↑

Properties

Label 15.3.283...000.2
Degree $15$
Signature $(3, 6)$
Discriminant $2.834\times 10^{24}$
Root discriminant \(42.67\)
Ramified primes $2,5,11$
Class number $1$
Class group trivial
Galois group $C_3^4:(S_3\times F_5)$ (as 15T64)

Related objects

Downloads

Learn more

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^15 - 5*x^14 - 5*x^13 + 35*x^12 + 145*x^11 - 263*x^10 - 540*x^9 - 180*x^8 + 1900*x^7 + 800*x^6 - 977*x^5 - 925*x^4 - 965*x^3 + 835*x^2 - 5*x + 89)
 
Copy content gp:K = bnfinit(y^15 - 5*y^14 - 5*y^13 + 35*y^12 + 145*y^11 - 263*y^10 - 540*y^9 - 180*y^8 + 1900*y^7 + 800*y^6 - 977*y^5 - 925*y^4 - 965*y^3 + 835*y^2 - 5*y + 89, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^14 - 5*x^13 + 35*x^12 + 145*x^11 - 263*x^10 - 540*x^9 - 180*x^8 + 1900*x^7 + 800*x^6 - 977*x^5 - 925*x^4 - 965*x^3 + 835*x^2 - 5*x + 89);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^15 - 5*x^14 - 5*x^13 + 35*x^12 + 145*x^11 - 263*x^10 - 540*x^9 - 180*x^8 + 1900*x^7 + 800*x^6 - 977*x^5 - 925*x^4 - 965*x^3 + 835*x^2 - 5*x + 89)
 

\( x^{15} - 5 x^{14} - 5 x^{13} + 35 x^{12} + 145 x^{11} - 263 x^{10} - 540 x^{9} - 180 x^{8} + 1900 x^{7} + \cdots + 89 \) 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:  $15$
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:  $(3, 6)$
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:   \(2834497600000000000000000\) \(\medspace = 2^{21}\cdot 5^{17}\cdot 11^{6}\) 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:  \(42.67\)
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:  not computed
Ramified primes:   \(2\), \(5\), \(11\) 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:  \(\Q(\sqrt{10}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{51\cdots 18}a^{14}+\frac{10\cdots 23}{25\cdots 59}a^{13}-\frac{10\cdots 69}{51\cdots 18}a^{12}+\frac{85\cdots 50}{25\cdots 59}a^{11}+\frac{98\cdots 93}{51\cdots 18}a^{10}-\frac{10\cdots 31}{25\cdots 59}a^{9}-\frac{33\cdots 97}{25\cdots 59}a^{8}-\frac{12\cdots 01}{25\cdots 59}a^{7}-\frac{40\cdots 04}{25\cdots 59}a^{6}-\frac{72\cdots 78}{25\cdots 59}a^{5}-\frac{45\cdots 73}{51\cdots 18}a^{4}+\frac{30\cdots 57}{25\cdots 59}a^{3}-\frac{17\cdots 13}{51\cdots 18}a^{2}+\frac{11\cdots 57}{25\cdots 59}a+\frac{18\cdots 53}{51\cdots 18}$ 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:   \( -1 \)  (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:   $\frac{72\cdots 03}{25\cdots 59}a^{14}-\frac{42\cdots 57}{25\cdots 59}a^{13}-\frac{54\cdots 00}{25\cdots 59}a^{12}+\frac{29\cdots 24}{25\cdots 59}a^{11}+\frac{83\cdots 74}{25\cdots 59}a^{10}-\frac{28\cdots 20}{25\cdots 59}a^{9}-\frac{23\cdots 01}{25\cdots 59}a^{8}+\frac{24\cdots 70}{25\cdots 59}a^{7}+\frac{15\cdots 26}{25\cdots 59}a^{6}-\frac{65\cdots 52}{25\cdots 59}a^{5}-\frac{13\cdots 13}{25\cdots 59}a^{4}+\frac{44\cdots 95}{25\cdots 59}a^{3}+\frac{83\cdots 24}{25\cdots 59}a^{2}+\frac{11\cdots 08}{25\cdots 59}a-\frac{58\cdots 72}{25\cdots 59}$, $\frac{19\cdots 43}{25\cdots 59}a^{14}-\frac{67\cdots 66}{25\cdots 59}a^{13}-\frac{23\cdots 26}{25\cdots 59}a^{12}+\frac{45\cdots 08}{25\cdots 59}a^{11}+\frac{37\cdots 66}{25\cdots 59}a^{10}-\frac{18\cdots 23}{25\cdots 59}a^{9}-\frac{14\cdots 97}{25\cdots 59}a^{8}-\frac{19\cdots 72}{25\cdots 59}a^{7}+\frac{20\cdots 24}{25\cdots 59}a^{6}+\frac{56\cdots 47}{25\cdots 59}a^{5}+\frac{15\cdots 19}{25\cdots 59}a^{4}-\frac{21\cdots 97}{25\cdots 59}a^{3}-\frac{56\cdots 36}{25\cdots 59}a^{2}-\frac{20\cdots 95}{25\cdots 59}a-\frac{72\cdots 24}{25\cdots 59}$, $\frac{73\cdots 82}{25\cdots 59}a^{14}-\frac{39\cdots 20}{25\cdots 59}a^{13}-\frac{36\cdots 78}{25\cdots 59}a^{12}+\frac{31\cdots 32}{25\cdots 59}a^{11}+\frac{10\cdots 60}{25\cdots 59}a^{10}-\frac{25\cdots 02}{25\cdots 59}a^{9}-\frac{52\cdots 48}{25\cdots 59}a^{8}+\frac{50\cdots 43}{25\cdots 59}a^{7}+\frac{19\cdots 61}{25\cdots 59}a^{6}+\frac{88\cdots 34}{25\cdots 59}a^{5}-\frac{16\cdots 52}{25\cdots 59}a^{4}-\frac{11\cdots 32}{25\cdots 59}a^{3}-\frac{29\cdots 82}{25\cdots 59}a^{2}+\frac{64\cdots 37}{25\cdots 59}a+\frac{99\cdots 44}{25\cdots 59}$, $\frac{28\cdots 92}{25\cdots 59}a^{14}-\frac{99\cdots 64}{25\cdots 59}a^{13}-\frac{32\cdots 91}{25\cdots 59}a^{12}+\frac{59\cdots 41}{25\cdots 59}a^{11}+\frac{54\cdots 21}{25\cdots 59}a^{10}+\frac{11\cdots 40}{25\cdots 59}a^{9}-\frac{21\cdots 49}{25\cdots 59}a^{8}-\frac{38\cdots 89}{25\cdots 59}a^{7}+\frac{27\cdots 26}{25\cdots 59}a^{6}+\frac{97\cdots 30}{25\cdots 59}a^{5}+\frac{67\cdots 75}{25\cdots 59}a^{4}-\frac{47\cdots 72}{25\cdots 59}a^{3}-\frac{81\cdots 53}{25\cdots 59}a^{2}-\frac{11\cdots 01}{25\cdots 59}a-\frac{74\cdots 47}{25\cdots 59}$, $\frac{48\cdots 89}{25\cdots 59}a^{14}-\frac{52\cdots 23}{25\cdots 59}a^{13}+\frac{14\cdots 42}{25\cdots 59}a^{12}+\frac{22\cdots 61}{25\cdots 59}a^{11}+\frac{11\cdots 50}{25\cdots 59}a^{10}-\frac{35\cdots 41}{25\cdots 59}a^{9}+\frac{60\cdots 59}{25\cdots 59}a^{8}-\frac{11\cdots 68}{25\cdots 59}a^{7}+\frac{27\cdots 05}{25\cdots 59}a^{6}-\frac{93\cdots 49}{25\cdots 59}a^{5}+\frac{69\cdots 89}{25\cdots 59}a^{4}-\frac{15\cdots 57}{25\cdots 59}a^{3}-\frac{71\cdots 72}{25\cdots 59}a^{2}+\frac{10\cdots 25}{25\cdots 59}a+\frac{11\cdots 39}{25\cdots 59}$, $\frac{16\cdots 19}{25\cdots 59}a^{14}-\frac{15\cdots 23}{25\cdots 59}a^{13}+\frac{15\cdots 12}{25\cdots 59}a^{12}+\frac{17\cdots 06}{25\cdots 59}a^{11}-\frac{99\cdots 15}{25\cdots 59}a^{10}-\frac{17\cdots 88}{25\cdots 59}a^{9}+\frac{30\cdots 18}{25\cdots 59}a^{8}+\frac{73\cdots 62}{25\cdots 59}a^{7}+\frac{15\cdots 50}{25\cdots 59}a^{6}-\frac{58\cdots 08}{25\cdots 59}a^{5}-\frac{72\cdots 89}{25\cdots 59}a^{4}-\frac{61\cdots 68}{25\cdots 59}a^{3}+\frac{19\cdots 75}{25\cdots 59}a^{2}-\frac{31\cdots 42}{25\cdots 59}a+\frac{22\cdots 32}{25\cdots 59}$, $\frac{11\cdots 51}{25\cdots 59}a^{14}-\frac{43\cdots 07}{25\cdots 59}a^{13}-\frac{10\cdots 40}{25\cdots 59}a^{12}+\frac{27\cdots 21}{25\cdots 59}a^{11}+\frac{19\cdots 19}{25\cdots 59}a^{10}-\frac{66\cdots 09}{25\cdots 59}a^{9}-\frac{69\cdots 33}{25\cdots 59}a^{8}-\frac{10\cdots 59}{25\cdots 59}a^{7}+\frac{93\cdots 58}{25\cdots 59}a^{6}+\frac{20\cdots 75}{25\cdots 59}a^{5}+\frac{14\cdots 41}{25\cdots 59}a^{4}+\frac{61\cdots 76}{25\cdots 59}a^{3}-\frac{60\cdots 50}{25\cdots 59}a^{2}+\frac{15\cdots 73}{25\cdots 59}a-\frac{10\cdots 57}{25\cdots 59}$, $\frac{94\cdots 66}{25\cdots 59}a^{14}+\frac{58\cdots 06}{25\cdots 59}a^{13}-\frac{34\cdots 71}{25\cdots 59}a^{12}+\frac{20\cdots 78}{25\cdots 59}a^{11}+\frac{35\cdots 38}{25\cdots 59}a^{10}+\frac{39\cdots 61}{25\cdots 59}a^{9}-\frac{24\cdots 97}{25\cdots 59}a^{8}-\frac{22\cdots 83}{25\cdots 59}a^{7}+\frac{34\cdots 87}{25\cdots 59}a^{6}+\frac{10\cdots 28}{25\cdots 59}a^{5}-\frac{44\cdots 30}{25\cdots 59}a^{4}-\frac{10\cdots 14}{25\cdots 59}a^{3}+\frac{51\cdots 10}{25\cdots 59}a^{2}-\frac{56\cdots 41}{25\cdots 59}a+\frac{58\cdots 63}{25\cdots 59}$ 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:  \( 19160065.1923 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 3 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 19160065.1923 \cdot 1}{2\cdot\sqrt{2834497600000000000000000}}\cr\approx \mathstrut & 2.80090344099 \end{aligned}\]

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^15 - 5*x^14 - 5*x^13 + 35*x^12 + 145*x^11 - 263*x^10 - 540*x^9 - 180*x^8 + 1900*x^7 + 800*x^6 - 977*x^5 - 925*x^4 - 965*x^3 + 835*x^2 - 5*x + 89) 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^15 - 5*x^14 - 5*x^13 + 35*x^12 + 145*x^11 - 263*x^10 - 540*x^9 - 180*x^8 + 1900*x^7 + 800*x^6 - 977*x^5 - 925*x^4 - 965*x^3 + 835*x^2 - 5*x + 89, 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^15 - 5*x^14 - 5*x^13 + 35*x^12 + 145*x^11 - 263*x^10 - 540*x^9 - 180*x^8 + 1900*x^7 + 800*x^6 - 977*x^5 - 925*x^4 - 965*x^3 + 835*x^2 - 5*x + 89); 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^15 - 5*x^14 - 5*x^13 + 35*x^12 + 145*x^11 - 263*x^10 - 540*x^9 - 180*x^8 + 1900*x^7 + 800*x^6 - 977*x^5 - 925*x^4 - 965*x^3 + 835*x^2 - 5*x + 89); 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

$C_3^4:(S_3\times F_5)$ (as 15T64):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 9720
The 45 conjugacy class representatives for $C_3^4:(S_3\times F_5)$
Character table for $C_3^4:(S_3\times F_5)$

Intermediate fields

5.1.200000.1

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

Degree 15 sibling: data not computed
Degree 30 siblings: data not computed
Degree 45 siblings: data not computed
Minimal sibling: 15.3.2834497600000000000000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ R ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.3.0.1}{3} }$ R ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ $15$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $15$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.1.2.3a1.3$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$$[3]$$
2.2.2.6a1.6$x^{4} + 2 x^{3} + 7 x^{2} + 14 x + 7$$2$$2$$6$$C_4$$$[3]^{2}$$
2.4.2.12a1.9$x^{8} + 2 x^{5} + 6 x^{4} + x^{2} + 6 x + 7$$2$$4$$12$$C_4\times C_2$$$[3]^{4}$$
\(5\) Copy content Toggle raw display 5.1.15.17a1.4$x^{15} + 20 x^{3} + 5$$15$$1$$17$$F_5 \times S_3$$$[\frac{5}{4}]_{12}^{2}$$
\(11\) Copy content Toggle raw display 11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.2.1.0a1.1$x^{2} + 7 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
11.1.3.2a1.1$x^{3} + 11$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
11.2.3.4a1.1$x^{6} + 21 x^{5} + 153 x^{4} + 427 x^{3} + 306 x^{2} + 95 x + 8$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$

Spectrum of ring of integers

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