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Properties

Label 18.14.115...424.2
Degree $18$
Signature $[14, 2]$
Discriminant $1.152\times 10^{25}$
Root discriminant \(24.68\)
Ramified primes $2,37,16361$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_4^3.S_4$ (as 18T883)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^16 + 13*x^14 + 29*x^12 - 85*x^10 + 30*x^8 + 58*x^6 - 51*x^4 + 13*x^2 - 1)
 
Copy content gp:K = bnfinit(y^18 - 8*y^16 + 13*y^14 + 29*y^12 - 85*y^10 + 30*y^8 + 58*y^6 - 51*y^4 + 13*y^2 - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 8*x^16 + 13*x^14 + 29*x^12 - 85*x^10 + 30*x^8 + 58*x^6 - 51*x^4 + 13*x^2 - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 8*x^16 + 13*x^14 + 29*x^12 - 85*x^10 + 30*x^8 + 58*x^6 - 51*x^4 + 13*x^2 - 1)
 

\( x^{18} - 8x^{16} + 13x^{14} + 29x^{12} - 85x^{10} + 30x^{8} + 58x^{6} - 51x^{4} + 13x^{2} - 1 \) 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:  $18$
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:  $[14, 2]$
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:   \(11522585241472304020455424\) \(\medspace = 2^{24}\cdot 37^{6}\cdot 16361^{2}\) 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:  \(24.68\)
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:  not computed
Ramified primes:   \(2\), \(37\), \(16361\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}+\frac{3}{8}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}+\frac{3}{8}a+\frac{3}{8}$, $\frac{1}{8}a^{16}-\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{1}{4}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}+\frac{3}{8}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{8}a^{17}-\frac{1}{8}a^{14}-\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{8}a^{9}+\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-\frac{3}{8}a^{4}+\frac{1}{4}a^{3}-\frac{3}{8}a^{2}+\frac{1}{8}a+\frac{3}{8}$ 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:  Trivial group, which has order $1$ (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);
 

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:  $15$
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:   $a$, $a^{17}-\frac{15}{2}a^{15}+\frac{19}{2}a^{13}+32a^{11}-\frac{135}{2}a^{9}+5a^{7}+48a^{5}-\frac{65}{2}a^{3}+7a$, $\frac{11}{4}a^{16}-\frac{41}{2}a^{14}+\frac{99}{4}a^{12}+92a^{10}-\frac{731}{4}a^{8}-10a^{6}+\frac{291}{2}a^{4}-69a^{2}+\frac{33}{4}$, $\frac{11}{4}a^{17}-21a^{15}+\frac{113}{4}a^{13}+89a^{11}-\frac{801}{4}a^{9}+15a^{7}+\frac{311}{2}a^{5}-\frac{175}{2}a^{3}+\frac{53}{4}a$, $\frac{5}{4}a^{17}-11a^{15}+\frac{93}{4}a^{13}+\frac{61}{2}a^{11}-\frac{571}{4}a^{9}+\frac{175}{2}a^{7}+\frac{203}{2}a^{5}-\frac{213}{2}a^{3}+\frac{81}{4}a$, $\frac{7}{4}a^{17}-13a^{15}+\frac{61}{4}a^{13}+60a^{11}-\frac{461}{4}a^{9}-15a^{7}+\frac{195}{2}a^{5}-\frac{73}{2}a^{3}+\frac{1}{4}a$, $\frac{3}{2}a^{17}-\frac{21}{2}a^{15}+\frac{35}{4}a^{13}+54a^{11}-75a^{9}-\frac{77}{2}a^{7}+\frac{237}{4}a^{5}-\frac{19}{2}a^{3}+\frac{5}{4}a$, $\frac{1}{2}a^{17}+\frac{35}{8}a^{16}-\frac{21}{4}a^{15}-\frac{261}{8}a^{14}+\frac{121}{8}a^{13}+\frac{79}{2}a^{12}+8a^{11}+146a^{10}-\frac{351}{4}a^{9}-\frac{2327}{8}a^{8}+73a^{7}-\frac{115}{8}a^{6}+\frac{521}{8}a^{5}+\frac{1839}{8}a^{4}-\frac{143}{2}a^{3}-\frac{861}{8}a^{2}+\frac{89}{8}a+\frac{47}{4}$, $\frac{27}{8}a^{17}-\frac{17}{8}a^{16}-25a^{15}+\frac{63}{4}a^{14}+\frac{117}{4}a^{13}-\frac{147}{8}a^{12}+\frac{455}{4}a^{11}-\frac{289}{4}a^{10}-\frac{1741}{8}a^{9}+\frac{1101}{8}a^{8}-\frac{83}{4}a^{7}+\frac{65}{4}a^{6}+\frac{1359}{8}a^{5}-\frac{441}{4}a^{4}-75a^{3}+45a^{2}+10a-\frac{33}{8}$, $\frac{3}{8}a^{17}+\frac{9}{8}a^{16}-3a^{15}-\frac{35}{4}a^{14}+\frac{19}{4}a^{13}+\frac{101}{8}a^{12}+\frac{47}{4}a^{11}+\frac{143}{4}a^{10}-\frac{261}{8}a^{9}-\frac{701}{8}a^{8}+\frac{27}{4}a^{7}+\frac{49}{4}a^{6}+\frac{227}{8}a^{5}+\frac{139}{2}a^{4}-\frac{31}{2}a^{3}-\frac{81}{2}a^{2}-a+\frac{39}{8}$, $\frac{3}{8}a^{17}+\frac{9}{8}a^{16}-3a^{15}-\frac{35}{4}a^{14}+\frac{19}{4}a^{13}+\frac{101}{8}a^{12}+\frac{47}{4}a^{11}+\frac{143}{4}a^{10}-\frac{261}{8}a^{9}-\frac{701}{8}a^{8}+\frac{27}{4}a^{7}+\frac{49}{4}a^{6}+\frac{227}{8}a^{5}+\frac{139}{2}a^{4}-\frac{31}{2}a^{3}-\frac{81}{2}a^{2}-a+\frac{31}{8}$, $\frac{15}{4}a^{17}-\frac{57}{2}a^{15}+\frac{151}{4}a^{13}+121a^{11}-\frac{1071}{4}a^{9}+20a^{7}+\frac{407}{2}a^{5}-119a^{3}+\frac{73}{4}a$, $\frac{11}{4}a^{17}+\frac{3}{4}a^{16}-\frac{171}{8}a^{15}-\frac{23}{4}a^{14}+\frac{249}{8}a^{13}+\frac{63}{8}a^{12}+\frac{341}{4}a^{11}+24a^{10}-\frac{853}{4}a^{9}-55a^{8}+\frac{335}{8}a^{7}+\frac{11}{2}a^{6}+\frac{1273}{8}a^{5}+\frac{329}{8}a^{4}-\frac{869}{8}a^{3}-25a^{2}+\frac{149}{8}a+\frac{31}{8}$, $5a^{17}+\frac{11}{8}a^{16}-\frac{293}{8}a^{15}-\frac{83}{8}a^{14}+\frac{81}{2}a^{13}+\frac{107}{8}a^{12}+\frac{683}{4}a^{11}+\frac{89}{2}a^{10}-\frac{1233}{4}a^{9}-\frac{765}{8}a^{8}-\frac{401}{8}a^{7}+\frac{41}{8}a^{6}+\frac{977}{4}a^{5}+\frac{291}{4}a^{4}-\frac{761}{8}a^{3}-\frac{337}{8}a^{2}+\frac{19}{2}a+\frac{45}{8}$, $\frac{9}{8}a^{17}+\frac{7}{4}a^{16}-\frac{37}{4}a^{15}-\frac{111}{8}a^{14}+\frac{131}{8}a^{13}+\frac{173}{8}a^{12}+\frac{125}{4}a^{11}+\frac{213}{4}a^{10}-\frac{841}{8}a^{9}-\frac{583}{4}a^{8}+\frac{185}{4}a^{7}+\frac{295}{8}a^{6}+\frac{299}{4}a^{5}+\frac{889}{8}a^{4}-69a^{3}-\frac{609}{8}a^{2}+\frac{105}{8}a+\frac{93}{8}$ 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:  \( 1828417.55183 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{2}\cdot 1828417.55183 \cdot 1}{2\cdot\sqrt{11522585241472304020455424}}\cr\approx \mathstrut & 0.174200780762 \end{aligned}\] (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^18 - 8*x^16 + 13*x^14 + 29*x^12 - 85*x^10 + 30*x^8 + 58*x^6 - 51*x^4 + 13*x^2 - 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 - 8*x^16 + 13*x^14 + 29*x^12 - 85*x^10 + 30*x^8 + 58*x^6 - 51*x^4 + 13*x^2 - 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 8*x^16 + 13*x^14 + 29*x^12 - 85*x^10 + 30*x^8 + 58*x^6 - 51*x^4 + 13*x^2 - 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 8*x^16 + 13*x^14 + 29*x^12 - 85*x^10 + 30*x^8 + 58*x^6 - 51*x^4 + 13*x^2 - 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

$S_4^3.S_4$ (as 18T883):

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 331776
The 165 conjugacy class representatives for $S_4^3.S_4$
Character table for $S_4^3.S_4$

Intermediate fields

3.3.148.1, 9.9.53038958912.1

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

Sibling fields

Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 siblings: data not computed
Minimal sibling: 18.14.9549155080976048926982931891617792.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.9.0.1}{9} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ R ${\href{/padicField/41.9.0.1}{9} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.3.6.24a3.14$x^{18} + 8 x^{16} + 8 x^{15} + 25 x^{14} + 50 x^{13} + 65 x^{12} + 120 x^{11} + 157 x^{10} + 182 x^{9} + 232 x^{8} + 232 x^{7} + 210 x^{6} + 196 x^{5} + 145 x^{4} + 90 x^{3} + 61 x^{2} + 26 x + 7$$6$$3$$24$18T463$$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$$
\(37\) Copy content Toggle raw display 37.1.2.1a1.1$x^{2} + 37$$2$$1$$1$$C_2$$$[\ ]_{2}$$
37.1.2.1a1.1$x^{2} + 37$$2$$1$$1$$C_2$$$[\ ]_{2}$$
37.3.1.0a1.1$x^{3} + 6 x + 35$$1$$3$$0$$C_3$$$[\ ]^{3}$$
37.3.1.0a1.1$x^{3} + 6 x + 35$$1$$3$$0$$C_3$$$[\ ]^{3}$$
37.4.2.4a1.2$x^{8} + 12 x^{6} + 48 x^{5} + 40 x^{4} + 288 x^{3} + 600 x^{2} + 96 x + 41$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(16361\) Copy content Toggle raw display $\Q_{16361}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{16361}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{16361}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{16361}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{16361}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{16361}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{16361}$$x$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{16361}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $4$$1$$4$$0$$C_4$$$[\ ]^{4}$$
Deg $4$$2$$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)