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Properties

Label 18.0.39032301703...5159.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 13^{15}$
Root discriminant $44.05$
Ramified primes $3, 13$
Class number $52$ (GRH)
Class group $[52]$ (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![6616, 2232, 8946, 4789, 2784, 2025, -1266, 1515, 2412, 1854, 1437, -27, -73, -72, -63, -4, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 - 4*x^15 - 63*x^14 - 72*x^13 - 73*x^12 - 27*x^11 + 1437*x^10 + 1854*x^9 + 2412*x^8 + 1515*x^7 - 1266*x^6 + 2025*x^5 + 2784*x^4 + 4789*x^3 + 8946*x^2 + 2232*x + 6616)
 
gp: K = bnfinit(x^18 + 6*x^16 - 4*x^15 - 63*x^14 - 72*x^13 - 73*x^12 - 27*x^11 + 1437*x^10 + 1854*x^9 + 2412*x^8 + 1515*x^7 - 1266*x^6 + 2025*x^5 + 2784*x^4 + 4789*x^3 + 8946*x^2 + 2232*x + 6616, 1)
 

Normalized defining polynomial

\( x^{18} + 6 x^{16} - 4 x^{15} - 63 x^{14} - 72 x^{13} - 73 x^{12} - 27 x^{11} + 1437 x^{10} + 1854 x^{9} + 2412 x^{8} + 1515 x^{7} - 1266 x^{6} + 2025 x^{5} + 2784 x^{4} + 4789 x^{3} + 8946 x^{2} + 2232 x + 6616 \)

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:  \(-390323017035064129531762965159=-\,3^{27}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.05$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(117=3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{117}(1,·)$, $\chi_{117}(77,·)$, $\chi_{117}(79,·)$, $\chi_{117}(16,·)$, $\chi_{117}(17,·)$, $\chi_{117}(22,·)$, $\chi_{117}(23,·)$, $\chi_{117}(94,·)$, $\chi_{117}(95,·)$, $\chi_{117}(100,·)$, $\chi_{117}(101,·)$, $\chi_{117}(38,·)$, $\chi_{117}(40,·)$, $\chi_{117}(116,·)$, $\chi_{117}(55,·)$, $\chi_{117}(56,·)$, $\chi_{117}(61,·)$, $\chi_{117}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{100} a^{15} + \frac{2}{25} a^{13} + \frac{23}{100} a^{12} - \frac{11}{50} a^{11} + \frac{3}{20} a^{10} - \frac{21}{100} a^{9} + \frac{9}{100} a^{8} - \frac{1}{2} a^{7} - \frac{1}{5} a^{6} + \frac{1}{20} a^{5} + \frac{29}{100} a^{3} + \frac{7}{20} a^{2} - \frac{2}{25} a - \frac{2}{25}$, $\frac{1}{35900} a^{16} - \frac{39}{8975} a^{15} + \frac{179}{17950} a^{14} - \frac{83}{718} a^{13} + \frac{197}{1795} a^{12} + \frac{1443}{8975} a^{11} - \frac{9}{8975} a^{10} - \frac{289}{3590} a^{9} + \frac{3573}{17950} a^{8} - \frac{1347}{3590} a^{7} + \frac{355}{718} a^{6} - \frac{589}{1795} a^{5} - \frac{1174}{8975} a^{4} + \frac{2434}{8975} a^{3} - \frac{6643}{35900} a^{2} + \frac{799}{3590} a + \frac{2437}{8975}$, $\frac{1}{120066688126434773453746905100} a^{17} - \frac{197935110482666439660563}{120066688126434773453746905100} a^{16} + \frac{9933500034946889067417871}{60033344063217386726873452550} a^{15} - \frac{1632892097789910672148578281}{120066688126434773453746905100} a^{14} - \frac{4147312974983058021809029637}{60033344063217386726873452550} a^{13} - \frac{25714399515877488625430923917}{120066688126434773453746905100} a^{12} - \frac{23021933194123869230463030339}{120066688126434773453746905100} a^{11} - \frac{9905117173111733644973767483}{120066688126434773453746905100} a^{10} - \frac{4854060364233644376757824664}{30016672031608693363436726275} a^{9} + \frac{10474658570382885319257266443}{60033344063217386726873452550} a^{8} - \frac{6867282381765756134374154267}{24013337625286954690749381020} a^{7} - \frac{183017321542041426333377641}{6003334406321738672687345255} a^{6} + \frac{55282141777683581446062327749}{120066688126434773453746905100} a^{5} - \frac{28643988793907955500856645267}{120066688126434773453746905100} a^{4} - \frac{2540402911170287327688789238}{30016672031608693363436726275} a^{3} + \frac{24309645026130771077735543861}{120066688126434773453746905100} a^{2} - \frac{12445583369756037513888297792}{30016672031608693363436726275} a - \frac{1015236347141955441720424093}{30016672031608693363436726275}$

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

Class group and class number

$C_{52}$, which has order $52$ (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:  \( 400417.136445 \) (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

\(\Q(\sqrt{-39}) \), \(\Q(\zeta_{9})^+\), 3.3.169.1, 3.3.13689.1, 3.3.13689.2, 6.0.43243551.1, 6.0.10024911.1, 6.0.7308160119.1, 6.0.7308160119.2, 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
13Data not computed