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Properties

Label 18.12.2889069193...3559.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,3^{9}\cdot 7^{12}\cdot 13^{9}$
Root discriminant $22.85$
Ramified primes $3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![29, 177, 175, -533, -678, 677, 630, -525, 15, 229, -300, 36, 156, -87, -26, 29, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 3*x^16 + 29*x^15 - 26*x^14 - 87*x^13 + 156*x^12 + 36*x^11 - 300*x^10 + 229*x^9 + 15*x^8 - 525*x^7 + 630*x^6 + 677*x^5 - 678*x^4 - 533*x^3 + 175*x^2 + 177*x + 29)
 
gp: K = bnfinit(x^18 - 3*x^17 - 3*x^16 + 29*x^15 - 26*x^14 - 87*x^13 + 156*x^12 + 36*x^11 - 300*x^10 + 229*x^9 + 15*x^8 - 525*x^7 + 630*x^6 + 677*x^5 - 678*x^4 - 533*x^3 + 175*x^2 + 177*x + 29, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 3 x^{16} + 29 x^{15} - 26 x^{14} - 87 x^{13} + 156 x^{12} + 36 x^{11} - 300 x^{10} + 229 x^{9} + 15 x^{8} - 525 x^{7} + 630 x^{6} + 677 x^{5} - 678 x^{4} - 533 x^{3} + 175 x^{2} + 177 x + 29 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2889069193792436475743559=-\,3^{9}\cdot 7^{12}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.85$
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:  $3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{21943959798169361} a^{17} + \frac{4566084300773495}{21943959798169361} a^{16} + \frac{1922567585244899}{21943959798169361} a^{15} + \frac{3716742364584949}{21943959798169361} a^{14} - \frac{10498548304915753}{21943959798169361} a^{13} + \frac{10380663318921925}{21943959798169361} a^{12} - \frac{10621520642121908}{21943959798169361} a^{11} + \frac{7235253205137945}{21943959798169361} a^{10} + \frac{2417333044859549}{21943959798169361} a^{9} + \frac{8821007888480732}{21943959798169361} a^{8} - \frac{891321915581479}{21943959798169361} a^{7} + \frac{4119553650250654}{21943959798169361} a^{6} + \frac{10523096242468589}{21943959798169361} a^{5} + \frac{3324674514109361}{21943959798169361} a^{4} - \frac{1518230305782792}{21943959798169361} a^{3} - \frac{9495325605635771}{21943959798169361} a^{2} + \frac{2369478809306670}{21943959798169361} a + \frac{5991795794759693}{21943959798169361}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 625728.498159 \) (assuming GRH)
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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{7})^+\), 6.6.5274997.1, 6.4.142424919.1

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 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.9.9.1$x^{9} + 54 x^{5} + 27 x^{3} + 189$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
7Data not computed
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$