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Properties

Label 18.18.389...112.1
Degree $18$
Signature $[18, 0]$
Discriminant $3.896\times 10^{140}$
Root discriminant \(6.465\times 10^{7}\)
Ramified primes $2,3,7,103,271,3030020153113$
Class number not computed
Class group not computed
Galois group $C_3^4:Q_{16}:C_2$ (as 18T395)

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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 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312)
 
Copy content gp:K = bnfinit(y^18 - 146687112*y^16 - 37892719296*y^15 + 8408618911888512*y^14 + 4325813678364681744*y^13 - 243604861180754026285992*y^12 - 192688273988213296086844512*y^11 + 3797668582351192771935287035158*y^10 + 4155203541072329873982601967541328*y^9 - 30824703601196378502940767667441818360*y^8 - 44117476415146499319729949379490251620992*y^7 + 112503359565037178406650855732715578314837728*y^6 + 209476226244754241024858216530145737787244951984*y^5 - 121787795931109896131166351794082151507555501395576*y^4 - 371916116695980755216640900185153166330681546329851680*y^3 - 57935826570177631099916455044945775522459483676836854543*y^2 + 201710684116538249174185560612387919784188057130327410730544*y + 93572448667401892834025834694441537328078001525771364657373312, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312)
 

\( x^{18} - 146687112 x^{16} - 37892719296 x^{15} + \cdots + 93\!\cdots\!12 \) 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:  $[18, 0]$
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:   \(389\!\cdots\!112\) \(\medspace = 2^{60}\cdot 3^{28}\cdot 7^{9}\cdot 103^{6}\cdot 271^{6}\cdot 3030020153113^{6}\) 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:  \(6.465\times 10^{7}\)
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:   \(2\), \(3\), \(7\), \(103\), \(271\), \(3030020153113\) 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:  \(\Q(\sqrt{7}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{6}a^{10}-\frac{1}{2}a^{2}+\frac{1}{3}a$, $\frac{1}{6}a^{11}-\frac{1}{2}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{12}a^{12}-\frac{1}{12}a^{11}-\frac{1}{12}a^{10}-\frac{1}{12}a^{9}+\frac{1}{4}a^{4}+\frac{5}{12}a^{3}+\frac{1}{12}a^{2}+\frac{1}{12}a+\frac{1}{3}$, $\frac{1}{12}a^{13}-\frac{1}{12}a^{9}+\frac{1}{4}a^{5}-\frac{1}{3}a^{4}-\frac{1}{4}a+\frac{1}{3}$, $\frac{1}{36}a^{14}+\frac{1}{36}a^{13}+\frac{1}{36}a^{12}-\frac{1}{36}a^{11}+\frac{1}{18}a^{10}+\frac{1}{18}a^{9}+\frac{1}{6}a^{8}+\frac{1}{6}a^{7}-\frac{1}{12}a^{6}-\frac{7}{36}a^{5}-\frac{7}{36}a^{4}+\frac{11}{36}a^{3}-\frac{2}{9}a^{2}+\frac{4}{9}a+\frac{4}{9}$, $\frac{1}{36}a^{15}+\frac{1}{36}a^{12}-\frac{1}{12}a^{10}+\frac{1}{36}a^{9}-\frac{1}{4}a^{7}-\frac{1}{9}a^{6}-\frac{1}{4}a^{4}-\frac{1}{9}a^{3}-\frac{1}{4}a^{2}+\frac{1}{12}a-\frac{1}{9}$, $\frac{1}{36}a^{16}+\frac{1}{36}a^{13}-\frac{1}{12}a^{11}+\frac{1}{36}a^{10}-\frac{1}{4}a^{8}-\frac{1}{9}a^{7}-\frac{1}{4}a^{5}-\frac{1}{9}a^{4}-\frac{1}{4}a^{3}+\frac{1}{12}a^{2}-\frac{1}{9}a$, $\frac{1}{15\cdots 56}a^{17}-\frac{23\cdots 83}{37\cdots 14}a^{16}-\frac{99\cdots 71}{75\cdots 28}a^{15}+\frac{60\cdots 97}{75\cdots 28}a^{14}+\frac{14\cdots 95}{83\cdots 92}a^{13}-\frac{68\cdots 25}{25\cdots 76}a^{12}+\frac{81\cdots 25}{12\cdots 38}a^{11}-\frac{13\cdots 55}{18\cdots 07}a^{10}+\frac{22\cdots 93}{75\cdots 28}a^{9}-\frac{13\cdots 32}{18\cdots 07}a^{8}+\frac{27\cdots 35}{75\cdots 28}a^{7}-\frac{30\cdots 27}{75\cdots 28}a^{6}-\frac{12\cdots 59}{57\cdots 56}a^{5}+\frac{44\cdots 55}{25\cdots 76}a^{4}-\frac{62\cdots 93}{12\cdots 38}a^{3}-\frac{29\cdots 21}{12\cdots 38}a^{2}-\frac{65\cdots 99}{15\cdots 56}a+\frac{68\cdots 23}{18\cdots 07}$ 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:  not computed
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:  not computed
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:  $17$
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:  not computed
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:  not computed
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{389573518709461277048874506777900581813141954677569649354154692963827395433931232826260909975966492327717495503836697626432438146795529306112}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312) 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 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312, 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 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312); 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 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312); 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

$C_3^4:Q_{16}:C_2$ (as 18T395):

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 2592
The 18 conjugacy class representatives for $C_3^4:Q_{16}:C_2$
Character table for $C_3^4:Q_{16}:C_2$

Intermediate fields

\(\Q(\sqrt{7}) \)

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: 18.18.101021491785353943121599220466066134823210682447466364273131349270310608615239543822518099454441648990964464532520525902670902484365127865556516192905764809189134409473593787699116277131264798611980696259642050064514324642078790284827764457472.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ R ${\href{/padicField/11.8.0.1}{8} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{3}$ ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.1.2.2a1.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$$[2]$$
2.1.16.58m1.524$x^{16} + 8 x^{15} + 4 x^{14} + 4 x^{12} + 8 x^{11} + 10 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 2$$16$$1$$58$16T50$$[2, 3, \frac{7}{2}, \frac{9}{2}]^{2}$$
\(3\) Copy content Toggle raw display 3.1.9.16a1.1$x^{9} + 3 x^{8} + 3$$9$$1$$16$$C_3^2:C_4$$$[2, 2]^{4}$$
3.1.9.12a1.1$x^{9} + 6 x^{4} + 3$$9$$1$$12$$C_3^2:C_4$$$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$$
\(7\) Copy content Toggle raw display 7.1.2.1a1.2$x^{2} + 21$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.4.2.4a1.2$x^{8} + 10 x^{6} + 8 x^{5} + 31 x^{4} + 40 x^{3} + 46 x^{2} + 24 x + 16$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
7.4.2.4a1.2$x^{8} + 10 x^{6} + 8 x^{5} + 31 x^{4} + 40 x^{3} + 46 x^{2} + 24 x + 16$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(103\) Copy content Toggle raw display $\Q_{103}$$x + 98$$1$$1$$0$Trivial$$[\ ]$$
103.2.1.0a1.1$x^{2} + 102 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
103.2.1.0a1.1$x^{2} + 102 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
103.2.1.0a1.1$x^{2} + 102 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
103.2.1.0a1.1$x^{2} + 102 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
103.3.3.6a1.3$x^{9} + 6 x^{7} + 294 x^{6} + 12 x^{5} + 1176 x^{4} + 28820 x^{3} + 1176 x^{2} + 57624 x + 941295$$3$$3$$6$$C_3^2$$$[\ ]_{3}^{3}$$
\(271\) Copy content Toggle raw display $\Q_{271}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $3$$3$$1$$2$
Deg $3$$3$$1$$2$
Deg $3$$3$$1$$2$
\(3030020153113\) Copy content Toggle raw display Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $9$$3$$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)