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Properties

Label 18.0.16924241871...6464.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 37^{14}\cdot 71^{9}$
Root discriminant $221.83$
Ramified primes $2, 37, 71$
Class number $81348624$ (GRH)
Class group $[6, 42, 42, 7686]$ (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![693932370643, -271819124605, 284676104674, -84577334312, 49602460501, -11724742015, 5002295088, -962984783, 330391428, -52229455, 15151936, -1957691, 492124, -50767, 11075, -850, 156, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 156*x^16 - 850*x^15 + 11075*x^14 - 50767*x^13 + 492124*x^12 - 1957691*x^11 + 15151936*x^10 - 52229455*x^9 + 330391428*x^8 - 962984783*x^7 + 5002295088*x^6 - 11724742015*x^5 + 49602460501*x^4 - 84577334312*x^3 + 284676104674*x^2 - 271819124605*x + 693932370643)
 
gp: K = bnfinit(x^18 - 7*x^17 + 156*x^16 - 850*x^15 + 11075*x^14 - 50767*x^13 + 492124*x^12 - 1957691*x^11 + 15151936*x^10 - 52229455*x^9 + 330391428*x^8 - 962984783*x^7 + 5002295088*x^6 - 11724742015*x^5 + 49602460501*x^4 - 84577334312*x^3 + 284676104674*x^2 - 271819124605*x + 693932370643, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 156 x^{16} - 850 x^{15} + 11075 x^{14} - 50767 x^{13} + 492124 x^{12} - 1957691 x^{11} + 15151936 x^{10} - 52229455 x^{9} + 330391428 x^{8} - 962984783 x^{7} + 5002295088 x^{6} - 11724742015 x^{5} + 49602460501 x^{4} - 84577334312 x^{3} + 284676104674 x^{2} - 271819124605 x + 693932370643 \)

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:  \(-1692424187101732222765092905977936491966464=-\,2^{12}\cdot 37^{14}\cdot 71^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $221.83$
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:  $2, 37, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{394140000715344465406271897484157022019697622188500121046448833193} a^{17} - \frac{42332369172948027708169205407147642366374203013006760245050388491}{394140000715344465406271897484157022019697622188500121046448833193} a^{16} + \frac{29870454567392562684822785906092963596182220535567963604706928111}{394140000715344465406271897484157022019697622188500121046448833193} a^{15} + \frac{64349622153127000054814574428029157908071647947977557814551914525}{394140000715344465406271897484157022019697622188500121046448833193} a^{14} - \frac{1957819553499347330148257906923806360665816304017709440736119658}{131380000238448155135423965828052340673232540729500040348816277731} a^{13} + \frac{2074213343537385871690333915751265866256444647474366158912417072}{17136521770232368061142256412354653131291200964717396567236905791} a^{12} - \frac{137261246532954845806190393386196672471727393666578924592750620075}{394140000715344465406271897484157022019697622188500121046448833193} a^{11} + \frac{64198294976400960772141544705509899265511363053716027986655639278}{131380000238448155135423965828052340673232540729500040348816277731} a^{10} + \frac{44072149048473626847783897456770258900195319628276773300212106964}{394140000715344465406271897484157022019697622188500121046448833193} a^{9} + \frac{79985287512137599763105431778786117169865166918741646750191350483}{394140000715344465406271897484157022019697622188500121046448833193} a^{8} - \frac{38260665189879072609307405235223258148981515277916098039921538661}{131380000238448155135423965828052340673232540729500040348816277731} a^{7} + \frac{5572136785384091798773888950399881269082583893781465064879491481}{17136521770232368061142256412354653131291200964717396567236905791} a^{6} - \frac{126464280681900171321276795406084923277049188889150306269724289100}{394140000715344465406271897484157022019697622188500121046448833193} a^{5} - \frac{1454609299363365936035593550214276778812704791332856421572045691}{394140000715344465406271897484157022019697622188500121046448833193} a^{4} - \frac{14274551359892652936860781241006981744295105533874215259268030405}{131380000238448155135423965828052340673232540729500040348816277731} a^{3} - \frac{94298443660975583784628603889855544465626430192548246789212896163}{394140000715344465406271897484157022019697622188500121046448833193} a^{2} - \frac{61039819400233224195737445536983124763438649389974134988051092946}{131380000238448155135423965828052340673232540729500040348816277731} a + \frac{1983259220167946180125223261616343001897936590665384307892845618}{394140000715344465406271897484157022019697622188500121046448833193}$

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

Class group and class number

$C_{6}\times C_{42}\times C_{42}\times C_{7686}$, which has order $81348624$ (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:  \( 615797.1340659427 \) (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

\(\Q(\sqrt{-71}) \), 3.3.1369.1, 3.3.148.1, 6.0.7839682544.2, 6.0.670782837671.2, 9.9.6075640136512.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
$71$71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
71.6.3.2$x^{6} - 5041 x^{2} + 715822$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$