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Properties

Label 18.0.51923518320...9867.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 107^{9}$
Root discriminant $44.76$
Ramified primes $3, 107$
Class number $144$ (GRH)
Class group $[2, 2, 2, 18]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![59273, 183555, 269637, 282511, 211056, 117033, 61893, 23070, 11598, 4550, 537, 624, -252, -192, 171, -28, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 12*x^16 - 28*x^15 + 171*x^14 - 192*x^13 - 252*x^12 + 624*x^11 + 537*x^10 + 4550*x^9 + 11598*x^8 + 23070*x^7 + 61893*x^6 + 117033*x^5 + 211056*x^4 + 282511*x^3 + 269637*x^2 + 183555*x + 59273)
 
gp: K = bnfinit(x^18 - 6*x^17 + 12*x^16 - 28*x^15 + 171*x^14 - 192*x^13 - 252*x^12 + 624*x^11 + 537*x^10 + 4550*x^9 + 11598*x^8 + 23070*x^7 + 61893*x^6 + 117033*x^5 + 211056*x^4 + 282511*x^3 + 269637*x^2 + 183555*x + 59273, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 12 x^{16} - 28 x^{15} + 171 x^{14} - 192 x^{13} - 252 x^{12} + 624 x^{11} + 537 x^{10} + 4550 x^{9} + 11598 x^{8} + 23070 x^{7} + 61893 x^{6} + 117033 x^{5} + 211056 x^{4} + 282511 x^{3} + 269637 x^{2} + 183555 x + 59273 \)

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:  \(-519235183203048555634413069867=-\,3^{24}\cdot 107^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.76$
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:  $3, 107$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{107} a^{12} + \frac{46}{107} a^{11} + \frac{25}{107} a^{10} + \frac{43}{107} a^{9} - \frac{44}{107} a^{8} - \frac{46}{107} a^{7} + \frac{1}{107} a^{6} + \frac{4}{107} a^{5} - \frac{18}{107} a^{4} + \frac{19}{107} a^{3} + \frac{53}{107} a^{2} + \frac{44}{107} a - \frac{21}{107}$, $\frac{1}{107} a^{13} + \frac{49}{107} a^{11} - \frac{37}{107} a^{10} + \frac{11}{107} a^{9} + \frac{52}{107} a^{8} - \frac{23}{107} a^{7} - \frac{42}{107} a^{6} + \frac{12}{107} a^{5} - \frac{9}{107} a^{4} + \frac{35}{107} a^{3} - \frac{40}{107} a^{2} - \frac{12}{107} a + \frac{3}{107}$, $\frac{1}{5671} a^{14} + \frac{23}{5671} a^{13} + \frac{21}{5671} a^{12} - \frac{32}{107} a^{11} + \frac{1777}{5671} a^{10} - \frac{21}{107} a^{9} - \frac{2517}{5671} a^{8} - \frac{2707}{5671} a^{7} - \frac{340}{5671} a^{6} + \frac{2081}{5671} a^{5} + \frac{2044}{5671} a^{4} + \frac{2266}{5671} a^{3} + \frac{687}{5671} a^{2} + \frac{421}{5671} a + \frac{2690}{5671}$, $\frac{1}{5671} a^{15} + \frac{22}{5671} a^{13} - \frac{6}{5671} a^{12} + \frac{2254}{5671} a^{11} - \frac{1598}{5671} a^{10} - \frac{2411}{5671} a^{9} - \frac{1526}{5671} a^{8} + \frac{812}{5671} a^{7} + \frac{1156}{5671} a^{6} - \frac{2412}{5671} a^{5} + \frac{2106}{5671} a^{4} + \frac{2735}{5671} a^{3} - \frac{805}{5671} a^{2} - \frac{2806}{5671} a + \frac{1836}{5671}$, $\frac{1}{54004933} a^{16} + \frac{1494}{54004933} a^{15} - \frac{3635}{54004933} a^{14} + \frac{121213}{54004933} a^{13} + \frac{249121}{54004933} a^{12} + \frac{10329442}{54004933} a^{11} - \frac{315670}{1018961} a^{10} - \frac{6419874}{54004933} a^{9} + \frac{23998739}{54004933} a^{8} + \frac{19441355}{54004933} a^{7} - \frac{25946224}{54004933} a^{6} + \frac{12762805}{54004933} a^{5} - \frac{20360429}{54004933} a^{4} + \frac{20653509}{54004933} a^{3} + \frac{23844497}{54004933} a^{2} + \frac{149132}{606797} a - \frac{24416442}{54004933}$, $\frac{1}{936179514684311635396950371} a^{17} - \frac{74423211307959538}{10518870951509119498842139} a^{16} - \frac{59445202069287496523767}{936179514684311635396950371} a^{15} + \frac{82240482988652163999221}{936179514684311635396950371} a^{14} + \frac{2701377512314310126073081}{936179514684311635396950371} a^{13} - \frac{2367696679022985779308196}{936179514684311635396950371} a^{12} + \frac{107111694770118192968091809}{936179514684311635396950371} a^{11} + \frac{267788882804267750754564352}{936179514684311635396950371} a^{10} + \frac{73411078751399217579983040}{936179514684311635396950371} a^{9} + \frac{1747469123399344122546948}{936179514684311635396950371} a^{8} - \frac{262126853307122838051116569}{936179514684311635396950371} a^{7} - \frac{223502080839154753387010059}{936179514684311635396950371} a^{6} - \frac{136361266237992469834637307}{936179514684311635396950371} a^{5} - \frac{135414611080302855343430273}{936179514684311635396950371} a^{4} + \frac{216873841669414098505705024}{936179514684311635396950371} a^{3} - \frac{238036567317240160288925653}{936179514684311635396950371} a^{2} + \frac{121240568315695823065139744}{936179514684311635396950371} a + \frac{280404023043560234861051592}{936179514684311635396950371}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{18}$, which has order $144$ (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:  \( 146101.088241 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-107}) \), 3.1.8667.1 x3, \(\Q(\zeta_{9})^+\), 6.0.8037507123.2, 6.0.99228483.1 x2, 6.0.8037507123.1, 9.3.651038076963.10 x3

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

Sibling fields

Degree 6 sibling: 6.0.99228483.1
Degree 9 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$107$107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$