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Properties

Label 18.0.10493915932...0343.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,17^{12}\cdot 23^{9}$
Root discriminant $31.71$
Ramified primes $17, 23$
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![13961, 149960, 306556, -557964, 511018, -258728, 178616, -63455, 25845, -3725, 2030, 560, 231, 80, -9, -34, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 3*x^16 - 34*x^15 - 9*x^14 + 80*x^13 + 231*x^12 + 560*x^11 + 2030*x^10 - 3725*x^9 + 25845*x^8 - 63455*x^7 + 178616*x^6 - 258728*x^5 + 511018*x^4 - 557964*x^3 + 306556*x^2 + 149960*x + 13961)
 
gp: K = bnfinit(x^18 - 3*x^17 + 3*x^16 - 34*x^15 - 9*x^14 + 80*x^13 + 231*x^12 + 560*x^11 + 2030*x^10 - 3725*x^9 + 25845*x^8 - 63455*x^7 + 178616*x^6 - 258728*x^5 + 511018*x^4 - 557964*x^3 + 306556*x^2 + 149960*x + 13961, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 3 x^{16} - 34 x^{15} - 9 x^{14} + 80 x^{13} + 231 x^{12} + 560 x^{11} + 2030 x^{10} - 3725 x^{9} + 25845 x^{8} - 63455 x^{7} + 178616 x^{6} - 258728 x^{5} + 511018 x^{4} - 557964 x^{3} + 306556 x^{2} + 149960 x + 13961 \)

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:  \(-1049391593213911389381400343=-\,17^{12}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.71$
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:  $17, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} + \frac{3}{11} a^{12} + \frac{3}{11} a^{10} - \frac{4}{11} a^{9} - \frac{1}{11} a^{8} + \frac{1}{11} a^{7} - \frac{4}{11} a^{6} - \frac{1}{11} a^{5} + \frac{4}{11} a^{4} - \frac{1}{11} a^{3} - \frac{2}{11} a^{2} - \frac{2}{11} a + \frac{2}{11}$, $\frac{1}{121} a^{14} + \frac{4}{121} a^{13} + \frac{58}{121} a^{12} + \frac{25}{121} a^{11} + \frac{43}{121} a^{10} - \frac{5}{121} a^{9} - \frac{14}{121} a^{7} + \frac{39}{121} a^{6} - \frac{8}{121} a^{5} + \frac{58}{121} a^{4} + \frac{41}{121} a^{3} - \frac{26}{121} a^{2} - \frac{4}{11} a - \frac{53}{121}$, $\frac{1}{25289} a^{15} - \frac{28}{25289} a^{14} - \frac{1038}{25289} a^{13} - \frac{5582}{25289} a^{12} - \frac{4387}{25289} a^{11} + \frac{9267}{25289} a^{10} + \frac{11050}{25289} a^{9} - \frac{5943}{25289} a^{8} - \frac{573}{1331} a^{7} + \frac{9029}{25289} a^{6} - \frac{8035}{25289} a^{5} + \frac{3}{209} a^{4} + \frac{10641}{25289} a^{3} - \frac{10465}{25289} a^{2} + \frac{3170}{25289} a - \frac{8105}{25289}$, $\frac{1}{2642422321} a^{16} - \frac{16571}{2642422321} a^{15} - \frac{441341}{2642422321} a^{14} - \frac{93875648}{2642422321} a^{13} + \frac{510209477}{2642422321} a^{12} + \frac{260728970}{2642422321} a^{11} + \frac{5465580}{12643169} a^{10} - \frac{1266676540}{2642422321} a^{9} + \frac{15746380}{240220211} a^{8} + \frac{370366415}{2642422321} a^{7} - \frac{2075553}{2642422321} a^{6} - \frac{655946296}{2642422321} a^{5} + \frac{677689049}{2642422321} a^{4} - \frac{15141505}{240220211} a^{3} - \frac{85229058}{2642422321} a^{2} + \frac{53777607}{114887927} a + \frac{54279535}{114887927}$, $\frac{1}{356650927810127138891898763762522405} a^{17} - \frac{458477568794627136800169}{71330185562025427778379752752504481} a^{16} + \frac{6660719158871213983272835251358}{356650927810127138891898763762522405} a^{15} + \frac{67457973228013407164103299889034}{71330185562025427778379752752504481} a^{14} - \frac{7353970281426998573508680167512939}{356650927810127138891898763762522405} a^{13} + \frac{217193179545909149580996174540414}{816134846247430523780088704262065} a^{12} - \frac{19498004750059366614774928338853106}{71330185562025427778379752752504481} a^{11} + \frac{13070686721984320731707420365605700}{71330185562025427778379752752504481} a^{10} + \frac{27387351518651625295335438516792200}{71330185562025427778379752752504481} a^{9} + \frac{144082636519219045398876515333997}{536317184676882915626915434229357} a^{8} + \frac{14162174550083690071415285583938321}{71330185562025427778379752752504481} a^{7} - \frac{29556316572002051394018170056735699}{71330185562025427778379752752504481} a^{6} + \frac{141257171619968365439174322401995966}{356650927810127138891898763762522405} a^{5} - \frac{35097648231858842193261992056433954}{71330185562025427778379752752504481} a^{4} + \frac{33277963941208010117452695866523948}{356650927810127138891898763762522405} a^{3} + \frac{254736896226121615466077893070630}{536317184676882915626915434229357} a^{2} + \frac{3048938950075829778014311944093552}{15506562078701179951821685380979235} a + \frac{6993349407512407723777107531433656}{15506562078701179951821685380979235}$

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:  \( 1645723.56403 \) (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

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.6754681446529.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{18}$

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
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$