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Properties

Label 18.0.13717212741...1799.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{12}\cdot 37^{9}$
Root discriminant $115.66$
Ramified primes $3, 7, 37$
Class number $3299616$ (GRH)
Class group $[2, 18, 91656]$ (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![64238725531, -2336691090, 34192143147, -863213550, 8399149179, -76884324, 1169457127, -40385748, 116128899, -9040888, 8360469, -823572, 395818, -35448, 11088, -734, 165, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 165*x^16 - 734*x^15 + 11088*x^14 - 35448*x^13 + 395818*x^12 - 823572*x^11 + 8360469*x^10 - 9040888*x^9 + 116128899*x^8 - 40385748*x^7 + 1169457127*x^6 - 76884324*x^5 + 8399149179*x^4 - 863213550*x^3 + 34192143147*x^2 - 2336691090*x + 64238725531)
 
gp: K = bnfinit(x^18 - 6*x^17 + 165*x^16 - 734*x^15 + 11088*x^14 - 35448*x^13 + 395818*x^12 - 823572*x^11 + 8360469*x^10 - 9040888*x^9 + 116128899*x^8 - 40385748*x^7 + 1169457127*x^6 - 76884324*x^5 + 8399149179*x^4 - 863213550*x^3 + 34192143147*x^2 - 2336691090*x + 64238725531, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 165 x^{16} - 734 x^{15} + 11088 x^{14} - 35448 x^{13} + 395818 x^{12} - 823572 x^{11} + 8360469 x^{10} - 9040888 x^{9} + 116128899 x^{8} - 40385748 x^{7} + 1169457127 x^{6} - 76884324 x^{5} + 8399149179 x^{4} - 863213550 x^{3} + 34192143147 x^{2} - 2336691090 x + 64238725531 \)

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:  \(-13717212741604370463308637102507521799=-\,3^{27}\cdot 7^{12}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.66$
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:  $3, 7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2331=3^{2}\cdot 7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2331}(1,·)$, $\chi_{2331}(1220,·)$, $\chi_{2331}(1222,·)$, $\chi_{2331}(778,·)$, $\chi_{2331}(1997,·)$, $\chi_{2331}(1999,·)$, $\chi_{2331}(1555,·)$, $\chi_{2331}(667,·)$, $\chi_{2331}(221,·)$, $\chi_{2331}(1444,·)$, $\chi_{2331}(998,·)$, $\chi_{2331}(554,·)$, $\chi_{2331}(2221,·)$, $\chi_{2331}(1775,·)$, $\chi_{2331}(1331,·)$, $\chi_{2331}(443,·)$, $\chi_{2331}(2108,·)$, $\chi_{2331}(445,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} + \frac{3}{20} a^{5} + \frac{1}{5} a^{4} - \frac{1}{4} a^{3} - \frac{1}{5} a^{2} + \frac{1}{4} a + \frac{3}{10}$, $\frac{1}{20} a^{10} - \frac{1}{20} a^{8} - \frac{1}{5} a^{7} - \frac{1}{20} a^{6} - \frac{1}{10} a^{5} + \frac{1}{10} a^{4} - \frac{1}{5} a^{3} - \frac{1}{10} a^{2} + \frac{3}{10} a - \frac{7}{20}$, $\frac{1}{20} a^{11} - \frac{1}{10} a^{8} - \frac{3}{20} a^{7} - \frac{1}{4} a^{5} + \frac{3}{20} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{3}{10}$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{10} - \frac{1}{40} a^{9} + \frac{1}{20} a^{7} - \frac{1}{20} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{9}{40} a^{3} - \frac{1}{10} a^{2} - \frac{1}{8} a + \frac{3}{40}$, $\frac{1}{80} a^{13} - \frac{1}{80} a^{12} - \frac{1}{80} a^{11} - \frac{1}{40} a^{10} + \frac{1}{80} a^{9} + \frac{1}{20} a^{8} + \frac{1}{20} a^{7} + \frac{9}{80} a^{6} - \frac{3}{40} a^{5} - \frac{1}{10} a^{4} - \frac{27}{80} a^{3} + \frac{3}{80} a^{2} - \frac{1}{20} a + \frac{11}{80}$, $\frac{1}{80} a^{14} + \frac{1}{80} a^{11} + \frac{1}{80} a^{10} - \frac{1}{80} a^{9} + \frac{1}{10} a^{8} - \frac{3}{80} a^{7} - \frac{13}{80} a^{6} - \frac{1}{20} a^{5} + \frac{7}{80} a^{4} - \frac{13}{40} a^{3} + \frac{27}{80} a^{2} + \frac{33}{80} a - \frac{31}{80}$, $\frac{1}{121040} a^{15} + \frac{49}{12104} a^{14} - \frac{689}{121040} a^{13} - \frac{1}{1780} a^{12} + \frac{6}{7565} a^{11} + \frac{197}{24208} a^{10} + \frac{671}{121040} a^{9} + \frac{14957}{121040} a^{8} - \frac{6911}{121040} a^{7} + \frac{28709}{121040} a^{6} + \frac{25983}{121040} a^{5} - \frac{797}{3560} a^{4} + \frac{200}{1513} a^{3} - \frac{2967}{15130} a^{2} - \frac{25807}{121040} a - \frac{26623}{121040}$, $\frac{1}{484160} a^{16} + \frac{701}{121040} a^{14} - \frac{137}{48416} a^{13} + \frac{789}{121040} a^{12} - \frac{823}{48416} a^{11} + \frac{1847}{242080} a^{10} + \frac{5731}{242080} a^{9} - \frac{18973}{484160} a^{8} + \frac{2183}{15130} a^{7} + \frac{8503}{121040} a^{6} + \frac{22407}{121040} a^{5} - \frac{112673}{484160} a^{4} + \frac{66991}{242080} a^{3} - \frac{29851}{121040} a^{2} - \frac{519}{60520} a + \frac{22879}{484160}$, $\frac{1}{32815217810020054518913417097588553519592433861454369229174223360} a^{17} + \frac{9798241916327400886667094158265296490973592510455576830669}{32815217810020054518913417097588553519592433861454369229174223360} a^{16} + \frac{15952122761569289370725214387790883544369629399587359227191}{8203804452505013629728354274397138379898108465363592307293555840} a^{15} - \frac{2216545075773250202698591103755363022202062044232041547687157}{16407608905010027259456708548794276759796216930727184614587111680} a^{14} - \frac{58408131480158229114277027646476841476258866491629255049991447}{16407608905010027259456708548794276759796216930727184614587111680} a^{13} + \frac{13713881912022741526260684189363220103467513593495323726814767}{16407608905010027259456708548794276759796216930727184614587111680} a^{12} - \frac{118952633734299138918767510453865431130477906674925966350410591}{8203804452505013629728354274397138379898108465363592307293555840} a^{11} - \frac{93612081199106293295428308104293122345985287797412820882220993}{4101902226252506814864177137198569189949054232681796153646777920} a^{10} - \frac{550695168274601764898772681729159263553278611790551690219652291}{32815217810020054518913417097588553519592433861454369229174223360} a^{9} - \frac{592827801948309293535134640152635369787228879852857360554759261}{6563043562004010903782683419517710703918486772290873845834844672} a^{8} - \frac{32505508225870671142644612865413356184886173861125282291596183}{205095111312625340743208856859928459497452711634089807682338896} a^{7} - \frac{1443258939308755042694042475344174514875555357633087906252186061}{8203804452505013629728354274397138379898108465363592307293555840} a^{6} + \frac{614522322608828692095740461820467932979280623903311801137046703}{6563043562004010903782683419517710703918486772290873845834844672} a^{5} + \frac{249235050397851624545724254636844989317956491962408793008412381}{1930306930001179677583142182211091383505437285967904072304366080} a^{4} - \frac{418209147075765604038978537011834501543362441160853612146919863}{965153465000589838791571091105545691752718642983952036152183040} a^{3} + \frac{403924421893175954367010863089913699210923911144232553031369093}{4101902226252506814864177137198569189949054232681796153646777920} a^{2} + \frac{87476890576833064816755494590054845673564691349766773138143879}{6563043562004010903782683419517710703918486772290873845834844672} a - \frac{6998833480491170407832980810647012147936370198654990441693477049}{32815217810020054518913417097588553519592433861454369229174223360}$

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

Class group and class number

$C_{2}\times C_{18}\times C_{91656}$, which has order $3299616$ (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:  \( 54408.48888868202 \) (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

\(\Q(\sqrt{-111}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.2, 3.3.3969.1, 6.0.997002999.1, 6.0.3283682031.3, 6.0.2393804200599.8, 6.0.2393804200599.7, 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$37$37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$