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Properties

Label 18.0.16053944057...5696.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{34}\cdot 3^{9}\cdot 7^{15}$
Root discriminant $32.47$
Ramified primes $2, 3, 7$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_2\times S_3^2$ (as 18T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![918, -5832, 20178, -45720, 73686, -89688, 87448, -71112, 50122, -30804, 16814, -8124, 3523, -1344, 449, -126, 31, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 31*x^16 - 126*x^15 + 449*x^14 - 1344*x^13 + 3523*x^12 - 8124*x^11 + 16814*x^10 - 30804*x^9 + 50122*x^8 - 71112*x^7 + 87448*x^6 - 89688*x^5 + 73686*x^4 - 45720*x^3 + 20178*x^2 - 5832*x + 918)
 
gp: K = bnfinit(x^18 - 6*x^17 + 31*x^16 - 126*x^15 + 449*x^14 - 1344*x^13 + 3523*x^12 - 8124*x^11 + 16814*x^10 - 30804*x^9 + 50122*x^8 - 71112*x^7 + 87448*x^6 - 89688*x^5 + 73686*x^4 - 45720*x^3 + 20178*x^2 - 5832*x + 918, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 31 x^{16} - 126 x^{15} + 449 x^{14} - 1344 x^{13} + 3523 x^{12} - 8124 x^{11} + 16814 x^{10} - 30804 x^{9} + 50122 x^{8} - 71112 x^{7} + 87448 x^{6} - 89688 x^{5} + 73686 x^{4} - 45720 x^{3} + 20178 x^{2} - 5832 x + 918 \)

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:  \(-1605394405714515987001245696=-\,2^{34}\cdot 3^{9}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.47$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{4}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} + \frac{1}{9} a^{10} - \frac{4}{9} a^{8} + \frac{1}{3} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{4}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{27} a^{16} + \frac{1}{27} a^{15} + \frac{1}{27} a^{14} + \frac{4}{27} a^{12} + \frac{4}{27} a^{11} + \frac{4}{27} a^{10} + \frac{1}{9} a^{9} + \frac{7}{27} a^{8} + \frac{10}{27} a^{7} + \frac{4}{27} a^{6} - \frac{1}{3} a^{5} + \frac{2}{9} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{816644438032751577576513} a^{17} + \frac{203343161024354004277}{17375413575164927182479} a^{16} + \frac{32665163409244581439433}{816644438032751577576513} a^{15} - \frac{12569346018753755072201}{816644438032751577576513} a^{14} + \frac{36293866153780044174670}{816644438032751577576513} a^{13} - \frac{44948620477689954612337}{816644438032751577576513} a^{12} - \frac{127788023851258703702902}{816644438032751577576513} a^{11} + \frac{126517988335129613899000}{816644438032751577576513} a^{10} + \frac{107302526147557266444484}{816644438032751577576513} a^{9} - \frac{56720256327321201897355}{816644438032751577576513} a^{8} + \frac{334883066490571856711282}{816644438032751577576513} a^{7} + \frac{124518869815401157504036}{816644438032751577576513} a^{6} - \frac{84412073530623539416136}{272214812677583859192171} a^{5} - \frac{1478647058846456368754}{272214812677583859192171} a^{4} + \frac{14451233311629081506026}{90738270892527953064057} a^{3} - \frac{43263266885380874001989}{90738270892527953064057} a^{2} + \frac{262585700060045833958}{30246090297509317688019} a - \frac{6628926050990376483788}{30246090297509317688019}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 1486658.4572439173 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 18T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{-21}) \), 3.1.1176.1, 3.1.588.1, 6.0.464679936.1, 6.0.116169984.1, 9.1.78066229248.1

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

Sibling fields

Degree 12 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.26.99$x^{12} + 4 x^{11} + 6 x^{10} + 6 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 6$$12$$1$$26$$S_3 \times C_2^2$$[2, 3]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7Data not computed