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Properties

Label 18.18.2980200459...9769.1
Degree $18$
Signature $[18, 0]$
Discriminant $1129^{9}$
Root discriminant $33.60$
Ramified prime $1129$
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![1, 6, -78, -250, 1876, 1871, -12300, 646, 18736, -5875, -10538, 4629, 2473, -1395, -196, 178, -6, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 6*x^16 + 178*x^15 - 196*x^14 - 1395*x^13 + 2473*x^12 + 4629*x^11 - 10538*x^10 - 5875*x^9 + 18736*x^8 + 646*x^7 - 12300*x^6 + 1871*x^5 + 1876*x^4 - 250*x^3 - 78*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^18 - 8*x^17 - 6*x^16 + 178*x^15 - 196*x^14 - 1395*x^13 + 2473*x^12 + 4629*x^11 - 10538*x^10 - 5875*x^9 + 18736*x^8 + 646*x^7 - 12300*x^6 + 1871*x^5 + 1876*x^4 - 250*x^3 - 78*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 6 x^{16} + 178 x^{15} - 196 x^{14} - 1395 x^{13} + 2473 x^{12} + 4629 x^{11} - 10538 x^{10} - 5875 x^{9} + 18736 x^{8} + 646 x^{7} - 12300 x^{6} + 1871 x^{5} + 1876 x^{4} - 250 x^{3} - 78 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2980200459393400813138329769=1129^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.60$
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:  $1129$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{15} + \frac{1}{6} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{4} a^{10} + \frac{5}{12} a^{9} + \frac{1}{4} a^{8} - \frac{1}{6} a^{7} + \frac{1}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{1}{6} a^{3} + \frac{1}{12} a^{2} + \frac{1}{4} a - \frac{5}{12}$, $\frac{1}{12} a^{16} + \frac{1}{12} a^{14} + \frac{1}{12} a^{13} + \frac{1}{12} a^{12} - \frac{1}{12} a^{11} + \frac{5}{12} a^{10} - \frac{1}{12} a^{9} + \frac{1}{3} a^{8} - \frac{1}{12} a^{7} + \frac{1}{4} a^{6} + \frac{1}{12} a^{5} - \frac{1}{2} a^{4} + \frac{5}{12} a^{3} - \frac{5}{12} a^{2} + \frac{1}{12} a - \frac{1}{6}$, $\frac{1}{223538873207001418236} a^{17} - \frac{22081643227235317}{6574672741382394654} a^{16} + \frac{722227832395771969}{74512957735667139412} a^{15} + \frac{1002732891792113383}{11765203853000074644} a^{14} + \frac{8418886488431352697}{223538873207001418236} a^{13} - \frac{49517936121115762063}{223538873207001418236} a^{12} - \frac{48093243462640941455}{223538873207001418236} a^{11} + \frac{28078036208914600639}{74512957735667139412} a^{10} + \frac{10554582549292539221}{37256478867833569706} a^{9} - \frac{3416377133792710069}{13149345482764789308} a^{8} - \frac{12078940926636684961}{74512957735667139412} a^{7} + \frac{27826378383991755605}{74512957735667139412} a^{6} - \frac{353311326720248803}{1095778790230399109} a^{5} + \frac{14364806889695934213}{74512957735667139412} a^{4} - \frac{41795032118591496143}{223538873207001418236} a^{3} - \frac{24021133165271853451}{223538873207001418236} a^{2} - \frac{11246757648953867425}{37256478867833569706} a + \frac{2243642177003939128}{55884718301750354559}$

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:  $17$
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:  \( 67435935.459 \) (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

\(\Q(\sqrt{1129}) \), 3.3.1129.1 x3, 6.6.1439069689.1, 9.9.1624709678881.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
1129Data not computed