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Properties

Label 18.0.27824830371...4375.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 5^{12}\cdot 7^{9}$
Root discriminant $33.47$
Ramified primes $3, 5, 7$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![343, -1029, 3087, -5684, 10143, -11844, 13756, -12180, 10080, -6552, 4200, -1764, 1132, -336, 189, -35, 21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 21*x^16 - 35*x^15 + 189*x^14 - 336*x^13 + 1132*x^12 - 1764*x^11 + 4200*x^10 - 6552*x^9 + 10080*x^8 - 12180*x^7 + 13756*x^6 - 11844*x^5 + 10143*x^4 - 5684*x^3 + 3087*x^2 - 1029*x + 343)
 
gp: K = bnfinit(x^18 + 21*x^16 - 35*x^15 + 189*x^14 - 336*x^13 + 1132*x^12 - 1764*x^11 + 4200*x^10 - 6552*x^9 + 10080*x^8 - 12180*x^7 + 13756*x^6 - 11844*x^5 + 10143*x^4 - 5684*x^3 + 3087*x^2 - 1029*x + 343, 1)
 

Normalized defining polynomial

\( x^{18} + 21 x^{16} - 35 x^{15} + 189 x^{14} - 336 x^{13} + 1132 x^{12} - 1764 x^{11} + 4200 x^{10} - 6552 x^{9} + 10080 x^{8} - 12180 x^{7} + 13756 x^{6} - 11844 x^{5} + 10143 x^{4} - 5684 x^{3} + 3087 x^{2} - 1029 x + 343 \)

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:  \(-2782483037193954337646484375=-\,3^{24}\cdot 5^{12}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.47$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{14} a^{12} - \frac{1}{7} a^{11} + \frac{1}{14} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{6} - \frac{5}{14} a^{5} + \frac{3}{7} a^{3} - \frac{5}{14} a^{2}$, $\frac{1}{14} a^{13} + \frac{3}{14} a^{11} + \frac{1}{14} a^{10} + \frac{3}{14} a^{8} - \frac{1}{14} a^{7} - \frac{1}{2} a^{6} + \frac{2}{7} a^{5} - \frac{1}{14} a^{4} - \frac{1}{2} a^{3} + \frac{2}{7} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{28} a^{14} - \frac{1}{14} a^{8} + \frac{2}{7} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{3780} a^{15} - \frac{1}{70} a^{14} - \frac{1}{126} a^{13} + \frac{43}{1890} a^{12} + \frac{1}{210} a^{11} + \frac{1}{21} a^{10} - \frac{1}{30} a^{9} - \frac{1}{30} a^{8} - \frac{22}{315} a^{7} + \frac{31}{210} a^{6} + \frac{41}{210} a^{5} + \frac{43}{126} a^{4} + \frac{271}{945} a^{3} + \frac{1}{70} a^{2} + \frac{89}{180} a - \frac{59}{270}$, $\frac{1}{371129891580} a^{16} + \frac{604889}{17672851980} a^{15} - \frac{252900157}{17672851980} a^{14} + \frac{345558272}{13254638985} a^{13} - \frac{82649767}{8836425990} a^{12} + \frac{35990471}{210391095} a^{11} + \frac{166064869}{10309163655} a^{10} + \frac{78272638}{1472737665} a^{9} - \frac{26726758}{4418212995} a^{8} + \frac{33127201}{490912555} a^{7} - \frac{97416737}{210391095} a^{6} - \frac{753414043}{8836425990} a^{5} + \frac{36890912746}{92782472895} a^{4} - \frac{456323356}{4418212995} a^{3} - \frac{800345989}{17672851980} a^{2} - \frac{858100261}{7574079420} a - \frac{1117047509}{2524693140}$, $\frac{1}{45648976664340} a^{17} + \frac{17}{22824488332170} a^{16} - \frac{78318769}{3260641190310} a^{15} + \frac{115942987511}{6521282380620} a^{14} - \frac{10123866791}{3260641190310} a^{13} + \frac{4616932708}{326064119031} a^{12} - \frac{312661603963}{1268027129565} a^{11} + \frac{36525293821}{1268027129565} a^{10} - \frac{25915580632}{543440198385} a^{9} - \frac{54165967117}{217376079354} a^{8} + \frac{157979728579}{1086880396770} a^{7} - \frac{260775538789}{1086880396770} a^{6} - \frac{552118063712}{11412244166085} a^{5} - \frac{4380424858826}{11412244166085} a^{4} + \frac{310998794201}{1304256476124} a^{3} + \frac{193890942005}{652128238062} a^{2} - \frac{84991924234}{232902942165} a - \frac{312183096259}{931611768660}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( 1284239.07252 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.14175.1 x3, 3.1.567.1 x3, 3.1.14175.2 x3, 3.1.175.1 x3, 6.0.1406514375.2, 6.0.2250423.3, 6.0.1406514375.1, 6.0.214375.1, 9.1.19937341265625.3 x9

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

Sibling fields

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
3.6.8.5$x^{6} + 9 x^{2} + 9$$3$$2$$8$$S_3$$[2]^{2}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$