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Properties

Label 16.0.858...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $8.584\times 10^{21}$
Root discriminant \(23.49\)
Ramified primes $5,181$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $D_4:C_4$ (as 16T26)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 + 20*x^13 - x^12 - 12*x^11 - 42*x^10 - 80*x^9 + 681*x^8 - 482*x^7 - 1002*x^6 + 1879*x^5 - 386*x^4 - 1089*x^3 + 1089*x^2 - 459*x + 81)
 
Copy content gp:K = bnfinit(y^16 - 2*y^15 - 5*y^14 + 20*y^13 - y^12 - 12*y^11 - 42*y^10 - 80*y^9 + 681*y^8 - 482*y^7 - 1002*y^6 + 1879*y^5 - 386*y^4 - 1089*y^3 + 1089*y^2 - 459*y + 81, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 5*x^14 + 20*x^13 - x^12 - 12*x^11 - 42*x^10 - 80*x^9 + 681*x^8 - 482*x^7 - 1002*x^6 + 1879*x^5 - 386*x^4 - 1089*x^3 + 1089*x^2 - 459*x + 81);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 5*x^14 + 20*x^13 - x^12 - 12*x^11 - 42*x^10 - 80*x^9 + 681*x^8 - 482*x^7 - 1002*x^6 + 1879*x^5 - 386*x^4 - 1089*x^3 + 1089*x^2 - 459*x + 81)
 

\( x^{16} - 2 x^{15} - 5 x^{14} + 20 x^{13} - x^{12} - 12 x^{11} - 42 x^{10} - 80 x^{9} + 681 x^{8} + \cdots + 81 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[0, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(8584430743916259765625\) \(\medspace = 5^{12}\cdot 181^{6}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(23.49\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $5^{3/4}181^{1/2}\approx 44.98490324139099$
Ramified primes:   \(5\), \(181\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{13}-\frac{2}{9}a^{12}-\frac{4}{9}a^{11}-\frac{4}{9}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{9}a^{7}+\frac{4}{9}a^{5}-\frac{2}{9}a^{3}+\frac{4}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{26\cdots 67}a^{15}+\frac{13\cdots 98}{26\cdots 67}a^{14}-\frac{24\cdots 79}{26\cdots 67}a^{13}+\frac{11\cdots 13}{26\cdots 67}a^{12}+\frac{79\cdots 32}{26\cdots 67}a^{11}+\frac{67\cdots 30}{29\cdots 63}a^{10}-\frac{22\cdots 71}{89\cdots 89}a^{9}+\frac{46\cdots 08}{26\cdots 67}a^{8}+\frac{35\cdots 43}{89\cdots 89}a^{7}+\frac{55\cdots 46}{26\cdots 67}a^{6}-\frac{14\cdots 44}{89\cdots 89}a^{5}+\frac{43\cdots 79}{26\cdots 67}a^{4}-\frac{78\cdots 53}{26\cdots 67}a^{3}-\frac{18\cdots 92}{89\cdots 89}a^{2}+\frac{58\cdots 28}{29\cdots 63}a-\frac{17\cdots 03}{99\cdots 21}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  $C_{2}$, which has order $2$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}$, which has order $2$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   $2$ (assuming GRH)

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $7$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( \frac{3433281994457538368}{269726363277673430367} a^{15} - \frac{5503829630558682961}{269726363277673430367} a^{14} - \frac{18731391854287311340}{269726363277673430367} a^{13} + \frac{60419009814054795973}{269726363277673430367} a^{12} + \frac{17663468919411794296}{269726363277673430367} a^{11} - \frac{8651463567167375684}{89908787759224476789} a^{10} - \frac{50706496390001297032}{89908787759224476789} a^{9} - \frac{324273628477408971217}{269726363277673430367} a^{8} + \frac{726863904665304290032}{89908787759224476789} a^{7} - \frac{852643893918418065052}{269726363277673430367} a^{6} - \frac{1153218169136850692249}{89908787759224476789} a^{5} + \frac{4957067474367841681604}{269726363277673430367} a^{4} + \frac{502252115630327093093}{269726363277673430367} a^{3} - \frac{118387237231008419325}{9989865306580497421} a^{2} + \frac{91294933617019402024}{9989865306580497421} a - \frac{17019425584767442571}{9989865306580497421} \)  (order $10$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{19\cdots 75}{89\cdots 89}a^{15}-\frac{28\cdots 39}{89\cdots 89}a^{14}-\frac{11\cdots 54}{89\cdots 89}a^{13}+\frac{32\cdots 67}{89\cdots 89}a^{12}+\frac{14\cdots 90}{89\cdots 89}a^{11}-\frac{46\cdots 20}{29\cdots 63}a^{10}-\frac{92\cdots 54}{99\cdots 21}a^{9}-\frac{20\cdots 38}{89\cdots 89}a^{8}+\frac{39\cdots 57}{29\cdots 63}a^{7}-\frac{30\cdots 14}{89\cdots 89}a^{6}-\frac{67\cdots 76}{29\cdots 63}a^{5}+\frac{26\cdots 31}{89\cdots 89}a^{4}+\frac{44\cdots 63}{89\cdots 89}a^{3}-\frac{20\cdots 41}{99\cdots 21}a^{2}+\frac{45\cdots 46}{29\cdots 63}a-\frac{28\cdots 09}{99\cdots 21}$, $\frac{13\cdots 53}{89\cdots 89}a^{15}-\frac{19\cdots 05}{89\cdots 89}a^{14}-\frac{75\cdots 15}{89\cdots 89}a^{13}+\frac{22\cdots 39}{89\cdots 89}a^{12}+\frac{10\cdots 56}{89\cdots 89}a^{11}-\frac{11\cdots 18}{99\cdots 21}a^{10}-\frac{66\cdots 26}{99\cdots 21}a^{9}-\frac{13\cdots 96}{89\cdots 89}a^{8}+\frac{90\cdots 20}{99\cdots 21}a^{7}-\frac{18\cdots 06}{89\cdots 89}a^{6}-\frac{15\cdots 03}{99\cdots 21}a^{5}+\frac{17\cdots 51}{89\cdots 89}a^{4}+\frac{40\cdots 14}{89\cdots 89}a^{3}-\frac{40\cdots 34}{29\cdots 63}a^{2}+\frac{26\cdots 35}{29\cdots 63}a-\frac{21\cdots 99}{99\cdots 21}$, $\frac{20\cdots 99}{89\cdots 89}a^{15}+\frac{26\cdots 86}{29\cdots 63}a^{14}-\frac{18\cdots 21}{99\cdots 21}a^{13}+\frac{14\cdots 21}{89\cdots 89}a^{12}+\frac{23\cdots 56}{29\cdots 63}a^{11}+\frac{99\cdots 94}{89\cdots 89}a^{10}-\frac{76\cdots 84}{99\cdots 21}a^{9}-\frac{39\cdots 14}{89\cdots 89}a^{8}+\frac{87\cdots 89}{89\cdots 89}a^{7}+\frac{18\cdots 50}{89\cdots 89}a^{6}-\frac{27\cdots 94}{89\cdots 89}a^{5}+\frac{26\cdots 86}{89\cdots 89}a^{4}+\frac{42\cdots 38}{99\cdots 21}a^{3}-\frac{15\cdots 59}{89\cdots 89}a^{2}+\frac{20\cdots 77}{29\cdots 63}a+\frac{49\cdots 56}{99\cdots 21}$, $\frac{41\cdots 41}{89\cdots 89}a^{15}-\frac{86\cdots 68}{89\cdots 89}a^{14}-\frac{21\cdots 53}{89\cdots 89}a^{13}+\frac{96\cdots 72}{99\cdots 21}a^{12}-\frac{42\cdots 24}{89\cdots 89}a^{11}-\frac{74\cdots 35}{89\cdots 89}a^{10}-\frac{20\cdots 35}{99\cdots 21}a^{9}-\frac{29\cdots 71}{89\cdots 89}a^{8}+\frac{29\cdots 15}{89\cdots 89}a^{7}-\frac{21\cdots 38}{89\cdots 89}a^{6}-\frac{48\cdots 41}{89\cdots 89}a^{5}+\frac{83\cdots 09}{89\cdots 89}a^{4}-\frac{66\cdots 26}{89\cdots 89}a^{3}-\frac{61\cdots 11}{89\cdots 89}a^{2}+\frac{47\cdots 61}{99\cdots 21}a-\frac{98\cdots 78}{99\cdots 21}$, $\frac{27\cdots 51}{89\cdots 89}a^{15}-\frac{48\cdots 45}{89\cdots 89}a^{14}-\frac{15\cdots 71}{89\cdots 89}a^{13}+\frac{52\cdots 88}{89\cdots 89}a^{12}+\frac{13\cdots 41}{89\cdots 89}a^{11}-\frac{47\cdots 34}{99\cdots 21}a^{10}-\frac{40\cdots 65}{29\cdots 63}a^{9}-\frac{25\cdots 04}{89\cdots 89}a^{8}+\frac{61\cdots 15}{29\cdots 63}a^{7}-\frac{81\cdots 05}{89\cdots 89}a^{6}-\frac{11\cdots 96}{29\cdots 63}a^{5}+\frac{47\cdots 44}{89\cdots 89}a^{4}+\frac{41\cdots 36}{89\cdots 89}a^{3}-\frac{12\cdots 84}{29\cdots 63}a^{2}+\frac{74\cdots 61}{29\cdots 63}a-\frac{59\cdots 16}{99\cdots 21}$, $\frac{13\cdots 56}{26\cdots 67}a^{15}-\frac{34\cdots 74}{26\cdots 67}a^{14}-\frac{50\cdots 53}{26\cdots 67}a^{13}+\frac{29\cdots 85}{26\cdots 67}a^{12}-\frac{13\cdots 92}{26\cdots 67}a^{11}-\frac{46\cdots 98}{89\cdots 89}a^{10}-\frac{22\cdots 90}{89\cdots 89}a^{9}-\frac{42\cdots 57}{26\cdots 67}a^{8}+\frac{32\cdots 74}{89\cdots 89}a^{7}-\frac{10\cdots 74}{26\cdots 67}a^{6}-\frac{32\cdots 93}{89\cdots 89}a^{5}+\frac{26\cdots 26}{26\cdots 67}a^{4}-\frac{49\cdots 18}{26\cdots 67}a^{3}-\frac{37\cdots 80}{99\cdots 21}a^{2}+\frac{11\cdots 23}{29\cdots 63}a-\frac{72\cdots 85}{99\cdots 21}$, $\frac{87\cdots 15}{89\cdots 89}a^{15}-\frac{10\cdots 63}{89\cdots 89}a^{14}-\frac{53\cdots 09}{89\cdots 89}a^{13}+\frac{44\cdots 86}{29\cdots 63}a^{12}+\frac{10\cdots 86}{89\cdots 89}a^{11}-\frac{42\cdots 80}{89\cdots 89}a^{10}-\frac{14\cdots 65}{29\cdots 63}a^{9}-\frac{10\cdots 34}{89\cdots 89}a^{8}+\frac{52\cdots 94}{89\cdots 89}a^{7}+\frac{45\cdots 15}{89\cdots 89}a^{6}-\frac{94\cdots 94}{89\cdots 89}a^{5}+\frac{87\cdots 15}{89\cdots 89}a^{4}+\frac{49\cdots 67}{89\cdots 89}a^{3}-\frac{66\cdots 68}{89\cdots 89}a^{2}+\frac{10\cdots 02}{29\cdots 63}a-\frac{60\cdots 94}{99\cdots 21}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 43749.6863738 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 43749.6863738 \cdot 2}{10\cdot\sqrt{8584430743916259765625}}\cr\approx \mathstrut & 0.229397212322 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 5*x^14 + 20*x^13 - x^12 - 12*x^11 - 42*x^10 - 80*x^9 + 681*x^8 - 482*x^7 - 1002*x^6 + 1879*x^5 - 386*x^4 - 1089*x^3 + 1089*x^2 - 459*x + 81) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - 2*x^15 - 5*x^14 + 20*x^13 - x^12 - 12*x^11 - 42*x^10 - 80*x^9 + 681*x^8 - 482*x^7 - 1002*x^6 + 1879*x^5 - 386*x^4 - 1089*x^3 + 1089*x^2 - 459*x + 81, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 5*x^14 + 20*x^13 - x^12 - 12*x^11 - 42*x^10 - 80*x^9 + 681*x^8 - 482*x^7 - 1002*x^6 + 1879*x^5 - 386*x^4 - 1089*x^3 + 1089*x^2 - 459*x + 81); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 - 5*x^14 + 20*x^13 - x^12 - 12*x^11 - 42*x^10 - 80*x^9 + 681*x^8 - 482*x^7 - 1002*x^6 + 1879*x^5 - 386*x^4 - 1089*x^3 + 1089*x^2 - 459*x + 81); 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

$D_4:C_4$ (as 16T26):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 32
The 14 conjugacy class representatives for $D_4:C_4$
Character table for $D_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.4525.1, 4.0.22625.1, 8.0.3706088125.1, 8.8.92652203125.1, 8.0.511890625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 32
Degree 16 sibling: deg 16
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
\(5\) Copy content Toggle raw display 5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(181\) Copy content Toggle raw display $\Q_{181}$$x + 179$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{181}$$x + 179$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{181}$$x + 179$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{181}$$x + 179$$1$$1$$0$Trivial$$[\ ]$$
181.1.2.1a1.1$x^{2} + 181$$2$$1$$1$$C_2$$$[\ ]_{2}$$
181.1.2.1a1.1$x^{2} + 181$$2$$1$$1$$C_2$$$[\ ]_{2}$$
181.1.2.1a1.1$x^{2} + 181$$2$$1$$1$$C_2$$$[\ ]_{2}$$
181.1.2.1a1.1$x^{2} + 181$$2$$1$$1$$C_2$$$[\ ]_{2}$$
181.1.2.1a1.1$x^{2} + 181$$2$$1$$1$$C_2$$$[\ ]_{2}$$
181.1.2.1a1.1$x^{2} + 181$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)