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Properties

Label 18.0.52468237577...4768.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{24}\cdot 7^{12}$
Root discriminant $44.78$
Ramified primes $2, 3, 7$
Class number $324$ (GRH)
Class group $[18, 18]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![386513, 90444, 531078, 87920, 331017, 44268, 114732, 13782, 21948, 3532, 2376, 438, 246, 96, 117, -8, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^16 - 8*x^15 + 117*x^14 + 96*x^13 + 246*x^12 + 438*x^11 + 2376*x^10 + 3532*x^9 + 21948*x^8 + 13782*x^7 + 114732*x^6 + 44268*x^5 + 331017*x^4 + 87920*x^3 + 531078*x^2 + 90444*x + 386513)
 
gp: K = bnfinit(x^18 - 12*x^16 - 8*x^15 + 117*x^14 + 96*x^13 + 246*x^12 + 438*x^11 + 2376*x^10 + 3532*x^9 + 21948*x^8 + 13782*x^7 + 114732*x^6 + 44268*x^5 + 331017*x^4 + 87920*x^3 + 531078*x^2 + 90444*x + 386513, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{16} - 8 x^{15} + 117 x^{14} + 96 x^{13} + 246 x^{12} + 438 x^{11} + 2376 x^{10} + 3532 x^{9} + 21948 x^{8} + 13782 x^{7} + 114732 x^{6} + 44268 x^{5} + 331017 x^{4} + 87920 x^{3} + 531078 x^{2} + 90444 x + 386513 \)

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:  \(-524682375772545974113841184768=-\,2^{27}\cdot 3^{24}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.78$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(67,·)$, $\chi_{504}(193,·)$, $\chi_{504}(457,·)$, $\chi_{504}(331,·)$, $\chi_{504}(403,·)$, $\chi_{504}(337,·)$, $\chi_{504}(211,·)$, $\chi_{504}(235,·)$, $\chi_{504}(25,·)$, $\chi_{504}(289,·)$, $\chi_{504}(163,·)$, $\chi_{504}(169,·)$, $\chi_{504}(43,·)$, $\chi_{504}(499,·)$, $\chi_{504}(361,·)$, $\chi_{504}(121,·)$, $\chi_{504}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{7} + \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{8} + \frac{1}{4} a$, $\frac{1}{2032} a^{16} - \frac{69}{1016} a^{15} + \frac{45}{2032} a^{14} + \frac{209}{508} a^{13} - \frac{201}{1016} a^{12} - \frac{59}{254} a^{11} + \frac{235}{508} a^{10} - \frac{261}{1016} a^{9} + \frac{67}{254} a^{8} - \frac{99}{1016} a^{7} - \frac{17}{127} a^{6} + \frac{93}{254} a^{5} - \frac{253}{508} a^{4} - \frac{213}{508} a^{3} - \frac{227}{2032} a^{2} - \frac{235}{1016} a + \frac{739}{2032}$, $\frac{1}{3021945331191994060617427135288301909728208} a^{17} - \frac{215562849930157677497354095706655280979}{1510972665595997030308713567644150954864104} a^{16} + \frac{232371366537956378947417931779380902864237}{3021945331191994060617427135288301909728208} a^{15} - \frac{62460563221523657203856468932030151860437}{755486332797998515154356783822075477432052} a^{14} + \frac{197371894531375346347517753585665886470999}{1510972665595997030308713567644150954864104} a^{13} + \frac{151044555135211603993672320060460770080033}{377743166398999257577178391911037738716026} a^{12} + \frac{335766744836429055038773446952199321916735}{755486332797998515154356783822075477432052} a^{11} - \frac{694659697557326419687915812515768004292093}{1510972665595997030308713567644150954864104} a^{10} + \frac{20720391532434307082760727341924220249256}{188871583199499628788589195955518869358013} a^{9} - \frac{311458809774434245773698200231429029061315}{1510972665595997030308713567644150954864104} a^{8} - \frac{25082947150873066725027793304809915023682}{188871583199499628788589195955518869358013} a^{7} + \frac{110203117922342760168360637535597513563585}{377743166398999257577178391911037738716026} a^{6} - \frac{320442781717083557528142111908869199965269}{755486332797998515154356783822075477432052} a^{5} + \frac{87195312287742891396987369404115250390971}{755486332797998515154356783822075477432052} a^{4} - \frac{185753948336770130809072931324844395214739}{3021945331191994060617427135288301909728208} a^{3} - \frac{503591646587135715205117352184710139592161}{1510972665595997030308713567644150954864104} a^{2} + \frac{983324736378191248398997025994625077839619}{3021945331191994060617427135288301909728208} a - \frac{61660109585797079429381656388328797889176}{188871583199499628788589195955518869358013}$

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

Class group and class number

$C_{18}\times C_{18}$, which has order $324$ (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:  \( 54408.4888887 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, 6.0.3359232.1, 6.0.8065516032.2, 6.0.1229312.1, 6.0.8065516032.1, 9.9.62523502209.1

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

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed