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Properties

Label 18.0.18793877302...1776.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{9}\cdot 13^{12}$
Root discriminant $152.02$
Ramified primes $2, 3, 7, 13$
Class number $2888704$ (GRH)
Class group $[2, 2, 2, 2, 8, 22568]$ (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![14079582973, 1517303154, 7631428449, -656223070, 1622202279, -402034464, 289649625, -74876916, 38364819, -8367384, 3260457, -588492, 180572, -24564, 6156, -578, 117, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 117*x^16 - 578*x^15 + 6156*x^14 - 24564*x^13 + 180572*x^12 - 588492*x^11 + 3260457*x^10 - 8367384*x^9 + 38364819*x^8 - 74876916*x^7 + 289649625*x^6 - 402034464*x^5 + 1622202279*x^4 - 656223070*x^3 + 7631428449*x^2 + 1517303154*x + 14079582973)
 
gp: K = bnfinit(x^18 - 6*x^17 + 117*x^16 - 578*x^15 + 6156*x^14 - 24564*x^13 + 180572*x^12 - 588492*x^11 + 3260457*x^10 - 8367384*x^9 + 38364819*x^8 - 74876916*x^7 + 289649625*x^6 - 402034464*x^5 + 1622202279*x^4 - 656223070*x^3 + 7631428449*x^2 + 1517303154*x + 14079582973, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 117 x^{16} - 578 x^{15} + 6156 x^{14} - 24564 x^{13} + 180572 x^{12} - 588492 x^{11} + 3260457 x^{10} - 8367384 x^{9} + 38364819 x^{8} - 74876916 x^{7} + 289649625 x^{6} - 402034464 x^{5} + 1622202279 x^{4} - 656223070 x^{3} + 7631428449 x^{2} + 1517303154 x + 14079582973 \)

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:  \(-1879387730226102112759375150760179531776=-\,2^{18}\cdot 3^{27}\cdot 7^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $152.02$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3276=2^{2}\cdot 3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{3276}(1,·)$, $\chi_{3276}(1093,·)$, $\chi_{3276}(2185,·)$, $\chi_{3276}(1847,·)$, $\chi_{3276}(1933,·)$, $\chi_{3276}(3025,·)$, $\chi_{3276}(1595,·)$, $\chi_{3276}(419,·)$, $\chi_{3276}(1511,·)$, $\chi_{3276}(2603,·)$, $\chi_{3276}(841,·)$, $\chi_{3276}(755,·)$, $\chi_{3276}(757,·)$, $\chi_{3276}(503,·)$, $\chi_{3276}(1849,·)$, $\chi_{3276}(2939,·)$, $\chi_{3276}(2941,·)$, $\chi_{3276}(2687,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{7} + \frac{3}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{14} a^{8} - \frac{1}{7} a^{5} - \frac{5}{14} a^{4} - \frac{1}{7} a^{3} + \frac{5}{14} a^{2} + \frac{3}{7} a - \frac{3}{14}$, $\frac{1}{14} a^{9} + \frac{5}{14} a^{5} - \frac{1}{7} a^{4} + \frac{3}{14} a^{3} - \frac{1}{14} a + \frac{1}{7}$, $\frac{1}{14} a^{10} - \frac{1}{14} a^{6} - \frac{2}{7} a^{5} + \frac{3}{14} a^{4} + \frac{3}{7} a^{3} + \frac{3}{14} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{14} a^{11} - \frac{1}{14} a^{7} - \frac{5}{14} a^{5} + \frac{3}{7} a^{4} - \frac{1}{14} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{98} a^{12} + \frac{3}{98} a^{11} - \frac{3}{98} a^{10} - \frac{1}{49} a^{9} - \frac{1}{49} a^{8} - \frac{1}{14} a^{7} + \frac{3}{49} a^{6} - \frac{47}{98} a^{5} - \frac{1}{14} a^{4} + \frac{13}{98} a^{3} + \frac{15}{49} a^{2} - \frac{13}{49} a + \frac{43}{98}$, $\frac{1}{98} a^{13} + \frac{1}{49} a^{11} - \frac{3}{98} a^{9} - \frac{1}{98} a^{8} - \frac{1}{98} a^{7} - \frac{1}{49} a^{6} + \frac{1}{98} a^{5} + \frac{27}{98} a^{4} + \frac{13}{49} a^{3} + \frac{31}{98} a^{2} + \frac{22}{49} a - \frac{17}{98}$, $\frac{1}{98} a^{14} + \frac{1}{98} a^{11} + \frac{3}{98} a^{10} + \frac{3}{98} a^{9} + \frac{3}{98} a^{8} + \frac{5}{98} a^{7} + \frac{3}{98} a^{6} - \frac{20}{49} a^{5} - \frac{8}{49} a^{4} - \frac{8}{49} a^{3} + \frac{13}{49} a^{2} + \frac{5}{14} a - \frac{22}{49}$, $\frac{1}{98} a^{15} - \frac{1}{98} a^{10} - \frac{1}{49} a^{9} - \frac{2}{49} a^{7} + \frac{3}{98} a^{6} + \frac{5}{49} a^{5} + \frac{33}{98} a^{4} - \frac{4}{49} a^{3} - \frac{9}{98} a^{2} - \frac{25}{98} a + \frac{10}{49}$, $\frac{1}{84049626348430617691108852} a^{16} - \frac{84896745871827108355287}{21012406587107654422777213} a^{15} - \frac{98569071697254826570734}{21012406587107654422777213} a^{14} + \frac{130492929329390521702203}{42024813174215308845554426} a^{13} + \frac{136970672361335377936387}{42024813174215308845554426} a^{12} + \frac{1324677370312512050085079}{42024813174215308845554426} a^{11} + \frac{27421661347292329921245}{3001772369586807774682459} a^{10} - \frac{553121160946025671024745}{21012406587107654422777213} a^{9} - \frac{221690581964769277828523}{84049626348430617691108852} a^{8} + \frac{207575908102073950151084}{21012406587107654422777213} a^{7} + \frac{176310392137402198291208}{3001772369586807774682459} a^{6} - \frac{2958886292607534013912879}{6003544739173615549364918} a^{5} + \frac{16027053611097957550342867}{84049626348430617691108852} a^{4} + \frac{8675004182144761116482758}{21012406587107654422777213} a^{3} - \frac{6891106946874981180575587}{42024813174215308845554426} a^{2} + \frac{19402231194682801918628241}{42024813174215308845554426} a - \frac{15133420929218777100916237}{84049626348430617691108852}$, $\frac{1}{199934848462784138231435870463664214447284} a^{17} - \frac{210484013931830}{49983712115696034557858967615916053611821} a^{16} - \frac{144486201328292686166505842316133100821}{49983712115696034557858967615916053611821} a^{15} - \frac{22327992129140357974153956065497920885}{49983712115696034557858967615916053611821} a^{14} - \frac{29242404837190202279892715025531223110}{7140530302242290651122709659416579087403} a^{13} + \frac{113043787992668005236465791226784771331}{99967424231392069115717935231832107223642} a^{12} + \frac{138820486659602998825272229777800093905}{99967424231392069115717935231832107223642} a^{11} + \frac{1935285035111075017546674512430628063725}{99967424231392069115717935231832107223642} a^{10} - \frac{6117526626660654282016893430243390537327}{199934848462784138231435870463664214447284} a^{9} - \frac{55628165090980523039825625556144629430}{7140530302242290651122709659416579087403} a^{8} + \frac{471277015158405739292952579506263741013}{49983712115696034557858967615916053611821} a^{7} + \frac{270805001512358894360096215707035011294}{49983712115696034557858967615916053611821} a^{6} + \frac{43622521297252114296541165308921841785297}{199934848462784138231435870463664214447284} a^{5} - \frac{4490901218111535494360082742113310251937}{99967424231392069115717935231832107223642} a^{4} - \frac{27764324091665949532907937793980063715965}{99967424231392069115717935231832107223642} a^{3} + \frac{11215989121647195929002011462148579830573}{49983712115696034557858967615916053611821} a^{2} + \frac{39424824364587811023623320786491649176613}{199934848462784138231435870463664214447284} a - \frac{27122918857416402115837639045203818389395}{99967424231392069115717935231832107223642}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{8}\times C_{22568}$, which has order $2888704$ (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:  \( 400417.1364448253 \) (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

\(\Q(\sqrt{-21}) \), 3.3.13689.2, \(\Q(\zeta_{9})^+\), 3.3.13689.1, 3.3.169.1, 6.0.12340671610176.4, 6.0.432081216.1, 6.0.12340671610176.3, 6.0.16928218944.4, 9.9.2565164201769.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13Data not computed