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Properties

Label 18.0.20614860325...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 5^{9}\cdot 7^{12}$
Root discriminant $42.52$
Ramified primes $3, 5, 7$
Class number $296$ (GRH)
Class group $[2, 2, 74]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![114211, -54930, 112827, -124326, 178851, -92796, 79951, -34212, 17883, -880, -123, 540, 82, -72, 72, -38, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 38*x^15 + 72*x^14 - 72*x^13 + 82*x^12 + 540*x^11 - 123*x^10 - 880*x^9 + 17883*x^8 - 34212*x^7 + 79951*x^6 - 92796*x^5 + 178851*x^4 - 124326*x^3 + 112827*x^2 - 54930*x + 114211)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 - 38*x^15 + 72*x^14 - 72*x^13 + 82*x^12 + 540*x^11 - 123*x^10 - 880*x^9 + 17883*x^8 - 34212*x^7 + 79951*x^6 - 92796*x^5 + 178851*x^4 - 124326*x^3 + 112827*x^2 - 54930*x + 114211, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} - 38 x^{15} + 72 x^{14} - 72 x^{13} + 82 x^{12} + 540 x^{11} - 123 x^{10} - 880 x^{9} + 17883 x^{8} - 34212 x^{7} + 79951 x^{6} - 92796 x^{5} + 178851 x^{4} - 124326 x^{3} + 112827 x^{2} - 54930 x + 114211 \)

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:  \(-206148603259625688967552734375=-\,3^{27}\cdot 5^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.52$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(134,·)$, $\chi_{315}(74,·)$, $\chi_{315}(16,·)$, $\chi_{315}(211,·)$, $\chi_{315}(149,·)$, $\chi_{315}(151,·)$, $\chi_{315}(284,·)$, $\chi_{315}(29,·)$, $\chi_{315}(226,·)$, $\chi_{315}(106,·)$, $\chi_{315}(44,·)$, $\chi_{315}(46,·)$, $\chi_{315}(239,·)$, $\chi_{315}(179,·)$, $\chi_{315}(121,·)$, $\chi_{315}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} - \frac{1}{4} a^{7} + \frac{1}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} + \frac{7}{16} a^{2} + \frac{1}{4} a - \frac{1}{16}$, $\frac{1}{16} a^{14} + \frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{3}{16} a^{7} + \frac{3}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{16} a^{4} + \frac{3}{8} a^{3} + \frac{3}{16} a^{2} + \frac{1}{16} a - \frac{7}{16}$, $\frac{1}{272} a^{15} - \frac{11}{272} a^{12} - \frac{11}{272} a^{11} - \frac{9}{272} a^{10} - \frac{1}{68} a^{9} + \frac{1}{16} a^{8} + \frac{7}{272} a^{7} + \frac{5}{34} a^{6} + \frac{7}{272} a^{5} - \frac{11}{136} a^{4} - \frac{117}{272} a^{3} + \frac{41}{272} a^{2} - \frac{107}{272} a + \frac{1}{34}$, $\frac{1}{2176} a^{16} - \frac{1}{32} a^{14} - \frac{31}{1088} a^{13} + \frac{5}{272} a^{12} - \frac{13}{1088} a^{11} + \frac{15}{1088} a^{10} - \frac{3}{64} a^{9} - \frac{61}{2176} a^{8} - \frac{29}{272} a^{7} + \frac{5}{68} a^{6} - \frac{99}{544} a^{5} + \frac{359}{2176} a^{4} - \frac{515}{1088} a^{3} - \frac{3}{34} a^{2} + \frac{15}{34} a + \frac{43}{128}$, $\frac{1}{272204082615337736676706436313708544} a^{17} - \frac{24997155216530990721609933286203}{272204082615337736676706436313708544} a^{16} + \frac{124674184578249323924307947333107}{68051020653834434169176609078427136} a^{15} - \frac{2568271531554709675423402865598897}{136102041307668868338353218156854272} a^{14} - \frac{2586101773245230957702647264514295}{136102041307668868338353218156854272} a^{13} - \frac{3906971543333067002116806951365185}{136102041307668868338353218156854272} a^{12} + \frac{2921857908337161923836599874388399}{68051020653834434169176609078427136} a^{11} - \frac{233421556835817051605004752833341}{17012755163458608542294152269606784} a^{10} + \frac{212999423145050542891674523882005}{272204082615337736676706436313708544} a^{9} - \frac{9029232050204945013010076817533897}{272204082615337736676706436313708544} a^{8} + \frac{6225799801687172254633614949580159}{34025510326917217084588304539213568} a^{7} - \frac{10734883787076239354178945328716895}{68051020653834434169176609078427136} a^{6} - \frac{10524736187396765067720924334850309}{272204082615337736676706436313708544} a^{5} - \frac{27911369758688793834684381433230451}{272204082615337736676706436313708544} a^{4} + \frac{42455508170489972425659722465966777}{136102041307668868338353218156854272} a^{3} + \frac{1328544961985286320444926523950241}{4253188790864652135573538067401696} a^{2} + \frac{2481313398993795811985643882710475}{16012004859725749216276849194924032} a + \frac{379459917334113752787217133356795}{1503889959200760976114400200628224}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{74}$, which has order $296$ (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:  \( 54408.4888887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.1, 3.3.3969.2, 6.0.2460375.1, 6.0.8103375.1, 6.0.5907360375.1, 6.0.5907360375.2, 9.9.62523502209.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed