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Properties

Label 18.0.371...000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-3.720\times 10^{29}$
Root discriminant \(43.93\)
Ramified primes $2,3,5,7$
Class number $6$ (GRH)
Class group [6] (GRH)
Galois group $C_6:S_3$ (as 18T12)

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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 - 30*x^14 + 53*x^12 + 302*x^10 - 351*x^8 - 3179*x^6 + 6484*x^4 - 672*x^2 + 36)
 
Copy content gp:K = bnfinit(y^18 + y^16 - 30*y^14 + 53*y^12 + 302*y^10 - 351*y^8 - 3179*y^6 + 6484*y^4 - 672*y^2 + 36, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + x^16 - 30*x^14 + 53*x^12 + 302*x^10 - 351*x^8 - 3179*x^6 + 6484*x^4 - 672*x^2 + 36);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + x^16 - 30*x^14 + 53*x^12 + 302*x^10 - 351*x^8 - 3179*x^6 + 6484*x^4 - 672*x^2 + 36)
 

\( x^{18} + x^{16} - 30x^{14} + 53x^{12} + 302x^{10} - 351x^{8} - 3179x^{6} + 6484x^{4} - 672x^{2} + 36 \) 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:   \(-371968759094317056000000000000\) \(\medspace = -\,2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 7^{12}\) 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:  \(43.93\)
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:  $2^{4/3}3^{1/2}5^{2/3}7^{2/3}\approx 46.69954520953564$
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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_2$
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.
Maximal CM subfield:  \(\Q(\sqrt{-1}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{2}$, $\frac{1}{18}a^{9}-\frac{1}{6}a^{8}-\frac{1}{6}a^{6}+\frac{1}{3}a^{5}+\frac{1}{6}a^{4}-\frac{5}{18}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{18}a^{10}-\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}+\frac{1}{6}a^{5}+\frac{2}{9}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{18}a^{11}-\frac{1}{6}a^{7}+\frac{2}{9}a^{5}-\frac{1}{6}a^{3}$, $\frac{1}{18}a^{12}-\frac{1}{6}a^{8}-\frac{1}{9}a^{6}+\frac{1}{6}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{18}a^{13}-\frac{1}{6}a^{8}-\frac{1}{9}a^{7}-\frac{1}{6}a^{6}+\frac{1}{6}a^{5}+\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{162}a^{14}+\frac{1}{81}a^{12}-\frac{1}{81}a^{10}-\frac{5}{162}a^{8}-\frac{1}{6}a^{7}+\frac{1}{81}a^{6}+\frac{1}{6}a^{5}+\frac{17}{81}a^{4}-\frac{1}{6}a^{3}-\frac{7}{27}a^{2}+\frac{2}{9}$, $\frac{1}{162}a^{15}+\frac{1}{81}a^{13}-\frac{1}{81}a^{11}+\frac{2}{81}a^{9}+\frac{1}{81}a^{7}-\frac{37}{81}a^{5}+\frac{25}{54}a^{3}-\frac{4}{9}a$, $\frac{1}{692950302}a^{16}+\frac{17287}{20998494}a^{14}-\frac{2078251}{230983434}a^{12}+\frac{3767470}{346475151}a^{10}-\frac{897806}{38497239}a^{8}-\frac{1}{6}a^{7}+\frac{15440011}{115491717}a^{6}+\frac{1}{6}a^{5}-\frac{3170810}{6537267}a^{4}-\frac{1}{6}a^{3}+\frac{2003774}{115491717}a^{2}-\frac{6132124}{38497239}$, $\frac{1}{1385900604}a^{17}+\frac{17287}{41996988}a^{15}+\frac{5377081}{230983434}a^{13}-\frac{30962299}{1385900604}a^{11}-\frac{448903}{38497239}a^{9}-\frac{33282043}{461966868}a^{7}-\frac{1}{6}a^{6}-\frac{2709805}{26149068}a^{5}+\frac{1}{6}a^{4}-\frac{36493465}{230983434}a^{3}-\frac{1}{6}a^{2}-\frac{3066062}{38497239}a$ 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:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

Ideal class group:  $C_{6}$, which has order $6$ (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:  $C_{6}$, which has order $6$ (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:  $8$
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:   \( \frac{58985}{5726862} a^{17} + \frac{32213}{2863431} a^{15} - \frac{880346}{2863431} a^{13} + \frac{2972891}{5726862} a^{11} + \frac{9009443}{2863431} a^{9} - \frac{19010893}{5726862} a^{7} - \frac{1184461}{36018} a^{5} + \frac{40668959}{636318} a^{3} - \frac{211279}{106053} a \)  (order $4$) 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:   $\frac{81205}{11947419}a^{16}+\frac{875}{120681}a^{14}-\frac{1620745}{7964946}a^{12}+\frac{4111004}{11947419}a^{10}+\frac{16496225}{7964946}a^{8}-\frac{8725055}{3982473}a^{6}-\frac{9822455}{450846}a^{4}+\frac{168878035}{3982473}a^{2}-\frac{2208359}{1327491}$, $\frac{3396185}{1385900604}a^{17}+\frac{73031}{23894838}a^{16}+\frac{404389}{125990964}a^{15}+\frac{2729}{724086}a^{14}-\frac{50917585}{692950302}a^{13}-\frac{360139}{3982473}a^{12}+\frac{48036155}{461966868}a^{11}+\frac{3341383}{23894838}a^{10}+\frac{274145660}{346475151}a^{9}+\frac{413492}{442497}a^{8}-\frac{866433005}{1385900604}a^{7}-\frac{3271843}{3982473}a^{6}-\frac{214902709}{26149068}a^{5}-\frac{2199394}{225423}a^{4}+\frac{3003106505}{230983434}a^{3}+\frac{137373809}{7964946}a^{2}+\frac{183016285}{38497239}a+\frac{578461}{1327491}$, $\frac{4930967}{1385900604}a^{17}-\frac{21415}{13074534}a^{16}+\frac{184441}{41996988}a^{15}-\frac{1625}{1188594}a^{14}-\frac{1355080}{12832413}a^{13}+\frac{329618}{6537267}a^{12}+\frac{226339843}{1385900604}a^{11}-\frac{405775}{4358178}a^{10}+\frac{127650361}{115491717}a^{9}-\frac{6685351}{13074534}a^{8}-\frac{149549419}{153988956}a^{7}+\frac{8475295}{13074534}a^{6}-\frac{299064197}{26149068}a^{5}+\frac{70863679}{13074534}a^{4}+\frac{2359212329}{115491717}a^{3}-\frac{47935667}{4358178}a^{2}+\frac{62152604}{38497239}a+\frac{107276}{726363}$, $\frac{9343403}{1385900604}a^{17}+\frac{21415}{13074534}a^{16}+\frac{864049}{125990964}a^{15}+\frac{1625}{1188594}a^{14}-\frac{69934706}{346475151}a^{13}-\frac{329618}{6537267}a^{12}+\frac{164366593}{461966868}a^{11}+\frac{405775}{4358178}a^{10}+\frac{707191520}{346475151}a^{9}+\frac{6685351}{13074534}a^{8}-\frac{3254691335}{1385900604}a^{7}-\frac{8475295}{13074534}a^{6}-\frac{560854489}{26149068}a^{5}-\frac{70863679}{13074534}a^{4}+\frac{10044407459}{230983434}a^{3}+\frac{47935667}{4358178}a^{2}-\frac{138846881}{38497239}a-\frac{833639}{726363}$, $\frac{521269}{38497239}a^{17}+\frac{2241329}{692950302}a^{16}+\frac{133528}{10499247}a^{15}+\frac{66622}{10499247}a^{14}-\frac{95339809}{230983434}a^{13}-\frac{9736561}{115491717}a^{12}+\frac{166377709}{230983434}a^{11}+\frac{39873653}{346475151}a^{10}+\frac{960610711}{230983434}a^{9}+\frac{36220421}{38497239}a^{8}-\frac{1143433873}{230983434}a^{7}-\frac{37483108}{115491717}a^{6}-\frac{193042129}{4358178}a^{5}-\frac{112266317}{13074534}a^{4}+\frac{6757959431}{76994478}a^{3}+\frac{1799746900}{115491717}a^{2}-\frac{18123068}{4277471}a-\frac{54068450}{38497239}$, $\frac{708823}{692950302}a^{17}+\frac{143309}{346475151}a^{16}+\frac{12115}{10499247}a^{15}+\frac{84373}{62995482}a^{14}-\frac{1117216}{38497239}a^{13}-\frac{6208097}{692950302}a^{12}+\frac{18577684}{346475151}a^{11}+\frac{9932143}{692950302}a^{10}+\frac{32669242}{115491717}a^{9}+\frac{99592007}{692950302}a^{8}-\frac{8463023}{38497239}a^{7}-\frac{60853415}{692950302}a^{6}-\frac{39291049}{13074534}a^{5}-\frac{609577}{1452726}a^{4}+\frac{619918160}{115491717}a^{3}+\frac{12696307}{12832413}a^{2}-\frac{8431105}{38497239}a-\frac{926657}{12832413}$, $\frac{7356743}{692950302}a^{17}+\frac{264785}{230983434}a^{16}+\frac{32129}{3499749}a^{15}+\frac{110315}{62995482}a^{14}-\frac{74132005}{230983434}a^{13}-\frac{22971385}{692950302}a^{12}+\frac{208499747}{346475151}a^{11}+\frac{14543939}{346475151}a^{10}+\frac{725980571}{230983434}a^{9}+\frac{122931836}{346475151}a^{8}-\frac{958893325}{230983434}a^{7}-\frac{37577264}{346475151}a^{6}-\frac{436206971}{13074534}a^{5}-\frac{23253488}{6537267}a^{4}+\frac{16855122107}{230983434}a^{3}+\frac{569955719}{115491717}a^{2}-\frac{587209913}{38497239}a+\frac{117724664}{38497239}$, $\frac{12167171}{346475151}a^{17}+\frac{2386315}{230983434}a^{16}+\frac{1188002}{31497741}a^{15}+\frac{364712}{31497741}a^{14}-\frac{363842662}{346475151}a^{13}-\frac{212721305}{692950302}a^{12}+\frac{1236415801}{692950302}a^{11}+\frac{355995839}{692950302}a^{10}+\frac{3716648986}{346475151}a^{9}+\frac{1084983547}{346475151}a^{8}-\frac{3975319228}{346475151}a^{7}-\frac{2279130179}{692950302}a^{6}-\frac{489847109}{4358178}a^{5}-\frac{213171529}{6537267}a^{4}+\frac{2810791639}{12832413}a^{3}+\frac{7338362950}{115491717}a^{2}-\frac{33416822}{4277471}a-\frac{140166737}{38497239}$ 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:  \( 37185048.62613057 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 37185048.62613057 \cdot 6}{4\cdot\sqrt{371968759094317056000000000000}}\cr\approx \mathstrut & 1.39580643139099 \end{aligned}\] (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 + x^16 - 30*x^14 + 53*x^12 + 302*x^10 - 351*x^8 - 3179*x^6 + 6484*x^4 - 672*x^2 + 36) 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 - 30*x^14 + 53*x^12 + 302*x^10 - 351*x^8 - 3179*x^6 + 6484*x^4 - 672*x^2 + 36, 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 - 30*x^14 + 53*x^12 + 302*x^10 - 351*x^8 - 3179*x^6 + 6484*x^4 - 672*x^2 + 36); 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 - 30*x^14 + 53*x^12 + 302*x^10 - 351*x^8 - 3179*x^6 + 6484*x^4 - 672*x^2 + 36); 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

$C_6:S_3$ (as 18T12):

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 12 conjugacy class representatives for $C_6:S_3$
Character table for $C_6:S_3$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.588.1, 3.1.3675.1, 3.1.14700.1, 3.1.300.1, 6.0.5531904.1, 6.0.3457440000.3, 6.0.864360000.5, 6.0.1440000.1, 9.1.9529569000000.1

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 18 sibling: 18.2.1115906277282951168000000000000.1
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 R R R R ${\href{/padicField/11.2.0.1}{2} }^{9}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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 2.1.6.8a1.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$$[2]_{3}^{2}$$
2.2.6.16a1.5$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 143 x^{6} + 132 x^{5} + 102 x^{4} + 64 x^{3} + 33 x^{2} + 12 x + 5$$6$$2$$16$$D_6$$$[2]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.1.3.2a1.1$x^{3} + 5$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
5.1.3.2a1.1$x^{3} + 5$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
5.2.3.4a1.2$x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
5.2.3.4a1.2$x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
\(7\) Copy content Toggle raw display 7.6.3.12a1.3$x^{18} + 3 x^{16} + 15 x^{15} + 15 x^{14} + 48 x^{13} + 109 x^{12} + 171 x^{11} + 333 x^{10} + 497 x^{9} + 717 x^{8} + 1032 x^{7} + 1216 x^{6} + 1296 x^{5} + 1143 x^{4} + 783 x^{3} + 432 x^{2} + 162 x + 34$$3$$6$$12$$C_6 \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)