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Properties

Label 18.0.60894079274...7871.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,19^{12}\cdot 31^{7}$
Root discriminant $27.07$
Ramified primes $19, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times A_4$ (as 18T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, 256, 0, -704, 96, 336, 728, -152, 24, -183, 68, -39, 40, -14, 6, -8, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + x^16 - 8*x^15 + 6*x^14 - 14*x^13 + 40*x^12 - 39*x^11 + 68*x^10 - 183*x^9 + 24*x^8 - 152*x^7 + 728*x^6 + 336*x^5 + 96*x^4 - 704*x^3 + 256*x + 512)
 
gp: K = bnfinit(x^18 + x^16 - 8*x^15 + 6*x^14 - 14*x^13 + 40*x^12 - 39*x^11 + 68*x^10 - 183*x^9 + 24*x^8 - 152*x^7 + 728*x^6 + 336*x^5 + 96*x^4 - 704*x^3 + 256*x + 512, 1)
 

Normalized defining polynomial

\( x^{18} + x^{16} - 8 x^{15} + 6 x^{14} - 14 x^{13} + 40 x^{12} - 39 x^{11} + 68 x^{10} - 183 x^{9} + 24 x^{8} - 152 x^{7} + 728 x^{6} + 336 x^{5} + 96 x^{4} - 704 x^{3} + 256 x + 512 \)

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:  \(-60894079274386484071197871=-\,19^{12}\cdot 31^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.07$
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:  $19, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{3}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{3}{8} a^{7} - \frac{1}{8} a^{6} + \frac{3}{16} a^{5} - \frac{7}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{5}{16} a^{6} - \frac{5}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{11}{32} a^{7} - \frac{5}{32} a^{5} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{256} a^{15} - \frac{1}{128} a^{14} + \frac{5}{256} a^{13} + \frac{3}{128} a^{12} + \frac{5}{128} a^{11} + \frac{11}{128} a^{10} + \frac{3}{64} a^{9} + \frac{49}{256} a^{8} - \frac{7}{128} a^{7} + \frac{5}{256} a^{6} + \frac{19}{128} a^{5} - \frac{29}{64} a^{4} + \frac{15}{32} a^{3} - \frac{1}{16} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{3634688} a^{16} + \frac{493}{454336} a^{15} - \frac{38751}{3634688} a^{14} - \frac{347}{14656} a^{13} - \frac{34149}{1817344} a^{12} + \frac{13693}{1817344} a^{11} - \frac{19071}{454336} a^{10} - \frac{258679}{3634688} a^{9} - \frac{61777}{908672} a^{8} - \frac{642903}{3634688} a^{7} + \frac{156599}{454336} a^{6} - \frac{186993}{454336} a^{5} - \frac{73949}{227168} a^{4} - \frac{24725}{113584} a^{3} + \frac{12443}{28396} a^{2} + \frac{12677}{28396} a - \frac{10639}{28396}$, $\frac{1}{7727346688} a^{17} + \frac{273}{3863673344} a^{16} - \frac{14860263}{7727346688} a^{15} - \frac{51480559}{3863673344} a^{14} + \frac{25633975}{3863673344} a^{13} - \frac{41432221}{3863673344} a^{12} + \frac{24374335}{1931836672} a^{11} - \frac{761129207}{7727346688} a^{10} + \frac{82150387}{3863673344} a^{9} + \frac{1577269289}{7727346688} a^{8} + \frac{1780158585}{3863673344} a^{7} - \frac{203815329}{482959168} a^{6} + \frac{225020703}{482959168} a^{5} + \frac{44139403}{241479584} a^{4} - \frac{1338291}{30184948} a^{3} - \frac{1276343}{15092474} a^{2} + \frac{7046053}{60369896} a + \frac{11672363}{30184948}$

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:  $8$
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:  \( 569373.693221 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times A_4$ (as 18T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $S_3\times A_4$
Character table for $S_3\times A_4$

Intermediate fields

3.3.361.1, 3.1.31.1, 6.0.4039951.1, 9.3.1401543840871.1

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

Sibling fields

Degree 12 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
31Data not computed