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Properties

Label 18.2.80843281565...4433.1
Degree $18$
Signature $[2, 8]$
Discriminant $3^{24}\cdot 17^{15}$
Root discriminant $45.87$
Ramified primes $3, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_9$ (as 18T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![166, -549, -36, 1269, -783, -171, 834, -981, 441, 248, -549, 405, -126, -63, 117, -84, 36, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 84*x^15 + 117*x^14 - 63*x^13 - 126*x^12 + 405*x^11 - 549*x^10 + 248*x^9 + 441*x^8 - 981*x^7 + 834*x^6 - 171*x^5 - 783*x^4 + 1269*x^3 - 36*x^2 - 549*x + 166)
 
gp: K = bnfinit(x^18 - 9*x^17 + 36*x^16 - 84*x^15 + 117*x^14 - 63*x^13 - 126*x^12 + 405*x^11 - 549*x^10 + 248*x^9 + 441*x^8 - 981*x^7 + 834*x^6 - 171*x^5 - 783*x^4 + 1269*x^3 - 36*x^2 - 549*x + 166, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 36 x^{16} - 84 x^{15} + 117 x^{14} - 63 x^{13} - 126 x^{12} + 405 x^{11} - 549 x^{10} + 248 x^{9} + 441 x^{8} - 981 x^{7} + 834 x^{6} - 171 x^{5} - 783 x^{4} + 1269 x^{3} - 36 x^{2} - 549 x + 166 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(808432815650446861638683444433=3^{24}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.87$
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:  $3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{4}$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{12} a^{5} - \frac{5}{12} a^{4} + \frac{5}{12} a^{2} + \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} + \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{7} + \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{5}{12} a^{4} + \frac{5}{12} a^{3} + \frac{1}{12} a + \frac{1}{6}$, $\frac{1}{34116} a^{16} - \frac{2}{8529} a^{15} + \frac{61}{8529} a^{14} - \frac{392}{8529} a^{13} - \frac{691}{8529} a^{12} + \frac{622}{8529} a^{11} - \frac{553}{5686} a^{10} - \frac{236}{2843} a^{9} + \frac{408}{2843} a^{8} - \frac{3806}{8529} a^{7} - \frac{3157}{17058} a^{6} - \frac{6007}{17058} a^{5} - \frac{2591}{8529} a^{4} + \frac{368}{8529} a^{3} - \frac{247}{17058} a^{2} - \frac{3687}{11372} a + \frac{1689}{5686}$, $\frac{1}{3445716} a^{17} + \frac{7}{574286} a^{16} - \frac{2869}{574286} a^{15} + \frac{46189}{1148572} a^{14} + \frac{325385}{3445716} a^{13} + \frac{80677}{574286} a^{12} + \frac{470771}{3445716} a^{11} - \frac{404701}{3445716} a^{10} - \frac{258833}{1722858} a^{9} - \frac{1058303}{3445716} a^{8} + \frac{36991}{1148572} a^{7} - \frac{51475}{287143} a^{6} - \frac{411227}{1148572} a^{5} + \frac{930359}{3445716} a^{4} - \frac{115683}{287143} a^{3} + \frac{148435}{1722858} a^{2} - \frac{1410031}{3445716} a + \frac{79927}{1722858}$

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:  $9$
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:  \( 250225810.428 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_9$ (as 18T28):

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 $F_9$
Character table for $F_9$

Intermediate fields

\(\Q(\sqrt{17}) \), 9.1.218070794717793.1

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

Sibling fields

Degree 9 sibling: data not computed
Degree 12 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$