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Properties

Label 18.0.16905185765...6592.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 47^{9}\cdot 79^{14}$
Root discriminant $325.59$
Ramified primes $2, 47, 79$
Class number $254866500$ (GRH)
Class group $[15, 30, 566370]$ (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![122860198521, -8404855443, 37943392200, -3773174596, 5832541787, -640807131, 582912192, -66097021, 41593164, -4683867, 2230268, -246181, 90154, -9509, 2823, -258, 62, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 62*x^16 - 258*x^15 + 2823*x^14 - 9509*x^13 + 90154*x^12 - 246181*x^11 + 2230268*x^10 - 4683867*x^9 + 41593164*x^8 - 66097021*x^7 + 582912192*x^6 - 640807131*x^5 + 5832541787*x^4 - 3773174596*x^3 + 37943392200*x^2 - 8404855443*x + 122860198521)
 
gp: K = bnfinit(x^18 - 5*x^17 + 62*x^16 - 258*x^15 + 2823*x^14 - 9509*x^13 + 90154*x^12 - 246181*x^11 + 2230268*x^10 - 4683867*x^9 + 41593164*x^8 - 66097021*x^7 + 582912192*x^6 - 640807131*x^5 + 5832541787*x^4 - 3773174596*x^3 + 37943392200*x^2 - 8404855443*x + 122860198521, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 62 x^{16} - 258 x^{15} + 2823 x^{14} - 9509 x^{13} + 90154 x^{12} - 246181 x^{11} + 2230268 x^{10} - 4683867 x^{9} + 41593164 x^{8} - 66097021 x^{7} + 582912192 x^{6} - 640807131 x^{5} + 5832541787 x^{4} - 3773174596 x^{3} + 37943392200 x^{2} - 8404855443 x + 122860198521 \)

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:  \(-1690518576584469820319183508027997709797486592=-\,2^{12}\cdot 47^{9}\cdot 79^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $325.59$
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:  $2, 47, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{6} a^{14} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{15} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a$, $\frac{1}{3583422} a^{16} + \frac{3019}{1194474} a^{15} - \frac{120097}{3583422} a^{14} - \frac{40903}{1791711} a^{13} + \frac{146923}{1791711} a^{12} - \frac{122105}{1791711} a^{11} - \frac{262106}{1791711} a^{10} - \frac{221578}{597237} a^{9} + \frac{246100}{1791711} a^{8} + \frac{464543}{1791711} a^{7} - \frac{495356}{1791711} a^{6} + \frac{57145}{199079} a^{5} - \frac{38607}{199079} a^{4} + \frac{114362}{597237} a^{3} + \frac{533699}{3583422} a^{2} - \frac{368761}{1194474} a - \frac{85355}{398158}$, $\frac{1}{14158535685678127175856344236596067813636122972559522200963353457966} a^{17} + \frac{401149493870119349178902676054515325174661712760313046346601}{7079267842839063587928172118298033906818061486279761100481676728983} a^{16} - \frac{110537075826495201811756287324951656032352242685064326631773126012}{7079267842839063587928172118298033906818061486279761100481676728983} a^{15} - \frac{511094060208666310481600575763891526162269318743602245835016795897}{14158535685678127175856344236596067813636122972559522200963353457966} a^{14} + \frac{523709392626999325684419846969110386885220913403658545978519211903}{7079267842839063587928172118298033906818061486279761100481676728983} a^{13} + \frac{105228519819406246565747969214996198097408207141621926894872373965}{4719511895226042391952114745532022604545374324186507400321117819322} a^{12} - \frac{315090926190663365528520665154775732344264837051227953116673492576}{2359755947613021195976057372766011302272687162093253700160558909661} a^{11} + \frac{2020181233030477506429712917074457826850080143338367682390992682669}{14158535685678127175856344236596067813636122972559522200963353457966} a^{10} - \frac{3415929324998310409413208736802210055175958010430109954719736365465}{7079267842839063587928172118298033906818061486279761100481676728983} a^{9} - \frac{2867751812373432044216840847240817484320091739968955007236964693255}{14158535685678127175856344236596067813636122972559522200963353457966} a^{8} + \frac{1712161311438429519749295409360969956391957941071582351841910624805}{7079267842839063587928172118298033906818061486279761100481676728983} a^{7} - \frac{4430503213146450972117228548009787689266061347330898422311038757479}{14158535685678127175856344236596067813636122972559522200963353457966} a^{6} + \frac{360728939091750681619844683226751358569327384063932520212374356}{786585315871007065325352457588670434090895720697751233386852969887} a^{5} + \frac{465995344145000534077984967629730113007577589201577941384048398691}{1573170631742014130650704915177340868181791441395502466773705939774} a^{4} - \frac{1576666848905483885373524089355116109243260702122715397908421926201}{14158535685678127175856344236596067813636122972559522200963353457966} a^{3} + \frac{4810781193422308993490815869781365525671117742539953422586026972079}{14158535685678127175856344236596067813636122972559522200963353457966} a^{2} - \frac{354020630261669697194667310442227270806177464231215569174782865916}{786585315871007065325352457588670434090895720697751233386852969887} a + \frac{183935339504609233123736823233907012538570753391348820114619724167}{786585315871007065325352457588670434090895720697751233386852969887}$

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

Class group and class number

$C_{15}\times C_{30}\times C_{566370}$, which has order $254866500$ (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:  \( 9044146.559729666 \) (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

\(\Q(\sqrt{-47}) \), 3.3.6241.1, 3.3.316.1, 6.0.10367349488.1, 6.0.4043914259663.2, 9.9.1229050175114176.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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}$
2.6.6.3$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}$
$47$47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$79$79.3.2.1$x^{3} - 79$$3$$1$$2$$C_3$$[\ ]_{3}$
79.3.2.1$x^{3} - 79$$3$$1$$2$$C_3$$[\ ]_{3}$
79.6.5.1$x^{6} - 79$$6$$1$$5$$C_6$$[\ ]_{6}$
79.6.5.1$x^{6} - 79$$6$$1$$5$$C_6$$[\ ]_{6}$