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Properties

Label 18.18.1194556966...0352.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{27}\cdot 2113^{6}$
Root discriminant $36.29$
Ramified primes $2, 2113$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\wr C_2$ (as 18T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 10, -13, -190, -30, 1160, 677, -2968, -2369, 3262, 3065, -1462, -1599, 254, 350, -14, -32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 32*x^16 - 14*x^15 + 350*x^14 + 254*x^13 - 1599*x^12 - 1462*x^11 + 3065*x^10 + 3262*x^9 - 2369*x^8 - 2968*x^7 + 677*x^6 + 1160*x^5 - 30*x^4 - 190*x^3 - 13*x^2 + 10*x + 1)
 
gp: K = bnfinit(x^18 - 32*x^16 - 14*x^15 + 350*x^14 + 254*x^13 - 1599*x^12 - 1462*x^11 + 3065*x^10 + 3262*x^9 - 2369*x^8 - 2968*x^7 + 677*x^6 + 1160*x^5 - 30*x^4 - 190*x^3 - 13*x^2 + 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 32 x^{16} - 14 x^{15} + 350 x^{14} + 254 x^{13} - 1599 x^{12} - 1462 x^{11} + 3065 x^{10} + 3262 x^{9} - 2369 x^{8} - 2968 x^{7} + 677 x^{6} + 1160 x^{5} - 30 x^{4} - 190 x^{3} - 13 x^{2} + 10 x + 1 \)

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:  \(11945569661051635899513700352=2^{27}\cdot 2113^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.29$
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:  $2, 2113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{11} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{42184580541268} a^{17} - \frac{1240189123091}{10546145135317} a^{16} + \frac{97150950011}{42184580541268} a^{15} + \frac{2414558436597}{21092290270634} a^{14} - \frac{2265651415131}{10546145135317} a^{13} - \frac{5342918283937}{42184580541268} a^{12} + \frac{2512398317413}{10546145135317} a^{11} + \frac{5208435601179}{42184580541268} a^{10} - \frac{4332682311055}{21092290270634} a^{9} - \frac{19629054688967}{42184580541268} a^{8} + \frac{1708643348903}{21092290270634} a^{7} + \frac{8417341538591}{42184580541268} a^{6} - \frac{31031746962}{148537255427} a^{5} - \frac{19949537295535}{42184580541268} a^{4} + \frac{13435008255267}{42184580541268} a^{3} + \frac{12247995569599}{42184580541268} a^{2} - \frac{19957924815167}{42184580541268} a - \frac{19494798610611}{42184580541268}$

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:  \( 179925968.916 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 18T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $S_3\wr C_2$
Character table for $S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), 6.6.1081856.1 x2, 9.9.4830237131264.1

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

Sibling fields

Degree 6 siblings: 6.6.1081856.1, 6.6.75472455176.1
Degree 9 sibling: 9.9.4830237131264.1
Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12
Degree 18 siblings: Deg 18, Deg 18

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2113Data not computed