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Properties

Label 18.0.174...000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-1.744\times 10^{22}$
Root discriminant \(17.20\)
Ramified primes $2,3,5,7$
Class number $1$
Class group trivial
Galois group $S_3 \times C_6$ (as 18T6)

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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 + x^16 - 6*x^15 - 4*x^14 - 5*x^13 - 6*x^12 + 16*x^11 + 22*x^10 + 43*x^9 + 41*x^8 - 34*x^7 + 121*x^6 - 49*x^5 + 17*x^4 + 15*x^3 - 4*x^2 - x + 1)
 
Copy content gp:K = bnfinit(y^18 + y^16 - 6*y^15 - 4*y^14 - 5*y^13 - 6*y^12 + 16*y^11 + 22*y^10 + 43*y^9 + 41*y^8 - 34*y^7 + 121*y^6 - 49*y^5 + 17*y^4 + 15*y^3 - 4*y^2 - y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + x^16 - 6*x^15 - 4*x^14 - 5*x^13 - 6*x^12 + 16*x^11 + 22*x^10 + 43*x^9 + 41*x^8 - 34*x^7 + 121*x^6 - 49*x^5 + 17*x^4 + 15*x^3 - 4*x^2 - x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + x^16 - 6*x^15 - 4*x^14 - 5*x^13 - 6*x^12 + 16*x^11 + 22*x^10 + 43*x^9 + 41*x^8 - 34*x^7 + 121*x^6 - 49*x^5 + 17*x^4 + 15*x^3 - 4*x^2 - x + 1)
 

\( x^{18} + x^{16} - 6 x^{15} - 4 x^{14} - 5 x^{13} - 6 x^{12} + 16 x^{11} + 22 x^{10} + 43 x^{9} + \cdots + 1 \) 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:  $[0, 9]$
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:   \(-17436035582546112000000\) \(\medspace = -\,2^{12}\cdot 3^{9}\cdot 5^{6}\cdot 7^{12}\) 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:  \(17.20\)
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:  $2\cdot 3^{1/2}5^{1/2}7^{2/3}\approx 28.344860147201352$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_6$
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.
Maximal CM subfield:  6.0.64827.1

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2938}a^{15}+\frac{111}{1469}a^{13}-\frac{5}{2938}a^{12}+\frac{581}{2938}a^{11}-\frac{444}{1469}a^{10}-\frac{441}{1469}a^{9}+\frac{1195}{2938}a^{8}-\frac{411}{2938}a^{7}-\frac{582}{1469}a^{6}-\frac{1455}{2938}a^{5}+\frac{855}{2938}a^{4}-\frac{887}{2938}a^{3}-\frac{581}{2938}a^{2}+\frac{111}{1469}a-\frac{1}{2938}$, $\frac{1}{208598}a^{16}+\frac{29}{208598}a^{15}-\frac{27689}{208598}a^{14}+\frac{9827}{104299}a^{13}-\frac{21599}{208598}a^{12}+\frac{77659}{208598}a^{11}-\frac{10475}{208598}a^{10}-\frac{49651}{104299}a^{9}-\frac{61241}{208598}a^{8}+\frac{57429}{208598}a^{7}+\frac{36013}{104299}a^{6}+\frac{1453}{16046}a^{5}-\frac{14488}{104299}a^{4}-\frac{68905}{208598}a^{3}+\frac{4597}{16046}a^{2}-\frac{454}{104299}a-\frac{24253}{104299}$, $\frac{1}{208598}a^{17}+\frac{6}{104299}a^{15}-\frac{11757}{208598}a^{14}+\frac{4157}{104299}a^{13}+\frac{39825}{208598}a^{12}-\frac{36473}{104299}a^{11}+\frac{99535}{208598}a^{10}+\frac{34379}{104299}a^{9}-\frac{42047}{208598}a^{8}+\frac{26095}{208598}a^{7}-\frac{39691}{208598}a^{6}-\frac{36435}{104299}a^{5}+\frac{19375}{104299}a^{4}-\frac{1}{2938}a^{3}-\frac{32277}{104299}a^{2}-\frac{48089}{208598}a+\frac{22245}{208598}$ 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 $1$
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:  Trivial group, which has order $1$
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:  $8$
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:   \( -\frac{14}{113} a^{17} - \frac{305}{1469} a^{16} - \frac{19}{113} a^{15} + \frac{774}{1469} a^{14} + \frac{2513}{1469} a^{13} + \frac{2507}{1469} a^{12} + \frac{2975}{1469} a^{11} - \frac{805}{1469} a^{10} - \frac{8509}{1469} a^{9} - \frac{15565}{1469} a^{8} - \frac{22435}{1469} a^{7} - \frac{9210}{1469} a^{6} - \frac{14568}{1469} a^{5} - \frac{25462}{1469} a^{4} + \frac{5644}{1469} a^{3} - \frac{6389}{1469} a^{2} - \frac{1671}{1469} a + \frac{75}{113} \)  (order $6$) 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:   $\frac{590}{8023}a^{17}+\frac{248}{8023}a^{16}-\frac{3111}{104299}a^{15}-\frac{3942}{8023}a^{14}-\frac{63340}{104299}a^{13}+\frac{7079}{104299}a^{12}+\frac{33374}{104299}a^{11}+\frac{209186}{104299}a^{10}+\frac{324745}{104299}a^{9}+\frac{258912}{104299}a^{8}+\frac{59000}{104299}a^{7}-\frac{841917}{104299}a^{6}-\frac{20622}{104299}a^{5}+\frac{50998}{104299}a^{4}-\frac{920323}{104299}a^{3}-\frac{98}{1469}a^{2}+\frac{178}{104299}a-\frac{73108}{104299}$, $\frac{1218}{8023}a^{17}-\frac{248}{8023}a^{16}+\frac{4354}{104299}a^{15}-\frac{110}{113}a^{14}-\frac{52937}{104299}a^{13}+\frac{4334}{104299}a^{12}-\frac{8966}{104299}a^{11}+\frac{332310}{104299}a^{10}+\frac{340260}{104299}a^{9}+\frac{399426}{104299}a^{8}+\frac{160220}{104299}a^{7}-\frac{1202818}{104299}a^{6}+\frac{1574598}{104299}a^{5}-\frac{627637}{104299}a^{4}-\frac{531840}{104299}a^{3}+\frac{408560}{104299}a^{2}-\frac{67978}{104299}a-\frac{67690}{104299}$, $\frac{3515}{16046}a^{17}+\frac{9047}{104299}a^{16}+\frac{37105}{208598}a^{15}-\frac{3623}{2938}a^{14}-\frac{11761}{8023}a^{13}-\frac{253273}{208598}a^{12}-\frac{330919}{208598}a^{11}+\frac{703947}{208598}a^{10}+\frac{698234}{104299}a^{9}+\frac{175573}{16046}a^{8}+\frac{1242288}{104299}a^{7}-\frac{11509}{1846}a^{6}+\frac{4315333}{208598}a^{5}-\frac{63695}{104299}a^{4}-\frac{764584}{104299}a^{3}+\frac{688394}{104299}a^{2}-\frac{322577}{208598}a-\frac{219885}{208598}$, $\frac{97925}{208598}a^{17}+\frac{1913}{16046}a^{16}+\frac{81115}{208598}a^{15}-\frac{589101}{208598}a^{14}-\frac{282541}{104299}a^{13}-\frac{506043}{208598}a^{12}-\frac{483123}{208598}a^{11}+\frac{1663761}{208598}a^{10}+\frac{1383683}{104299}a^{9}+\frac{4583205}{208598}a^{8}+\frac{2112814}{104299}a^{7}-\frac{1911387}{104299}a^{6}+\frac{9128143}{208598}a^{5}-\frac{85385}{8023}a^{4}-\frac{170697}{104299}a^{3}+\frac{677}{226}a^{2}+\frac{180699}{208598}a+\frac{10667}{104299}$, $\frac{670}{8023}a^{17}+\frac{592}{8023}a^{16}+\frac{11935}{208598}a^{15}-\frac{99}{226}a^{14}-\frac{164137}{208598}a^{13}-\frac{112247}{208598}a^{12}-\frac{138311}{208598}a^{11}+\frac{206579}{208598}a^{10}+\frac{613927}{208598}a^{9}+\frac{946711}{208598}a^{8}+\frac{1165429}{208598}a^{7}-\frac{166173}{208598}a^{6}+\frac{753591}{104299}a^{5}+\frac{726900}{104299}a^{4}-\frac{271916}{104299}a^{3}+\frac{258481}{104299}a^{2}-\frac{46357}{208598}a-\frac{81615}{208598}$, $\frac{19207}{104299}a^{17}+\frac{52761}{208598}a^{16}+\frac{1554}{104299}a^{15}-\frac{105281}{104299}a^{14}-\frac{259085}{104299}a^{13}-\frac{211505}{208598}a^{12}-\frac{77174}{104299}a^{11}+\frac{347773}{104299}a^{10}+\frac{1024825}{104299}a^{9}+\frac{2365221}{208598}a^{8}+\frac{1240245}{104299}a^{7}-\frac{1714613}{208598}a^{6}-\frac{56753}{104299}a^{5}+\frac{4356945}{208598}a^{4}-\frac{2320704}{104299}a^{3}+\frac{5158}{1469}a^{2}+\frac{98362}{104299}a-\frac{18266}{104299}$, $\frac{963}{208598}a^{17}-\frac{50355}{208598}a^{16}-\frac{2264}{104299}a^{15}-\frac{23568}{104299}a^{14}+\frac{304013}{208598}a^{13}+\frac{3459}{2938}a^{12}+\frac{111862}{104299}a^{11}+\frac{119870}{104299}a^{10}-\frac{895465}{208598}a^{9}-\frac{11471}{1846}a^{8}-\frac{2140887}{208598}a^{7}-\frac{144039}{16046}a^{6}+\frac{2454527}{208598}a^{5}-\frac{4976831}{208598}a^{4}+\frac{999605}{104299}a^{3}+\frac{34729}{104299}a^{2}-\frac{241676}{104299}a+\frac{103281}{104299}$, $\frac{428939}{208598}a^{17}+\frac{120515}{208598}a^{16}+\frac{218615}{104299}a^{15}-\frac{1235690}{104299}a^{14}-\frac{2446189}{208598}a^{13}-\frac{2701499}{208598}a^{12}-\frac{119737}{8023}a^{11}+\frac{3129902}{104299}a^{10}+\frac{11510625}{208598}a^{9}+\frac{1647963}{16046}a^{8}+\frac{22771833}{208598}a^{7}-\frac{9925031}{208598}a^{6}+\frac{46936873}{208598}a^{5}-\frac{8259605}{208598}a^{4}+\frac{1109027}{104299}a^{3}+\frac{3143518}{104299}a^{2}-\frac{247888}{104299}a-\frac{536918}{104299}$ Copy content Toggle raw display
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:  \( 13584.372345087919 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 13584.372345087919 \cdot 1}{6\cdot\sqrt{17436035582546112000000}}\cr\approx \mathstrut & 0.261687674911187 \end{aligned}\]

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 + x^16 - 6*x^15 - 4*x^14 - 5*x^13 - 6*x^12 + 16*x^11 + 22*x^10 + 43*x^9 + 41*x^8 - 34*x^7 + 121*x^6 - 49*x^5 + 17*x^4 + 15*x^3 - 4*x^2 - x + 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 + x^16 - 6*x^15 - 4*x^14 - 5*x^13 - 6*x^12 + 16*x^11 + 22*x^10 + 43*x^9 + 41*x^8 - 34*x^7 + 121*x^6 - 49*x^5 + 17*x^4 + 15*x^3 - 4*x^2 - x + 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 + x^16 - 6*x^15 - 4*x^14 - 5*x^13 - 6*x^12 + 16*x^11 + 22*x^10 + 43*x^9 + 41*x^8 - 34*x^7 + 121*x^6 - 49*x^5 + 17*x^4 + 15*x^3 - 4*x^2 - x + 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 + x^16 - 6*x^15 - 4*x^14 - 5*x^13 - 6*x^12 + 16*x^11 + 22*x^10 + 43*x^9 + 41*x^8 - 34*x^7 + 121*x^6 - 49*x^5 + 17*x^4 + 15*x^3 - 4*x^2 - x + 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

$C_6\times S_3$ (as 18T6):

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 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.980.1, \(\Q(\zeta_{7})^+\), 6.0.25930800.1, 6.0.64827.1, 9.3.941192000.1

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

Galois closure: data not computed
Degree 12 sibling: 12.0.112021056000000.2
Degree 18 sibling: 18.6.139488284660368896000000000.1
Minimal sibling: 12.0.112021056000000.2

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 R ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.6.1.0a1.1$x^{6} + x^{4} + x^{3} + x + 1$$1$$6$$0$$C_6$$$[\ ]^{6}$$
2.6.2.12a1.1$x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 5 x^{6} + 2 x^{5} + 6 x^{4} + 4 x^{3} + x^{2} + 4 x + 5$$2$$6$$12$$C_6\times C_2$$$[2]^{6}$$
\(3\) Copy content Toggle raw display 3.3.2.3a1.2$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
3.3.2.3a1.2$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
3.3.2.3a1.2$x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(5\) Copy content Toggle raw display 5.6.1.0a1.1$x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
5.6.2.6a1.2$x^{12} + 2 x^{10} + 8 x^{9} + 3 x^{8} + 8 x^{7} + 22 x^{6} + 8 x^{5} + 5 x^{4} + 16 x^{3} + 4 x^{2} + 9$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(7\) Copy content Toggle raw display 7.3.3.6a1.3$x^{9} + 18 x^{8} + 108 x^{7} + 228 x^{6} + 144 x^{5} + 432 x^{4} + 48 x^{3} + 288 x^{2} + 71$$3$$3$$6$$C_3^2$$$[\ ]_{3}^{3}$$
7.3.3.6a1.3$x^{9} + 18 x^{8} + 108 x^{7} + 228 x^{6} + 144 x^{5} + 432 x^{4} + 48 x^{3} + 288 x^{2} + 71$$3$$3$$6$$C_3^2$$$[\ ]_{3}^{3}$$

Spectrum of ring of integers

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