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Properties

Label 18.0.41720403987...5392.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 293^{9}$
Root discriminant $34.23$
Ramified primes $2, 293$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![16384, 8192, 9216, -16384, 9216, -3968, 12688, -17408, 8884, -1400, 344, -328, 265, -122, 23, -4, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 3*x^16 - 4*x^15 + 23*x^14 - 122*x^13 + 265*x^12 - 328*x^11 + 344*x^10 - 1400*x^9 + 8884*x^8 - 17408*x^7 + 12688*x^6 - 3968*x^5 + 9216*x^4 - 16384*x^3 + 9216*x^2 + 8192*x + 16384)
 
gp: K = bnfinit(x^18 - 2*x^17 + 3*x^16 - 4*x^15 + 23*x^14 - 122*x^13 + 265*x^12 - 328*x^11 + 344*x^10 - 1400*x^9 + 8884*x^8 - 17408*x^7 + 12688*x^6 - 3968*x^5 + 9216*x^4 - 16384*x^3 + 9216*x^2 + 8192*x + 16384, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 3 x^{16} - 4 x^{15} + 23 x^{14} - 122 x^{13} + 265 x^{12} - 328 x^{11} + 344 x^{10} - 1400 x^{9} + 8884 x^{8} - 17408 x^{7} + 12688 x^{6} - 3968 x^{5} + 9216 x^{4} - 16384 x^{3} + 9216 x^{2} + 8192 x + 16384 \)

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:  \(-4172040398734878910471995392=-\,2^{18}\cdot 293^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.23$
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:  $2, 293$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} + \frac{1}{8} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{9} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{11} + \frac{1}{32} a^{7} - \frac{1}{8} a^{6} - \frac{1}{16} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{8} + \frac{1}{16} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{12} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{3}{128} a^{9} - \frac{5}{128} a^{8} + \frac{3}{32} a^{6} + \frac{1}{32} a^{5} + \frac{3}{32} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{128} a^{14} + \frac{1}{128} a^{12} + \frac{3}{128} a^{10} - \frac{1}{128} a^{8} + \frac{3}{32} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{1024} a^{15} - \frac{1}{512} a^{14} + \frac{3}{1024} a^{13} + \frac{1}{256} a^{12} + \frac{15}{1024} a^{11} - \frac{5}{512} a^{10} - \frac{7}{1024} a^{9} + \frac{1}{32} a^{8} + \frac{3}{64} a^{7} + \frac{1}{128} a^{6} + \frac{5}{256} a^{5} + \frac{5}{32} a^{4} + \frac{15}{64} a^{3} + \frac{3}{8} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{59392} a^{16} - \frac{1}{14848} a^{15} - \frac{33}{59392} a^{14} + \frac{3}{29696} a^{13} - \frac{393}{59392} a^{12} + \frac{21}{7424} a^{11} + \frac{517}{59392} a^{10} - \frac{85}{29696} a^{9} - \frac{103}{3712} a^{8} - \frac{191}{7424} a^{7} + \frac{1161}{14848} a^{6} + \frac{659}{7424} a^{5} - \frac{341}{3712} a^{4} - \frac{159}{1856} a^{3} + \frac{171}{464} a^{2} - \frac{71}{232} a + \frac{3}{58}$, $\frac{1}{6392975270068299407360} a^{17} - \frac{11640049233651623}{3196487635034149703680} a^{16} + \frac{1073892436060979827}{6392975270068299407360} a^{15} + \frac{441586813224751741}{799121908758537425920} a^{14} - \frac{16899671346548475569}{6392975270068299407360} a^{13} + \frac{8437079081441207497}{3196487635034149703680} a^{12} - \frac{10680281893855321751}{6392975270068299407360} a^{11} + \frac{47522199001436798919}{1598243817517074851840} a^{10} - \frac{1704827813631683663}{79912190875853742592} a^{9} + \frac{3301341473821207417}{159824381751707485184} a^{8} + \frac{64592582654973746581}{1598243817517074851840} a^{7} + \frac{33520898130950453301}{399560954379268712960} a^{6} + \frac{31254203425966014919}{399560954379268712960} a^{5} - \frac{6381878683361070741}{99890238594817178240} a^{4} - \frac{10783642648493718251}{49945119297408589120} a^{3} - \frac{223723637500353883}{6243139912176073640} a^{2} + \frac{784058024079639893}{3121569956088036820} a - \frac{285667496611293842}{780392489022009205}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 365341428.034 \) (assuming GRH)
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

\(\Q(\sqrt{-293}) \), 3.1.1172.1 x3, 6.0.1609840448.1, 9.1.1886733005056.1 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 R ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
293Data not computed