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Properties

Label 18.0.24235852840...1664.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 7^{15}\cdot 11^{9}\cdot 127^{6}$
Root discriminant $292.28$
Ramified primes $2, 3, 7, 11, 127$
Class number $367579296$ (GRH)
Class group $[2, 2, 2, 6, 7657902]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![706194494839, -1789405059, 212309051196, -6634420134, 30612761917, 317441289, 2871719318, 56268523, 196884574, -9139, 9966370, -240073, 365092, -12805, 9007, -298, 136, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 136*x^16 - 298*x^15 + 9007*x^14 - 12805*x^13 + 365092*x^12 - 240073*x^11 + 9966370*x^10 - 9139*x^9 + 196884574*x^8 + 56268523*x^7 + 2871719318*x^6 + 317441289*x^5 + 30612761917*x^4 - 6634420134*x^3 + 212309051196*x^2 - 1789405059*x + 706194494839)
 
gp: K = bnfinit(x^18 - 3*x^17 + 136*x^16 - 298*x^15 + 9007*x^14 - 12805*x^13 + 365092*x^12 - 240073*x^11 + 9966370*x^10 - 9139*x^9 + 196884574*x^8 + 56268523*x^7 + 2871719318*x^6 + 317441289*x^5 + 30612761917*x^4 - 6634420134*x^3 + 212309051196*x^2 - 1789405059*x + 706194494839, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 136 x^{16} - 298 x^{15} + 9007 x^{14} - 12805 x^{13} + 365092 x^{12} - 240073 x^{11} + 9966370 x^{10} - 9139 x^{9} + 196884574 x^{8} + 56268523 x^{7} + 2871719318 x^{6} + 317441289 x^{5} + 30612761917 x^{4} - 6634420134 x^{3} + 212309051196 x^{2} - 1789405059 x + 706194494839 \)

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:  \(-242358528402273672667539881228774768857841664=-\,2^{18}\cdot 3^{9}\cdot 7^{15}\cdot 11^{9}\cdot 127^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $292.28$
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:  $2, 3, 7, 11, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{1874} a^{12} + \frac{157}{937} a^{11} - \frac{175}{1874} a^{10} - \frac{143}{937} a^{9} - \frac{183}{1874} a^{8} + \frac{336}{937} a^{7} + \frac{243}{1874} a^{6} + \frac{389}{937} a^{5} - \frac{587}{1874} a^{4} - \frac{116}{937} a^{3} + \frac{833}{1874} a^{2} - \frac{138}{937} a - \frac{159}{1874}$, $\frac{1}{1874} a^{13} + \frac{551}{1874} a^{11} + \frac{159}{937} a^{10} - \frac{331}{1874} a^{9} + \frac{20}{937} a^{8} - \frac{877}{1874} a^{7} - \frac{282}{937} a^{6} + \frac{615}{1874} a^{5} + \frac{217}{937} a^{4} + \frac{595}{1874} a^{3} + \frac{261}{937} a^{2} + \frac{301}{1874} a - \frac{336}{937}$, $\frac{1}{54346} a^{14} + \frac{7}{54346} a^{13} - \frac{11}{54346} a^{12} - \frac{9255}{54346} a^{11} + \frac{12167}{54346} a^{10} - \frac{27071}{54346} a^{9} - \frac{17687}{54346} a^{8} + \frac{11047}{54346} a^{7} - \frac{18089}{54346} a^{6} - \frac{16469}{54346} a^{5} - \frac{16911}{54346} a^{4} + \frac{17009}{54346} a^{3} - \frac{18179}{54346} a^{2} - \frac{19609}{54346} a - \frac{5460}{27173}$, $\frac{1}{54346} a^{15} - \frac{1}{27173} a^{13} - \frac{7}{27173} a^{12} - \frac{1312}{27173} a^{11} - \frac{6385}{27173} a^{10} - \frac{11361}{27173} a^{9} - \frac{9074}{27173} a^{8} - \frac{10241}{27173} a^{7} + \frac{10881}{27173} a^{6} - \frac{9365}{27173} a^{5} - \frac{747}{27173} a^{4} - \frac{297}{27173} a^{3} + \frac{27}{27173} a^{2} + \frac{5761}{54346} a - \frac{3308}{27173}$, $\frac{1}{54346} a^{16} - \frac{7}{54346} a^{12} - \frac{8912}{27173} a^{11} - \frac{25445}{54346} a^{10} - \frac{5927}{27173} a^{9} + \frac{4667}{54346} a^{8} + \frac{11923}{27173} a^{7} - \frac{11437}{54346} a^{6} + \frac{3954}{27173} a^{5} - \frac{7475}{54346} a^{4} + \frac{9815}{27173} a^{3} - \frac{3075}{27173} a^{2} - \frac{6677}{27173} a - \frac{6673}{54346}$, $\frac{1}{4057745036140469287455790089622905818192617540263650593025057701283426} a^{17} + \frac{35250321588192250271576341043425784544817442126061833053315335825}{4057745036140469287455790089622905818192617540263650593025057701283426} a^{16} - \frac{18245144741626809278570374875268141581919225397902905051604960455}{2028872518070234643727895044811452909096308770131825296512528850641713} a^{15} + \frac{13179968961496650357969474341495219644200272800089478201554981864}{2028872518070234643727895044811452909096308770131825296512528850641713} a^{14} - \frac{874782773177325594926767336912680685993870367758754732303203265481}{4057745036140469287455790089622905818192617540263650593025057701283426} a^{13} - \frac{124027548403262325456017956333504678165653176878659070516211334716}{2028872518070234643727895044811452909096308770131825296512528850641713} a^{12} - \frac{842885614518220994833013573234192192489399522581436141945796240238463}{4057745036140469287455790089622905818192617540263650593025057701283426} a^{11} - \frac{6819587704805898293877908760656501477502951292825922412450196620349}{2028872518070234643727895044811452909096308770131825296512528850641713} a^{10} - \frac{810631085184411020284035247210332029876991816167160731123168690349}{4330571009755036592802337342180262345989986702522572671318097866898} a^{9} - \frac{884404640747952279035122872692990157373503658053148632085598954096352}{2028872518070234643727895044811452909096308770131825296512528850641713} a^{8} + \frac{826241522793552340670177541007479586977173459614972658393918125776867}{4057745036140469287455790089622905818192617540263650593025057701283426} a^{7} - \frac{886367490645751089964324361146432933038553496257426963188629929214988}{2028872518070234643727895044811452909096308770131825296512528850641713} a^{6} - \frac{264887032867587778183623678459158686317721075686987641426563365296993}{4057745036140469287455790089622905818192617540263650593025057701283426} a^{5} - \frac{266403846785180966130169790964080380475744939442075028875228956087792}{2028872518070234643727895044811452909096308770131825296512528850641713} a^{4} + \frac{455957893344808476426353386826515582146258573022730047196143667623193}{2028872518070234643727895044811452909096308770131825296512528850641713} a^{3} + \frac{1465100977500105306480398769930044000084373335599704518237428870455199}{4057745036140469287455790089622905818192617540263650593025057701283426} a^{2} - \frac{1770327481645434914250581459525907413099639021359158425378159671248227}{4057745036140469287455790089622905818192617540263650593025057701283426} a + \frac{728827100439886435638066451785035958585358030569090666363652189247175}{2028872518070234643727895044811452909096308770131825296512528850641713}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{7657902}$, which has order $367579296$ (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:  \( 3608180.2583334274 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-231}) \), \(\Q(\zeta_{7})^+\), 3.3.1016.1, Deg 6, 6.0.603993159.1, 9.9.123386988322304.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$127$127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.4.2.1$x^{4} + 635 x^{2} + 145161$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
127.4.2.1$x^{4} + 635 x^{2} + 145161$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
127.4.2.1$x^{4} + 635 x^{2} + 145161$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$