Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^31 - 19*x^30 + 146*x^29 + 236*x^28 - 1994*x^27 - 1515*x^26 + 17532*x^25 + 248*x^24 - 102760*x^23 + 62939*x^22 + 452232*x^21 - 621557*x^20 - 1332326*x^19 + 3689491*x^18 + 75956*x^17 - 11152569*x^16 + 15594526*x^15 + 12008686*x^14 - 58161950*x^13 + 9977818*x^12 + 131342300*x^11 - 70421448*x^10 - 200370404*x^9 + 221182489*x^8 + 64149492*x^7 - 233429156*x^6 + 127683800*x^5 + 51958176*x^4 - 115466320*x^3 + 71910480*x^2 - 21408800*x + 2671600)
gp: K = bnfinit(y^32 - 6*y^31 - 19*y^30 + 146*y^29 + 236*y^28 - 1994*y^27 - 1515*y^26 + 17532*y^25 + 248*y^24 - 102760*y^23 + 62939*y^22 + 452232*y^21 - 621557*y^20 - 1332326*y^19 + 3689491*y^18 + 75956*y^17 - 11152569*y^16 + 15594526*y^15 + 12008686*y^14 - 58161950*y^13 + 9977818*y^12 + 131342300*y^11 - 70421448*y^10 - 200370404*y^9 + 221182489*y^8 + 64149492*y^7 - 233429156*y^6 + 127683800*y^5 + 51958176*y^4 - 115466320*y^3 + 71910480*y^2 - 21408800*y + 2671600, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 6*x^31 - 19*x^30 + 146*x^29 + 236*x^28 - 1994*x^27 - 1515*x^26 + 17532*x^25 + 248*x^24 - 102760*x^23 + 62939*x^22 + 452232*x^21 - 621557*x^20 - 1332326*x^19 + 3689491*x^18 + 75956*x^17 - 11152569*x^16 + 15594526*x^15 + 12008686*x^14 - 58161950*x^13 + 9977818*x^12 + 131342300*x^11 - 70421448*x^10 - 200370404*x^9 + 221182489*x^8 + 64149492*x^7 - 233429156*x^6 + 127683800*x^5 + 51958176*x^4 - 115466320*x^3 + 71910480*x^2 - 21408800*x + 2671600);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 6*x^31 - 19*x^30 + 146*x^29 + 236*x^28 - 1994*x^27 - 1515*x^26 + 17532*x^25 + 248*x^24 - 102760*x^23 + 62939*x^22 + 452232*x^21 - 621557*x^20 - 1332326*x^19 + 3689491*x^18 + 75956*x^17 - 11152569*x^16 + 15594526*x^15 + 12008686*x^14 - 58161950*x^13 + 9977818*x^12 + 131342300*x^11 - 70421448*x^10 - 200370404*x^9 + 221182489*x^8 + 64149492*x^7 - 233429156*x^6 + 127683800*x^5 + 51958176*x^4 - 115466320*x^3 + 71910480*x^2 - 21408800*x + 2671600)
\( x^{32} - 6 x^{31} - 19 x^{30} + 146 x^{29} + 236 x^{28} - 1994 x^{27} - 1515 x^{26} + 17532 x^{25} + \cdots + 2671600 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(7811497343330649377718034300042226077532160000000000000000\)
\(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{16}\cdot 89^{8}\cdot 181^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(64.44\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2^{3/2}3^{1/2}5^{1/2}89^{1/2}181^{1/2}\approx 1390.3524732958906$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(89\), \(181\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{5}a^{22}-\frac{2}{5}a^{21}+\frac{1}{5}a^{20}+\frac{1}{5}a^{19}+\frac{2}{5}a^{18}+\frac{2}{5}a^{16}+\frac{1}{5}a^{15}-\frac{2}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}-\frac{2}{5}a^{11}+\frac{2}{5}a^{10}+\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{25}a^{23}-\frac{2}{25}a^{22}-\frac{4}{25}a^{21}+\frac{6}{25}a^{20}-\frac{8}{25}a^{19}+\frac{2}{5}a^{18}-\frac{8}{25}a^{17}+\frac{1}{25}a^{16}-\frac{7}{25}a^{15}-\frac{7}{25}a^{14}-\frac{9}{25}a^{13}+\frac{8}{25}a^{12}-\frac{8}{25}a^{11}+\frac{2}{25}a^{10}-\frac{1}{25}a^{9}-\frac{2}{5}a^{8}+\frac{8}{25}a^{7}+\frac{11}{25}a^{6}+\frac{7}{25}a^{5}+\frac{12}{25}a^{4}-\frac{11}{25}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{300}a^{24}-\frac{7}{75}a^{22}+\frac{23}{50}a^{21}+\frac{17}{150}a^{20}+\frac{2}{25}a^{19}+\frac{49}{100}a^{18}-\frac{3}{10}a^{17}-\frac{19}{60}a^{16}+\frac{7}{25}a^{15}+\frac{23}{75}a^{14}-\frac{2}{5}a^{13}+\frac{113}{300}a^{12}-\frac{2}{25}a^{11}-\frac{31}{150}a^{10}+\frac{4}{25}a^{9}-\frac{17}{300}a^{8}-\frac{4}{25}a^{7}+\frac{119}{300}a^{6}-\frac{12}{25}a^{5}-\frac{9}{100}a^{4}+\frac{3}{50}a^{3}+\frac{13}{60}a^{2}+\frac{2}{5}a-\frac{1}{6}$, $\frac{1}{300}a^{25}-\frac{1}{75}a^{23}-\frac{1}{10}a^{22}-\frac{61}{150}a^{21}+\frac{4}{25}a^{20}+\frac{9}{20}a^{19}-\frac{3}{10}a^{18}+\frac{13}{300}a^{17}-\frac{11}{25}a^{16}+\frac{26}{75}a^{15}-\frac{4}{25}a^{14}+\frac{137}{300}a^{13}+\frac{4}{25}a^{12}-\frac{7}{150}a^{11}-\frac{12}{25}a^{10}+\frac{19}{300}a^{9}+\frac{11}{25}a^{8}+\frac{11}{300}a^{7}+\frac{1}{5}a^{6}+\frac{7}{100}a^{5}+\frac{11}{50}a^{4}-\frac{139}{300}a^{3}+\frac{1}{5}a^{2}-\frac{11}{30}a$, $\frac{1}{300}a^{26}-\frac{1}{50}a^{23}+\frac{3}{50}a^{22}-\frac{8}{25}a^{21}+\frac{23}{60}a^{20}+\frac{19}{50}a^{19}-\frac{59}{300}a^{18}-\frac{7}{25}a^{17}+\frac{4}{25}a^{16}+\frac{2}{5}a^{15}+\frac{37}{300}a^{14}-\frac{4}{25}a^{13}+\frac{1}{10}a^{12}-\frac{11}{25}a^{11}+\frac{119}{300}a^{10}+\frac{1}{100}a^{8}+\frac{1}{5}a^{7}-\frac{139}{300}a^{6}-\frac{7}{50}a^{5}+\frac{41}{300}a^{4}-\frac{11}{25}a^{3}-\frac{1}{10}a^{2}+\frac{2}{5}a+\frac{1}{3}$, $\frac{1}{300}a^{27}-\frac{1}{50}a^{23}+\frac{2}{25}a^{22}-\frac{41}{300}a^{21}+\frac{19}{50}a^{20}-\frac{83}{300}a^{19}+\frac{23}{50}a^{18}+\frac{1}{50}a^{16}+\frac{49}{300}a^{15}-\frac{9}{25}a^{14}-\frac{9}{50}a^{13}-\frac{1}{50}a^{12}-\frac{13}{300}a^{11}+\frac{1}{5}a^{10}-\frac{7}{20}a^{9}-\frac{7}{50}a^{8}-\frac{19}{300}a^{7}-\frac{6}{25}a^{6}+\frac{149}{300}a^{5}-\frac{17}{50}a^{4}-\frac{13}{50}a^{3}-\frac{1}{2}a^{2}-\frac{1}{15}a$, $\frac{1}{1200}a^{28}-\frac{1}{600}a^{27}+\frac{1}{1200}a^{26}-\frac{1}{600}a^{25}+\frac{7}{600}a^{23}-\frac{71}{1200}a^{22}+\frac{31}{150}a^{21}+\frac{107}{300}a^{20}+\frac{34}{75}a^{19}+\frac{307}{1200}a^{18}+\frac{37}{300}a^{17}-\frac{5}{48}a^{16}-\frac{27}{200}a^{15}+\frac{119}{240}a^{14}+\frac{7}{150}a^{13}+\frac{179}{1200}a^{12}-\frac{13}{40}a^{11}+\frac{91}{600}a^{10}+\frac{281}{600}a^{9}-\frac{59}{600}a^{8}+\frac{31}{300}a^{7}-\frac{59}{300}a^{6}+\frac{11}{300}a^{5}+\frac{1}{240}a^{4}+\frac{43}{150}a^{3}-\frac{13}{60}a^{2}+\frac{7}{15}a+\frac{1}{12}$, $\frac{1}{1200}a^{29}+\frac{1}{1200}a^{27}-\frac{1}{600}a^{24}+\frac{13}{1200}a^{23}+\frac{49}{600}a^{22}+\frac{7}{15}a^{21}-\frac{7}{15}a^{20}+\frac{211}{1200}a^{19}+\frac{47}{200}a^{18}-\frac{61}{240}a^{17}+\frac{11}{60}a^{16}-\frac{173}{1200}a^{15}+\frac{79}{600}a^{14}-\frac{241}{1200}a^{13}+\frac{7}{150}a^{12}+\frac{59}{120}a^{11}+\frac{47}{600}a^{10}+\frac{259}{600}a^{9}-\frac{1}{6}a^{8}+\frac{79}{300}a^{7}+\frac{89}{300}a^{6}-\frac{91}{1200}a^{5}+\frac{19}{200}a^{4}+\frac{17}{150}a^{3}+\frac{7}{15}a^{2}-\frac{13}{60}a-\frac{1}{6}$, $\frac{1}{86\!\cdots\!00}a^{30}-\frac{5820942437981}{43\!\cdots\!00}a^{29}-\frac{6638752930589}{86\!\cdots\!00}a^{28}-\frac{17370695843337}{14\!\cdots\!00}a^{27}+\frac{51032783436887}{43\!\cdots\!00}a^{26}-\frac{26628195060203}{43\!\cdots\!00}a^{25}+\frac{12416357274011}{28\!\cdots\!00}a^{24}+\frac{7344780163191}{72\!\cdots\!00}a^{23}+\frac{13\!\cdots\!17}{14\!\cdots\!00}a^{22}-\frac{710437815436589}{21\!\cdots\!80}a^{21}-\frac{33\!\cdots\!09}{86\!\cdots\!00}a^{20}-\frac{780342777717007}{72\!\cdots\!00}a^{19}+\frac{21\!\cdots\!17}{86\!\cdots\!00}a^{18}-\frac{22\!\cdots\!21}{43\!\cdots\!00}a^{17}-\frac{24\!\cdots\!01}{57\!\cdots\!80}a^{16}-\frac{29\!\cdots\!31}{21\!\cdots\!00}a^{15}-\frac{22\!\cdots\!91}{86\!\cdots\!00}a^{14}-\frac{11\!\cdots\!87}{43\!\cdots\!00}a^{13}-\frac{218029292783111}{72\!\cdots\!00}a^{12}+\frac{42\!\cdots\!67}{43\!\cdots\!00}a^{11}+\frac{62\!\cdots\!31}{14\!\cdots\!00}a^{10}-\frac{567754072622903}{36\!\cdots\!00}a^{9}+\frac{10\!\cdots\!39}{72\!\cdots\!00}a^{8}-\frac{20\!\cdots\!21}{21\!\cdots\!00}a^{7}+\frac{13\!\cdots\!13}{86\!\cdots\!00}a^{6}-\frac{267399683667301}{36\!\cdots\!00}a^{5}-\frac{68\!\cdots\!53}{14\!\cdots\!00}a^{4}-\frac{141183167572397}{450505198195100}a^{3}+\frac{277224166149155}{864969980534592}a^{2}+\frac{51699598100017}{270303118917060}a-\frac{27035854512299}{144161663422432}$, $\frac{1}{32\!\cdots\!00}a^{31}-\frac{83\!\cdots\!71}{16\!\cdots\!00}a^{30}-\frac{40\!\cdots\!59}{10\!\cdots\!00}a^{29}-\frac{22\!\cdots\!79}{54\!\cdots\!00}a^{28}-\frac{47\!\cdots\!61}{32\!\cdots\!20}a^{27}-\frac{10\!\cdots\!53}{88\!\cdots\!00}a^{26}+\frac{39\!\cdots\!93}{32\!\cdots\!00}a^{25}+\frac{25\!\cdots\!29}{20\!\cdots\!00}a^{24}+\frac{50\!\cdots\!73}{54\!\cdots\!00}a^{23}-\frac{66\!\cdots\!57}{10\!\cdots\!00}a^{22}-\frac{68\!\cdots\!97}{32\!\cdots\!00}a^{21}-\frac{11\!\cdots\!47}{67\!\cdots\!00}a^{20}-\frac{16\!\cdots\!11}{32\!\cdots\!00}a^{19}-\frac{30\!\cdots\!15}{64\!\cdots\!84}a^{18}-\frac{42\!\cdots\!99}{32\!\cdots\!00}a^{17}-\frac{67\!\cdots\!79}{33\!\cdots\!00}a^{16}-\frac{25\!\cdots\!59}{32\!\cdots\!00}a^{15}+\frac{61\!\cdots\!71}{16\!\cdots\!00}a^{14}-\frac{14\!\cdots\!43}{81\!\cdots\!00}a^{13}+\frac{21\!\cdots\!53}{10\!\cdots\!40}a^{12}+\frac{66\!\cdots\!61}{16\!\cdots\!00}a^{11}+\frac{10\!\cdots\!25}{81\!\cdots\!48}a^{10}-\frac{19\!\cdots\!17}{10\!\cdots\!64}a^{9}-\frac{12\!\cdots\!57}{81\!\cdots\!00}a^{8}-\frac{55\!\cdots\!19}{32\!\cdots\!00}a^{7}+\frac{71\!\cdots\!63}{27\!\cdots\!00}a^{6}-\frac{19\!\cdots\!57}{54\!\cdots\!00}a^{5}+\frac{43\!\cdots\!59}{40\!\cdots\!00}a^{4}-\frac{14\!\cdots\!61}{81\!\cdots\!00}a^{3}+\frac{52\!\cdots\!81}{10\!\cdots\!16}a^{2}-\frac{20\!\cdots\!69}{81\!\cdots\!80}a-\frac{32\!\cdots\!45}{13\!\cdots\!08}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}\times C_{312}$, which has order $1248$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $1248$ (assuming GRH)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{474453096513316889160764798010772063847177627973586095268211}{151008976782753205773088548491452774160953082051778855019763289600} a^{31} + \frac{1287770220943942841705486787499871691251258127489461025454213}{75504488391376602886544274245726387080476541025889427509881644800} a^{30} + \frac{10482021725117286110517718582942638809418168576448048464563039}{151008976782753205773088548491452774160953082051778855019763289600} a^{29} - \frac{31621262224729027924923347969954842897139181348170897294167493}{75504488391376602886544274245726387080476541025889427509881644800} a^{28} - \frac{74028979747284123830088283136347666805234523138467035665661073}{75504488391376602886544274245726387080476541025889427509881644800} a^{27} + \frac{430245027200906136756342505459292191613415126433688306266163317}{75504488391376602886544274245726387080476541025889427509881644800} a^{26} + \frac{242034775076507505034177146778271415370842974742814685367915313}{30201795356550641154617709698290554832190616410355771003952657920} a^{25} - \frac{79298223741986699873001331323153457830255958200162633622030947}{1573010174820345893469672380119299730843261271372696406455867600} a^{24} - \frac{2234705616743396463879255486657979906914821772405341569551395729}{75504488391376602886544274245726387080476541025889427509881644800} a^{23} + \frac{95985231117272918767944121116201247700271838863538091187840029}{314602034964069178693934476023859946168652254274539281291173520} a^{22} - \frac{3466865453960095865940472974597079712656827932637517896517102909}{151008976782753205773088548491452774160953082051778855019763289600} a^{21} - \frac{6743074622728585039155490354790254800886331827366327534304362241}{4719030524461037680409017140357899192529783814118089219367602800} a^{20} + \frac{56956623987561470566256551440707776339597284507049492216923602819}{50336325594251068591029516163817591386984360683926285006587763200} a^{19} + \frac{363219622311415864904388982809823998770649124832993868914621482073}{75504488391376602886544274245726387080476541025889427509881644800} a^{18} - \frac{443324239676372803175277059447813260896290045520654713649587001497}{50336325594251068591029516163817591386984360683926285006587763200} a^{17} - \frac{16380365703558287084895829332626303550899975452957252802694617777}{3146020349640691786939344760238599461686522542745392812911735200} a^{16} + \frac{1604468441984370860915115931124840715980395292780020885579593133803}{50336325594251068591029516163817591386984360683926285006587763200} a^{15} - \frac{2325730819391701128158259677427321672896433607893976433866996682693}{75504488391376602886544274245726387080476541025889427509881644800} a^{14} - \frac{689284829143326886394425574129465954026247156154043000095957702133}{12584081398562767147757379040954397846746090170981571251646940800} a^{13} + \frac{151780659791576833521278210895761862814417902029708006078282729753}{1006726511885021371820590323276351827739687213678525700131755264} a^{12} + \frac{4092719069668285964984671354580444864720920268924736482633726879621}{75504488391376602886544274245726387080476541025889427509881644800} a^{11} - \frac{238321389146031587138880610110787540408131907531261802066292620283}{629204069928138357387868952047719892337304508549078562582347040} a^{10} + \frac{159945441721144267890518525040823046354649332568651630058454868037}{37752244195688301443272137122863193540238270512944713754940822400} a^{9} + \frac{7852325936439545657990753760811237502165201776503115888203788662397}{12584081398562767147757379040954397846746090170981571251646940800} a^{8} - \frac{50644716345486541128734922088807477908681555676957152434812507876739}{151008976782753205773088548491452774160953082051778855019763289600} a^{7} - \frac{14300551625966205426425811713426742438745965879629338030279807571783}{37752244195688301443272137122863193540238270512944713754940822400} a^{6} + \frac{38475862001156357958524034978677356469045029370982040456075390316473}{75504488391376602886544274245726387080476541025889427509881644800} a^{5} - \frac{150875598463597763716486352229211162789303388870238171030077851519}{1258408139856276714775737904095439784674609017098157125164694080} a^{4} - \frac{8432530058491071554000260496583155121723436355037030546447852202769}{37752244195688301443272137122863193540238270512944713754940822400} a^{3} + \frac{1768646137081992573308033814236196103232266152894697664145040351}{7520367369659024191886879904952827398453838747598548556761120} a^{2} - \frac{120650510329753256474971963118325184519513959189081541480383541587}{1258408139856276714775737904095439784674609017098157125164694080} a + \frac{1022583735726069268370741351267939756835794478711850934430349719}{62920406992813835738786895204771989233730450854907856258234704} \)
(order $6$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{11\!\cdots\!37}{70\!\cdots\!00}a^{31}-\frac{29\!\cdots\!91}{35\!\cdots\!00}a^{30}-\frac{91\!\cdots\!63}{23\!\cdots\!00}a^{29}+\frac{71\!\cdots\!39}{35\!\cdots\!00}a^{28}+\frac{20\!\cdots\!03}{35\!\cdots\!00}a^{27}-\frac{32\!\cdots\!37}{11\!\cdots\!00}a^{26}-\frac{35\!\cdots\!27}{70\!\cdots\!00}a^{25}+\frac{74\!\cdots\!47}{31\!\cdots\!00}a^{24}+\frac{53\!\cdots\!37}{23\!\cdots\!60}a^{23}-\frac{21\!\cdots\!79}{14\!\cdots\!00}a^{22}-\frac{32\!\cdots\!43}{94\!\cdots\!04}a^{21}+\frac{62\!\cdots\!71}{88\!\cdots\!00}a^{20}-\frac{84\!\cdots\!33}{23\!\cdots\!00}a^{19}-\frac{88\!\cdots\!71}{35\!\cdots\!00}a^{18}+\frac{52\!\cdots\!17}{14\!\cdots\!60}a^{17}+\frac{31\!\cdots\!43}{88\!\cdots\!00}a^{16}-\frac{35\!\cdots\!57}{23\!\cdots\!00}a^{15}+\frac{41\!\cdots\!51}{35\!\cdots\!00}a^{14}+\frac{54\!\cdots\!49}{17\!\cdots\!00}a^{13}-\frac{23\!\cdots\!69}{35\!\cdots\!00}a^{12}-\frac{16\!\cdots\!99}{35\!\cdots\!00}a^{11}+\frac{95\!\cdots\!23}{55\!\cdots\!00}a^{10}+\frac{26\!\cdots\!67}{58\!\cdots\!00}a^{9}-\frac{16\!\cdots\!47}{58\!\cdots\!00}a^{8}+\frac{23\!\cdots\!19}{23\!\cdots\!00}a^{7}+\frac{34\!\cdots\!59}{17\!\cdots\!00}a^{6}-\frac{73\!\cdots\!71}{35\!\cdots\!00}a^{5}+\frac{15\!\cdots\!61}{88\!\cdots\!00}a^{4}+\frac{38\!\cdots\!91}{35\!\cdots\!40}a^{3}-\frac{80\!\cdots\!29}{88\!\cdots\!60}a^{2}+\frac{36\!\cdots\!21}{11\!\cdots\!88}a-\frac{80\!\cdots\!43}{29\!\cdots\!72}$, $\frac{55\!\cdots\!23}{81\!\cdots\!00}a^{31}+\frac{12\!\cdots\!11}{13\!\cdots\!00}a^{30}-\frac{72\!\cdots\!43}{16\!\cdots\!60}a^{29}-\frac{96\!\cdots\!53}{40\!\cdots\!00}a^{28}+\frac{14\!\cdots\!83}{13\!\cdots\!00}a^{27}+\frac{82\!\cdots\!61}{22\!\cdots\!00}a^{26}-\frac{11\!\cdots\!97}{81\!\cdots\!00}a^{25}-\frac{23\!\cdots\!23}{67\!\cdots\!00}a^{24}+\frac{47\!\cdots\!81}{40\!\cdots\!00}a^{23}+\frac{45\!\cdots\!27}{25\!\cdots\!90}a^{22}-\frac{77\!\cdots\!77}{10\!\cdots\!64}a^{21}-\frac{10\!\cdots\!99}{20\!\cdots\!00}a^{20}+\frac{58\!\cdots\!63}{16\!\cdots\!60}a^{19}-\frac{44\!\cdots\!33}{13\!\cdots\!00}a^{18}-\frac{72\!\cdots\!11}{54\!\cdots\!20}a^{17}+\frac{28\!\cdots\!09}{20\!\cdots\!00}a^{16}+\frac{18\!\cdots\!43}{81\!\cdots\!00}a^{15}-\frac{91\!\cdots\!59}{13\!\cdots\!00}a^{14}+\frac{26\!\cdots\!43}{67\!\cdots\!00}a^{13}+\frac{69\!\cdots\!89}{40\!\cdots\!00}a^{12}-\frac{38\!\cdots\!43}{13\!\cdots\!00}a^{11}-\frac{12\!\cdots\!43}{33\!\cdots\!00}a^{10}+\frac{14\!\cdots\!79}{20\!\cdots\!00}a^{9}+\frac{43\!\cdots\!17}{81\!\cdots\!48}a^{8}-\frac{28\!\cdots\!19}{27\!\cdots\!00}a^{7}+\frac{12\!\cdots\!83}{10\!\cdots\!00}a^{6}+\frac{49\!\cdots\!99}{81\!\cdots\!80}a^{5}-\frac{28\!\cdots\!39}{33\!\cdots\!00}a^{4}+\frac{13\!\cdots\!41}{81\!\cdots\!48}a^{3}+\frac{10\!\cdots\!33}{20\!\cdots\!80}a^{2}-\frac{17\!\cdots\!31}{40\!\cdots\!24}a+\frac{48\!\cdots\!89}{33\!\cdots\!52}$, $\frac{20\!\cdots\!41}{32\!\cdots\!00}a^{31}-\frac{41\!\cdots\!27}{16\!\cdots\!00}a^{30}-\frac{56\!\cdots\!73}{32\!\cdots\!00}a^{29}+\frac{33\!\cdots\!69}{54\!\cdots\!00}a^{28}+\frac{61\!\cdots\!53}{21\!\cdots\!28}a^{27}-\frac{41\!\cdots\!43}{53\!\cdots\!20}a^{26}-\frac{89\!\cdots\!43}{32\!\cdots\!00}a^{25}+\frac{27\!\cdots\!27}{40\!\cdots\!00}a^{24}+\frac{52\!\cdots\!63}{32\!\cdots\!20}a^{23}-\frac{87\!\cdots\!47}{20\!\cdots\!00}a^{22}-\frac{19\!\cdots\!13}{32\!\cdots\!00}a^{21}+\frac{18\!\cdots\!69}{81\!\cdots\!80}a^{20}+\frac{35\!\cdots\!17}{32\!\cdots\!00}a^{19}-\frac{51\!\cdots\!09}{54\!\cdots\!00}a^{18}+\frac{14\!\cdots\!69}{32\!\cdots\!00}a^{17}+\frac{30\!\cdots\!63}{13\!\cdots\!00}a^{16}-\frac{12\!\cdots\!79}{32\!\cdots\!00}a^{15}-\frac{41\!\cdots\!37}{16\!\cdots\!00}a^{14}+\frac{10\!\cdots\!81}{81\!\cdots\!00}a^{13}-\frac{69\!\cdots\!19}{54\!\cdots\!00}a^{12}-\frac{58\!\cdots\!67}{16\!\cdots\!00}a^{11}+\frac{26\!\cdots\!91}{67\!\cdots\!40}a^{10}+\frac{22\!\cdots\!37}{32\!\cdots\!92}a^{9}-\frac{10\!\cdots\!93}{16\!\cdots\!60}a^{8}-\frac{16\!\cdots\!59}{32\!\cdots\!00}a^{7}+\frac{52\!\cdots\!43}{81\!\cdots\!00}a^{6}-\frac{13\!\cdots\!67}{32\!\cdots\!20}a^{5}-\frac{25\!\cdots\!23}{81\!\cdots\!80}a^{4}+\frac{20\!\cdots\!91}{81\!\cdots\!00}a^{3}-\frac{49\!\cdots\!53}{10\!\cdots\!16}a^{2}-\frac{15\!\cdots\!07}{27\!\cdots\!60}a+\frac{53\!\cdots\!03}{40\!\cdots\!24}$, $\frac{39\!\cdots\!57}{64\!\cdots\!40}a^{31}-\frac{81\!\cdots\!53}{16\!\cdots\!60}a^{30}-\frac{34\!\cdots\!53}{64\!\cdots\!40}a^{29}+\frac{67\!\cdots\!01}{54\!\cdots\!20}a^{28}-\frac{29\!\cdots\!79}{32\!\cdots\!20}a^{27}-\frac{94\!\cdots\!49}{53\!\cdots\!20}a^{26}+\frac{14\!\cdots\!25}{12\!\cdots\!68}a^{25}+\frac{51\!\cdots\!17}{32\!\cdots\!20}a^{24}-\frac{19\!\cdots\!79}{10\!\cdots\!40}a^{23}-\frac{97\!\cdots\!03}{10\!\cdots\!64}a^{22}+\frac{97\!\cdots\!83}{64\!\cdots\!40}a^{21}+\frac{11\!\cdots\!27}{32\!\cdots\!20}a^{20}-\frac{20\!\cdots\!33}{21\!\cdots\!80}a^{19}-\frac{12\!\cdots\!73}{16\!\cdots\!60}a^{18}+\frac{28\!\cdots\!77}{64\!\cdots\!40}a^{17}-\frac{67\!\cdots\!39}{32\!\cdots\!20}a^{16}-\frac{71\!\cdots\!83}{64\!\cdots\!40}a^{15}+\frac{20\!\cdots\!63}{10\!\cdots\!60}a^{14}+\frac{23\!\cdots\!09}{81\!\cdots\!80}a^{13}-\frac{41\!\cdots\!61}{64\!\cdots\!84}a^{12}+\frac{53\!\cdots\!21}{10\!\cdots\!40}a^{11}+\frac{45\!\cdots\!13}{32\!\cdots\!92}a^{10}-\frac{89\!\cdots\!13}{54\!\cdots\!20}a^{9}-\frac{11\!\cdots\!29}{54\!\cdots\!20}a^{8}+\frac{22\!\cdots\!93}{64\!\cdots\!40}a^{7}+\frac{19\!\cdots\!47}{32\!\cdots\!20}a^{6}-\frac{10\!\cdots\!71}{32\!\cdots\!20}a^{5}+\frac{18\!\cdots\!25}{10\!\cdots\!64}a^{4}+\frac{11\!\cdots\!43}{16\!\cdots\!60}a^{3}-\frac{97\!\cdots\!89}{64\!\cdots\!96}a^{2}+\frac{13\!\cdots\!67}{16\!\cdots\!96}a-\frac{12\!\cdots\!59}{81\!\cdots\!48}$, $\frac{51\!\cdots\!69}{16\!\cdots\!60}a^{31}-\frac{29\!\cdots\!51}{16\!\cdots\!00}a^{30}-\frac{26\!\cdots\!31}{40\!\cdots\!00}a^{29}+\frac{73\!\cdots\!27}{16\!\cdots\!00}a^{28}+\frac{68\!\cdots\!39}{81\!\cdots\!00}a^{27}-\frac{82\!\cdots\!23}{13\!\cdots\!00}a^{26}-\frac{24\!\cdots\!99}{40\!\cdots\!00}a^{25}+\frac{29\!\cdots\!39}{54\!\cdots\!00}a^{24}+\frac{49\!\cdots\!51}{40\!\cdots\!24}a^{23}-\frac{89\!\cdots\!79}{27\!\cdots\!00}a^{22}+\frac{10\!\cdots\!07}{81\!\cdots\!00}a^{21}+\frac{24\!\cdots\!51}{16\!\cdots\!00}a^{20}-\frac{45\!\cdots\!03}{27\!\cdots\!00}a^{19}-\frac{76\!\cdots\!43}{16\!\cdots\!00}a^{18}+\frac{73\!\cdots\!83}{67\!\cdots\!00}a^{17}+\frac{15\!\cdots\!59}{54\!\cdots\!00}a^{16}-\frac{96\!\cdots\!03}{27\!\cdots\!00}a^{15}+\frac{68\!\cdots\!69}{16\!\cdots\!00}a^{14}+\frac{13\!\cdots\!91}{27\!\cdots\!00}a^{13}-\frac{12\!\cdots\!97}{67\!\cdots\!00}a^{12}-\frac{66\!\cdots\!51}{81\!\cdots\!00}a^{11}+\frac{11\!\cdots\!23}{27\!\cdots\!00}a^{10}-\frac{67\!\cdots\!31}{50\!\cdots\!00}a^{9}-\frac{94\!\cdots\!03}{13\!\cdots\!00}a^{8}+\frac{45\!\cdots\!07}{81\!\cdots\!00}a^{7}+\frac{60\!\cdots\!21}{16\!\cdots\!00}a^{6}-\frac{55\!\cdots\!81}{81\!\cdots\!80}a^{5}+\frac{12\!\cdots\!07}{54\!\cdots\!20}a^{4}+\frac{49\!\cdots\!59}{20\!\cdots\!00}a^{3}-\frac{10\!\cdots\!11}{32\!\cdots\!80}a^{2}+\frac{99\!\cdots\!77}{67\!\cdots\!40}a-\frac{69\!\cdots\!07}{27\!\cdots\!16}$, $\frac{24\!\cdots\!37}{24\!\cdots\!00}a^{31}-\frac{17\!\cdots\!31}{24\!\cdots\!20}a^{30}-\frac{10\!\cdots\!83}{74\!\cdots\!00}a^{29}+\frac{13\!\cdots\!89}{74\!\cdots\!60}a^{28}+\frac{26\!\cdots\!19}{24\!\cdots\!20}a^{27}-\frac{15\!\cdots\!21}{60\!\cdots\!00}a^{26}+\frac{11\!\cdots\!47}{74\!\cdots\!00}a^{25}+\frac{70\!\cdots\!09}{30\!\cdots\!00}a^{24}-\frac{17\!\cdots\!13}{12\!\cdots\!00}a^{23}-\frac{62\!\cdots\!79}{46\!\cdots\!00}a^{22}+\frac{11\!\cdots\!53}{74\!\cdots\!00}a^{21}+\frac{52\!\cdots\!53}{92\!\cdots\!00}a^{20}-\frac{27\!\cdots\!51}{24\!\cdots\!00}a^{19}-\frac{55\!\cdots\!33}{37\!\cdots\!00}a^{18}+\frac{41\!\cdots\!07}{74\!\cdots\!00}a^{17}-\frac{94\!\cdots\!59}{92\!\cdots\!00}a^{16}-\frac{23\!\cdots\!89}{14\!\cdots\!20}a^{15}+\frac{88\!\cdots\!01}{37\!\cdots\!00}a^{14}+\frac{44\!\cdots\!59}{37\!\cdots\!80}a^{13}-\frac{63\!\cdots\!63}{74\!\cdots\!60}a^{12}+\frac{14\!\cdots\!59}{37\!\cdots\!00}a^{11}+\frac{11\!\cdots\!27}{58\!\cdots\!50}a^{10}-\frac{96\!\cdots\!87}{61\!\cdots\!00}a^{9}-\frac{61\!\cdots\!11}{18\!\cdots\!00}a^{8}+\frac{10\!\cdots\!15}{29\!\cdots\!44}a^{7}+\frac{29\!\cdots\!89}{18\!\cdots\!00}a^{6}-\frac{45\!\cdots\!87}{12\!\cdots\!00}a^{5}+\frac{13\!\cdots\!71}{92\!\cdots\!00}a^{4}+\frac{71\!\cdots\!07}{61\!\cdots\!00}a^{3}-\frac{63\!\cdots\!29}{37\!\cdots\!20}a^{2}+\frac{14\!\cdots\!89}{18\!\cdots\!40}a-\frac{12\!\cdots\!95}{92\!\cdots\!92}$, $\frac{23\!\cdots\!23}{54\!\cdots\!00}a^{31}-\frac{18\!\cdots\!41}{81\!\cdots\!00}a^{30}-\frac{50\!\cdots\!43}{54\!\cdots\!00}a^{29}+\frac{93\!\cdots\!49}{16\!\cdots\!60}a^{28}+\frac{10\!\cdots\!87}{81\!\cdots\!00}a^{27}-\frac{34\!\cdots\!49}{44\!\cdots\!00}a^{26}-\frac{34\!\cdots\!67}{32\!\cdots\!20}a^{25}+\frac{18\!\cdots\!99}{27\!\cdots\!60}a^{24}+\frac{31\!\cdots\!43}{81\!\cdots\!00}a^{23}-\frac{28\!\cdots\!99}{67\!\cdots\!00}a^{22}+\frac{21\!\cdots\!41}{54\!\cdots\!00}a^{21}+\frac{53\!\cdots\!29}{27\!\cdots\!60}a^{20}-\frac{25\!\cdots\!71}{16\!\cdots\!00}a^{19}-\frac{53\!\cdots\!73}{81\!\cdots\!00}a^{18}+\frac{19\!\cdots\!61}{16\!\cdots\!00}a^{17}+\frac{97\!\cdots\!71}{13\!\cdots\!00}a^{16}-\frac{71\!\cdots\!63}{16\!\cdots\!00}a^{15}+\frac{11\!\cdots\!99}{27\!\cdots\!00}a^{14}+\frac{30\!\cdots\!43}{40\!\cdots\!00}a^{13}-\frac{56\!\cdots\!83}{27\!\cdots\!00}a^{12}-\frac{59\!\cdots\!03}{81\!\cdots\!00}a^{11}+\frac{10\!\cdots\!39}{20\!\cdots\!00}a^{10}-\frac{27\!\cdots\!07}{40\!\cdots\!00}a^{9}-\frac{23\!\cdots\!51}{27\!\cdots\!60}a^{8}+\frac{24\!\cdots\!51}{54\!\cdots\!00}a^{7}+\frac{37\!\cdots\!37}{67\!\cdots\!00}a^{6}-\frac{55\!\cdots\!83}{81\!\cdots\!00}a^{5}+\frac{93\!\cdots\!54}{63\!\cdots\!75}a^{4}+\frac{12\!\cdots\!63}{40\!\cdots\!00}a^{3}-\frac{64\!\cdots\!69}{20\!\cdots\!80}a^{2}+\frac{17\!\cdots\!69}{13\!\cdots\!80}a-\frac{10\!\cdots\!75}{50\!\cdots\!78}$, $\frac{42\!\cdots\!37}{29\!\cdots\!60}a^{31}+\frac{19\!\cdots\!59}{99\!\cdots\!72}a^{30}-\frac{11\!\cdots\!97}{14\!\cdots\!00}a^{29}-\frac{30\!\cdots\!31}{49\!\cdots\!60}a^{28}+\frac{11\!\cdots\!07}{74\!\cdots\!00}a^{27}+\frac{33\!\cdots\!37}{24\!\cdots\!80}a^{26}-\frac{53\!\cdots\!47}{29\!\cdots\!60}a^{25}-\frac{88\!\cdots\!27}{62\!\cdots\!00}a^{24}+\frac{10\!\cdots\!91}{74\!\cdots\!00}a^{23}+\frac{19\!\cdots\!37}{31\!\cdots\!00}a^{22}-\frac{38\!\cdots\!71}{49\!\cdots\!00}a^{21}+\frac{15\!\cdots\!61}{37\!\cdots\!20}a^{20}+\frac{10\!\cdots\!29}{29\!\cdots\!60}a^{19}-\frac{67\!\cdots\!67}{29\!\cdots\!16}a^{18}-\frac{17\!\cdots\!67}{14\!\cdots\!00}a^{17}+\frac{20\!\cdots\!87}{12\!\cdots\!40}a^{16}+\frac{23\!\cdots\!09}{14\!\cdots\!00}a^{15}-\frac{45\!\cdots\!33}{74\!\cdots\!00}a^{14}+\frac{19\!\cdots\!13}{37\!\cdots\!00}a^{13}+\frac{33\!\cdots\!73}{24\!\cdots\!00}a^{12}-\frac{24\!\cdots\!31}{74\!\cdots\!00}a^{11}-\frac{12\!\cdots\!73}{46\!\cdots\!00}a^{10}+\frac{31\!\cdots\!01}{37\!\cdots\!00}a^{9}+\frac{56\!\cdots\!61}{12\!\cdots\!00}a^{8}-\frac{63\!\cdots\!01}{49\!\cdots\!00}a^{7}-\frac{26\!\cdots\!57}{37\!\cdots\!00}a^{6}+\frac{24\!\cdots\!07}{24\!\cdots\!00}a^{5}-\frac{90\!\cdots\!11}{18\!\cdots\!00}a^{4}-\frac{51\!\cdots\!99}{24\!\cdots\!80}a^{3}+\frac{97\!\cdots\!47}{24\!\cdots\!20}a^{2}-\frac{13\!\cdots\!53}{74\!\cdots\!04}a+\frac{46\!\cdots\!71}{18\!\cdots\!76}$, $\frac{32\!\cdots\!23}{64\!\cdots\!40}a^{31}-\frac{32\!\cdots\!81}{16\!\cdots\!00}a^{30}-\frac{92\!\cdots\!71}{64\!\cdots\!40}a^{29}+\frac{79\!\cdots\!49}{16\!\cdots\!00}a^{28}+\frac{37\!\cdots\!21}{16\!\cdots\!00}a^{27}-\frac{16\!\cdots\!41}{26\!\cdots\!00}a^{26}-\frac{15\!\cdots\!73}{64\!\cdots\!40}a^{25}+\frac{37\!\cdots\!63}{67\!\cdots\!00}a^{24}+\frac{75\!\cdots\!27}{54\!\cdots\!00}a^{23}-\frac{36\!\cdots\!81}{10\!\cdots\!00}a^{22}-\frac{11\!\cdots\!21}{21\!\cdots\!80}a^{21}+\frac{12\!\cdots\!17}{67\!\cdots\!00}a^{20}+\frac{35\!\cdots\!31}{32\!\cdots\!00}a^{19}-\frac{43\!\cdots\!31}{54\!\cdots\!00}a^{18}+\frac{19\!\cdots\!27}{64\!\cdots\!40}a^{17}+\frac{51\!\cdots\!09}{25\!\cdots\!00}a^{16}-\frac{33\!\cdots\!11}{10\!\cdots\!00}a^{15}-\frac{26\!\cdots\!27}{32\!\cdots\!20}a^{14}+\frac{18\!\cdots\!39}{16\!\cdots\!60}a^{13}-\frac{15\!\cdots\!99}{16\!\cdots\!00}a^{12}-\frac{20\!\cdots\!33}{64\!\cdots\!84}a^{11}+\frac{65\!\cdots\!51}{20\!\cdots\!00}a^{10}+\frac{17\!\cdots\!61}{27\!\cdots\!00}a^{9}-\frac{91\!\cdots\!79}{16\!\cdots\!60}a^{8}-\frac{61\!\cdots\!31}{10\!\cdots\!00}a^{7}+\frac{16\!\cdots\!69}{27\!\cdots\!00}a^{6}+\frac{29\!\cdots\!89}{54\!\cdots\!00}a^{5}-\frac{14\!\cdots\!91}{40\!\cdots\!00}a^{4}+\frac{18\!\cdots\!53}{81\!\cdots\!00}a^{3}+\frac{47\!\cdots\!55}{10\!\cdots\!16}a^{2}-\frac{64\!\cdots\!63}{81\!\cdots\!80}a+\frac{75\!\cdots\!35}{40\!\cdots\!24}$, $\frac{19\!\cdots\!57}{10\!\cdots\!00}a^{31}-\frac{15\!\cdots\!79}{16\!\cdots\!00}a^{30}-\frac{28\!\cdots\!71}{64\!\cdots\!40}a^{29}+\frac{12\!\cdots\!89}{54\!\cdots\!00}a^{28}+\frac{69\!\cdots\!91}{10\!\cdots\!40}a^{27}-\frac{16\!\cdots\!33}{53\!\cdots\!20}a^{26}-\frac{60\!\cdots\!63}{10\!\cdots\!00}a^{25}+\frac{73\!\cdots\!51}{27\!\cdots\!00}a^{24}+\frac{40\!\cdots\!57}{16\!\cdots\!00}a^{23}-\frac{67\!\cdots\!89}{40\!\cdots\!00}a^{22}-\frac{22\!\cdots\!39}{64\!\cdots\!40}a^{21}+\frac{65\!\cdots\!71}{81\!\cdots\!00}a^{20}-\frac{45\!\cdots\!27}{10\!\cdots\!00}a^{19}-\frac{15\!\cdots\!61}{54\!\cdots\!00}a^{18}+\frac{27\!\cdots\!27}{64\!\cdots\!40}a^{17}+\frac{32\!\cdots\!13}{81\!\cdots\!00}a^{16}-\frac{18\!\cdots\!39}{10\!\cdots\!00}a^{15}+\frac{21\!\cdots\!03}{16\!\cdots\!00}a^{14}+\frac{27\!\cdots\!21}{81\!\cdots\!00}a^{13}-\frac{12\!\cdots\!51}{16\!\cdots\!00}a^{12}-\frac{27\!\cdots\!03}{54\!\cdots\!00}a^{11}+\frac{80\!\cdots\!63}{40\!\cdots\!00}a^{10}+\frac{40\!\cdots\!03}{81\!\cdots\!00}a^{9}-\frac{26\!\cdots\!47}{81\!\cdots\!00}a^{8}+\frac{34\!\cdots\!27}{32\!\cdots\!00}a^{7}+\frac{18\!\cdots\!39}{81\!\cdots\!48}a^{6}-\frac{12\!\cdots\!71}{54\!\cdots\!00}a^{5}+\frac{38\!\cdots\!01}{20\!\cdots\!00}a^{4}+\frac{98\!\cdots\!01}{81\!\cdots\!00}a^{3}-\frac{81\!\cdots\!67}{80\!\cdots\!20}a^{2}+\frac{94\!\cdots\!63}{27\!\cdots\!60}a-\frac{97\!\cdots\!85}{20\!\cdots\!12}$, $\frac{31\!\cdots\!29}{21\!\cdots\!80}a^{31}-\frac{55\!\cdots\!69}{67\!\cdots\!00}a^{30}-\frac{98\!\cdots\!63}{32\!\cdots\!00}a^{29}+\frac{27\!\cdots\!73}{13\!\cdots\!00}a^{28}+\frac{22\!\cdots\!97}{54\!\cdots\!00}a^{27}-\frac{73\!\cdots\!31}{26\!\cdots\!00}a^{26}-\frac{99\!\cdots\!17}{32\!\cdots\!00}a^{25}+\frac{93\!\cdots\!09}{38\!\cdots\!80}a^{24}+\frac{43\!\cdots\!31}{54\!\cdots\!00}a^{23}-\frac{39\!\cdots\!47}{27\!\cdots\!00}a^{22}+\frac{14\!\cdots\!81}{32\!\cdots\!00}a^{21}+\frac{10\!\cdots\!59}{16\!\cdots\!00}a^{20}-\frac{75\!\cdots\!23}{10\!\cdots\!00}a^{19}-\frac{10\!\cdots\!37}{50\!\cdots\!80}a^{18}+\frac{15\!\cdots\!87}{32\!\cdots\!00}a^{17}+\frac{82\!\cdots\!07}{54\!\cdots\!00}a^{16}-\frac{20\!\cdots\!01}{12\!\cdots\!68}a^{15}+\frac{14\!\cdots\!59}{81\!\cdots\!00}a^{14}+\frac{92\!\cdots\!01}{40\!\cdots\!40}a^{13}-\frac{41\!\cdots\!89}{54\!\cdots\!00}a^{12}-\frac{14\!\cdots\!99}{16\!\cdots\!00}a^{11}+\frac{15\!\cdots\!37}{81\!\cdots\!00}a^{10}-\frac{12\!\cdots\!79}{27\!\cdots\!00}a^{9}-\frac{81\!\cdots\!07}{27\!\cdots\!00}a^{8}+\frac{74\!\cdots\!11}{32\!\cdots\!00}a^{7}+\frac{25\!\cdots\!79}{16\!\cdots\!00}a^{6}-\frac{15\!\cdots\!63}{54\!\cdots\!00}a^{5}+\frac{81\!\cdots\!99}{81\!\cdots\!00}a^{4}+\frac{28\!\cdots\!79}{27\!\cdots\!00}a^{3}-\frac{14\!\cdots\!79}{10\!\cdots\!60}a^{2}+\frac{52\!\cdots\!33}{81\!\cdots\!80}a-\frac{99\!\cdots\!59}{81\!\cdots\!48}$, $\frac{20\!\cdots\!77}{20\!\cdots\!00}a^{31}-\frac{74\!\cdots\!43}{13\!\cdots\!00}a^{30}-\frac{77\!\cdots\!43}{33\!\cdots\!00}a^{29}+\frac{55\!\cdots\!77}{40\!\cdots\!00}a^{28}+\frac{21\!\cdots\!33}{67\!\cdots\!00}a^{27}-\frac{20\!\cdots\!93}{11\!\cdots\!00}a^{26}-\frac{13\!\cdots\!87}{50\!\cdots\!00}a^{25}+\frac{66\!\cdots\!91}{40\!\cdots\!00}a^{24}+\frac{35\!\cdots\!31}{33\!\cdots\!00}a^{23}-\frac{67\!\cdots\!83}{67\!\cdots\!00}a^{22}+\frac{21\!\cdots\!31}{20\!\cdots\!00}a^{21}+\frac{19\!\cdots\!57}{40\!\cdots\!00}a^{20}-\frac{68\!\cdots\!59}{20\!\cdots\!00}a^{19}-\frac{21\!\cdots\!19}{13\!\cdots\!00}a^{18}+\frac{14\!\cdots\!53}{50\!\cdots\!00}a^{17}+\frac{79\!\cdots\!47}{40\!\cdots\!00}a^{16}-\frac{20\!\cdots\!83}{20\!\cdots\!00}a^{15}+\frac{37\!\cdots\!39}{40\!\cdots\!00}a^{14}+\frac{75\!\cdots\!09}{40\!\cdots\!40}a^{13}-\frac{97\!\cdots\!51}{20\!\cdots\!20}a^{12}-\frac{14\!\cdots\!59}{67\!\cdots\!00}a^{11}+\frac{25\!\cdots\!03}{20\!\cdots\!00}a^{10}+\frac{17\!\cdots\!85}{16\!\cdots\!26}a^{9}-\frac{70\!\cdots\!23}{33\!\cdots\!00}a^{8}+\frac{17\!\cdots\!99}{20\!\cdots\!00}a^{7}+\frac{10\!\cdots\!59}{81\!\cdots\!80}a^{6}-\frac{35\!\cdots\!33}{25\!\cdots\!00}a^{5}+\frac{37\!\cdots\!07}{13\!\cdots\!80}a^{4}+\frac{32\!\cdots\!49}{50\!\cdots\!00}a^{3}-\frac{18\!\cdots\!71}{26\!\cdots\!40}a^{2}+\frac{37\!\cdots\!99}{12\!\cdots\!95}a-\frac{10\!\cdots\!61}{20\!\cdots\!12}$, $\frac{80\!\cdots\!47}{14\!\cdots\!00}a^{31}-\frac{22\!\cdots\!77}{74\!\cdots\!00}a^{30}-\frac{16\!\cdots\!91}{14\!\cdots\!00}a^{29}+\frac{56\!\cdots\!57}{74\!\cdots\!00}a^{28}+\frac{46\!\cdots\!09}{29\!\cdots\!16}a^{27}-\frac{12\!\cdots\!61}{12\!\cdots\!00}a^{26}-\frac{17\!\cdots\!61}{14\!\cdots\!00}a^{25}+\frac{17\!\cdots\!79}{18\!\cdots\!00}a^{24}+\frac{25\!\cdots\!21}{74\!\cdots\!00}a^{23}-\frac{17\!\cdots\!87}{31\!\cdots\!00}a^{22}+\frac{43\!\cdots\!29}{29\!\cdots\!60}a^{21}+\frac{15\!\cdots\!71}{62\!\cdots\!00}a^{20}-\frac{12\!\cdots\!27}{49\!\cdots\!00}a^{19}-\frac{12\!\cdots\!61}{14\!\cdots\!80}a^{18}+\frac{85\!\cdots\!37}{49\!\cdots\!00}a^{17}+\frac{26\!\cdots\!09}{37\!\cdots\!20}a^{16}-\frac{29\!\cdots\!03}{49\!\cdots\!00}a^{15}+\frac{15\!\cdots\!23}{24\!\cdots\!00}a^{14}+\frac{11\!\cdots\!37}{12\!\cdots\!00}a^{13}-\frac{21\!\cdots\!47}{74\!\cdots\!00}a^{12}-\frac{40\!\cdots\!53}{74\!\cdots\!00}a^{11}+\frac{20\!\cdots\!93}{29\!\cdots\!75}a^{10}-\frac{46\!\cdots\!81}{37\!\cdots\!00}a^{9}-\frac{14\!\cdots\!77}{12\!\cdots\!00}a^{8}+\frac{11\!\cdots\!51}{14\!\cdots\!00}a^{7}+\frac{89\!\cdots\!39}{12\!\cdots\!00}a^{6}-\frac{79\!\cdots\!21}{74\!\cdots\!00}a^{5}+\frac{15\!\cdots\!37}{62\!\cdots\!00}a^{4}+\frac{16\!\cdots\!57}{37\!\cdots\!00}a^{3}-\frac{69\!\cdots\!29}{14\!\cdots\!52}a^{2}+\frac{22\!\cdots\!31}{12\!\cdots\!40}a-\frac{46\!\cdots\!67}{18\!\cdots\!76}$, $\frac{15\!\cdots\!29}{32\!\cdots\!00}a^{31}-\frac{41\!\cdots\!49}{16\!\cdots\!60}a^{30}-\frac{34\!\cdots\!89}{32\!\cdots\!00}a^{29}+\frac{20\!\cdots\!47}{32\!\cdots\!92}a^{28}+\frac{24\!\cdots\!13}{16\!\cdots\!00}a^{27}-\frac{22\!\cdots\!89}{26\!\cdots\!00}a^{26}-\frac{40\!\cdots\!31}{32\!\cdots\!00}a^{25}+\frac{12\!\cdots\!97}{16\!\cdots\!00}a^{24}+\frac{79\!\cdots\!79}{16\!\cdots\!00}a^{23}-\frac{36\!\cdots\!41}{81\!\cdots\!00}a^{22}+\frac{24\!\cdots\!43}{32\!\cdots\!00}a^{21}+\frac{34\!\cdots\!79}{16\!\cdots\!00}a^{20}-\frac{17\!\cdots\!61}{10\!\cdots\!00}a^{19}-\frac{19\!\cdots\!71}{27\!\cdots\!00}a^{18}+\frac{41\!\cdots\!17}{32\!\cdots\!00}a^{17}+\frac{14\!\cdots\!93}{16\!\cdots\!00}a^{16}-\frac{10\!\cdots\!69}{21\!\cdots\!80}a^{15}+\frac{14\!\cdots\!01}{33\!\cdots\!00}a^{14}+\frac{68\!\cdots\!37}{81\!\cdots\!80}a^{13}-\frac{71\!\cdots\!41}{32\!\cdots\!20}a^{12}-\frac{15\!\cdots\!17}{16\!\cdots\!00}a^{11}+\frac{45\!\cdots\!29}{81\!\cdots\!00}a^{10}+\frac{24\!\cdots\!13}{81\!\cdots\!00}a^{9}-\frac{76\!\cdots\!03}{81\!\cdots\!00}a^{8}+\frac{14\!\cdots\!33}{32\!\cdots\!00}a^{7}+\frac{98\!\cdots\!59}{16\!\cdots\!00}a^{6}-\frac{11\!\cdots\!23}{16\!\cdots\!00}a^{5}+\frac{10\!\cdots\!27}{81\!\cdots\!00}a^{4}+\frac{90\!\cdots\!33}{27\!\cdots\!00}a^{3}-\frac{35\!\cdots\!83}{10\!\cdots\!60}a^{2}+\frac{33\!\cdots\!57}{27\!\cdots\!60}a-\frac{51\!\cdots\!33}{27\!\cdots\!16}$, $\frac{76\!\cdots\!59}{64\!\cdots\!40}a^{31}-\frac{13\!\cdots\!97}{16\!\cdots\!00}a^{30}-\frac{58\!\cdots\!07}{32\!\cdots\!00}a^{29}+\frac{32\!\cdots\!21}{16\!\cdots\!00}a^{28}+\frac{28\!\cdots\!21}{16\!\cdots\!00}a^{27}-\frac{75\!\cdots\!01}{26\!\cdots\!00}a^{26}-\frac{48\!\cdots\!47}{10\!\cdots\!00}a^{25}+\frac{10\!\cdots\!73}{40\!\cdots\!00}a^{24}-\frac{18\!\cdots\!07}{16\!\cdots\!00}a^{23}-\frac{30\!\cdots\!53}{20\!\cdots\!00}a^{22}+\frac{32\!\cdots\!91}{21\!\cdots\!80}a^{21}+\frac{52\!\cdots\!23}{81\!\cdots\!80}a^{20}-\frac{12\!\cdots\!51}{10\!\cdots\!00}a^{19}-\frac{96\!\cdots\!43}{54\!\cdots\!00}a^{18}+\frac{19\!\cdots\!39}{32\!\cdots\!00}a^{17}-\frac{77\!\cdots\!43}{13\!\cdots\!00}a^{16}-\frac{55\!\cdots\!49}{32\!\cdots\!00}a^{15}+\frac{13\!\cdots\!79}{54\!\cdots\!00}a^{14}+\frac{25\!\cdots\!91}{16\!\cdots\!60}a^{13}-\frac{49\!\cdots\!09}{54\!\cdots\!00}a^{12}+\frac{52\!\cdots\!11}{16\!\cdots\!00}a^{11}+\frac{22\!\cdots\!57}{10\!\cdots\!00}a^{10}-\frac{12\!\cdots\!69}{81\!\cdots\!00}a^{9}-\frac{29\!\cdots\!99}{81\!\cdots\!00}a^{8}+\frac{40\!\cdots\!77}{10\!\cdots\!00}a^{7}+\frac{14\!\cdots\!21}{81\!\cdots\!00}a^{6}-\frac{12\!\cdots\!01}{32\!\cdots\!20}a^{5}+\frac{58\!\cdots\!79}{40\!\cdots\!00}a^{4}+\frac{35\!\cdots\!47}{27\!\cdots\!00}a^{3}-\frac{29\!\cdots\!29}{16\!\cdots\!40}a^{2}+\frac{63\!\cdots\!89}{81\!\cdots\!80}a-\frac{13\!\cdots\!65}{13\!\cdots\!08}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 36960168098647.56 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 36960168098647.56 \cdot 1248}{6\cdot\sqrt{7811497343330649377718034300042226077532160000000000000000}}\cr\approx \mathstrut & 0.513225258062427 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 6*x^31 - 19*x^30 + 146*x^29 + 236*x^28 - 1994*x^27 - 1515*x^26 + 17532*x^25 + 248*x^24 - 102760*x^23 + 62939*x^22 + 452232*x^21 - 621557*x^20 - 1332326*x^19 + 3689491*x^18 + 75956*x^17 - 11152569*x^16 + 15594526*x^15 + 12008686*x^14 - 58161950*x^13 + 9977818*x^12 + 131342300*x^11 - 70421448*x^10 - 200370404*x^9 + 221182489*x^8 + 64149492*x^7 - 233429156*x^6 + 127683800*x^5 + 51958176*x^4 - 115466320*x^3 + 71910480*x^2 - 21408800*x + 2671600)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^32 - 6*x^31 - 19*x^30 + 146*x^29 + 236*x^28 - 1994*x^27 - 1515*x^26 + 17532*x^25 + 248*x^24 - 102760*x^23 + 62939*x^22 + 452232*x^21 - 621557*x^20 - 1332326*x^19 + 3689491*x^18 + 75956*x^17 - 11152569*x^16 + 15594526*x^15 + 12008686*x^14 - 58161950*x^13 + 9977818*x^12 + 131342300*x^11 - 70421448*x^10 - 200370404*x^9 + 221182489*x^8 + 64149492*x^7 - 233429156*x^6 + 127683800*x^5 + 51958176*x^4 - 115466320*x^3 + 71910480*x^2 - 21408800*x + 2671600, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 6*x^31 - 19*x^30 + 146*x^29 + 236*x^28 - 1994*x^27 - 1515*x^26 + 17532*x^25 + 248*x^24 - 102760*x^23 + 62939*x^22 + 452232*x^21 - 621557*x^20 - 1332326*x^19 + 3689491*x^18 + 75956*x^17 - 11152569*x^16 + 15594526*x^15 + 12008686*x^14 - 58161950*x^13 + 9977818*x^12 + 131342300*x^11 - 70421448*x^10 - 200370404*x^9 + 221182489*x^8 + 64149492*x^7 - 233429156*x^6 + 127683800*x^5 + 51958176*x^4 - 115466320*x^3 + 71910480*x^2 - 21408800*x + 2671600);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 6*x^31 - 19*x^30 + 146*x^29 + 236*x^28 - 1994*x^27 - 1515*x^26 + 17532*x^25 + 248*x^24 - 102760*x^23 + 62939*x^22 + 452232*x^21 - 621557*x^20 - 1332326*x^19 + 3689491*x^18 + 75956*x^17 - 11152569*x^16 + 15594526*x^15 + 12008686*x^14 - 58161950*x^13 + 9977818*x^12 + 131342300*x^11 - 70421448*x^10 - 200370404*x^9 + 221182489*x^8 + 64149492*x^7 - 233429156*x^6 + 127683800*x^5 + 51958176*x^4 - 115466320*x^3 + 71910480*x^2 - 21408800*x + 2671600);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$D_4^2:C_2^3$ (as 32T12882):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 512 |
| The 80 conjugacy class representatives for $D_4^2:C_2^3$ |
| Character table for $D_4^2:C_2^3$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 32 siblings: | data not computed |
| Minimal sibling: | not computed |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.8.0.1}{8} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.8.0.1}{8} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| 2.8.12.1 | $x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(89\)
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.2.1 | $x^{4} + 12268 x^{3} + 38122404 x^{2} + 3045212032 x + 156142232$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 89.4.0.1 | $x^{4} + 4 x^{2} + 72 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(181\)
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 181.4.2.1 | $x^{4} + 52120 x^{3} + 683705257 x^{2} + 119397981420 x + 2183938221$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 181.4.2.1 | $x^{4} + 52120 x^{3} + 683705257 x^{2} + 119397981420 x + 2183938221$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |