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Properties

Label 18.18.1241861436...8841.1
Degree $18$
Signature $[18, 0]$
Discriminant $19^{6}\cdot 1129^{8}$
Root discriminant $60.67$
Ramified primes $19, 1129$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2:D_9$ (as 18T38)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![239, -571, -5619, 13242, 16950, -52544, -7806, 77348, -19406, -48198, 21330, 12378, -6930, -1343, 908, 62, -51, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 51*x^16 + 62*x^15 + 908*x^14 - 1343*x^13 - 6930*x^12 + 12378*x^11 + 21330*x^10 - 48198*x^9 - 19406*x^8 + 77348*x^7 - 7806*x^6 - 52544*x^5 + 16950*x^4 + 13242*x^3 - 5619*x^2 - 571*x + 239)
 
gp: K = bnfinit(x^18 - x^17 - 51*x^16 + 62*x^15 + 908*x^14 - 1343*x^13 - 6930*x^12 + 12378*x^11 + 21330*x^10 - 48198*x^9 - 19406*x^8 + 77348*x^7 - 7806*x^6 - 52544*x^5 + 16950*x^4 + 13242*x^3 - 5619*x^2 - 571*x + 239, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 51 x^{16} + 62 x^{15} + 908 x^{14} - 1343 x^{13} - 6930 x^{12} + 12378 x^{11} + 21330 x^{10} - 48198 x^{9} - 19406 x^{8} + 77348 x^{7} - 7806 x^{6} - 52544 x^{5} + 16950 x^{4} + 13242 x^{3} - 5619 x^{2} - 571 x + 239 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(124186143639297844854038174358841=19^{6}\cdot 1129^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.67$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} + \frac{9}{19} a^{15} - \frac{5}{19} a^{14} - \frac{4}{19} a^{13} + \frac{5}{19} a^{12} + \frac{4}{19} a^{11} - \frac{4}{19} a^{10} + \frac{2}{19} a^{9} - \frac{1}{19} a^{8} + \frac{2}{19} a^{7} + \frac{6}{19} a^{5} + \frac{6}{19} a^{4} - \frac{4}{19} a^{3} + \frac{2}{19} a^{2} + \frac{5}{19} a + \frac{5}{19}$, $\frac{1}{712350484986941727440574863} a^{17} + \frac{16117437729884356890216634}{712350484986941727440574863} a^{16} + \frac{13558810403264297286474600}{712350484986941727440574863} a^{15} - \frac{105039673783981827805049417}{712350484986941727440574863} a^{14} + \frac{35441365745081414811242601}{712350484986941727440574863} a^{13} - \frac{170321598749969230421809723}{712350484986941727440574863} a^{12} - \frac{71677285459147185216446581}{712350484986941727440574863} a^{11} + \frac{328268098355803040402334121}{712350484986941727440574863} a^{10} - \frac{22921992601531987563294181}{64759134998812884312779533} a^{9} - \frac{103073716122290014816280045}{712350484986941727440574863} a^{8} + \frac{241166215980487670817302533}{712350484986941727440574863} a^{7} + \frac{26519277649121516802767712}{712350484986941727440574863} a^{6} + \frac{99666264337129175657380051}{712350484986941727440574863} a^{5} - \frac{308676925844925030195777803}{712350484986941727440574863} a^{4} - \frac{97448259069923855262622818}{712350484986941727440574863} a^{3} + \frac{314024068701723683476292749}{712350484986941727440574863} a^{2} + \frac{13444611168480436001783097}{64759134998812884312779533} a - \frac{223310542920247245748743677}{712350484986941727440574863}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 18254353547.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_9$ (as 18T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_2^2:D_9$
Character table for $C_2^2:D_9$

Intermediate fields

3.3.1129.1, 6.6.460145401.1, 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

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