantom{-}90975356804502,-949633381885839,-624050367056385,5750894189811100,3607649713387208,-29508660802301222$,
|
$-17639571023194681,128602029542560426,73114526892928284,-476661570350174302,-257209659492690357$,
|
$\phantom{-}1503149344492126347,768122531281080170,-4030993135188740266,-1945953639652130387,9180641646273875633$,
|
$\phantom{-}4175544021158828515,-17720410438845095950,-7570210080186966239,28904111895399788759,11556972210324882738$,
|
$-39694010615804869681,-14790824727072361901,45687502119081734033,15779507887426374013,-43834207774512448337$,
|
$-13933215738460480507,34832294570965230505,10092292269108676458,-22752966081378874965,-5930089941589565108$,
|
$\phantom{-}12110606741300452292,2787350977545282191,-5198631736503715705,-1029729865708380737,1777818076567389569$,
|
$\phantom{-}292332297736928516,-477200723807927472,-61925190559965783,98688594363464519,9404904692311374$,
|
$-15354866035626205,-967602418893197,1742477355890453,62117408755516,-138523447421396,-2288943125790$,
|
$\phantom{-}7331249803128,67426164550,-243056280171,-3889298793,4696185844,157596564,-45517649,-2527289,140716,12298,218,1$
$83,167^{82}$ |  $1,1,-82,-81,3240,3160,-82160,-79079,1502501,1426425,-21111090,-19757815,237093780,218618940,-2186189400$,
(183) |  $-1984829850,16871053725,15071474661,-110524147514,-97082021465,621324937376,536211932256,-3022285436352$,
|
$-2560547383576,12802736917880,10638894058520,-47465835030320,-38650751381832,154603005527328,123234279768160$,
|
$-443643407165376,-345780890878896,1123787895356412,855420636763836,-2515943049305400,-1867897112363100$,
|
$\phantom{-}4981058966301600,3601688791018080,-8719878125622720,-6131164307078475,13488561475572645,9206478467454345$,
|
$-18412956934908690,-12176310231149295,22138745874816900,14154280149473100,-23385332420868600$,
|
$-14420954992868970,21631432489303455,12832205713993575,-17451799771031262,-9929472283517787$,
|
$\phantom{-}12220888964329584,6646448384109072,-7384942649010080,-3824345300380220,3824345300380220,1877405874732108$,
|
$-1683191473897752,-779255311989700,623404249591760,270533919634160,-191991813933920,-77535155627160$,
|
$\phantom{-}48459472266975,18053528883775,-9847379391150,-3348108992991,1575580702584,482320623240,-192928249296$,
|
$-52251400851,17417133617,4076350421,-1101716330,-215553195,45379620,7059052,-1086008,-123410,12341,861,-42,-1$
$89,179^{88}$ |  $1,1,-88,-87,3741,3655,-102340,-98770,2024785,1929501,-30872016,-29034396,377447148,350161812,-3801756816$,
(199) |  $-3477216600,32164253550,28987537150,-231900297200,-205811513765,1440680596355,1258315963905,-7778680504140$,
|
$-6681687099710,36749279048405,31022118677225,-152724276564800,-126600387152400,560658857389200,456002537343216$,
|
$-1824010149372864,-1454278362337824,5271759063474612,4116305022165108,-13559593014190944,-10358022441395860$,
|
$\phantom{-}31074067324187580,23196134763125940,-63484158299081520,-46252743903616536,115631859759041340,82115378669464140$,
|
$-187692294101632320,-129728497393775280,271250494550621040,182183167981760400,-348524321356411200$,
|
$-227068876035237600,397370533061665800,250649105469666120,-401038568751465792,-244382877832924467$,
|
$\phantom{-}357174975294274221,209769429934732479,-279692573246309972,-157890968768078210,191724747789809255$,
|
$\phantom{-}103719945525634515,-114449595062769120,-59132290782430712,59132290782430712,29065024282889672$,
|
$-26252279997448736,-12220888964329584,9929472283517787,4355031703297275,-3167295784216200$,
|
$-1300853625660225,841728816603675,321387366339585,-183649923622620,-64617565719070,32308782859535$,
|
$\phantom{-}10363194502115,-4481381406320,-1292706174900,476260169700,121399651100,-37353738800,-8217822536$,
|
$\phantom{-}2054455634,377348994,-73629072,-10737573,1533939,163185,-15180,-990,45,1$
$97,389^{96}$ |  $1,1,-192,-187,17578,16664,-1021929,-942362,42389952,37997337,-1336462731,-1163680149,33314897667,28156698311$,
(249) |  $-674408368288,-552838527076,11301013598482,8977909375093,-158994886726653,-122308729912222,1898473578664977$,
|
$\phantom{-}1412901222447415,-19399951044988978,-13955275813688939,170762291665699972,118614446384719822$,
|
$-1301309882049061094,-871945780122664925,8619377666392013922,5565181768869578253,-49771732711929844788$,
|
$-30930479393262936612,251112071197806320739,150020160792137684755,-1108677862726332708568$,
|
$-635934910830427844783,4287605330953710413992,2358093640258069799412,-14530612000190559702170$,
|
$-7651494316042985990505,43150759530518584124075,21722212320251222385725,-112228827609141155647105$,
|
$-53922076560098598529529,255396998183926539104038,116914878588220059051162,-507829409247920514083495$,
|
$-221083325574365962264596,880677517892705249616378,363892311994091133884903,-1329003780593599646666617$,
|
$-520077115693458007818368,1740436761368591469099430,643542366602780582216934,-1971574632911812727476760$,
|
$-687085946667407137439777,1924701872709234943427512,630420911347065941491567,-1612252793611361847320163$,
|
$-494786845657867267549369,1153114327395849183265703,330399291621281386173050,-700204009654308355534634$,
|
$-186550016549016488099855,358661632153989150215180,88424127745933995583848,-153831904737635556704045$,
|
$-34895855606206393994169,54780945797382119086423,11357456950456450852662,-16039377858795390694256$,
|
$-3015740835790853975136,3817489549458070523980,645389203032607765111,-728761616821001045732$,
|
$-109825647572795780485,109819500917589950691,14644442153531449279,-12813785741765692684$,
|
$-1506241051689218153,1130523247638411169,117451448799052895,-73215396010936265,-6798178894924984$,
|
$\phantom{-}3351149496877315,283215594877129,-103171823466943,-8071372448439,1999976788345,145004553360$,
|  $-22289089940,-1467714628,122419905,7058361,-228410,-10990,73,1$