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Properties

Label 16.8.546...000.3
Degree $16$
Signature $[8, 4]$
Discriminant $5.466\times 10^{25}$
Root discriminant \(40.61\)
Ramified primes $2,5,19$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_2^3:\OD_{16}$ (as 16T252)

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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 - 16*x^14 + 122*x^12 - 868*x^10 + 3915*x^8 - 5572*x^6 + 2042*x^4 - 2924*x^2 + 1681)
 
Copy content gp:K = bnfinit(y^16 - 16*y^14 + 122*y^12 - 868*y^10 + 3915*y^8 - 5572*y^6 + 2042*y^4 - 2924*y^2 + 1681, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 16*x^14 + 122*x^12 - 868*x^10 + 3915*x^8 - 5572*x^6 + 2042*x^4 - 2924*x^2 + 1681);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 16*x^14 + 122*x^12 - 868*x^10 + 3915*x^8 - 5572*x^6 + 2042*x^4 - 2924*x^2 + 1681)
 

\( x^{16} - 16x^{14} + 122x^{12} - 868x^{10} + 3915x^{8} - 5572x^{6} + 2042x^{4} - 2924x^{2} + 1681 \) 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:  $[8, 4]$
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:   \(54660589158400000000000000\) \(\medspace = 2^{36}\cdot 5^{14}\cdot 19^{4}\) 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:  \(40.61\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{43/16}5^{7/8}19^{1/2}\approx 114.81368805634027$
Ramified primes:   \(2\), \(5\), \(19\) 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 not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{44}a^{12}+\frac{7}{44}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{3}{11}a^{4}-\frac{1}{2}a^{3}+\frac{15}{44}a^{2}-\frac{1}{2}a-\frac{3}{44}$, $\frac{1}{44}a^{13}+\frac{7}{44}a^{11}-\frac{1}{4}a^{7}+\frac{5}{22}a^{5}-\frac{1}{2}a^{4}-\frac{7}{44}a^{3}-\frac{1}{2}a^{2}+\frac{19}{44}a-\frac{1}{2}$, $\frac{1}{963963075724}a^{14}+\frac{8613982647}{963963075724}a^{12}+\frac{36537633837}{240990768931}a^{10}-\frac{3224379539}{87633006884}a^{8}+\frac{111125144861}{481981537862}a^{6}-\frac{1}{2}a^{5}-\frac{251823021841}{963963075724}a^{4}-\frac{1}{2}a^{3}-\frac{52314280861}{963963075724}a^{2}-\frac{1}{2}a+\frac{37332405051}{481981537862}$, $\frac{1}{39522486104684}a^{15}-\frac{35066777121}{3592953282244}a^{13}-\frac{1015058448771}{9880621526171}a^{11}+\frac{522573661765}{3592953282244}a^{9}+\frac{417048725827}{9880621526171}a^{7}+\frac{56773367909}{3592953282244}a^{5}+\frac{17649553109707}{39522486104684}a^{3}-\frac{9251766324653}{19761243052342}a$ 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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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);
 

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:  $11$
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:   \( -1 \)  (order $2$) 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:   $\frac{9439688}{21908251721}a^{14}-\frac{150962762}{21908251721}a^{12}+\frac{1143570372}{21908251721}a^{10}-\frac{8075092131}{21908251721}a^{8}+\frac{36128249588}{21908251721}a^{6}-\frac{47302517150}{21908251721}a^{4}-\frac{4916418844}{21908251721}a^{2}-\frac{1182831482}{21908251721}$, $\frac{119653208}{240990768931}a^{14}-\frac{166490498}{21908251721}a^{12}+\frac{13250095982}{240990768931}a^{10}-\frac{17102475507}{43816503442}a^{8}+\frac{400396569778}{240990768931}a^{6}-\frac{68504760009}{43816503442}a^{4}+\frac{17202465321}{240990768931}a^{2}-\frac{768665406165}{481981537862}$, $\frac{1406393145}{39522486104684}a^{15}-\frac{9439688}{21908251721}a^{14}-\frac{2478380805}{9880621526171}a^{13}+\frac{150962762}{21908251721}a^{12}-\frac{15569597965}{39522486104684}a^{11}-\frac{1143570372}{21908251721}a^{10}+\frac{14331494729}{3592953282244}a^{9}+\frac{8075092131}{21908251721}a^{8}-\frac{4545389840621}{39522486104684}a^{7}-\frac{36128249588}{21908251721}a^{6}+\frac{34597604560647}{39522486104684}a^{5}+\frac{47302517150}{21908251721}a^{4}-\frac{2255015352481}{1796476641122}a^{3}+\frac{4916418844}{21908251721}a^{2}+\frac{41626617582315}{39522486104684}a+\frac{1182831482}{21908251721}$, $\frac{6142248539}{39522486104684}a^{15}-\frac{102364811053}{39522486104684}a^{13}+\frac{389759951419}{19761243052342}a^{11}-\frac{488241633297}{3592953282244}a^{9}+\frac{12248877114299}{19761243052342}a^{7}-\frac{27470971672739}{39522486104684}a^{5}-\frac{3713714925623}{3592953282244}a^{3}-\frac{5011944752513}{19761243052342}a+1$, $\frac{3265147201}{3592953282244}a^{15}-\frac{556272136539}{39522486104684}a^{13}+\frac{2060949138607}{19761243052342}a^{11}-\frac{2672484545747}{3592953282244}a^{9}+\frac{5815825368393}{1796476641122}a^{7}-\frac{151748297952693}{39522486104684}a^{5}+\frac{61501723957231}{39522486104684}a^{3}-\frac{70826470836003}{19761243052342}a-1$, $\frac{3265147201}{3592953282244}a^{15}-\frac{15816640}{240990768931}a^{14}-\frac{556272136539}{39522486104684}a^{13}+\frac{15527736}{21908251721}a^{12}+\frac{2060949138607}{19761243052342}a^{11}-\frac{670821890}{240990768931}a^{10}-\frac{2672484545747}{3592953282244}a^{9}+\frac{952291245}{43816503442}a^{8}+\frac{5815825368393}{1796476641122}a^{7}-\frac{2985824310}{240990768931}a^{6}-\frac{151748297952693}{39522486104684}a^{5}-\frac{26100274291}{43816503442}a^{4}+\frac{61501723957231}{39522486104684}a^{3}-\frac{71283072605}{240990768931}a^{2}-\frac{70826470836003}{19761243052342}a+\frac{1224624651423}{481981537862}$, $\frac{6142248539}{39522486104684}a^{15}+\frac{15816640}{240990768931}a^{14}-\frac{102364811053}{39522486104684}a^{13}-\frac{15527736}{21908251721}a^{12}+\frac{389759951419}{19761243052342}a^{11}+\frac{670821890}{240990768931}a^{10}-\frac{488241633297}{3592953282244}a^{9}-\frac{952291245}{43816503442}a^{8}+\frac{12248877114299}{19761243052342}a^{7}+\frac{2985824310}{240990768931}a^{6}-\frac{27470971672739}{39522486104684}a^{5}+\frac{26100274291}{43816503442}a^{4}-\frac{3713714925623}{3592953282244}a^{3}+\frac{71283072605}{240990768931}a^{2}-\frac{5011944752513}{19761243052342}a-\frac{260661575699}{481981537862}$, $\frac{328473028}{898238320561}a^{15}+\frac{37144647}{240990768931}a^{14}-\frac{101388285469}{19761243052342}a^{13}-\frac{1271230693}{481981537862}a^{12}+\frac{339427860097}{9880621526171}a^{11}+\frac{5880587579}{240990768931}a^{10}-\frac{220434909810}{898238320561}a^{9}-\frac{7918316189}{43816503442}a^{8}+\frac{824881086423}{898238320561}a^{7}+\frac{438929927621}{481981537862}a^{6}-\frac{1015564743283}{19761243052342}a^{5}-\frac{1350783988657}{481981537862}a^{4}-\frac{8482385061329}{19761243052342}a^{3}+\frac{707256835933}{240990768931}a^{2}-\frac{29661641445538}{9880621526171}a+\frac{75698528991}{481981537862}$, $\frac{28452391501}{9880621526171}a^{15}+\frac{14159532}{21908251721}a^{14}-\frac{441834175385}{9880621526171}a^{13}-\frac{226444143}{21908251721}a^{12}+\frac{3261756274882}{9880621526171}a^{11}+\frac{1715355558}{21908251721}a^{10}-\frac{4208846550457}{1796476641122}a^{9}-\frac{24225276393}{43816503442}a^{8}+\frac{200875028943181}{19761243052342}a^{7}+\frac{54192374382}{21908251721}a^{6}-\frac{110726903153341}{9880621526171}a^{5}-\frac{70953775725}{21908251721}a^{4}+\frac{10880889232165}{19761243052342}a^{3}+\frac{7158995189}{43816503442}a^{2}-\frac{175882467529661}{19761243052342}a-\frac{67499002386}{21908251721}$, $\frac{10863602181}{39522486104684}a^{15}-\frac{769610407}{963963075724}a^{14}-\frac{16819321837}{3592953282244}a^{13}+\frac{2983044798}{240990768931}a^{12}+\frac{361419616909}{9880621526171}a^{11}-\frac{86402133635}{963963075724}a^{10}-\frac{926749963389}{3592953282244}a^{9}+\frac{56226580939}{87633006884}a^{8}+\frac{24076001678557}{19761243052342}a^{7}-\frac{2653464276205}{963963075724}a^{6}-\frac{6812497552973}{3592953282244}a^{5}+\frac{2607771764335}{963963075724}a^{4}+\frac{6066305150043}{39522486104684}a^{3}-\frac{610924001607}{481981537862}a^{2}-\frac{27889729635423}{19761243052342}a+\frac{1577013247785}{963963075724}$, $\frac{655251663201}{39522486104684}a^{15}+\frac{12285206635}{963963075724}a^{14}-\frac{2523603658291}{9880621526171}a^{13}-\frac{188559199633}{963963075724}a^{12}+\frac{74015049997615}{39522486104684}a^{11}+\frac{689638451995}{481981537862}a^{10}-\frac{47787696847399}{3592953282244}a^{9}-\frac{891195589903}{87633006884}a^{8}+\frac{22\cdots 93}{39522486104684}a^{7}+\frac{20978609066251}{481981537862}a^{6}-\frac{23\cdots 11}{39522486104684}a^{5}-\frac{42987056436531}{963963075724}a^{4}+\frac{17070896034888}{9880621526171}a^{3}+\frac{2138462287961}{963963075724}a^{2}-\frac{18\cdots 41}{39522486104684}a-\frac{17358153185743}{481981537862}$ 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:  \( 5126884.12499 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 5126884.12499 \cdot 2}{2\cdot\sqrt{54660589158400000000000000}}\cr\approx \mathstrut & 0.276678738201 \end{aligned}\] (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 - 16*x^14 + 122*x^12 - 868*x^10 + 3915*x^8 - 5572*x^6 + 2042*x^4 - 2924*x^2 + 1681) 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 - 16*x^14 + 122*x^12 - 868*x^10 + 3915*x^8 - 5572*x^6 + 2042*x^4 - 2924*x^2 + 1681, 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 - 16*x^14 + 122*x^12 - 868*x^10 + 3915*x^8 - 5572*x^6 + 2042*x^4 - 2924*x^2 + 1681); 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 = PolynomialRing(QQ); K, a = NumberField(x^16 - 16*x^14 + 122*x^12 - 868*x^10 + 3915*x^8 - 5572*x^6 + 2042*x^4 - 2924*x^2 + 1681); 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

$C_2^3:\OD_{16}$ (as 16T252):

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); G, transitive_group_identification(G)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^3:\OD_{16}$
Character table for $C_2^3:\OD_{16}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.4.115520000000.2, 8.8.5120000000.1, 8.4.369664000000.12

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.8.54660589158400000000000000.4

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ R ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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

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
\(2\) Copy content Toggle raw display 2.2.8.36b15.29$x^{16} + 8 x^{15} + 38 x^{14} + 128 x^{13} + 336 x^{12} + 712 x^{11} + 1254 x^{10} + 1860 x^{9} + 2351 x^{8} + 2540 x^{7} + 2346 x^{6} + 1836 x^{5} + 1198 x^{4} + 632 x^{3} + 260 x^{2} + 80 x + 17$$8$$2$$36$16T252$$[2, 2, 2, 3, 3]^{4}$$
\(5\) Copy content Toggle raw display 5.2.8.14a1.3$x^{16} + 32 x^{15} + 464 x^{14} + 4032 x^{13} + 23408 x^{12} + 95872 x^{11} + 285376 x^{10} + 627456 x^{9} + 1027168 x^{8} + 1254912 x^{7} + 1141504 x^{6} + 766976 x^{5} + 374528 x^{4} + 129024 x^{3} + 29696 x^{2} + 4096 x + 261$$8$$2$$14$$C_8: C_2$$$[\ ]_{8}^{2}$$
\(19\) Copy content Toggle raw display $\Q_{19}$$x + 17$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{19}$$x + 17$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{19}$$x + 17$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{19}$$x + 17$$1$$1$$0$Trivial$$[\ ]$$
19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.1.2.1a1.2$x^{2} + 38$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.1.2.1a1.2$x^{2} + 38$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.2.1.0a1.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
19.2.2.2a1.2$x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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