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Properties

Label 18.0.46779596406...7427.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,1187^{9}$
Root discriminant $34.45$
Ramified prime $1187$
Class number $1$ (GRH)
Class group Trivial (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![10000, -59000, 158800, -240390, 240121, -178442, 103815, -51770, 23952, -9634, 4216, -1860, 696, -334, 112, -40, 15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 15*x^16 - 40*x^15 + 112*x^14 - 334*x^13 + 696*x^12 - 1860*x^11 + 4216*x^10 - 9634*x^9 + 23952*x^8 - 51770*x^7 + 103815*x^6 - 178442*x^5 + 240121*x^4 - 240390*x^3 + 158800*x^2 - 59000*x + 10000)
 
gp: K = bnfinit(x^18 - 2*x^17 + 15*x^16 - 40*x^15 + 112*x^14 - 334*x^13 + 696*x^12 - 1860*x^11 + 4216*x^10 - 9634*x^9 + 23952*x^8 - 51770*x^7 + 103815*x^6 - 178442*x^5 + 240121*x^4 - 240390*x^3 + 158800*x^2 - 59000*x + 10000, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 15 x^{16} - 40 x^{15} + 112 x^{14} - 334 x^{13} + 696 x^{12} - 1860 x^{11} + 4216 x^{10} - 9634 x^{9} + 23952 x^{8} - 51770 x^{7} + 103815 x^{6} - 178442 x^{5} + 240121 x^{4} - 240390 x^{3} + 158800 x^{2} - 59000 x + 10000 \)

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:  \(-4677959640662633651334007427=-\,1187^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.45$
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:  $1187$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{112} a^{12} - \frac{3}{56} a^{11} + \frac{5}{112} a^{9} + \frac{1}{28} a^{8} - \frac{5}{28} a^{7} - \frac{3}{16} a^{6} + \frac{13}{56} a^{5} - \frac{1}{4} a^{4} + \frac{55}{112} a^{3} - \frac{5}{14} a^{2} - \frac{1}{7} a - \frac{3}{14}$, $\frac{1}{224} a^{13} - \frac{1}{224} a^{12} + \frac{3}{56} a^{11} - \frac{9}{224} a^{10} + \frac{1}{224} a^{9} + \frac{47}{224} a^{7} - \frac{23}{224} a^{6} + \frac{1}{56} a^{5} - \frac{15}{224} a^{4} + \frac{95}{224} a^{3} - \frac{3}{14} a^{2} + \frac{1}{28} a - \frac{1}{28}$, $\frac{1}{1120} a^{14} + \frac{1}{560} a^{13} + \frac{3}{1120} a^{12} + \frac{1}{160} a^{11} + \frac{3}{112} a^{10} - \frac{139}{1120} a^{9} + \frac{27}{224} a^{8} - \frac{3}{16} a^{7} + \frac{61}{1120} a^{6} - \frac{43}{224} a^{5} + \frac{81}{560} a^{4} + \frac{243}{1120} a^{3} + \frac{69}{140} a^{2} + \frac{3}{10} a + \frac{3}{28}$, $\frac{1}{1120} a^{15} - \frac{1}{1120} a^{13} + \frac{1}{1120} a^{12} + \frac{1}{70} a^{11} - \frac{59}{1120} a^{10} - \frac{1}{160} a^{9} + \frac{1}{14} a^{8} - \frac{79}{1120} a^{7} + \frac{223}{1120} a^{6} + \frac{1}{35} a^{5} - \frac{221}{1120} a^{4} - \frac{37}{560} a^{3} + \frac{11}{35} a^{2} + \frac{1}{140} a - \frac{3}{14}$, $\frac{1}{11200} a^{16} + \frac{3}{11200} a^{15} - \frac{1}{400} a^{12} + \frac{1}{700} a^{11} + \frac{9}{800} a^{10} - \frac{61}{1120} a^{9} - \frac{23}{1400} a^{8} + \frac{237}{1400} a^{7} + \frac{313}{2800} a^{6} - \frac{17}{140} a^{5} - \frac{339}{2240} a^{4} + \frac{2943}{11200} a^{3} - \frac{353}{1400} a^{2} - \frac{61}{280} a + \frac{3}{56}$, $\frac{1}{42926107452393032000} a^{17} - \frac{7046178958823}{670720428943641125} a^{16} - \frac{289440839551487}{1226460212925515200} a^{15} + \frac{19135095648237}{153307526615689400} a^{14} + \frac{6781068782503231}{21463053726196516000} a^{13} - \frac{47348514570434337}{21463053726196516000} a^{12} + \frac{473148546702926863}{21463053726196516000} a^{11} + \frac{138856906797618717}{4292610745239303200} a^{10} - \frac{1131545442133942771}{10731526863098258000} a^{9} - \frac{347283275889660513}{5365763431549129000} a^{8} + \frac{2169303646842427041}{21463053726196516000} a^{7} + \frac{953999168937164689}{4292610745239303200} a^{6} - \frac{441889536179557667}{8585221490478606400} a^{5} + \frac{4774034922697355279}{21463053726196516000} a^{4} + \frac{16957569833610007961}{42926107452393032000} a^{3} - \frac{25446175701335371}{268288171577456450} a^{2} - \frac{8226196827223184}{26828817157745645} a + \frac{7972255319240017}{42926107452393032}$

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:  $8$
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:  \( 22636075.4911 \) (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

\(\Q(\sqrt{-1187}) \), 3.1.1187.1 x3, 6.0.1672446203.1, 9.1.1985193642961.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
1187Data not computed