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Properties

Label 18.0.16585685619...4011.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,491^{9}$
Root discriminant $22.16$
Ramified prime $491$
Class number $4$
Class group $[2, 2]$
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -1120, 2372, -3378, 3757, -3298, 1880, -228, -739, 738, -191, -156, 153, 2, -47, 14, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 4*x^16 + 14*x^15 - 47*x^14 + 2*x^13 + 153*x^12 - 156*x^11 - 191*x^10 + 738*x^9 - 739*x^8 - 228*x^7 + 1880*x^6 - 3298*x^5 + 3757*x^4 - 3378*x^3 + 2372*x^2 - 1120*x + 256)
 
gp: K = bnfinit(x^18 - 4*x^17 + 4*x^16 + 14*x^15 - 47*x^14 + 2*x^13 + 153*x^12 - 156*x^11 - 191*x^10 + 738*x^9 - 739*x^8 - 228*x^7 + 1880*x^6 - 3298*x^5 + 3757*x^4 - 3378*x^3 + 2372*x^2 - 1120*x + 256, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 4 x^{16} + 14 x^{15} - 47 x^{14} + 2 x^{13} + 153 x^{12} - 156 x^{11} - 191 x^{10} + 738 x^{9} - 739 x^{8} - 228 x^{7} + 1880 x^{6} - 3298 x^{5} + 3757 x^{4} - 3378 x^{3} + 2372 x^{2} - 1120 x + 256 \)

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:  \(-1658568561963902101824011=-\,491^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.16$
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:  $491$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{16} a^{5} - \frac{1}{16} a^{4} + \frac{1}{16} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{3}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{5}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} + \frac{1}{16} a^{10} - \frac{3}{32} a^{9} - \frac{1}{8} a^{8} + \frac{1}{32} a^{7} - \frac{3}{16} a^{6} + \frac{7}{32} a^{5} - \frac{1}{4} a^{4} - \frac{7}{32} a^{3} + \frac{7}{16} a^{2} - \frac{3}{8} a$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{12} - \frac{1}{32} a^{10} + \frac{1}{16} a^{9} + \frac{1}{32} a^{8} - \frac{1}{16} a^{7} - \frac{5}{32} a^{6} + \frac{3}{16} a^{5} - \frac{5}{32} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{224} a^{15} + \frac{1}{112} a^{14} + \frac{3}{224} a^{13} - \frac{1}{112} a^{12} + \frac{5}{224} a^{11} + \frac{3}{112} a^{10} - \frac{3}{32} a^{9} - \frac{1}{28} a^{8} + \frac{1}{32} a^{7} + \frac{1}{14} a^{6} - \frac{19}{224} a^{5} + \frac{3}{28} a^{4} + \frac{11}{112} a^{3} - \frac{23}{56} a^{2} - \frac{3}{28} a + \frac{1}{7}$, $\frac{1}{3584} a^{16} - \frac{1}{3584} a^{15} - \frac{3}{448} a^{14} + \frac{31}{3584} a^{13} + \frac{51}{1792} a^{12} - \frac{51}{3584} a^{11} + \frac{117}{1792} a^{10} - \frac{323}{3584} a^{9} + \frac{159}{1792} a^{8} + \frac{135}{3584} a^{7} - \frac{127}{896} a^{6} + \frac{473}{3584} a^{5} + \frac{671}{3584} a^{4} + \frac{789}{1792} a^{3} + \frac{55}{128} a^{2} - \frac{23}{112} a - \frac{3}{14}$, $\frac{1}{2297197353472} a^{17} + \frac{7038735}{143574834592} a^{16} + \frac{269927015}{2297197353472} a^{15} + \frac{31740419223}{2297197353472} a^{14} + \frac{775249939}{328171050496} a^{13} + \frac{6308996835}{2297197353472} a^{12} + \frac{32384335831}{2297197353472} a^{11} - \frac{186121891369}{2297197353472} a^{10} - \frac{51659698565}{2297197353472} a^{9} + \frac{24385848405}{2297197353472} a^{8} - \frac{18822803139}{328171050496} a^{7} + \frac{14718040205}{2297197353472} a^{6} - \frac{52574434145}{287149669184} a^{5} - \frac{445463196663}{2297197353472} a^{4} + \frac{307067241319}{1148598676736} a^{3} - \frac{12519172705}{82042762624} a^{2} - \frac{29372729163}{71787417296} a - \frac{400321381}{1281918166}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 79186.8820128 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-491}) \), 3.1.491.1 x3, 6.0.118370771.1, 9.1.58120048561.1 x9

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
491Data not computed