Properties

Label 20.0.362...561.1
Degree $20$
Signature $[0, 10]$
Discriminant $3.624\times 10^{20}$
Root discriminant \(10.66\)
Ramified primes $47,83$
Class number $1$
Class group trivial
Galois group $C_2\wr D_5$ (as 20T81)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 7*x^15 + 8*x^14 - x^13 + 8*x^12 - 3*x^11 + 3*x^10 + 3*x^9 - 13*x^8 + 17*x^7 + 8*x^6 - 8*x^5 + x^4 + 4*x^3 + 1)
 
gp: K = bnfinit(y^20 - y^19 + 3*y^18 - 4*y^17 + 5*y^16 - 7*y^15 + 8*y^14 - y^13 + 8*y^12 - 3*y^11 + 3*y^10 + 3*y^9 - 13*y^8 + 17*y^7 + 8*y^6 - 8*y^5 + y^4 + 4*y^3 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 7*x^15 + 8*x^14 - x^13 + 8*x^12 - 3*x^11 + 3*x^10 + 3*x^9 - 13*x^8 + 17*x^7 + 8*x^6 - 8*x^5 + x^4 + 4*x^3 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 7*x^15 + 8*x^14 - x^13 + 8*x^12 - 3*x^11 + 3*x^10 + 3*x^9 - 13*x^8 + 17*x^7 + 8*x^6 - 8*x^5 + x^4 + 4*x^3 + 1)
 

\( x^{20} - x^{19} + 3 x^{18} - 4 x^{17} + 5 x^{16} - 7 x^{15} + 8 x^{14} - x^{13} + 8 x^{12} - 3 x^{11} + 3 x^{10} + 3 x^{9} - 13 x^{8} + 17 x^{7} + 8 x^{6} - 8 x^{5} + x^{4} + 4 x^{3} + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(362355421972633207561\) \(\medspace = 47^{10}\cdot 83^{2}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(10.66\)
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:  $47^{1/2}83^{1/2}\approx 62.457985878508765$
Ramified primes:   \(47\), \(83\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}-\frac{1}{5}a^{17}-\frac{1}{5}a^{16}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{2}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}$, $\frac{1}{2194608515}a^{19}+\frac{74558294}{2194608515}a^{18}-\frac{1017294346}{2194608515}a^{17}+\frac{6939284}{39901973}a^{16}+\frac{856912424}{2194608515}a^{15}-\frac{994013257}{2194608515}a^{14}-\frac{586393183}{2194608515}a^{13}-\frac{382352308}{2194608515}a^{12}+\frac{9694604}{438921703}a^{11}-\frac{654463201}{2194608515}a^{10}+\frac{509560233}{2194608515}a^{9}+\frac{127710497}{2194608515}a^{8}-\frac{19417716}{438921703}a^{7}+\frac{941830149}{2194608515}a^{6}-\frac{675824012}{2194608515}a^{5}-\frac{811533444}{2194608515}a^{4}-\frac{1050222631}{2194608515}a^{3}+\frac{195415228}{438921703}a^{2}+\frac{339804044}{2194608515}a-\frac{29307092}{438921703}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:   $\frac{17167678}{39901973}a^{19}-\frac{198448696}{199509865}a^{18}+\frac{459542381}{199509865}a^{17}-\frac{840311969}{199509865}a^{16}+\frac{260122109}{39901973}a^{15}-\frac{1849657884}{199509865}a^{14}+\frac{2532135297}{199509865}a^{13}-\frac{2478167597}{199509865}a^{12}+\frac{2760348783}{199509865}a^{11}-\frac{578017673}{39901973}a^{10}+\frac{2728324706}{199509865}a^{9}-\frac{1951319303}{199509865}a^{8}+\frac{521684558}{199509865}a^{7}+\frac{368598185}{39901973}a^{6}-\frac{1381145139}{199509865}a^{5}+\frac{302967502}{199509865}a^{4}+\frac{53721484}{199509865}a^{3}+\frac{280222406}{199509865}a^{2}-\frac{56123045}{39901973}a+\frac{294207701}{199509865}$, $\frac{35747331}{438921703}a^{19}-\frac{145326186}{2194608515}a^{18}+\frac{403294346}{2194608515}a^{17}-\frac{16979709}{199509865}a^{16}+\frac{12943137}{438921703}a^{15}+\frac{338271231}{2194608515}a^{14}-\frac{1035587648}{2194608515}a^{13}+\frac{2580202278}{2194608515}a^{12}-\frac{2523174922}{2194608515}a^{11}+\frac{644125245}{438921703}a^{10}-\frac{1462277224}{2194608515}a^{9}+\frac{4268504467}{2194608515}a^{8}-\frac{4522480687}{2194608515}a^{7}+\frac{416180518}{438921703}a^{6}+\frac{2431179411}{2194608515}a^{5}-\frac{9009900908}{2194608515}a^{4}+\frac{6052739344}{2194608515}a^{3}+\frac{2462644731}{2194608515}a^{2}-\frac{267045956}{438921703}a-\frac{225933434}{2194608515}$, $\frac{140164190}{438921703}a^{19}-\frac{3029118987}{2194608515}a^{18}+\frac{5495443702}{2194608515}a^{17}-\frac{1056094673}{199509865}a^{16}+\frac{3543454491}{438921703}a^{15}-\frac{24383649848}{2194608515}a^{14}+\frac{34129248594}{2194608515}a^{13}-\frac{36148677594}{2194608515}a^{12}+\frac{30310058261}{2194608515}a^{11}-\frac{7978649180}{438921703}a^{10}+\frac{33598390482}{2194608515}a^{9}-\frac{25547226321}{2194608515}a^{8}+\frac{7023841916}{2194608515}a^{7}+\frac{6150886199}{438921703}a^{6}-\frac{34791286373}{2194608515}a^{5}-\frac{2801703951}{2194608515}a^{4}+\frac{9638984958}{2194608515}a^{3}+\frac{2035043012}{2194608515}a^{2}-\frac{595370537}{438921703}a+\frac{2421876982}{2194608515}$, $\frac{599914055}{438921703}a^{19}-\frac{6328764117}{2194608515}a^{18}+\frac{14781313397}{2194608515}a^{17}-\frac{2359285278}{199509865}a^{16}+\frac{7763200751}{438921703}a^{15}-\frac{54448405703}{2194608515}a^{14}+\frac{72218992069}{2194608515}a^{13}-\frac{65019050674}{2194608515}a^{12}+\frac{72818839881}{2194608515}a^{11}-\frac{14485773913}{438921703}a^{10}+\frac{68365953842}{2194608515}a^{9}-\frac{42559432106}{2194608515}a^{8}-\frac{5470702269}{2194608515}a^{7}+\frac{14242725327}{438921703}a^{6}-\frac{43343911258}{2194608515}a^{5}-\frac{4552730366}{2194608515}a^{4}+\frac{15215114478}{2194608515}a^{3}+\frac{4852902172}{2194608515}a^{2}-\frac{1309620302}{438921703}a+\frac{5352449662}{2194608515}$, $\frac{5352449662}{2194608515}a^{19}-\frac{8352019937}{2194608515}a^{18}+\frac{22386113103}{2194608515}a^{17}-\frac{658020219}{39901973}a^{16}+\frac{52714386368}{2194608515}a^{15}-\frac{76283151389}{2194608515}a^{14}+\frac{97268002999}{2194608515}a^{13}-\frac{77571441731}{2194608515}a^{12}+\frac{21567729594}{438921703}a^{11}-\frac{88876188867}{2194608515}a^{10}+\frac{88486218551}{2194608515}a^{9}-\frac{52308604856}{2194608515}a^{8}-\frac{5404482700}{438921703}a^{7}+\frac{96462346523}{2194608515}a^{6}-\frac{28394029339}{2194608515}a^{5}+\frac{524313962}{2194608515}a^{4}+\frac{9905180028}{2194608515}a^{3}+\frac{1238936834}{438921703}a^{2}-\frac{4852902172}{2194608515}a+\frac{1309620302}{438921703}$, $\frac{2707494034}{2194608515}a^{19}-\frac{5555157859}{2194608515}a^{18}+\frac{12429096146}{2194608515}a^{17}-\frac{400515119}{39901973}a^{16}+\frac{31963004506}{2194608515}a^{15}-\frac{44760304288}{2194608515}a^{14}+\frac{59543687813}{2194608515}a^{13}-\frac{50880774287}{2194608515}a^{12}+\frac{11611246756}{438921703}a^{11}-\frac{61335888844}{2194608515}a^{10}+\frac{53948495212}{2194608515}a^{9}-\frac{33498872742}{2194608515}a^{8}-\frac{1695935381}{438921703}a^{7}+\frac{61586534701}{2194608515}a^{6}-\frac{26555629888}{2194608515}a^{5}-\frac{15417470496}{2194608515}a^{4}+\frac{8340201046}{2194608515}a^{3}+\frac{2034866506}{438921703}a^{2}-\frac{6407575579}{2194608515}a+\frac{507156570}{438921703}$, $\frac{2529411972}{2194608515}a^{19}-\frac{5532330713}{2194608515}a^{18}+\frac{12279972399}{2194608515}a^{17}-\frac{1995362534}{199509865}a^{16}+\frac{32003869343}{2194608515}a^{15}-\frac{44568041303}{2194608515}a^{14}+\frac{58830908611}{2194608515}a^{13}-\frac{51404782113}{2194608515}a^{12}+\frac{55323097553}{2194608515}a^{11}-\frac{57669217082}{2194608515}a^{10}+\frac{50985256052}{2194608515}a^{9}-\frac{30691585729}{2194608515}a^{8}-\frac{11694648927}{2194608515}a^{7}+\frac{65504322668}{2194608515}a^{6}-\frac{38687198478}{2194608515}a^{5}-\frac{11043805931}{2194608515}a^{4}+\frac{15212489607}{2194608515}a^{3}+\frac{4352691061}{2194608515}a^{2}-\frac{6269809532}{2194608515}a+\frac{3185400421}{2194608515}$, $\frac{3769389187}{2194608515}a^{19}-\frac{5479438171}{2194608515}a^{18}+\frac{14363617537}{2194608515}a^{17}-\frac{2064386926}{199509865}a^{16}+\frac{31905425633}{2194608515}a^{15}-\frac{9127974501}{438921703}a^{14}+\frac{57768211187}{2194608515}a^{13}-\frac{40322926184}{2194608515}a^{12}+\frac{61538631897}{2194608515}a^{11}-\frac{51031028432}{2194608515}a^{10}+\frac{9741648969}{438921703}a^{9}-\frac{23774998318}{2194608515}a^{8}-\frac{25925501378}{2194608515}a^{7}+\frac{65878753533}{2194608515}a^{6}-\frac{622046301}{438921703}a^{5}-\frac{3489893156}{438921703}a^{4}+\frac{5043566969}{2194608515}a^{3}+\frac{13483188839}{2194608515}a^{2}-\frac{4861363692}{2194608515}a+\frac{2302714179}{2194608515}$, $\frac{904504175}{438921703}a^{19}-\frac{7684902578}{2194608515}a^{18}+\frac{19536109453}{2194608515}a^{17}-\frac{2933794162}{199509865}a^{16}+\frac{9416070745}{438921703}a^{15}-\frac{66918843862}{2194608515}a^{14}+\frac{86368648006}{2194608515}a^{13}-\frac{69506968761}{2194608515}a^{12}+\frac{90524548749}{2194608515}a^{11}-\frac{15815417419}{438921703}a^{10}+\frac{75705925198}{2194608515}a^{9}-\frac{42994422364}{2194608515}a^{8}-\frac{23685749551}{2194608515}a^{7}+\frac{18733010240}{438921703}a^{6}-\frac{32577094552}{2194608515}a^{5}-\frac{4564214254}{2194608515}a^{4}+\frac{12290055387}{2194608515}a^{3}+\frac{3849438258}{2194608515}a^{2}-\frac{756814606}{438921703}a+\frac{5327087823}{2194608515}$ Copy content Toggle raw display
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:  \( 59.602445011 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 59.602445011 \cdot 1}{2\cdot\sqrt{362355421972633207561}}\cr\approx \mathstrut & 0.15012927302 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 7*x^15 + 8*x^14 - x^13 + 8*x^12 - 3*x^11 + 3*x^10 + 3*x^9 - 13*x^8 + 17*x^7 + 8*x^6 - 8*x^5 + x^4 + 4*x^3 + 1)
 
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^20 - x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 7*x^15 + 8*x^14 - x^13 + 8*x^12 - 3*x^11 + 3*x^10 + 3*x^9 - 13*x^8 + 17*x^7 + 8*x^6 - 8*x^5 + x^4 + 4*x^3 + 1, 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^20 - x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 7*x^15 + 8*x^14 - x^13 + 8*x^12 - 3*x^11 + 3*x^10 + 3*x^9 - 13*x^8 + 17*x^7 + 8*x^6 - 8*x^5 + x^4 + 4*x^3 + 1);
 
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^20 - x^19 + 3*x^18 - 4*x^17 + 5*x^16 - 7*x^15 + 8*x^14 - x^13 + 8*x^12 - 3*x^11 + 3*x^10 + 3*x^9 - 13*x^8 + 17*x^7 + 8*x^6 - 8*x^5 + x^4 + 4*x^3 + 1);
 
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

$C_2\wr D_5$ (as 20T81):

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 320
The 20 conjugacy class representatives for $C_2\wr D_5$
Character table for $C_2\wr D_5$

Intermediate fields

\(\Q(\sqrt{-47}) \), 5.1.2209.1 x5, 10.0.405013523.1, 10.2.19035635581.1, 10.0.229345007.1

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 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 siblings: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 10.0.405013523.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{2}$ ${\href{/padicField/3.5.0.1}{5} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ ${\href{/padicField/7.5.0.1}{5} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.5.0.1}{5} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{10}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ R ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(47\) Copy content Toggle raw display 47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(83\) Copy content Toggle raw display $\Q_{83}$$x + 81$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 81$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 81$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 81$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 81$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 81$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 81$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 81$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 83$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.0.1$x^{2} + 82 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} + 82 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} + 82 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} + 82 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.1.2$x^{2} + 83$$2$$1$$1$$C_2$$[\ ]_{2}$