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Properties

Label 18.10.2799928238...0217.6
Degree $18$
Signature $[10, 4]$
Discriminant $7^{12}\cdot 53^{6}\cdot 97^{3}$
Root discriminant $29.46$
Ramified primes $7, 53, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T367

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 48, 150, -1051, 1227, 102, -151, -468, -177, 227, 8, 259, -166, -25, 40, -26, 14, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 14*x^16 - 26*x^15 + 40*x^14 - 25*x^13 - 166*x^12 + 259*x^11 + 8*x^10 + 227*x^9 - 177*x^8 - 468*x^7 - 151*x^6 + 102*x^5 + 1227*x^4 - 1051*x^3 + 150*x^2 + 48*x - 8)
 
gp: K = bnfinit(x^18 - 5*x^17 + 14*x^16 - 26*x^15 + 40*x^14 - 25*x^13 - 166*x^12 + 259*x^11 + 8*x^10 + 227*x^9 - 177*x^8 - 468*x^7 - 151*x^6 + 102*x^5 + 1227*x^4 - 1051*x^3 + 150*x^2 + 48*x - 8, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 14 x^{16} - 26 x^{15} + 40 x^{14} - 25 x^{13} - 166 x^{12} + 259 x^{11} + 8 x^{10} + 227 x^{9} - 177 x^{8} - 468 x^{7} - 151 x^{6} + 102 x^{5} + 1227 x^{4} - 1051 x^{3} + 150 x^{2} + 48 x - 8 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(279992823820843547402730217=7^{12}\cdot 53^{6}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.46$
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:  $7, 53, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{48971740892519244791044} a^{17} + \frac{2653451139968366874405}{48971740892519244791044} a^{16} + \frac{4729794720860703593730}{12242935223129811197761} a^{15} + \frac{6652970109744347599269}{24485870446259622395522} a^{14} + \frac{5781383068694510761324}{12242935223129811197761} a^{13} + \frac{15146231032675381286183}{48971740892519244791044} a^{12} + \frac{2811965081828796254636}{12242935223129811197761} a^{11} + \frac{22915216731349503397399}{48971740892519244791044} a^{10} - \frac{9425130472372937074007}{24485870446259622395522} a^{9} - \frac{10162967748823684106585}{48971740892519244791044} a^{8} - \frac{4855615446720450759531}{48971740892519244791044} a^{7} - \frac{10437786662511585191431}{24485870446259622395522} a^{6} + \frac{11712055337021803453097}{48971740892519244791044} a^{5} + \frac{3417735543732961907566}{12242935223129811197761} a^{4} - \frac{8660023876658963172725}{48971740892519244791044} a^{3} - \frac{413195623925891729629}{48971740892519244791044} a^{2} - \frac{2191365683920573238918}{12242935223129811197761} a + \frac{74620563472301234750}{12242935223129811197761}$

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:  $13$
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:  \( 4339003.92097 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T367:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n367
Character table for t18n367 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
7Data not computed
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
97Data not computed