↑

Properties

Label 18.0.24300817552...6851.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 13^{15}$
Root discriminant $42.91$
Ramified primes $7, 13$
Class number $56$ (GRH)
Class group $[2, 2, 14]$ (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![9619, 4141, 10718, 2715, 5605, 8175, 3898, 3887, 3265, 816, 1159, -82, -154, -76, -4, -17, 12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 12*x^16 - 17*x^15 - 4*x^14 - 76*x^13 - 154*x^12 - 82*x^11 + 1159*x^10 + 816*x^9 + 3265*x^8 + 3887*x^7 + 3898*x^6 + 8175*x^5 + 5605*x^4 + 2715*x^3 + 10718*x^2 + 4141*x + 9619)
 
gp: K = bnfinit(x^18 - x^17 + 12*x^16 - 17*x^15 - 4*x^14 - 76*x^13 - 154*x^12 - 82*x^11 + 1159*x^10 + 816*x^9 + 3265*x^8 + 3887*x^7 + 3898*x^6 + 8175*x^5 + 5605*x^4 + 2715*x^3 + 10718*x^2 + 4141*x + 9619, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 12 x^{16} - 17 x^{15} - 4 x^{14} - 76 x^{13} - 154 x^{12} - 82 x^{11} + 1159 x^{10} + 816 x^{9} + 3265 x^{8} + 3887 x^{7} + 3898 x^{6} + 8175 x^{5} + 5605 x^{4} + 2715 x^{3} + 10718 x^{2} + 4141 x + 9619 \)

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:  \(-243008175525757569678159896851=-\,7^{15}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.91$
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:  $7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(91=7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{91}(1,·)$, $\chi_{91}(69,·)$, $\chi_{91}(9,·)$, $\chi_{91}(10,·)$, $\chi_{91}(75,·)$, $\chi_{91}(12,·)$, $\chi_{91}(79,·)$, $\chi_{91}(16,·)$, $\chi_{91}(17,·)$, $\chi_{91}(82,·)$, $\chi_{91}(22,·)$, $\chi_{91}(90,·)$, $\chi_{91}(29,·)$, $\chi_{91}(38,·)$, $\chi_{91}(81,·)$, $\chi_{91}(53,·)$, $\chi_{91}(74,·)$, $\chi_{91}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5039} a^{15} + \frac{1771}{5039} a^{14} + \frac{2270}{5039} a^{13} + \frac{1163}{5039} a^{12} + \frac{973}{5039} a^{11} + \frac{628}{5039} a^{10} + \frac{1663}{5039} a^{9} - \frac{1993}{5039} a^{8} - \frac{924}{5039} a^{7} + \frac{1261}{5039} a^{6} - \frac{2358}{5039} a^{5} + \frac{1183}{5039} a^{4} - \frac{75}{5039} a^{3} - \frac{2117}{5039} a^{2} - \frac{840}{5039} a + \frac{971}{5039}$, $\frac{1}{1698143} a^{16} + \frac{133}{1698143} a^{15} - \frac{127178}{1698143} a^{14} + \frac{702106}{1698143} a^{13} + \frac{46072}{1698143} a^{12} + \frac{765106}{1698143} a^{11} + \frac{86618}{1698143} a^{10} - \frac{413086}{1698143} a^{9} + \frac{330912}{1698143} a^{8} - \frac{692309}{1698143} a^{7} - \frac{87549}{1698143} a^{6} + \frac{391716}{1698143} a^{5} + \frac{203746}{1698143} a^{4} - \frac{70749}{1698143} a^{3} - \frac{438419}{1698143} a^{2} + \frac{832679}{1698143} a - \frac{194695}{1698143}$, $\frac{1}{908515410611947997992191397403} a^{17} + \frac{124384888204068054943576}{908515410611947997992191397403} a^{16} + \frac{88596552002020052130245650}{908515410611947997992191397403} a^{15} - \frac{243401284138517113194394693833}{908515410611947997992191397403} a^{14} + \frac{149390836022515112394949583845}{908515410611947997992191397403} a^{13} - \frac{317917991797020188446026621092}{908515410611947997992191397403} a^{12} - \frac{427616493419743565804081833537}{908515410611947997992191397403} a^{11} - \frac{380756766663800673608982955000}{908515410611947997992191397403} a^{10} + \frac{102674902285183186281912746507}{908515410611947997992191397403} a^{9} - \frac{238854646088076789490037337117}{908515410611947997992191397403} a^{8} - \frac{250310787849268068986313738875}{908515410611947997992191397403} a^{7} + \frac{83690913881475950388853786747}{908515410611947997992191397403} a^{6} - \frac{190176143272571604224978118008}{908515410611947997992191397403} a^{5} - \frac{347624456767148117291246807731}{908515410611947997992191397403} a^{4} + \frac{106544045884263233980743002584}{908515410611947997992191397403} a^{3} - \frac{323424063289605625830939064601}{908515410611947997992191397403} a^{2} - \frac{289160458484631781526103116260}{908515410611947997992191397403} a + \frac{3921881911628605139183786940}{908515410611947997992191397403}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{14}$, which has order $56$ (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:  \( 205236.825908 \) (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

\(\Q(\sqrt{-91}) \), 3.3.8281.1, 3.3.169.1, \(\Q(\zeta_{7})^+\), 3.3.8281.2, 6.0.6240321451.1, 6.0.127353499.1, 6.0.36924979.1, 6.0.6240321451.2, 9.9.567869252041.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
7Data not computed
$13$13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$