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Properties

Label 18.0.38394761924...0511.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{45}\cdot 37^{9}$
Root discriminant $94.82$
Ramified primes $3, 37$
Class number $317984$ (GRH)
Class group $[2, 2, 79496]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8082504811, -4912168212, 3486784401, -1819321560, 2582803260, -181932156, 736577226, -6738228, 105225318, -83188, 8444007, 0, 398034, 0, 10935, 0, 162, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 162*x^16 + 10935*x^14 + 398034*x^12 + 8444007*x^10 - 83188*x^9 + 105225318*x^8 - 6738228*x^7 + 736577226*x^6 - 181932156*x^5 + 2582803260*x^4 - 1819321560*x^3 + 3486784401*x^2 - 4912168212*x + 8082504811)
 
gp: K = bnfinit(x^18 + 162*x^16 + 10935*x^14 + 398034*x^12 + 8444007*x^10 - 83188*x^9 + 105225318*x^8 - 6738228*x^7 + 736577226*x^6 - 181932156*x^5 + 2582803260*x^4 - 1819321560*x^3 + 3486784401*x^2 - 4912168212*x + 8082504811, 1)
 

Normalized defining polynomial

\( x^{18} + 162 x^{16} + 10935 x^{14} + 398034 x^{12} + 8444007 x^{10} - 83188 x^{9} + 105225318 x^{8} - 6738228 x^{7} + 736577226 x^{6} - 181932156 x^{5} + 2582803260 x^{4} - 1819321560 x^{3} + 3486784401 x^{2} - 4912168212 x + 8082504811 \)

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:  \(-383947619242049123161763424792980511=-\,3^{45}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.82$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(999=3^{3}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{999}(1,·)$, $\chi_{999}(776,·)$, $\chi_{999}(778,·)$, $\chi_{999}(332,·)$, $\chi_{999}(334,·)$, $\chi_{999}(665,·)$, $\chi_{999}(667,·)$, $\chi_{999}(221,·)$, $\chi_{999}(223,·)$, $\chi_{999}(998,·)$, $\chi_{999}(554,·)$, $\chi_{999}(556,·)$, $\chi_{999}(110,·)$, $\chi_{999}(112,·)$, $\chi_{999}(887,·)$, $\chi_{999}(889,·)$, $\chi_{999}(443,·)$, $\chi_{999}(445,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{15130} a^{9} + \frac{81}{15130} a^{7} + \frac{2187}{15130} a^{5} + \frac{674}{1513} a^{3} - \frac{1471}{15130} a - \frac{3769}{15130}$, $\frac{1}{15130} a^{10} + \frac{81}{15130} a^{8} + \frac{2187}{15130} a^{6} + \frac{674}{1513} a^{4} - \frac{1471}{15130} a^{2} - \frac{3769}{15130} a$, $\frac{1}{15130} a^{11} - \frac{2187}{7565} a^{7} - \frac{3977}{15130} a^{5} - \frac{2731}{15130} a^{3} - \frac{3769}{15130} a^{2} - \frac{1889}{15130} a + \frac{2689}{15130}$, $\frac{1}{15130} a^{12} - \frac{2187}{7565} a^{8} - \frac{3977}{15130} a^{6} - \frac{2731}{15130} a^{4} - \frac{3769}{15130} a^{3} - \frac{1889}{15130} a^{2} + \frac{2689}{15130} a$, $\frac{1}{15130} a^{13} + \frac{2327}{15130} a^{7} + \frac{1047}{15130} a^{5} - \frac{3769}{15130} a^{4} + \frac{5631}{15130} a^{3} + \frac{2689}{15130} a^{2} - \frac{1952}{7565} a + \frac{3047}{7565}$, $\frac{1}{110564953793290} a^{14} + \frac{2346783663}{110564953793290} a^{13} + \frac{63}{55282476896645} a^{12} - \frac{3117537083}{110564953793290} a^{11} + \frac{6237}{110564953793290} a^{10} - \frac{721777954}{55282476896645} a^{9} + \frac{15309}{11056495379329} a^{8} + \frac{7661841651721}{22112990758658} a^{7} + \frac{964467}{55282476896645} a^{6} + \frac{15061080019597}{110564953793290} a^{5} - \frac{7936117135836}{55282476896645} a^{4} + \frac{5780293960931}{110564953793290} a^{3} + \frac{32957589927439}{110564953793290} a^{2} - \frac{16975746882541}{55282476896645} a + \frac{25598755972937}{110564953793290}$, $\frac{1}{110564953793290} a^{15} + \frac{27}{22112990758658} a^{13} + \frac{400969266}{55282476896645} a^{12} + \frac{729}{11056495379329} a^{11} - \frac{6368262}{650382081137} a^{10} + \frac{40095}{22112990758658} a^{9} - \frac{1797685053354}{55282476896645} a^{8} + \frac{295245}{11056495379329} a^{7} - \frac{6087274655833}{110564953793290} a^{6} + \frac{11160261}{55282476896645} a^{5} + \frac{20220345132241}{55282476896645} a^{4} - \frac{50342421743797}{110564953793290} a^{3} + \frac{25876680208861}{110564953793290} a^{2} - \frac{3365172020643}{11056495379329} a + \frac{394674491034}{55282476896645}$, $\frac{1}{110564953793290} a^{16} - \frac{892155577}{55282476896645} a^{13} - \frac{972}{11056495379329} a^{12} + \frac{1624031592}{55282476896645} a^{11} - \frac{64152}{11056495379329} a^{10} - \frac{2426376783}{110564953793290} a^{9} - \frac{1771470}{11056495379329} a^{8} - \frac{13789459664383}{55282476896645} a^{7} - \frac{119042784}{55282476896645} a^{6} + \frac{8236798410701}{55282476896645} a^{5} + \frac{11779210081396}{55282476896645} a^{4} - \frac{1257930112381}{3251910405685} a^{3} + \frac{6539670382999}{22112990758658} a^{2} + \frac{15147452901098}{55282476896645} a + \frac{10806743407377}{110564953793290}$, $\frac{1}{110564953793290} a^{17} - \frac{324}{3251910405685} a^{13} + \frac{18043409}{1300764162274} a^{12} - \frac{23328}{3251910405685} a^{11} - \frac{1624863472}{55282476896645} a^{10} - \frac{144342}{650382081137} a^{9} + \frac{3266450198647}{110564953793290} a^{8} - \frac{11337408}{3251910405685} a^{7} + \frac{915136273399}{3251910405685} a^{6} - \frac{89282088}{3251910405685} a^{5} - \frac{47431121437917}{110564953793290} a^{4} + \frac{1395457682087}{3251910405685} a^{3} - \frac{3927399179917}{55282476896645} a^{2} - \frac{35474337701147}{110564953793290} a - \frac{1550649812251}{55282476896645}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{79496}$, which has order $317984$ (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:  \( 40934.0329443 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-111}) \), \(\Q(\zeta_{9})^+\), 6.0.997002999.1, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ R $18$ $18$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$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}$