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Properties

Label 18.6.74441721379...7216.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{26}\cdot 3^{21}\cdot 13^{9}$
Root discriminant $35.35$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times S_3\wr C_2$ (as 18T150)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![768, 2304, 1152, -1344, -1824, 384, 1956, -24, -708, -272, -126, 180, 56, 0, 15, -12, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 - 12*x^15 + 15*x^14 + 56*x^12 + 180*x^11 - 126*x^10 - 272*x^9 - 708*x^8 - 24*x^7 + 1956*x^6 + 384*x^5 - 1824*x^4 - 1344*x^3 + 1152*x^2 + 2304*x + 768)
 
gp: K = bnfinit(x^18 - 6*x^16 - 12*x^15 + 15*x^14 + 56*x^12 + 180*x^11 - 126*x^10 - 272*x^9 - 708*x^8 - 24*x^7 + 1956*x^6 + 384*x^5 - 1824*x^4 - 1344*x^3 + 1152*x^2 + 2304*x + 768, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} - 12 x^{15} + 15 x^{14} + 56 x^{12} + 180 x^{11} - 126 x^{10} - 272 x^{9} - 708 x^{8} - 24 x^{7} + 1956 x^{6} + 384 x^{5} - 1824 x^{4} - 1344 x^{3} + 1152 x^{2} + 2304 x + 768 \)

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:  \(7444172137965405517922697216=2^{26}\cdot 3^{21}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.35$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{11} + \frac{1}{20} a^{10} + \frac{1}{5} a^{9} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{40} a^{13} + \frac{1}{20} a^{11} - \frac{1}{10} a^{10} - \frac{1}{8} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{10} a^{6} - \frac{9}{20} a^{5} - \frac{1}{10} a^{3} - \frac{2}{5} a^{2} - \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{80} a^{14} - \frac{1}{40} a^{12} + \frac{1}{20} a^{11} - \frac{9}{80} a^{10} - \frac{1}{10} a^{9} - \frac{1}{4} a^{8} - \frac{3}{20} a^{7} + \frac{1}{40} a^{6} + \frac{3}{10} a^{5} - \frac{1}{4} a^{4} - \frac{3}{10} a^{3} + \frac{1}{4} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{800} a^{15} + \frac{1}{200} a^{14} - \frac{1}{80} a^{13} - \frac{1}{40} a^{12} + \frac{63}{800} a^{11} + \frac{23}{200} a^{10} + \frac{41}{200} a^{9} - \frac{27}{200} a^{8} - \frac{47}{400} a^{7} + \frac{1}{20} a^{6} + \frac{29}{200} a^{5} + \frac{29}{100} a^{4} + \frac{61}{200} a^{3} - \frac{3}{50} a^{2} - \frac{7}{50} a - \frac{2}{25}$, $\frac{1}{1600} a^{16} - \frac{3}{800} a^{14} - \frac{1}{80} a^{13} - \frac{17}{1600} a^{12} + \frac{1}{20} a^{11} + \frac{7}{200} a^{10} - \frac{1}{400} a^{9} - \frac{111}{800} a^{8} + \frac{11}{100} a^{7} - \frac{41}{400} a^{6} - \frac{89}{200} a^{5} - \frac{191}{400} a^{4} + \frac{13}{50} a^{3} + \frac{3}{10} a^{2} - \frac{13}{50} a - \frac{6}{25}$, $\frac{1}{78313922610272000} a^{17} + \frac{6583733133653}{39156961305136000} a^{16} + \frac{1331348239651}{7831392261027200} a^{15} + \frac{11000591793709}{2447310081571000} a^{14} + \frac{473056070732623}{78313922610272000} a^{13} - \frac{588070128508581}{39156961305136000} a^{12} + \frac{1240095348675781}{19578480652568000} a^{11} - \frac{237423196171889}{19578480652568000} a^{10} + \frac{1291568099595949}{39156961305136000} a^{9} - \frac{2271461143000611}{19578480652568000} a^{8} + \frac{33541984267361}{1151675332504000} a^{7} + \frac{699571806165649}{4894620163142000} a^{6} + \frac{5056317556621}{230335066500800} a^{5} - \frac{4155982618140187}{9789240326284000} a^{4} + \frac{364330601092107}{978924032628400} a^{3} - \frac{701675434928367}{2447310081571000} a^{2} + \frac{327566080268687}{1223655040785500} a + \frac{26327410880259}{611827520392750}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 41801638.6224 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times S_3\wr C_2$ (as 18T150):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 27 conjugacy class representatives for $S_3\times S_3\wr C_2$
Character table for $S_3\times S_3\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.2808.1, 6.6.102503232.1, 6.2.949104.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.16.10$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
$3$3.9.9.6$x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
3.9.12.24$x^{9} + 3 x^{4} + 3$$9$$1$$12$$S_3^2$$[3/2, 3/2]_{2}^{2}$
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$