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Properties

Label 18.18.2029685690...3664.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 7^{16}\cdot 13^{12}$
Root discriminant $62.35$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2.A_4$ (as 18T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-28561, 0, 1012817, 0, -1946373, 0, 1557465, 0, -660569, 0, 159978, 0, -22316, 0, 1726, 0, -67, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 67*x^16 + 1726*x^14 - 22316*x^12 + 159978*x^10 - 660569*x^8 + 1557465*x^6 - 1946373*x^4 + 1012817*x^2 - 28561)
 
gp: K = bnfinit(x^18 - 67*x^16 + 1726*x^14 - 22316*x^12 + 159978*x^10 - 660569*x^8 + 1557465*x^6 - 1946373*x^4 + 1012817*x^2 - 28561, 1)
 

Normalized defining polynomial

\( x^{18} - 67 x^{16} + 1726 x^{14} - 22316 x^{12} + 159978 x^{10} - 660569 x^{8} + 1557465 x^{6} - 1946373 x^{4} + 1012817 x^{2} - 28561 \)

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:  \(202968569028297381165216584433664=2^{18}\cdot 7^{16}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.35$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} - \frac{2}{13} a^{10} - \frac{3}{13} a^{8} + \frac{5}{13} a^{6}$, $\frac{1}{13} a^{13} - \frac{2}{13} a^{11} - \frac{3}{13} a^{9} + \frac{5}{13} a^{7}$, $\frac{1}{169} a^{14} - \frac{2}{169} a^{12} + \frac{75}{169} a^{10} - \frac{34}{169} a^{8} - \frac{6}{13} a^{6} + \frac{4}{13} a^{4} - \frac{3}{13} a^{2}$, $\frac{1}{169} a^{15} - \frac{2}{169} a^{13} + \frac{75}{169} a^{11} - \frac{34}{169} a^{9} - \frac{6}{13} a^{7} + \frac{4}{13} a^{5} - \frac{3}{13} a^{3}$, $\frac{1}{375690481048037051} a^{16} + \frac{766066019718816}{375690481048037051} a^{14} + \frac{13385029233088965}{375690481048037051} a^{12} - \frac{154960989490188279}{375690481048037051} a^{10} + \frac{13456083693000839}{28899267772925927} a^{8} - \frac{11330852607860509}{28899267772925927} a^{6} + \frac{11284083430935858}{28899267772925927} a^{4} + \frac{845246956203571}{2223020597917379} a^{2} - \frac{26440068549789}{171001584455183}$, $\frac{1}{375690481048037051} a^{17} + \frac{766066019718816}{375690481048037051} a^{15} + \frac{13385029233088965}{375690481048037051} a^{13} - \frac{154960989490188279}{375690481048037051} a^{11} + \frac{13456083693000839}{28899267772925927} a^{9} - \frac{11330852607860509}{28899267772925927} a^{7} + \frac{11284083430935858}{28899267772925927} a^{5} + \frac{845246956203571}{2223020597917379} a^{3} - \frac{26440068549789}{171001584455183} a$

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:  \( 20486548774.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2.A_4$ (as 18T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_3^2.A_4$
Character table for $C_3^2.A_4$

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.25969216.1, 9.9.164648481361.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
$7$7.9.8.2$x^{9} - 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
7.9.8.2$x^{9} - 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
$13$13.6.5.4$x^{6} + 26$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$