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Properties

Label 18.6.52370069148...9344.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{18}\cdot 3^{36}\cdot 11^{3}$
Root discriminant $26.84$
Ramified primes $2, 3, 11$
Class number $1$
Class group Trivial
Galois group $C_3\times S_3\wr C_2$ (as 18T93)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37, -90, -261, 378, 171, -72, -453, 126, 9, 40, 171, 180, -36, -90, -54, -24, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^16 - 24*x^15 - 54*x^14 - 90*x^13 - 36*x^12 + 180*x^11 + 171*x^10 + 40*x^9 + 9*x^8 + 126*x^7 - 453*x^6 - 72*x^5 + 171*x^4 + 378*x^3 - 261*x^2 - 90*x + 37)
 
gp: K = bnfinit(x^18 - 9*x^16 - 24*x^15 - 54*x^14 - 90*x^13 - 36*x^12 + 180*x^11 + 171*x^10 + 40*x^9 + 9*x^8 + 126*x^7 - 453*x^6 - 72*x^5 + 171*x^4 + 378*x^3 - 261*x^2 - 90*x + 37, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{16} - 24 x^{15} - 54 x^{14} - 90 x^{13} - 36 x^{12} + 180 x^{11} + 171 x^{10} + 40 x^{9} + 9 x^{8} + 126 x^{7} - 453 x^{6} - 72 x^{5} + 171 x^{4} + 378 x^{3} - 261 x^{2} - 90 x + 37 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(52370069148219691512889344=2^{18}\cdot 3^{36}\cdot 11^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.84$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{26} a^{15} - \frac{3}{13} a^{14} + \frac{1}{26} a^{13} + \frac{5}{26} a^{12} + \frac{5}{26} a^{11} + \frac{3}{13} a^{10} - \frac{1}{26} a^{9} - \frac{3}{26} a^{8} - \frac{2}{13} a^{7} + \frac{3}{26} a^{6} - \frac{5}{13} a^{5} + \frac{7}{26} a^{4} - \frac{1}{2} a^{3} - \frac{3}{13} a^{2} - \frac{3}{26} a + \frac{3}{13}$, $\frac{1}{3796} a^{16} + \frac{12}{949} a^{15} - \frac{136}{949} a^{14} - \frac{393}{1898} a^{13} - \frac{58}{949} a^{12} - \frac{35}{949} a^{11} + \frac{110}{949} a^{10} + \frac{381}{1898} a^{9} + \frac{263}{3796} a^{8} + \frac{693}{1898} a^{7} + \frac{193}{1898} a^{6} - \frac{69}{146} a^{5} - \frac{987}{3796} a^{4} - \frac{47}{949} a^{3} - \frac{235}{1898} a^{2} + \frac{19}{73} a - \frac{105}{3796}$, $\frac{1}{1306436691293502028} a^{17} + \frac{153813287570291}{1306436691293502028} a^{16} - \frac{1853262445786202}{326609172823375507} a^{15} - \frac{33710658802418015}{653218345646751014} a^{14} + \frac{137204457585262281}{653218345646751014} a^{13} + \frac{4155387986894184}{25123782524875039} a^{12} + \frac{154600478177556247}{653218345646751014} a^{11} - \frac{58102526457479314}{326609172823375507} a^{10} + \frac{160102061464051795}{1306436691293502028} a^{9} + \frac{3188617275839229}{100495130099500156} a^{8} - \frac{151421487251786179}{653218345646751014} a^{7} + \frac{306322641387391775}{653218345646751014} a^{6} + \frac{206384755929114219}{1306436691293502028} a^{5} + \frac{317398090439883291}{1306436691293502028} a^{4} + \frac{20681034589793961}{653218345646751014} a^{3} + \frac{100497182706507415}{653218345646751014} a^{2} - \frac{23695024858187963}{76849217134911884} a - \frac{313299932034105}{2716084597283788}$

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:  $11$
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:  \( 1406421.43532 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3\wr C_2$ (as 18T93):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 27 conjugacy class representatives for $C_3\times S_3\wr C_2$
Character table for $C_3\times S_3\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{36})^+\), 6.2.41570496.2

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 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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$11$11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$