Properties

Label 18.0.92836019722...0243.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 31^{6}$
Root discriminant $11.32$
Ramified primes $3, 31$
Class number $1$
Class group Trivial
Galois group $C_3.S_3^2$ (as 18T57)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, 39, 103, 201, 303, 342, 231, -6, -205, -219, -75, 54, 69, 21, -11, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^16 - 11*x^15 + 21*x^14 + 69*x^13 + 54*x^12 - 75*x^11 - 219*x^10 - 205*x^9 - 6*x^8 + 231*x^7 + 342*x^6 + 303*x^5 + 201*x^4 + 103*x^3 + 39*x^2 + 9*x + 1)
 
gp: K = bnfinit(x^18 - 9*x^16 - 11*x^15 + 21*x^14 + 69*x^13 + 54*x^12 - 75*x^11 - 219*x^10 - 205*x^9 - 6*x^8 + 231*x^7 + 342*x^6 + 303*x^5 + 201*x^4 + 103*x^3 + 39*x^2 + 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{16} - 11 x^{15} + 21 x^{14} + 69 x^{13} + 54 x^{12} - 75 x^{11} - 219 x^{10} - 205 x^{9} - 6 x^{8} + 231 x^{7} + 342 x^{6} + 303 x^{5} + 201 x^{4} + 103 x^{3} + 39 x^{2} + 9 x + 1 \)

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:  \(-9283601972222640243=-\,3^{21}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.32$
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:  $3, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
$|\Aut(K/\Q)|$:  $6$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{179} a^{15} - \frac{1}{179} a^{14} - \frac{39}{179} a^{13} + \frac{29}{179} a^{12} - \frac{22}{179} a^{11} + \frac{4}{179} a^{10} + \frac{41}{179} a^{9} + \frac{62}{179} a^{8} - \frac{61}{179} a^{7} - \frac{74}{179} a^{6} - \frac{80}{179} a^{5} - \frac{40}{179} a^{4} + \frac{70}{179} a^{3} - \frac{65}{179} a^{2} + \frac{12}{179} a + \frac{6}{179}$, $\frac{1}{179} a^{16} - \frac{40}{179} a^{14} - \frac{10}{179} a^{13} + \frac{7}{179} a^{12} - \frac{18}{179} a^{11} + \frac{45}{179} a^{10} - \frac{76}{179} a^{9} + \frac{1}{179} a^{8} + \frac{44}{179} a^{7} + \frac{25}{179} a^{6} + \frac{59}{179} a^{5} + \frac{30}{179} a^{4} + \frac{5}{179} a^{3} - \frac{53}{179} a^{2} + \frac{18}{179} a + \frac{6}{179}$, $\frac{1}{119393} a^{17} + \frac{122}{119393} a^{16} + \frac{201}{119393} a^{15} - \frac{835}{119393} a^{14} + \frac{32885}{119393} a^{13} + \frac{13374}{119393} a^{12} - \frac{41821}{119393} a^{11} + \frac{17655}{119393} a^{10} - \frac{40739}{119393} a^{9} + \frac{788}{119393} a^{8} - \frac{320}{667} a^{7} + \frac{68}{5191} a^{6} + \frac{28402}{119393} a^{5} - \frac{29066}{119393} a^{4} + \frac{12594}{119393} a^{3} + \frac{47160}{119393} a^{2} - \frac{59346}{119393} a - \frac{47942}{119393}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{100290}{119393} a^{17} - \frac{26081}{119393} a^{16} - \frac{889562}{119393} a^{15} - \frac{856728}{119393} a^{14} + \frac{2230237}{119393} a^{13} + \frac{6188582}{119393} a^{12} + \frac{4102693}{119393} a^{11} - \frac{7808820}{119393} a^{10} - \frac{19646959}{119393} a^{9} - \frac{16689982}{119393} a^{8} + \frac{1534437}{119393} a^{7} + \frac{948902}{5191} a^{6} + \frac{29930698}{119393} a^{5} + \frac{25103525}{119393} a^{4} + \frac{15893659}{119393} a^{3} + \frac{7431791}{119393} a^{2} + \frac{2550686}{119393} a + \frac{457899}{119393} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 296.277259881 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3.S_3^2$ (as 18T57):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 11 conjugacy class representatives for $C_3.S_3^2$
Character table for $C_3.S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.31.1, 6.0.25947.1, 9.1.586376253.1 x3

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

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.12.14.5$x^{12} - 12 x^{11} - 3 x^{10} - 9 x^{7} + 9 x^{5} + 9 x^{2} + 9$$6$$2$$14$$S_3^2$$[3/2, 3/2]_{2}^{2}$
$31$31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$