↑

Properties

Label 18.0.42875986703...5819.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 19^{15}$
Root discriminant $50.33$
Ramified primes $3, 19$
Class number $28$ (GRH)
Class group $[2, 14]$ (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![11863, -15837, 16641, 21719, -17880, -5832, 44148, 18246, -3918, 1335, -474, -975, 307, -138, 0, 53, -9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 9*x^16 + 53*x^15 - 138*x^13 + 307*x^12 - 975*x^11 - 474*x^10 + 1335*x^9 - 3918*x^8 + 18246*x^7 + 44148*x^6 - 5832*x^5 - 17880*x^4 + 21719*x^3 + 16641*x^2 - 15837*x + 11863)
 
gp: K = bnfinit(x^18 - 3*x^17 - 9*x^16 + 53*x^15 - 138*x^13 + 307*x^12 - 975*x^11 - 474*x^10 + 1335*x^9 - 3918*x^8 + 18246*x^7 + 44148*x^6 - 5832*x^5 - 17880*x^4 + 21719*x^3 + 16641*x^2 - 15837*x + 11863, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 9 x^{16} + 53 x^{15} - 138 x^{13} + 307 x^{12} - 975 x^{11} - 474 x^{10} + 1335 x^{9} - 3918 x^{8} + 18246 x^{7} + 44148 x^{6} - 5832 x^{5} - 17880 x^{4} + 21719 x^{3} + 16641 x^{2} - 15837 x + 11863 \)

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:  \(-4287598670306719523049937245819=-\,3^{24}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.33$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(171=3^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{171}(64,·)$, $\chi_{171}(1,·)$, $\chi_{171}(7,·)$, $\chi_{171}(145,·)$, $\chi_{171}(151,·)$, $\chi_{171}(88,·)$, $\chi_{171}(94,·)$, $\chi_{171}(31,·)$, $\chi_{171}(160,·)$, $\chi_{171}(163,·)$, $\chi_{171}(37,·)$, $\chi_{171}(103,·)$, $\chi_{171}(106,·)$, $\chi_{171}(46,·)$, $\chi_{171}(49,·)$, $\chi_{171}(115,·)$, $\chi_{171}(121,·)$, $\chi_{171}(58,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{77} a^{15} - \frac{5}{77} a^{14} - \frac{23}{77} a^{13} - \frac{15}{77} a^{12} - \frac{19}{77} a^{11} + \frac{3}{11} a^{10} - \frac{4}{77} a^{9} - \frac{4}{11} a^{8} + \frac{17}{77} a^{6} - \frac{18}{77} a^{5} + \frac{10}{77} a^{4} + \frac{3}{77} a^{3} - \frac{18}{77} a^{2} - \frac{5}{77} a - \frac{24}{77}$, $\frac{1}{77} a^{16} + \frac{29}{77} a^{14} + \frac{24}{77} a^{13} - \frac{17}{77} a^{12} + \frac{3}{77} a^{11} + \frac{24}{77} a^{10} + \frac{29}{77} a^{9} + \frac{2}{11} a^{8} + \frac{17}{77} a^{7} - \frac{10}{77} a^{6} - \frac{3}{77} a^{5} - \frac{24}{77} a^{4} - \frac{3}{77} a^{3} - \frac{18}{77} a^{2} + \frac{4}{11} a + \frac{34}{77}$, $\frac{1}{1279952263965005579562301583487876051099563} a^{17} + \frac{416732929014614590988595683570294891279}{116359296724091416323845598498897822827233} a^{16} - \frac{1900502070208468883379721494424060615590}{1279952263965005579562301583487876051099563} a^{15} - \frac{610574147612320075010066716596716057049239}{1279952263965005579562301583487876051099563} a^{14} - \frac{329782972762003857829973159580369016700960}{1279952263965005579562301583487876051099563} a^{13} + \frac{82092005012537853304568345463408115391474}{1279952263965005579562301583487876051099563} a^{12} + \frac{393149719193387180188765241983469560276464}{1279952263965005579562301583487876051099563} a^{11} - \frac{585353602371240626213388921036436391820942}{1279952263965005579562301583487876051099563} a^{10} - \frac{63045877420539180507819816358918433498721}{1279952263965005579562301583487876051099563} a^{9} - \frac{5050929697305024691371094375430299584474}{116359296724091416323845598498897822827233} a^{8} + \frac{86640160660020752992645761715917419556927}{1279952263965005579562301583487876051099563} a^{7} - \frac{455446546738848713739974216868922807819634}{1279952263965005579562301583487876051099563} a^{6} + \frac{590192462029624389322477448630430419220654}{1279952263965005579562301583487876051099563} a^{5} - \frac{76353456140586957925031719500778504793411}{1279952263965005579562301583487876051099563} a^{4} - \frac{379810371787207023121418528913362725475278}{1279952263965005579562301583487876051099563} a^{3} + \frac{286435639127889574223662346860348120585988}{1279952263965005579562301583487876051099563} a^{2} - \frac{617662804584481744641051028136645182574522}{1279952263965005579562301583487876051099563} a + \frac{270160747596924795642411143349991208273985}{1279952263965005579562301583487876051099563}$

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

Class group and class number

$C_{2}\times C_{14}$, which has order $28$ (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:  \( 1472619.0824 \) (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

\(\Q(\sqrt{-19}) \), 3.3.29241.1, 3.3.361.1, \(\Q(\zeta_{9})^+\), 3.3.29241.2, 6.0.16245685539.2, 6.0.2476099.1, 6.0.45001899.1, 6.0.16245685539.1, 9.9.25002110044521.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$19$19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$