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Properties

Label 18.0.86100382362...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{18}\cdot 5^{9}\cdot 7^{14}$
Root discriminant $76.79$
Ramified primes $2, 3, 5, 7$
Class number $13608$ (GRH)
Class group $[6, 6, 378]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![236250749, 126029568, 139982661, 39093488, 27436416, 3283728, 3618320, 375408, 559842, 47424, 49218, -1680, 3928, 240, 360, -16, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 16*x^15 + 360*x^14 + 240*x^13 + 3928*x^12 - 1680*x^11 + 49218*x^10 + 47424*x^9 + 559842*x^8 + 375408*x^7 + 3618320*x^6 + 3283728*x^5 + 27436416*x^4 + 39093488*x^3 + 139982661*x^2 + 126029568*x + 236250749)
 
gp: K = bnfinit(x^18 + 9*x^16 - 16*x^15 + 360*x^14 + 240*x^13 + 3928*x^12 - 1680*x^11 + 49218*x^10 + 47424*x^9 + 559842*x^8 + 375408*x^7 + 3618320*x^6 + 3283728*x^5 + 27436416*x^4 + 39093488*x^3 + 139982661*x^2 + 126029568*x + 236250749, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 16 x^{15} + 360 x^{14} + 240 x^{13} + 3928 x^{12} - 1680 x^{11} + 49218 x^{10} + 47424 x^{9} + 559842 x^{8} + 375408 x^{7} + 3618320 x^{6} + 3283728 x^{5} + 27436416 x^{4} + 39093488 x^{3} + 139982661 x^{2} + 126029568 x + 236250749 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8610038236258048513179648000000000=-\,2^{24}\cdot 3^{18}\cdot 5^{9}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.79$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{8} + \frac{3}{16} a^{4} - \frac{5}{16}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{16} a + \frac{3}{8}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{10} - \frac{1}{8} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{4} + \frac{1}{16} a^{2} + \frac{3}{8}$, $\frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{32} a^{12} + \frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{3}{32} a^{7} - \frac{3}{32} a^{6} - \frac{3}{32} a^{5} - \frac{3}{32} a^{4} - \frac{5}{32} a^{3} + \frac{5}{32} a^{2} + \frac{5}{32} a + \frac{5}{32}$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{8} - \frac{1}{4} a^{4} + \frac{5}{32}$, $\frac{1}{16775730791919220110012737596826913695923546528} a^{17} - \frac{106820407968028326517768844645377510361988437}{8387865395959610055006368798413456847961773264} a^{16} + \frac{3351752324860183228765419616641907012359181}{1048483174494951256875796099801682105995221658} a^{15} - \frac{50406294069344825594400578746773996871344483}{8387865395959610055006368798413456847961773264} a^{14} - \frac{48330514033466900801089913073024095804433733}{4193932697979805027503184399206728423980886632} a^{13} + \frac{44431205467647721733986485992034717022241653}{4193932697979805027503184399206728423980886632} a^{12} + \frac{42163177152005547940646631281968765462274403}{4193932697979805027503184399206728423980886632} a^{11} - \frac{381493818139190139976172632285864620264612319}{8387865395959610055006368798413456847961773264} a^{10} - \frac{59647598557775674491557417471749705631095145}{8387865395959610055006368798413456847961773264} a^{9} - \frac{138690521716023813062310291600359041721214523}{4193932697979805027503184399206728423980886632} a^{8} + \frac{204755160994134859652066673087526175904436425}{2096966348989902513751592199603364211990443316} a^{7} - \frac{172514477598462588775126839455422179882496437}{8387865395959610055006368798413456847961773264} a^{6} + \frac{692848950115682669129625970635058985782292623}{4193932697979805027503184399206728423980886632} a^{5} + \frac{760578839535056928309489222519978236487124615}{4193932697979805027503184399206728423980886632} a^{4} - \frac{233707856584695550955230781545931248306760897}{4193932697979805027503184399206728423980886632} a^{3} + \frac{3259140932086318963782562598993828584471502295}{8387865395959610055006368798413456847961773264} a^{2} + \frac{5158552604473357032226056010493286052899268713}{16775730791919220110012737596826913695923546528} a + \frac{2105932758339427727124473271289162147879316067}{8387865395959610055006368798413456847961773264}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{6}\times C_{6}\times C_{378}$, which has order $13608$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 296124.35954857944 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{-5}) \), \(\Q(\zeta_{7})^+\), 3.3.756.1, 6.0.1143072000.8, 6.0.19208000.1, 9.9.1037426999616.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: data not computed

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.9.9.1$x^{9} + 54 x^{5} + 27 x^{3} + 189$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
3.9.9.1$x^{9} + 54 x^{5} + 27 x^{3} + 189$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$