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Properties

Label 18.0.84535014172...0064.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{12}\cdot 13^{12}$
Root discriminant $40.46$
Ramified primes $2, 7, 13$
Class number $108$ (GRH)
Class group $[3, 6, 6]$ (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![64, 0, 4112, 0, 20600, 0, 40081, 0, 37370, 0, 17745, 0, 4498, 0, 606, 0, 40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 40*x^16 + 606*x^14 + 4498*x^12 + 17745*x^10 + 37370*x^8 + 40081*x^6 + 20600*x^4 + 4112*x^2 + 64)
 
gp: K = bnfinit(x^18 + 40*x^16 + 606*x^14 + 4498*x^12 + 17745*x^10 + 37370*x^8 + 40081*x^6 + 20600*x^4 + 4112*x^2 + 64, 1)
 

Normalized defining polynomial

\( x^{18} + 40 x^{16} + 606 x^{14} + 4498 x^{12} + 17745 x^{10} + 37370 x^{8} + 40081 x^{6} + 20600 x^{4} + 4112 x^{2} + 64 \)

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:  \(-84535014172552012147112280064=-\,2^{18}\cdot 7^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.46$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(364=2^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{364}(1,·)$, $\chi_{364}(107,·)$, $\chi_{364}(261,·)$, $\chi_{364}(263,·)$, $\chi_{364}(9,·)$, $\chi_{364}(79,·)$, $\chi_{364}(81,·)$, $\chi_{364}(211,·)$, $\chi_{364}(347,·)$, $\chi_{364}(29,·)$, $\chi_{364}(289,·)$, $\chi_{364}(165,·)$, $\chi_{364}(295,·)$, $\chi_{364}(235,·)$, $\chi_{364}(113,·)$, $\chi_{364}(53,·)$, $\chi_{364}(183,·)$, $\chi_{364}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{8} - \frac{1}{2} a^{6} - \frac{1}{4} a^{4} + \frac{5}{12} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{9} - \frac{1}{4} a^{5} - \frac{1}{12} a^{3} - \frac{1}{6} a$, $\frac{1}{12} a^{10} - \frac{1}{4} a^{6} - \frac{1}{12} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{4} a^{7} - \frac{1}{12} a^{5} - \frac{1}{6} a^{3}$, $\frac{1}{12} a^{12} + \frac{5}{12} a^{6} + \frac{1}{12} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{7} + \frac{1}{12} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{12} a^{14} - \frac{5}{12} a^{6} - \frac{1}{2} a^{4} - \frac{1}{12} a^{2} + \frac{1}{3}$, $\frac{1}{24} a^{15} - \frac{5}{24} a^{7} + \frac{1}{4} a^{5} - \frac{1}{24} a^{3} + \frac{1}{6} a$, $\frac{1}{729036144} a^{16} - \frac{654617}{60753012} a^{14} + \frac{231569}{121506024} a^{12} - \frac{1500241}{364518072} a^{10} + \frac{15288161}{729036144} a^{8} + \frac{12708849}{40502008} a^{6} + \frac{261642421}{729036144} a^{4} + \frac{36107299}{182259036} a^{2} + \frac{14601484}{45564759}$, $\frac{1}{1458072288} a^{17} - \frac{654617}{121506024} a^{15} + \frac{231569}{243012048} a^{13} + \frac{28876265}{729036144} a^{11} + \frac{15288161}{1458072288} a^{9} + \frac{2583347}{81004016} a^{7} - \frac{528146735}{1458072288} a^{5} + \frac{5730793}{364518072} a^{3} - \frac{30963275}{91129518} a$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}$, which has order $108$ (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:  \( \frac{2275}{335652} a^{17} + \frac{178643}{671304} a^{15} + \frac{656345}{167826} a^{13} + \frac{9260633}{335652} a^{11} + \frac{16725835}{167826} a^{9} + \frac{39655853}{223768} a^{7} + \frac{10840237}{83913} a^{5} + \frac{13272287}{671304} a^{3} - \frac{1238579}{167826} a \) (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 (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

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{7})^+\), 3.3.8281.1, 3.3.169.1, 3.3.8281.2, 6.0.153664.1, 6.0.4388797504.2, 6.0.1827904.1, 6.0.4388797504.1, 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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[2]^{3}$
7Data not computed
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$