↑

Properties

Label 18.0.907...368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-9.073\times 10^{20}$
Root discriminant \(14.60\)
Ramified primes $2,3,7$
Class number $1$
Class group trivial
Galois group $S_3 \times C_6$ (as 18T6)

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^18 - 6*x^17 + 17*x^16 - 36*x^15 + 76*x^14 - 140*x^13 + 228*x^12 - 348*x^11 + 465*x^10 - 542*x^9 + 551*x^8 - 488*x^7 + 373*x^6 - 234*x^5 + 127*x^4 - 52*x^3 + 17*x^2 - 2*x + 1)
 
Copy content gp:K = bnfinit(y^18 - 6*y^17 + 17*y^16 - 36*y^15 + 76*y^14 - 140*y^13 + 228*y^12 - 348*y^11 + 465*y^10 - 542*y^9 + 551*y^8 - 488*y^7 + 373*y^6 - 234*y^5 + 127*y^4 - 52*y^3 + 17*y^2 - 2*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 17*x^16 - 36*x^15 + 76*x^14 - 140*x^13 + 228*x^12 - 348*x^11 + 465*x^10 - 542*x^9 + 551*x^8 - 488*x^7 + 373*x^6 - 234*x^5 + 127*x^4 - 52*x^3 + 17*x^2 - 2*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 17*x^16 - 36*x^15 + 76*x^14 - 140*x^13 + 228*x^12 - 348*x^11 + 465*x^10 - 542*x^9 + 551*x^8 - 488*x^7 + 373*x^6 - 234*x^5 + 127*x^4 - 52*x^3 + 17*x^2 - 2*x + 1)
 

\( x^{18} - 6 x^{17} + 17 x^{16} - 36 x^{15} + 76 x^{14} - 140 x^{13} + 228 x^{12} - 348 x^{11} + 465 x^{10} + \cdots + 1 \) 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:   \(-907273133293160890368\) \(\medspace = -\,2^{18}\cdot 3^{6}\cdot 7^{15}\) 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:  \(14.60\)
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:  $2^{3/2}3^{1/2}7^{5/6}\approx 24.794421938893013$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q(\sqrt{-7}) \)
$\Aut(K/\Q)$:   $C_6$
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:  \(\Q(\zeta_{7})\)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}+\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{104}a^{15}-\frac{5}{104}a^{14}-\frac{1}{13}a^{13}-\frac{21}{104}a^{12}+\frac{1}{52}a^{11}-\frac{3}{26}a^{10}-\frac{1}{104}a^{9}-\frac{3}{104}a^{8}-\frac{3}{13}a^{7}-\frac{1}{8}a^{6}+\frac{1}{104}a^{5}-\frac{17}{104}a^{4}+\frac{37}{104}a^{3}-\frac{19}{52}a^{2}+\frac{1}{13}a-\frac{27}{104}$, $\frac{1}{104}a^{16}-\frac{7}{104}a^{14}-\frac{9}{104}a^{13}+\frac{1}{104}a^{12}+\frac{3}{13}a^{11}+\frac{17}{104}a^{10}+\frac{9}{52}a^{9}+\frac{1}{8}a^{8}-\frac{3}{104}a^{7}+\frac{7}{52}a^{6}+\frac{5}{13}a^{5}-\frac{11}{52}a^{4}+\frac{43}{104}a^{3}-\frac{1}{2}a^{2}+\frac{3}{8}a-\frac{5}{104}$, $\frac{1}{48152}a^{17}+\frac{183}{48152}a^{16}-\frac{121}{48152}a^{15}+\frac{2669}{24076}a^{14}+\frac{1661}{24076}a^{13}+\frac{4057}{48152}a^{12}+\frac{6755}{48152}a^{11}+\frac{5875}{48152}a^{10}-\frac{11009}{48152}a^{9}-\frac{955}{24076}a^{8}-\frac{5781}{48152}a^{7}+\frac{9283}{24076}a^{6}-\frac{545}{6019}a^{5}-\frac{18191}{48152}a^{4}+\frac{23879}{48152}a^{3}-\frac{15987}{48152}a^{2}-\frac{6939}{24076}a+\frac{4581}{48152}$ 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:  $2$

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:   \( -\frac{1173}{48152} a^{17} - \frac{2605}{48152} a^{16} + \frac{25257}{48152} a^{15} - \frac{24007}{24076} a^{14} + \frac{15949}{12038} a^{13} - \frac{194607}{48152} a^{12} + \frac{315007}{48152} a^{11} - \frac{405671}{48152} a^{10} + \frac{680171}{48152} a^{9} - \frac{14309}{926} a^{8} + \frac{675069}{48152} a^{7} - \frac{83487}{6019} a^{6} + \frac{80907}{6019} a^{5} - \frac{544263}{48152} a^{4} + \frac{25967}{3704} a^{3} - \frac{323775}{48152} a^{2} + \frac{4979}{1852} a - \frac{45309}{48152} \)  (order $14$) 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:   $\frac{369}{24076}a^{17}-\frac{12643}{48152}a^{16}+\frac{26653}{24076}a^{15}-\frac{121529}{48152}a^{14}+\frac{244515}{48152}a^{13}-\frac{521493}{48152}a^{12}+\frac{217743}{12038}a^{11}-\frac{1354983}{48152}a^{10}+\frac{1007529}{24076}a^{9}-\frac{2433273}{48152}a^{8}+\frac{2649453}{48152}a^{7}-\frac{316847}{6019}a^{6}+\frac{252472}{6019}a^{5}-\frac{169550}{6019}a^{4}+\frac{694033}{48152}a^{3}-\frac{37996}{6019}a^{2}+\frac{17653}{48152}a-\frac{7919}{48152}$, $\frac{11241}{48152}a^{17}-\frac{70845}{48152}a^{16}+\frac{106325}{24076}a^{15}-\frac{473065}{48152}a^{14}+\frac{504107}{24076}a^{13}-\frac{939577}{24076}a^{12}+\frac{3113241}{48152}a^{11}-\frac{4849981}{48152}a^{10}+\frac{826669}{6019}a^{9}-\frac{7890983}{48152}a^{8}+\frac{8286455}{48152}a^{7}-\frac{7421049}{48152}a^{6}+\frac{5664447}{48152}a^{5}-\frac{1786563}{24076}a^{4}+\frac{945935}{24076}a^{3}-\frac{714993}{48152}a^{2}+\frac{60501}{24076}a-\frac{12345}{12038}$, $\frac{9643}{48152}a^{17}-\frac{22599}{24076}a^{16}+\frac{45121}{24076}a^{15}-\frac{19118}{6019}a^{14}+\frac{358827}{48152}a^{13}-\frac{582451}{48152}a^{12}+\frac{813875}{48152}a^{11}-\frac{153202}{6019}a^{10}+\frac{689917}{24076}a^{9}-\frac{165174}{6019}a^{8}+\frac{155040}{6019}a^{7}-\frac{83445}{3704}a^{6}+\frac{818745}{48152}a^{5}-\frac{243271}{24076}a^{4}+\frac{384771}{48152}a^{3}-\frac{149395}{48152}a^{2}+\frac{95807}{48152}a-\frac{1173}{48152}$, $\frac{8013}{48152}a^{17}-\frac{729}{926}a^{16}+\frac{41413}{24076}a^{15}-\frac{85391}{24076}a^{14}+\frac{31977}{3704}a^{13}-\frac{702249}{48152}a^{12}+\frac{1097647}{48152}a^{11}-\frac{901499}{24076}a^{10}+\frac{578471}{12038}a^{9}-\frac{1361171}{24076}a^{8}+\frac{1538779}{24076}a^{7}-\frac{2848789}{48152}a^{6}+\frac{2248239}{48152}a^{5}-\frac{750777}{24076}a^{4}+\frac{892933}{48152}a^{3}-\frac{369539}{48152}a^{2}+\frac{109583}{48152}a-\frac{39831}{48152}$, $\frac{1167}{48152}a^{17}-\frac{1793}{24076}a^{16}+\frac{8805}{48152}a^{15}-\frac{42815}{48152}a^{14}+\frac{128789}{48152}a^{13}-\frac{27737}{6019}a^{12}+\frac{431563}{48152}a^{11}-\frac{203599}{12038}a^{10}+\frac{1186953}{48152}a^{9}-\frac{1721985}{48152}a^{8}+\frac{552577}{12038}a^{7}-\frac{1129733}{24076}a^{6}+\frac{494315}{12038}a^{5}-\frac{1414349}{48152}a^{4}+\frac{99803}{6019}a^{3}-\frac{245171}{48152}a^{2}+\frac{21875}{48152}a+\frac{1244}{6019}$, $\frac{21673}{48152}a^{17}-\frac{25787}{12038}a^{16}+\frac{212053}{48152}a^{15}-\frac{356309}{48152}a^{14}+\frac{776731}{48152}a^{13}-\frac{599375}{24076}a^{12}+\frac{1632127}{48152}a^{11}-\frac{1143293}{24076}a^{10}+\frac{2111939}{48152}a^{9}-\frac{112615}{3704}a^{8}+\frac{228899}{24076}a^{7}+\frac{327169}{24076}a^{6}-\frac{41551}{1852}a^{5}+\frac{1303099}{48152}a^{4}-\frac{392753}{24076}a^{3}+\frac{430389}{48152}a^{2}-\frac{6029}{3704}a+\frac{25769}{24076}$, $\frac{12653}{12038}a^{17}-\frac{288305}{48152}a^{16}+\frac{95442}{6019}a^{15}-\frac{1547009}{48152}a^{14}+\frac{3283303}{48152}a^{13}-\frac{454219}{3704}a^{12}+\frac{4667155}{24076}a^{11}-\frac{14134509}{48152}a^{10}+\frac{4559075}{12038}a^{9}-\frac{20347537}{48152}a^{8}+\frac{19994903}{48152}a^{7}-\frac{4194733}{12038}a^{6}+\frac{1506750}{6019}a^{5}-\frac{3469251}{24076}a^{4}+\frac{3473567}{48152}a^{3}-\frac{576835}{24076}a^{2}+\frac{249445}{48152}a-\frac{51553}{48152}$, $\frac{19}{6019}a^{17}-\frac{3223}{24076}a^{16}+\frac{24603}{24076}a^{15}-\frac{82777}{24076}a^{14}+\frac{172373}{24076}a^{13}-\frac{161137}{12038}a^{12}+\frac{621259}{24076}a^{11}-\frac{38565}{926}a^{10}+\frac{713693}{12038}a^{9}-\frac{472436}{6019}a^{8}+\frac{1010513}{12038}a^{7}-\frac{432494}{6019}a^{6}+\frac{1208115}{24076}a^{5}-\frac{167837}{6019}a^{4}+\frac{61540}{6019}a^{3}-\frac{1489}{24076}a^{2}-\frac{12971}{12038}a-\frac{15299}{24076}$ 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:  \( 7362.5561247660435 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 7362.5561247660435 \cdot 1}{14\cdot\sqrt{907273133293160890368}}\cr\approx \mathstrut & 0.266471258319181 \end{aligned}\]

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 - 6*x^17 + 17*x^16 - 36*x^15 + 76*x^14 - 140*x^13 + 228*x^12 - 348*x^11 + 465*x^10 - 542*x^9 + 551*x^8 - 488*x^7 + 373*x^6 - 234*x^5 + 127*x^4 - 52*x^3 + 17*x^2 - 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 - 6*x^17 + 17*x^16 - 36*x^15 + 76*x^14 - 140*x^13 + 228*x^12 - 348*x^11 + 465*x^10 - 542*x^9 + 551*x^8 - 488*x^7 + 373*x^6 - 234*x^5 + 127*x^4 - 52*x^3 + 17*x^2 - 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 - 6*x^17 + 17*x^16 - 36*x^15 + 76*x^14 - 140*x^13 + 228*x^12 - 348*x^11 + 465*x^10 - 542*x^9 + 551*x^8 - 488*x^7 + 373*x^6 - 234*x^5 + 127*x^4 - 52*x^3 + 17*x^2 - 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 - 6*x^17 + 17*x^16 - 36*x^15 + 76*x^14 - 140*x^13 + 228*x^12 - 348*x^11 + 465*x^10 - 542*x^9 + 551*x^8 - 488*x^7 + 373*x^6 - 234*x^5 + 127*x^4 - 52*x^3 + 17*x^2 - 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

\(\Q(\sqrt{-7}) \), 3.1.1176.1, \(\Q(\zeta_{7})^+\), 6.0.9680832.1, \(\Q(\zeta_{7})\), 9.3.1626379776.1

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: 12.0.1101670627147776.7
Degree 18 sibling: 18.6.12542143794644656148447232.1
Minimal sibling: 12.0.1101670627147776.7

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 ${\href{/padicField/5.6.0.1}{6} }^{3}$ R ${\href{/padicField/11.3.0.1}{3} }^{6}$ ${\href{/padicField/13.2.0.1}{2} }^{9}$ ${\href{/padicField/17.6.0.1}{6} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{3}$ ${\href{/padicField/53.3.0.1}{3} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{3}$

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 2.3.1.0a1.1$x^{3} + x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
2.3.1.0a1.1$x^{3} + x + 1$$1$$3$$0$$C_3$$$[\ ]^{3}$$
2.3.2.9a1.6$x^{6} + 2 x^{4} + 6 x^{3} + x^{2} + 6 x + 15$$2$$3$$9$$C_6$$$[3]^{3}$$
2.3.2.9a1.6$x^{6} + 2 x^{4} + 6 x^{3} + x^{2} + 6 x + 15$$2$$3$$9$$C_6$$$[3]^{3}$$
\(3\) Copy content Toggle raw display 3.6.1.0a1.1$x^{6} + 2 x^{4} + x^{2} + 2 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
3.6.2.6a1.2$x^{12} + 4 x^{10} + 6 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} + 9 x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$6$$6$$C_6\times C_2$$$[\ ]_{2}^{6}$$
\(7\) Copy content Toggle raw display 7.3.6.15a1.3$x^{18} + 36 x^{17} + 540 x^{16} + 4344 x^{15} + 20160 x^{14} + 55296 x^{13} + 98736 x^{12} + 161280 x^{11} + 238464 x^{10} + 208640 x^{9} + 334080 x^{8} + 138240 x^{7} + 280320 x^{6} + 46080 x^{5} + 138240 x^{4} + 6144 x^{3} + 36864 x^{2} + 4103$$6$$3$$15$$C_6 \times C_3$$$[\ ]_{6}^{3}$$

Spectrum of ring of integers

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