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Properties

Label 18.0.56829724891...8323.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,563^{9}$
Root discriminant $23.73$
Ramified prime $563$
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![2396, -1896, 2409, -4448, 3530, -2556, 2873, -2036, 1190, -860, 644, -336, 194, -124, 65, -28, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 10*x^16 - 28*x^15 + 65*x^14 - 124*x^13 + 194*x^12 - 336*x^11 + 644*x^10 - 860*x^9 + 1190*x^8 - 2036*x^7 + 2873*x^6 - 2556*x^5 + 3530*x^4 - 4448*x^3 + 2409*x^2 - 1896*x + 2396)
 
gp: K = bnfinit(x^18 - 4*x^17 + 10*x^16 - 28*x^15 + 65*x^14 - 124*x^13 + 194*x^12 - 336*x^11 + 644*x^10 - 860*x^9 + 1190*x^8 - 2036*x^7 + 2873*x^6 - 2556*x^5 + 3530*x^4 - 4448*x^3 + 2409*x^2 - 1896*x + 2396, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 10 x^{16} - 28 x^{15} + 65 x^{14} - 124 x^{13} + 194 x^{12} - 336 x^{11} + 644 x^{10} - 860 x^{9} + 1190 x^{8} - 2036 x^{7} + 2873 x^{6} - 2556 x^{5} + 3530 x^{4} - 4448 x^{3} + 2409 x^{2} - 1896 x + 2396 \)

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:  \(-5682972489147397363698323=-\,563^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.73$
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:  $563$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{11} - \frac{1}{16} a^{10} - \frac{1}{32} a^{9} - \frac{3}{32} a^{8} + \frac{3}{32} a^{6} - \frac{5}{32} a^{5} + \frac{3}{16} a^{4} + \frac{9}{32} a^{3} + \frac{7}{32} a^{2} - \frac{1}{8} a - \frac{3}{8}$, $\frac{1}{32} a^{13} + \frac{1}{32} a^{11} + \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{1}{32} a^{8} + \frac{3}{32} a^{7} - \frac{1}{4} a^{6} + \frac{7}{32} a^{5} + \frac{3}{32} a^{4} + \frac{7}{16} a^{3} - \frac{7}{32} a^{2} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{32} a^{14} - \frac{1}{16} a^{8} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{9}{32} a^{2} + \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{1}{32} a^{9} + \frac{1}{32} a^{8} + \frac{1}{16} a^{7} - \frac{3}{16} a^{6} + \frac{1}{8} a^{4} - \frac{23}{64} a^{3} - \frac{1}{64} a^{2} - \frac{3}{16} a - \frac{7}{16}$, $\frac{1}{256} a^{16} + \frac{3}{256} a^{14} - \frac{1}{64} a^{12} - \frac{3}{64} a^{11} + \frac{7}{128} a^{10} - \frac{1}{64} a^{9} - \frac{7}{128} a^{8} - \frac{3}{32} a^{7} - \frac{7}{32} a^{6} + \frac{5}{64} a^{5} + \frac{49}{256} a^{4} - \frac{17}{64} a^{3} - \frac{125}{256} a^{2} - \frac{5}{32} a + \frac{1}{64}$, $\frac{1}{6632875309212160} a^{17} - \frac{1910818711985}{1326575061842432} a^{16} + \frac{1779752211311}{1326575061842432} a^{15} + \frac{7580204958297}{6632875309212160} a^{14} - \frac{13641392243733}{1658218827303040} a^{13} + \frac{831411258161}{829109413651520} a^{12} + \frac{73863331152893}{3316437654606080} a^{11} - \frac{163153636502701}{3316437654606080} a^{10} - \frac{120008179422357}{3316437654606080} a^{9} - \frac{104831497081433}{3316437654606080} a^{8} - \frac{18433316233109}{414554706825760} a^{7} - \frac{172711004174093}{1658218827303040} a^{6} - \frac{205656691710151}{1326575061842432} a^{5} - \frac{173392505663481}{6632875309212160} a^{4} + \frac{2994755382279071}{6632875309212160} a^{3} + \frac{1355395503289281}{6632875309212160} a^{2} + \frac{130385858139707}{1658218827303040} a + \frac{338512193044099}{1658218827303040}$

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:  \( 168278.897914 \)
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

\(\Q(\sqrt{-563}) \), 3.1.563.1 x3, 6.0.178453547.1, 9.1.100469346961.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
563Data not computed