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Properties

Label 18.2.38000833928...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 5^{9}\cdot 41^{6}$
Root discriminant $12.24$
Ramified primes $2, 5, 41$
Class number $1$
Class group Trivial
Galois group $S_3\wr C_2$ (as 18T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, -2, -3, -2, -4, -2, -10, -2, -3, 2, -10, 2, -4, 2, -3, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 2*x^16 - 3*x^15 + 2*x^14 - 4*x^13 + 2*x^12 - 10*x^11 + 2*x^10 - 3*x^9 - 2*x^8 - 10*x^7 - 2*x^6 - 4*x^5 - 2*x^4 - 3*x^3 - 2*x^2 - 2*x - 1)
 
gp: K = bnfinit(x^18 - 2*x^17 + 2*x^16 - 3*x^15 + 2*x^14 - 4*x^13 + 2*x^12 - 10*x^11 + 2*x^10 - 3*x^9 - 2*x^8 - 10*x^7 - 2*x^6 - 4*x^5 - 2*x^4 - 3*x^3 - 2*x^2 - 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 2 x^{16} - 3 x^{15} + 2 x^{14} - 4 x^{13} + 2 x^{12} - 10 x^{11} + 2 x^{10} - 3 x^{9} - 2 x^{8} - 10 x^{7} - 2 x^{6} - 4 x^{5} - 2 x^{4} - 3 x^{3} - 2 x^{2} - 2 x - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(38000833928000000000=2^{12}\cdot 5^{9}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12.24$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{15} a^{14} - \frac{4}{15} a^{13} + \frac{1}{15} a^{12} + \frac{1}{5} a^{11} - \frac{7}{15} a^{10} + \frac{4}{15} a^{9} + \frac{2}{15} a^{8} - \frac{7}{15} a^{7} - \frac{2}{15} a^{6} + \frac{4}{15} a^{5} + \frac{7}{15} a^{4} + \frac{1}{5} a^{3} - \frac{1}{15} a^{2} - \frac{4}{15} a - \frac{1}{15}$, $\frac{1}{15} a^{15} + \frac{7}{15} a^{12} + \frac{1}{3} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{15} a^{8} - \frac{4}{15} a^{6} - \frac{7}{15} a^{5} + \frac{1}{15} a^{4} - \frac{4}{15} a^{3} + \frac{7}{15} a^{2} - \frac{2}{15} a - \frac{4}{15}$, $\frac{1}{75} a^{16} + \frac{2}{75} a^{15} + \frac{1}{75} a^{14} + \frac{11}{25} a^{13} - \frac{1}{3} a^{12} - \frac{26}{75} a^{11} + \frac{23}{75} a^{10} - \frac{34}{75} a^{9} + \frac{34}{75} a^{8} - \frac{11}{75} a^{7} - \frac{2}{75} a^{6} + \frac{2}{25} a^{5} - \frac{2}{15} a^{4} + \frac{2}{75} a^{3} - \frac{4}{75} a^{2} - \frac{4}{25} a + \frac{12}{25}$, $\frac{1}{75} a^{17} + \frac{2}{75} a^{15} + \frac{1}{75} a^{14} + \frac{29}{75} a^{13} + \frac{29}{75} a^{12} + \frac{2}{15} a^{11} + \frac{2}{15} a^{10} - \frac{1}{25} a^{9} + \frac{16}{75} a^{8} + \frac{1}{15} a^{7} - \frac{1}{3} a^{6} - \frac{9}{25} a^{5} - \frac{11}{25} a^{4} + \frac{32}{75} a^{3} - \frac{14}{75} a^{2} + \frac{4}{15} a + \frac{13}{75}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 225.62524734 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 18T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $S_3\wr C_2$
Character table for $S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 6.2.82000.1 x2, 9.1.551368000.1

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

Sibling fields

Degree 6 siblings: 6.2.82000.1, 6.2.1378420.1
Degree 9 sibling: 9.1.551368000.1
Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12
Degree 18 siblings: Deg 18, Deg 18

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$