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Properties

Label 18.2.46109508133...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{18}\cdot 5^{10}\cdot 23^{9}$
Root discriminant $23.45$
Ramified primes $2, 5, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3:S_3:S_4$ (as 18T155)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-56, -152, -28, 320, 418, -30, -583, -416, 216, 562, 30, -462, 136, 64, -47, 24, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 24*x^15 - 47*x^14 + 64*x^13 + 136*x^12 - 462*x^11 + 30*x^10 + 562*x^9 + 216*x^8 - 416*x^7 - 583*x^6 - 30*x^5 + 418*x^4 + 320*x^3 - 28*x^2 - 152*x - 56)
 
gp: K = bnfinit(x^18 - 4*x^17 + 24*x^15 - 47*x^14 + 64*x^13 + 136*x^12 - 462*x^11 + 30*x^10 + 562*x^9 + 216*x^8 - 416*x^7 - 583*x^6 - 30*x^5 + 418*x^4 + 320*x^3 - 28*x^2 - 152*x - 56, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 24 x^{15} - 47 x^{14} + 64 x^{13} + 136 x^{12} - 462 x^{11} + 30 x^{10} + 562 x^{9} + 216 x^{8} - 416 x^{7} - 583 x^{6} - 30 x^{5} + 418 x^{4} + 320 x^{3} - 28 x^{2} - 152 x - 56 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4610950813345280000000000=2^{18}\cdot 5^{10}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.45$
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:  $2, 5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{14} + \frac{1}{8} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{8} a^{15} + \frac{1}{8} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{16} - \frac{1}{32} a^{15} + \frac{1}{32} a^{14} - \frac{1}{4} a^{13} - \frac{3}{64} a^{12} - \frac{3}{32} a^{11} + \frac{5}{32} a^{10} + \frac{1}{32} a^{9} - \frac{1}{32} a^{8} - \frac{3}{32} a^{7} - \frac{1}{16} a^{5} + \frac{17}{64} a^{4} + \frac{3}{16} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a - \frac{3}{16}$, $\frac{1}{10926539242029206336} a^{17} - \frac{735675832186509}{118766730891621808} a^{16} + \frac{8291655957214745}{237533461783243616} a^{15} + \frac{33778116529941907}{2731634810507301584} a^{14} + \frac{367573200371569981}{10926539242029206336} a^{13} - \frac{271219558860517841}{1365817405253650792} a^{12} - \frac{61419669957163529}{5463269621014603168} a^{11} + \frac{1237800092148275943}{5463269621014603168} a^{10} + \frac{379249311609791685}{5463269621014603168} a^{9} + \frac{362703235771738847}{5463269621014603168} a^{8} + \frac{201437350950052971}{2731634810507301584} a^{7} + \frac{1049837289505292011}{2731634810507301584} a^{6} - \frac{3541381762384959527}{10926539242029206336} a^{5} - \frac{537070764057618863}{5463269621014603168} a^{4} - \frac{125175271059990125}{682908702626825396} a^{3} + \frac{470846991334295301}{1365817405253650792} a^{2} + \frac{302955233764730629}{2731634810507301584} a - \frac{89286443594129237}{1365817405253650792}$

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:  $9$
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:  \( 171769.930082 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T155):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.1.23.1, 6.2.19467200.4, 9.1.486680000.1

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

Sibling fields

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.10.2$x^{12} + 15 x^{6} + 100$$6$$2$$10$$C_6\times S_3$$[\ ]_{6}^{6}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.8.6.2$x^{8} - 1633 x^{4} + 1270129$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$