↑

Properties

Label 18.18.3514962009...9952.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{24}\cdot 7^{15}$
Root discriminant $43.80$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (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![1, -24, -327, 2802, -4209, -8406, 20201, 5016, -25785, 494, 14433, -792, -4060, 132, 576, -6, -39, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 39*x^16 - 6*x^15 + 576*x^14 + 132*x^13 - 4060*x^12 - 792*x^11 + 14433*x^10 + 494*x^9 - 25785*x^8 + 5016*x^7 + 20201*x^6 - 8406*x^5 - 4209*x^4 + 2802*x^3 - 327*x^2 - 24*x + 1)
 
gp: K = bnfinit(x^18 - 39*x^16 - 6*x^15 + 576*x^14 + 132*x^13 - 4060*x^12 - 792*x^11 + 14433*x^10 + 494*x^9 - 25785*x^8 + 5016*x^7 + 20201*x^6 - 8406*x^5 - 4209*x^4 + 2802*x^3 - 327*x^2 - 24*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 39 x^{16} - 6 x^{15} + 576 x^{14} + 132 x^{13} - 4060 x^{12} - 792 x^{11} + 14433 x^{10} + 494 x^{9} - 25785 x^{8} + 5016 x^{7} + 20201 x^{6} - 8406 x^{5} - 4209 x^{4} + 2802 x^{3} - 327 x^{2} - 24 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(351496200956998572502045949952=2^{18}\cdot 3^{24}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.80$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(252=2^{2}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(193,·)$, $\chi_{252}(139,·)$, $\chi_{252}(205,·)$, $\chi_{252}(19,·)$, $\chi_{252}(85,·)$, $\chi_{252}(25,·)$, $\chi_{252}(223,·)$, $\chi_{252}(37,·)$, $\chi_{252}(103,·)$, $\chi_{252}(31,·)$, $\chi_{252}(169,·)$, $\chi_{252}(199,·)$, $\chi_{252}(109,·)$, $\chi_{252}(115,·)$, $\chi_{252}(55,·)$, $\chi_{252}(121,·)$, $\chi_{252}(187,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{185890316} a^{16} + \frac{9611670}{46472579} a^{15} + \frac{6365726}{46472579} a^{14} - \frac{22608723}{92945158} a^{13} + \frac{3974991}{46472579} a^{12} - \frac{9714415}{46472579} a^{11} + \frac{4908627}{46472579} a^{10} + \frac{19288699}{92945158} a^{9} + \frac{9405847}{185890316} a^{8} - \frac{43654763}{92945158} a^{7} + \frac{5989193}{92945158} a^{6} + \frac{21322593}{92945158} a^{5} - \frac{82717403}{185890316} a^{4} + \frac{20792532}{46472579} a^{3} + \frac{7312037}{92945158} a^{2} - \frac{7188199}{46472579} a - \frac{90036405}{185890316}$, $\frac{1}{190224720498172} a^{17} + \frac{51767}{190224720498172} a^{16} - \frac{8408807421033}{47556180124543} a^{15} - \frac{20503221142855}{95112360249086} a^{14} + \frac{14328663060047}{95112360249086} a^{13} - \frac{8844445783931}{95112360249086} a^{12} + \frac{10831114226953}{95112360249086} a^{11} + \frac{2097557173323}{47556180124543} a^{10} - \frac{647482695153}{190224720498172} a^{9} - \frac{47464515895857}{190224720498172} a^{8} - \frac{7352755760193}{95112360249086} a^{7} - \frac{9272313300594}{47556180124543} a^{6} + \frac{85313479850335}{190224720498172} a^{5} + \frac{52548759849537}{190224720498172} a^{4} - \frac{37419762891199}{95112360249086} a^{3} - \frac{20221526928181}{95112360249086} a^{2} + \frac{34670745269761}{190224720498172} a + \frac{84942862143299}{190224720498172}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 875332910.208 \) (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

\(\Q(\sqrt{7}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, \(\Q(\zeta_{7})^+\), 3.3.3969.2, 6.6.144027072.1, 6.6.7057326528.2, \(\Q(\zeta_{28})^+\), 6.6.7057326528.1, 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed