↑

Properties

Label 18.0.21005238210...5344.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{15}\cdot 19^{12}$
Root discriminant $374.51$
Ramified primes $2, 3, 7, 19$
Class number $111476736$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 4, 12, 36, 252]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2149375249792, 121672015488, 441096494976, 25422376576, 53542445592, 787839768, 3951456340, -196704744, 54773046, -27557386, 5940777, -653382, 112168, -5478, 3546, -426, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 30*x^16 - 426*x^15 + 3546*x^14 - 5478*x^13 + 112168*x^12 - 653382*x^11 + 5940777*x^10 - 27557386*x^9 + 54773046*x^8 - 196704744*x^7 + 3951456340*x^6 + 787839768*x^5 + 53542445592*x^4 + 25422376576*x^3 + 441096494976*x^2 + 121672015488*x + 2149375249792)
 
gp: K = bnfinit(x^18 - 30*x^16 - 426*x^15 + 3546*x^14 - 5478*x^13 + 112168*x^12 - 653382*x^11 + 5940777*x^10 - 27557386*x^9 + 54773046*x^8 - 196704744*x^7 + 3951456340*x^6 + 787839768*x^5 + 53542445592*x^4 + 25422376576*x^3 + 441096494976*x^2 + 121672015488*x + 2149375249792, 1)
 

Normalized defining polynomial

\( x^{18} - 30 x^{16} - 426 x^{15} + 3546 x^{14} - 5478 x^{13} + 112168 x^{12} - 653382 x^{11} + 5940777 x^{10} - 27557386 x^{9} + 54773046 x^{8} - 196704744 x^{7} + 3951456340 x^{6} + 787839768 x^{5} + 53542445592 x^{4} + 25422376576 x^{3} + 441096494976 x^{2} + 121672015488 x + 2149375249792 \)

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:  \(-21005238210476463669563930923405113491534905344=-\,2^{18}\cdot 3^{27}\cdot 7^{15}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $374.51$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(2243,·)$, $\chi_{4788}(961,·)$, $\chi_{4788}(2063,·)$, $\chi_{4788}(1873,·)$, $\chi_{4788}(3155,·)$, $\chi_{4788}(3649,·)$, $\chi_{4788}(923,·)$, $\chi_{4788}(2015,·)$, $\chi_{4788}(4453,·)$, $\chi_{4788}(4561,·)$, $\chi_{4788}(4225,·)$, $\chi_{4788}(1223,·)$, $\chi_{4788}(3313,·)$, $\chi_{4788}(83,·)$, $\chi_{4788}(2101,·)$, $\chi_{4788}(311,·)$, $\chi_{4788}(1151,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{4} a^{5} - \frac{3}{16} a^{4} + \frac{7}{16} a^{3} + \frac{1}{8} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{32} a^{9} - \frac{1}{32} a^{8} - \frac{7}{32} a^{5} + \frac{3}{32} a^{4} - \frac{5}{16} a^{3} + \frac{1}{16} a^{2} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} + \frac{1}{32} a^{6} + \frac{1}{8} a^{5} + \frac{1}{32} a^{4} - \frac{1}{2} a^{3} + \frac{7}{16} a^{2} + \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{8} + \frac{1}{32} a^{7} - \frac{1}{16} a^{5} + \frac{7}{32} a^{4} - \frac{1}{2} a^{3} + \frac{3}{16} a^{2} - \frac{3}{8} a$, $\frac{1}{128} a^{12} - \frac{1}{64} a^{11} - \frac{1}{128} a^{10} - \frac{1}{128} a^{8} - \frac{1}{64} a^{7} - \frac{7}{128} a^{6} + \frac{7}{32} a^{5} + \frac{5}{32} a^{4} + \frac{1}{8} a^{3} - \frac{5}{32} a^{2} - \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{256} a^{13} - \frac{1}{256} a^{12} - \frac{3}{256} a^{11} + \frac{3}{256} a^{10} - \frac{1}{256} a^{9} + \frac{1}{256} a^{8} + \frac{15}{256} a^{7} - \frac{7}{256} a^{6} - \frac{1}{4} a^{5} - \frac{1}{16} a^{4} - \frac{3}{64} a^{3} + \frac{9}{64} a^{2} + \frac{1}{8} a - \frac{7}{16}$, $\frac{1}{256} a^{14} - \frac{1}{128} a^{10} + \frac{1}{64} a^{8} + \frac{1}{32} a^{7} - \frac{3}{256} a^{6} + \frac{3}{16} a^{5} + \frac{3}{64} a^{4} + \frac{7}{32} a^{3} + \frac{25}{64} a^{2} - \frac{3}{16} a - \frac{3}{16}$, $\frac{1}{1732989952} a^{15} - \frac{204131}{433247488} a^{14} - \frac{50251}{27077968} a^{13} - \frac{1710187}{866494976} a^{12} - \frac{1064023}{866494976} a^{11} - \frac{1468281}{866494976} a^{10} - \frac{125779}{433247488} a^{9} + \frac{14624579}{866494976} a^{8} - \frac{105478751}{1732989952} a^{7} + \frac{30132535}{866494976} a^{6} + \frac{87052009}{433247488} a^{5} - \frac{18240139}{216623744} a^{4} - \frac{41727979}{433247488} a^{3} - \frac{8879495}{216623744} a^{2} - \frac{23440885}{108311872} a + \frac{4608655}{54155936}$, $\frac{1}{1732989952} a^{16} - \frac{58159}{108311872} a^{14} - \frac{1301619}{866494976} a^{13} + \frac{1015749}{866494976} a^{12} + \frac{8357931}{866494976} a^{11} - \frac{4946801}{433247488} a^{10} + \frac{11406451}{866494976} a^{9} - \frac{17144215}{1732989952} a^{8} - \frac{17319803}{866494976} a^{7} - \frac{6365581}{433247488} a^{6} - \frac{20257049}{216623744} a^{5} - \frac{108011059}{433247488} a^{4} + \frac{51752319}{216623744} a^{3} + \frac{28066249}{108311872} a^{2} - \frac{20990275}{54155936} a - \frac{3873165}{13538984}$, $\frac{1}{562918176691859681007075188944409227989076230749470069067124330819584} a^{17} + \frac{18050898557162777601967591300723304021516548858400978351621}{281459088345929840503537594472204613994538115374735034533562165409792} a^{16} - \frac{651326579213532205174395202885257998167753193076363830321}{281459088345929840503537594472204613994538115374735034533562165409792} a^{15} - \frac{176830346374904001467871383365814111718833387058260426213136761027}{281459088345929840503537594472204613994538115374735034533562165409792} a^{14} - \frac{390258047714208750552570628745466945305029545713174921195854330041}{281459088345929840503537594472204613994538115374735034533562165409792} a^{13} + \frac{741526540213788009402649433724296430822131374587642415646829807187}{281459088345929840503537594472204613994538115374735034533562165409792} a^{12} - \frac{709793594686355296329193252861800207236957032248096098951378842891}{140729544172964920251768797236102306997269057687367517266781082704896} a^{11} + \frac{641115924055412635701402431578514766505971735625531240798399189705}{281459088345929840503537594472204613994538115374735034533562165409792} a^{10} + \frac{2773090331928618284309063849870238204298879161412063095807917561213}{562918176691859681007075188944409227989076230749470069067124330819584} a^{9} + \frac{68584020404499473380726518631246520336093557601499780454938510199}{70364772086482460125884398618051153498634528843683758633390541352448} a^{8} - \frac{9163144285441105636096784096201354522905963749313054775317604594625}{281459088345929840503537594472204613994538115374735034533562165409792} a^{7} + \frac{465161089736814348698483453696754330462026974362356161663628359143}{140729544172964920251768797236102306997269057687367517266781082704896} a^{6} - \frac{33604645361568216115573510377673135996686357773184515279101236175289}{140729544172964920251768797236102306997269057687367517266781082704896} a^{5} - \frac{4298250985382799647722296480197292057819822110111279561610461992819}{35182386043241230062942199309025576749317264421841879316695270676224} a^{4} - \frac{24840435096957833700736432270918093170465980947543779760510077540989}{70364772086482460125884398618051153498634528843683758633390541352448} a^{3} - \frac{5899598130856439644227704100603604365802930817350563539191654465283}{35182386043241230062942199309025576749317264421841879316695270676224} a^{2} + \frac{7583016708016271108944680642886286544965651188087640684423808048119}{17591193021620615031471099654512788374658632210920939658347635338112} a - \frac{48752900224664918496548362057756871182613378621364923793833468321}{8795596510810307515735549827256394187329316105460469829173817669056}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{12}\times C_{36}\times C_{252}$, which has order $111476736$ (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:  \( 10681224266.072006 \) (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

\(\Q(\sqrt{-21}) \), 3.3.1432809.2, 3.3.29241.1, 3.3.3969.1, 3.3.17689.2, 6.0.2759153551366464.5, 6.0.56309256150336.3, 6.0.21171979584.2, 6.0.3784847121216.2, 9.9.2941473244627851129.8

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
3Data not computed
7Data not computed
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$