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Properties

Label 18.18.1481633261...7648.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{26}\cdot 13^{9}\cdot 113^{6}$
Root discriminant $47.44$
Ramified primes $2, 13, 113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times S_3\wr C_2$ (as 18T63)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, 17, 277, -722, -1472, 5262, 656, -12678, 7541, 8477, -8951, -790, 3334, -700, -412, 164, 5, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 5*x^16 + 164*x^15 - 412*x^14 - 700*x^13 + 3334*x^12 - 790*x^11 - 8951*x^10 + 8477*x^9 + 7541*x^8 - 12678*x^7 + 656*x^6 + 5262*x^5 - 1472*x^4 - 722*x^3 + 277*x^2 + 17*x - 9)
 
gp: K = bnfinit(x^18 - 9*x^17 + 5*x^16 + 164*x^15 - 412*x^14 - 700*x^13 + 3334*x^12 - 790*x^11 - 8951*x^10 + 8477*x^9 + 7541*x^8 - 12678*x^7 + 656*x^6 + 5262*x^5 - 1472*x^4 - 722*x^3 + 277*x^2 + 17*x - 9, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 5 x^{16} + 164 x^{15} - 412 x^{14} - 700 x^{13} + 3334 x^{12} - 790 x^{11} - 8951 x^{10} + 8477 x^{9} + 7541 x^{8} - 12678 x^{7} + 656 x^{6} + 5262 x^{5} - 1472 x^{4} - 722 x^{3} + 277 x^{2} + 17 x - 9 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1481633261190001001293202587648=2^{26}\cdot 13^{9}\cdot 113^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.44$
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:  $2, 13, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{8} a^{3} - \frac{1}{8} a$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{13} - \frac{1}{24} a^{12} + \frac{1}{48} a^{11} - \frac{1}{48} a^{10} - \frac{1}{24} a^{9} + \frac{1}{48} a^{8} + \frac{5}{48} a^{7} - \frac{3}{16} a^{6} - \frac{1}{8} a^{5} + \frac{5}{48} a^{4} - \frac{1}{16} a^{3} - \frac{7}{24} a^{2} + \frac{7}{48} a + \frac{5}{16}$, $\frac{1}{48} a^{15} - \frac{1}{16} a^{13} - \frac{1}{48} a^{12} - \frac{1}{16} a^{10} - \frac{1}{48} a^{9} - \frac{1}{8} a^{8} - \frac{1}{12} a^{7} + \frac{3}{16} a^{6} - \frac{1}{48} a^{5} - \frac{5}{24} a^{4} - \frac{17}{48} a^{3} - \frac{7}{48} a^{2} + \frac{11}{24} a - \frac{7}{16}$, $\frac{1}{6672} a^{16} - \frac{1}{834} a^{15} + \frac{5}{1112} a^{14} - \frac{35}{3336} a^{13} + \frac{115}{3336} a^{12} - \frac{1}{8} a^{11} + \frac{319}{3336} a^{10} + \frac{101}{1668} a^{9} - \frac{559}{6672} a^{8} - \frac{773}{3336} a^{7} + \frac{307}{3336} a^{6} + \frac{145}{1668} a^{5} + \frac{27}{278} a^{4} - \frac{551}{1112} a^{3} - \frac{61}{278} a^{2} - \frac{1015}{3336} a - \frac{935}{2224}$, $\frac{1}{487056} a^{17} + \frac{7}{121764} a^{16} + \frac{2939}{487056} a^{15} + \frac{50}{10147} a^{14} + \frac{13417}{487056} a^{13} - \frac{22717}{487056} a^{12} + \frac{1281}{40588} a^{11} + \frac{9889}{487056} a^{10} - \frac{2210}{30441} a^{9} - \frac{9357}{81176} a^{8} + \frac{5369}{30441} a^{7} + \frac{119845}{487056} a^{6} + \frac{60031}{487056} a^{5} - \frac{8617}{81176} a^{4} + \frac{177953}{487056} a^{3} + \frac{1447}{162352} a^{2} + \frac{16961}{162352} a - \frac{31019}{162352}$

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:  $17$
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:  \( 6497548324.32 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\wr C_2$ (as 18T63):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times S_3\wr C_2$
Character table for $C_2\times S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{13}) \), 9.9.337597081329664.1

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 siblings: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.22.60$x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$$6$$2$$22$$D_6$$[3]_{3}^{2}$
$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}$
$113$113.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
113.3.0.1$x^{3} - x + 5$$1$$3$$0$$C_3$$[\ ]^{3}$
113.6.3.1$x^{6} - 226 x^{4} + 12769 x^{2} - 36072425$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
113.6.3.1$x^{6} - 226 x^{4} + 12769 x^{2} - 36072425$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$