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Properties

Label 18.0.16277118145...7632.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{44}\cdot 7^{9}$
Root discriminant $61.59$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6112, 12816, 39240, -186948, 209844, -97992, 47298, -50346, 41706, -27063, 15741, -8334, 4008, -1764, 684, -216, 54, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 54*x^16 - 216*x^15 + 684*x^14 - 1764*x^13 + 4008*x^12 - 8334*x^11 + 15741*x^10 - 27063*x^9 + 41706*x^8 - 50346*x^7 + 47298*x^6 - 97992*x^5 + 209844*x^4 - 186948*x^3 + 39240*x^2 + 12816*x + 6112)
 
gp: K = bnfinit(x^18 - 9*x^17 + 54*x^16 - 216*x^15 + 684*x^14 - 1764*x^13 + 4008*x^12 - 8334*x^11 + 15741*x^10 - 27063*x^9 + 41706*x^8 - 50346*x^7 + 47298*x^6 - 97992*x^5 + 209844*x^4 - 186948*x^3 + 39240*x^2 + 12816*x + 6112, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 54 x^{16} - 216 x^{15} + 684 x^{14} - 1764 x^{13} + 4008 x^{12} - 8334 x^{11} + 15741 x^{10} - 27063 x^{9} + 41706 x^{8} - 50346 x^{7} + 47298 x^{6} - 97992 x^{5} + 209844 x^{4} - 186948 x^{3} + 39240 x^{2} + 12816 x + 6112 \)

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:  \(-162771181452299835524978080837632=-\,2^{12}\cdot 3^{44}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.59$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} + \frac{1}{12} a^{8} + \frac{1}{12} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} + \frac{1}{12} a^{9} + \frac{1}{12} a^{7} + \frac{1}{6} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{13} - \frac{1}{24} a^{12} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{5}{24} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{12} a^{5} - \frac{5}{12} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{24} a^{15} - \frac{1}{6} a^{9} - \frac{1}{4} a^{8} + \frac{1}{8} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a$, $\frac{1}{816} a^{16} - \frac{7}{408} a^{15} + \frac{1}{68} a^{14} + \frac{5}{204} a^{12} + \frac{5}{102} a^{11} - \frac{4}{51} a^{10} + \frac{77}{408} a^{9} - \frac{55}{272} a^{8} + \frac{1}{24} a^{7} + \frac{5}{102} a^{6} - \frac{79}{408} a^{5} - \frac{20}{51} a^{4} - \frac{15}{34} a^{3} + \frac{31}{68} a^{2} - \frac{31}{102} a + \frac{11}{51}$, $\frac{1}{224184129421895892453970639746192} a^{17} - \frac{1477723759206815134051857749}{14011508088868493278373164984137} a^{16} + \frac{781686552665472052286694035239}{112092064710947946226985319873096} a^{15} - \frac{1583993474909812050450325093043}{112092064710947946226985319873096} a^{14} + \frac{749444079963541556224806637567}{112092064710947946226985319873096} a^{13} + \frac{1686772144597428149664389144717}{112092064710947946226985319873096} a^{12} + \frac{805655225310597898779002010655}{37364021570315982075661773291032} a^{11} + \frac{156150212105972565757095573887}{18682010785157991037830886645516} a^{10} - \frac{5806024126966088947208259920981}{74728043140631964151323546582064} a^{9} + \frac{5312825229032048655517148639491}{37364021570315982075661773291032} a^{8} - \frac{13250371373947649621575948675021}{56046032355473973113492659936548} a^{7} + \frac{3877847508969403970430224920925}{37364021570315982075661773291032} a^{6} + \frac{1787636410906869954548658664621}{28023016177736986556746329968274} a^{5} - \frac{6390644861081852801267046831706}{14011508088868493278373164984137} a^{4} - \frac{15459295266747598684757563052483}{56046032355473973113492659936548} a^{3} - \frac{5763267344333421226092741726499}{28023016177736986556746329968274} a^{2} - \frac{964476282188284216483262275266}{4670502696289497759457721661379} a + \frac{4268125847491912104625344268658}{14011508088868493278373164984137}$

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:  $8$
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:  \( 2929716169.8555565 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

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 $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.243.1, 6.0.20253807.1, 9.1.2008387814976.4

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$