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Properties

Label 18.0.386...896.4
Degree $18$
Signature $[0, 9]$
Discriminant $-3.868\times 10^{27}$
Root discriminant \(34.09\)
Ramified primes $2,3$
Class number $4$ (GRH)
Class group [2, 2] (GRH)
Galois group $C_6:S_3$ (as 18T12)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 + 90*x^12 + 108*x^6 + 216)
 
Copy content gp:K = bnfinit(y^18 + 90*y^12 + 108*y^6 + 216, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 90*x^12 + 108*x^6 + 216);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + 90*x^12 + 108*x^6 + 216)
 

\( x^{18} + 90x^{12} + 108x^{6} + 216 \) 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:   \(-3867909299433950829814480896\) \(\medspace = -\,2^{33}\cdot 3^{37}\) 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:  \(34.09\)
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:  $2^{11/6}3^{37/18}\approx 34.0908297010423$
Ramified primes:   \(2\), \(3\) 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:  \(\Q(\sqrt{-6}) \)
$\Aut(K/\Q)$:   $C_2$
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:  \(\Q(\sqrt{-6}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}$, $\frac{1}{6}a^{7}$, $\frac{1}{12}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{36}a^{10}+\frac{1}{6}a^{4}$, $\frac{1}{36}a^{11}+\frac{1}{6}a^{5}$, $\frac{1}{360}a^{12}+\frac{1}{30}a^{6}-\frac{3}{10}$, $\frac{1}{1080}a^{13}+\frac{1}{36}a^{9}+\frac{1}{90}a^{7}+\frac{1}{3}a^{5}+\frac{1}{6}a^{3}+\frac{7}{30}a$, $\frac{1}{1080}a^{14}+\frac{1}{90}a^{8}+\frac{7}{30}a^{2}$, $\frac{1}{1080}a^{15}+\frac{1}{90}a^{9}+\frac{7}{30}a^{3}$, $\frac{1}{2160}a^{16}+\frac{1}{180}a^{10}-\frac{23}{60}a^{4}$, $\frac{1}{2160}a^{17}+\frac{1}{180}a^{11}-\frac{23}{60}a^{5}$ 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:  $C_{2}\times C_{2}$, which has order $4$ (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:  $C_{2}\times C_{2}$, which has order $4$ (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:   \( -1 \)  (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:   $\frac{13}{2160}a^{16}+\frac{49}{90}a^{10}+\frac{17}{20}a^{4}$, $\frac{1}{432}a^{16}+\frac{1}{120}a^{12}+\frac{7}{36}a^{10}+\frac{23}{30}a^{6}-\frac{11}{12}a^{4}+\frac{11}{10}$, $\frac{1}{432}a^{16}-\frac{1}{90}a^{12}+\frac{7}{36}a^{10}-\frac{29}{30}a^{6}-\frac{11}{12}a^{4}+\frac{1}{5}$, $\frac{31}{2160}a^{16}-\frac{1}{90}a^{13}+\frac{77}{60}a^{10}-\frac{29}{30}a^{7}+\frac{47}{60}a^{4}+\frac{6}{5}a$, $\frac{13}{2160}a^{17}-\frac{1}{135}a^{16}+\frac{1}{135}a^{15}-\frac{1}{1080}a^{14}-\frac{1}{360}a^{12}+\frac{49}{90}a^{11}-\frac{121}{180}a^{10}+\frac{121}{180}a^{9}-\frac{17}{180}a^{8}-\frac{1}{5}a^{6}+\frac{17}{20}a^{5}-\frac{41}{30}a^{4}+\frac{41}{30}a^{3}-\frac{11}{15}a^{2}+\frac{13}{10}$, $\frac{13}{2160}a^{17}-\frac{1}{1080}a^{16}-\frac{1}{180}a^{15}-\frac{1}{72}a^{14}-\frac{1}{120}a^{13}+\frac{1}{360}a^{12}+\frac{49}{90}a^{11}-\frac{1}{15}a^{10}-\frac{29}{60}a^{9}-\frac{5}{4}a^{8}-\frac{23}{30}a^{7}+\frac{1}{5}a^{6}+\frac{17}{20}a^{5}+\frac{43}{30}a^{4}+\frac{11}{10}a^{3}-a^{2}-\frac{11}{10}a-\frac{13}{10}$, $\frac{1}{108}a^{17}-\frac{11}{540}a^{16}-\frac{11}{1080}a^{15}+\frac{5}{216}a^{14}+\frac{7}{180}a^{13}+\frac{1}{360}a^{12}+\frac{29}{36}a^{11}-\frac{329}{180}a^{10}-\frac{157}{180}a^{9}+\frac{19}{9}a^{8}+\frac{52}{15}a^{7}+\frac{1}{5}a^{6}-\frac{3}{2}a^{5}-\frac{49}{30}a^{4}+\frac{44}{15}a^{3}+\frac{29}{6}a^{2}+\frac{4}{5}a-\frac{43}{10}$, $\frac{19}{432}a^{17}+\frac{13}{240}a^{16}+\frac{1}{30}a^{15}-\frac{1}{40}a^{14}-\frac{5}{72}a^{13}-\frac{1}{15}a^{12}+\frac{47}{12}a^{11}+\frac{877}{180}a^{10}+\frac{46}{15}a^{9}-\frac{32}{15}a^{8}-\frac{37}{6}a^{7}-\frac{179}{30}a^{6}+\frac{11}{12}a^{5}+\frac{329}{60}a^{4}+\frac{47}{5}a^{3}+\frac{77}{10}a^{2}+\frac{1}{2}a-\frac{29}{5}$ 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:  \( 3129330.11622241 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 3129330.11622241 \cdot 4}{2\cdot\sqrt{3867909299433950829814480896}}\cr\approx \mathstrut & 1.53589685054807 \end{aligned}\] (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 + 90*x^12 + 108*x^6 + 216) 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 + 90*x^12 + 108*x^6 + 216, 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 + 90*x^12 + 108*x^6 + 216); 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 + 90*x^12 + 108*x^6 + 216); 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

\(\Q(\sqrt{-6}) \), 3.1.243.1, 3.1.972.2, 3.1.972.1, 3.1.108.1, 6.0.90699264.1, 6.0.362797056.5, 6.0.362797056.3, 6.0.4478976.2, 9.1.24794911296.1

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: 18.2.1289303099811316943271493632.4
Minimal sibling: 18.2.1289303099811316943271493632.4

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 ${\href{/padicField/5.2.0.1}{2} }^{8}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.3.0.1}{3} }^{6}$ ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.1.6.11a1.6$x^{6} + 4 x^{3} + 10$$6$$1$$11$$D_{6}$$$[3]_{3}^{2}$$
2.2.6.22a1.9$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 145 x^{6} + 138 x^{5} + 114 x^{4} + 78 x^{3} + 45 x^{2} + 18 x + 7$$6$$2$$22$$D_6$$$[3]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.18.37c4.77$x^{18} + 9 x^{8} + 3 x^{6} + 18 x^{4} + 9 x^{2} + 6$$18$$1$$37$not computednot computed

Spectrum of ring of integers

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