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Properties

Label 18.0.18105575687...2299.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,19^{3}\cdot 1129^{8}$
Root discriminant $37.14$
Ramified primes $19, 1129$
Class number $16$ (GRH)
Class group $[2, 8]$ (GRH)
Galois group $C_2\times C_2^2:D_9$ (as 18T67)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, -1174, 4547, -11300, 19355, -24262, 23343, -17934, 11467, -6363, 3296, -1712, 911, -468, 207, -71, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 71*x^15 + 207*x^14 - 468*x^13 + 911*x^12 - 1712*x^11 + 3296*x^10 - 6363*x^9 + 11467*x^8 - 17934*x^7 + 23343*x^6 - 24262*x^5 + 19355*x^4 - 11300*x^3 + 4547*x^2 - 1174*x + 169)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 - 71*x^15 + 207*x^14 - 468*x^13 + 911*x^12 - 1712*x^11 + 3296*x^10 - 6363*x^9 + 11467*x^8 - 17934*x^7 + 23343*x^6 - 24262*x^5 + 19355*x^4 - 11300*x^3 + 4547*x^2 - 1174*x + 169, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} - 71 x^{15} + 207 x^{14} - 468 x^{13} + 911 x^{12} - 1712 x^{11} + 3296 x^{10} - 6363 x^{9} + 11467 x^{8} - 17934 x^{7} + 23343 x^{6} - 24262 x^{5} + 19355 x^{4} - 11300 x^{3} + 4547 x^{2} - 1174 x + 169 \)

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:  \(-18105575687315621060510012299=-\,19^{3}\cdot 1129^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.14$
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:  $19, 1129$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{266771380063264350533361} a^{17} + \frac{12120599567599891661107}{266771380063264350533361} a^{16} + \frac{8942079570791597298796}{266771380063264350533361} a^{15} + \frac{33449055858027814014995}{266771380063264350533361} a^{14} + \frac{29372353574905677420395}{266771380063264350533361} a^{13} + \frac{36381709486363205796278}{266771380063264350533361} a^{12} + \frac{108003046351578310012834}{266771380063264350533361} a^{11} - \frac{75635320262036703431176}{266771380063264350533361} a^{10} + \frac{100275229095692286292402}{266771380063264350533361} a^{9} + \frac{16296165620409332680939}{266771380063264350533361} a^{8} + \frac{87210799845956997122591}{266771380063264350533361} a^{7} + \frac{8089670082916578947138}{266771380063264350533361} a^{6} - \frac{61584270018567760382455}{266771380063264350533361} a^{5} + \frac{4211488093718748488872}{266771380063264350533361} a^{4} - \frac{28164011969789516149412}{88923793354421450177787} a^{3} - \frac{60307265108411081039828}{266771380063264350533361} a^{2} + \frac{1702570792533699207664}{88923793354421450177787} a + \frac{51266221130158077424943}{266771380063264350533361}$

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

Class group and class number

$C_{2}\times C_{8}$, which has order $16$ (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:  \( 290542.983381 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:D_9$ (as 18T67):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times C_2^2:D_9$
Character table for $C_2\times C_2^2:D_9$

Intermediate fields

3.3.1129.1, 6.0.24218179.1, 9.9.1624709678881.1

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

Sibling fields

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 $18$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1129Data not computed