↑

Properties

Label 18.0.10247702651...6064.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 7^{12}$
Root discriminant $31.67$
Ramified primes $2, 3, 7$
Class number $28$
Class group $[2, 14]$
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, 0, 54, 0, 561, 0, 2316, 0, 4500, 0, 4164, 0, 1712, 0, 333, 0, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 30*x^16 + 333*x^14 + 1712*x^12 + 4164*x^10 + 4500*x^8 + 2316*x^6 + 561*x^4 + 54*x^2 + 1)
 
gp: K = bnfinit(x^18 + 30*x^16 + 333*x^14 + 1712*x^12 + 4164*x^10 + 4500*x^8 + 2316*x^6 + 561*x^4 + 54*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 30 x^{16} + 333 x^{14} + 1712 x^{12} + 4164 x^{10} + 4500 x^{8} + 2316 x^{6} + 561 x^{4} + 54 x^{2} + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1024770265180753855691096064=-\,2^{18}\cdot 3^{24}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.67$
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$$\chi_{252}(1,·)$, $\chi_{252}(67,·)$, $\chi_{252}(193,·)$, $\chi_{252}(151,·)$, $\chi_{252}(205,·)$, $\chi_{252}(79,·)$, $\chi_{252}(211,·)$, $\chi_{252}(85,·)$, $\chi_{252}(235,·)$, $\chi_{252}(25,·)$, $\chi_{252}(163,·)$, $\chi_{252}(37,·)$, $\chi_{252}(169,·)$, $\chi_{252}(43,·)$, $\chi_{252}(109,·)$, $\chi_{252}(247,·)$, $\chi_{252}(121,·)$, $\chi_{252}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4} a^{14} + \frac{1}{4}$, $\frac{1}{4} a^{15} + \frac{1}{4} a$, $\frac{1}{36212272} a^{16} + \frac{2271617}{36212272} a^{14} - \frac{43059}{2263267} a^{12} - \frac{654587}{2263267} a^{10} - \frac{2972711}{9053068} a^{8} + \frac{572662}{2263267} a^{6} + \frac{1483799}{9053068} a^{4} + \frac{14451477}{36212272} a^{2} - \frac{13350283}{36212272}$, $\frac{1}{36212272} a^{17} + \frac{2271617}{36212272} a^{15} - \frac{43059}{2263267} a^{13} - \frac{654587}{2263267} a^{11} - \frac{2972711}{9053068} a^{9} + \frac{572662}{2263267} a^{7} + \frac{1483799}{9053068} a^{5} + \frac{14451477}{36212272} a^{3} - \frac{13350283}{36212272} a$

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

Class group and class number

$C_{2}\times C_{14}$, which has order $28$

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:  \( \frac{90657}{142568} a^{17} + \frac{2712125}{142568} a^{15} + \frac{3744774}{17821} a^{13} + \frac{19073916}{17821} a^{11} + \frac{90900497}{35642} a^{9} + \frac{46535655}{17821} a^{7} + \frac{42137451}{35642} a^{5} + \frac{30849621}{142568} a^{3} + \frac{1568145}{142568} a \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 54408.4888887 \)
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

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, 6.0.153664.1, 6.0.419904.1, 6.0.1008189504.1, 6.0.1008189504.2, 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$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}$
3Data not computed
7Data not computed