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Properties

Label 18.0.32108153175...1696.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{32}\cdot 31^{9}$
Root discriminant $72.69$
Ramified primes $2, 3, 31$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![132669478, -155589642, 232468164, -175402506, 146696742, -82922184, 49269396, -21843180, 9919386, -3484347, 1242963, -342828, 96528, -20286, 4446, -660, 108, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 108*x^16 - 660*x^15 + 4446*x^14 - 20286*x^13 + 96528*x^12 - 342828*x^11 + 1242963*x^10 - 3484347*x^9 + 9919386*x^8 - 21843180*x^7 + 49269396*x^6 - 82922184*x^5 + 146696742*x^4 - 175402506*x^3 + 232468164*x^2 - 155589642*x + 132669478)
 
gp: K = bnfinit(x^18 - 9*x^17 + 108*x^16 - 660*x^15 + 4446*x^14 - 20286*x^13 + 96528*x^12 - 342828*x^11 + 1242963*x^10 - 3484347*x^9 + 9919386*x^8 - 21843180*x^7 + 49269396*x^6 - 82922184*x^5 + 146696742*x^4 - 175402506*x^3 + 232468164*x^2 - 155589642*x + 132669478, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 108 x^{16} - 660 x^{15} + 4446 x^{14} - 20286 x^{13} + 96528 x^{12} - 342828 x^{11} + 1242963 x^{10} - 3484347 x^{9} + 9919386 x^{8} - 21843180 x^{7} + 49269396 x^{6} - 82922184 x^{5} + 146696742 x^{4} - 175402506 x^{3} + 232468164 x^{2} - 155589642 x + 132669478 \)

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:  \(-3210815317563008405151163699101696=-\,2^{16}\cdot 3^{32}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.69$
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:  $2, 3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{1460557071226893686344501580976058620111517192790065887} a^{17} + \frac{17809283667576084815408732329452980586076898985426781}{1460557071226893686344501580976058620111517192790065887} a^{16} + \frac{453572986237189445147987543910574187136438520366811}{1460557071226893686344501580976058620111517192790065887} a^{15} + \frac{59405109250615796415809240345645921506472182007589117}{486852357075631228781500526992019540037172397596688629} a^{14} - \frac{1807427732471206787573229866345760856839404085781502}{85915121836876099196735387116238742359501011340592111} a^{13} + \frac{53270080339425185378762821072571961906107152276498923}{486852357075631228781500526992019540037172397596688629} a^{12} + \frac{15237337041418666013232944783577129013812526657857361}{486852357075631228781500526992019540037172397596688629} a^{11} - \frac{46441902562002030655139128618806525875883162865234105}{1460557071226893686344501580976058620111517192790065887} a^{10} + \frac{20303822252121597657372601222489498305996273047402860}{486852357075631228781500526992019540037172397596688629} a^{9} + \frac{171734246473407584203903170055598501004771195661280045}{486852357075631228781500526992019540037172397596688629} a^{8} - \frac{19804862975580543143190625298847914123264586399249214}{1460557071226893686344501580976058620111517192790065887} a^{7} - \frac{115280885620947836389389208559730016071818002475356192}{486852357075631228781500526992019540037172397596688629} a^{6} + \frac{658911445375773103334926528991045597276542829672716400}{1460557071226893686344501580976058620111517192790065887} a^{5} - \frac{126408164048127182046565714564077846467869543458882032}{1460557071226893686344501580976058620111517192790065887} a^{4} - \frac{571699664099276718933484591442419323150658442240746664}{1460557071226893686344501580976058620111517192790065887} a^{3} + \frac{173251842898863616004908091239153456311988322803079547}{486852357075631228781500526992019540037172397596688629} a^{2} - \frac{168568420597295723748344307751095733366699554547482721}{486852357075631228781500526992019540037172397596688629} a - \frac{88561406157122736638251405725816202008646228346208096}{486852357075631228781500526992019540037172397596688629}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 1519167484.847329 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-31}) \), 3.1.108.1, 6.0.347482224.4, 9.1.11019960576.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ R $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
3Data not computed
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$