↑

Properties

Label 18.0.14176142707...8912.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{6}\cdot 7^{15}$
Root discriminant $11.59$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 58, -171, 378, -673, 1016, -1316, 1488, -1479, 1301, -1007, 684, -403, 204, -86, 29, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 29*x^16 - 86*x^15 + 204*x^14 - 403*x^13 + 684*x^12 - 1007*x^11 + 1301*x^10 - 1479*x^9 + 1488*x^8 - 1316*x^7 + 1016*x^6 - 673*x^5 + 378*x^4 - 171*x^3 + 58*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^18 - 7*x^17 + 29*x^16 - 86*x^15 + 204*x^14 - 403*x^13 + 684*x^12 - 1007*x^11 + 1301*x^10 - 1479*x^9 + 1488*x^8 - 1316*x^7 + 1016*x^6 - 673*x^5 + 378*x^4 - 171*x^3 + 58*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 29 x^{16} - 86 x^{15} + 204 x^{14} - 403 x^{13} + 684 x^{12} - 1007 x^{11} + 1301 x^{10} - 1479 x^{9} + 1488 x^{8} - 1316 x^{7} + 1016 x^{6} - 673 x^{5} + 378 x^{4} - 171 x^{3} + 58 x^{2} - 10 x + 1 \)

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:  \(-14176142707705638912=-\,2^{12}\cdot 3^{6}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $11.59$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{2} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{67976} a^{17} - \frac{449}{8497} a^{16} - \frac{4103}{67976} a^{15} - \frac{7635}{67976} a^{14} + \frac{391}{2344} a^{13} - \frac{535}{33988} a^{12} + \frac{14989}{33988} a^{11} + \frac{31907}{67976} a^{10} - \frac{7837}{33988} a^{9} - \frac{26341}{67976} a^{8} - \frac{18679}{67976} a^{7} + \frac{6539}{67976} a^{6} - \frac{23567}{67976} a^{5} + \frac{13421}{33988} a^{4} + \frac{3228}{8497} a^{3} - \frac{29887}{67976} a^{2} + \frac{14777}{67976} a - \frac{22251}{67976}$

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:  \( \frac{3605}{8497} a^{17} - \frac{92395}{33988} a^{16} + \frac{182361}{16994} a^{15} - \frac{1037705}{33988} a^{14} + \frac{41177}{586} a^{13} - \frac{1146807}{8497} a^{12} + \frac{7594203}{33988} a^{11} - \frac{5444885}{16994} a^{10} + \frac{6857639}{16994} a^{9} - \frac{7581487}{16994} a^{8} + \frac{7402747}{16994} a^{7} - \frac{6308437}{16994} a^{6} + \frac{9314087}{33988} a^{5} - \frac{1459910}{8497} a^{4} + \frac{3025987}{33988} a^{3} - \frac{1183783}{33988} a^{2} + \frac{310963}{33988} a - \frac{4203}{33988} \) (order $14$)
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:  \( 720.4866908744744 \)
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

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), 3.1.588.1, \(\Q(\zeta_{7})\), 6.0.2420208.1, 9.3.203297472.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed