↑

Properties

Label 18.0.76258165114...3984.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 11^{6}\cdot 37^{6}$
Root discriminant $18.67$
Ramified primes $2, 11, 37$
Class number $1$
Class group Trivial
Galois group $C_2\times S_3^2$ (as 18T29)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -8, 0, 12, 0, 55, 0, -112, 0, 40, 0, 39, 0, -20, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 20*x^14 + 39*x^12 + 40*x^10 - 112*x^8 + 55*x^6 + 12*x^4 - 8*x^2 + 1)
 
gp: K = bnfinit(x^18 - 20*x^14 + 39*x^12 + 40*x^10 - 112*x^8 + 55*x^6 + 12*x^4 - 8*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 20 x^{14} + 39 x^{12} + 40 x^{10} - 112 x^{8} + 55 x^{6} + 12 x^{4} - 8 x^{2} + 1 \)

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:  \(-76258165114191147433984=-\,2^{24}\cdot 11^{6}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.67$
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:  $2, 11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} - \frac{1}{3} a$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{12}$, $\frac{1}{12} a^{15} + \frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{20688} a^{16} + \frac{455}{20688} a^{14} + \frac{125}{20688} a^{12} - \frac{851}{10344} a^{10} + \frac{813}{3448} a^{8} - \frac{4015}{10344} a^{6} + \frac{1285}{20688} a^{4} - \frac{491}{6896} a^{2} + \frac{5585}{20688}$, $\frac{1}{41376} a^{17} - \frac{1}{41376} a^{16} + \frac{455}{41376} a^{15} - \frac{455}{41376} a^{14} + \frac{125}{41376} a^{13} - \frac{125}{41376} a^{12} + \frac{4321}{20688} a^{11} - \frac{4321}{20688} a^{10} - \frac{911}{6896} a^{9} + \frac{911}{6896} a^{8} - \frac{9187}{20688} a^{7} - \frac{1157}{20688} a^{6} - \frac{9059}{41376} a^{5} + \frac{9059}{41376} a^{4} + \frac{2957}{13792} a^{3} - \frac{2957}{13792} a^{2} - \frac{4759}{41376} a - \frac{15929}{41376}$

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:  \( \frac{3995}{3448} a^{17} + \frac{629}{3448} a^{15} - \frac{79889}{3448} a^{13} + \frac{71529}{1724} a^{11} + \frac{91991}{1724} a^{9} - \frac{209295}{1724} a^{7} + \frac{146043}{3448} a^{5} + \frac{71785}{3448} a^{3} - \frac{15449}{3448} a \) (order $4$)
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:  \( 36250.2959789 \)
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

\(\Q(\sqrt{-1}) \), 3.3.148.1, 3.1.44.1, 6.0.350464.1, 6.0.30976.1, 9.3.4314825152.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$37$37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$