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Properties

Label 18.18.3245833493...8709.1
Degree $18$
Signature $[18, 0]$
Discriminant $37^{4}\cdot 229^{9}$
Root discriminant $33.76$
Ramified primes $37, 229$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\wr S_3$ (as 18T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-53, -1065, -82, 15830, -8341, -40257, 19585, 43078, -18450, -23692, 9135, 7131, -2559, -1165, 401, 95, -32, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 32*x^16 + 95*x^15 + 401*x^14 - 1165*x^13 - 2559*x^12 + 7131*x^11 + 9135*x^10 - 23692*x^9 - 18450*x^8 + 43078*x^7 + 19585*x^6 - 40257*x^5 - 8341*x^4 + 15830*x^3 - 82*x^2 - 1065*x - 53)
 
gp: K = bnfinit(x^18 - 3*x^17 - 32*x^16 + 95*x^15 + 401*x^14 - 1165*x^13 - 2559*x^12 + 7131*x^11 + 9135*x^10 - 23692*x^9 - 18450*x^8 + 43078*x^7 + 19585*x^6 - 40257*x^5 - 8341*x^4 + 15830*x^3 - 82*x^2 - 1065*x - 53, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 32 x^{16} + 95 x^{15} + 401 x^{14} - 1165 x^{13} - 2559 x^{12} + 7131 x^{11} + 9135 x^{10} - 23692 x^{9} - 18450 x^{8} + 43078 x^{7} + 19585 x^{6} - 40257 x^{5} - 8341 x^{4} + 15830 x^{3} - 82 x^{2} - 1065 x - 53 \)

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:  \(3245833493018808094352478709=37^{4}\cdot 229^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.76$
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:  $37, 229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{28370867936217364196446} a^{17} + \frac{955703149317955917340}{14185433968108682098223} a^{16} + \frac{4969954635433931025363}{28370867936217364196446} a^{15} - \frac{2012345163627267424607}{28370867936217364196446} a^{14} + \frac{4075644293531228180437}{28370867936217364196446} a^{13} - \frac{4912355588207422347303}{28370867936217364196446} a^{12} - \frac{5511054030158443010647}{28370867936217364196446} a^{11} + \frac{2095203049305795379373}{14185433968108682098223} a^{10} - \frac{4187752459777754854289}{28370867936217364196446} a^{9} - \frac{1311526639800061830277}{28370867936217364196446} a^{8} + \frac{12427993240572868851419}{28370867936217364196446} a^{7} + \frac{6200375724890021515812}{14185433968108682098223} a^{6} + \frac{4968719910118342749180}{14185433968108682098223} a^{5} - \frac{6542166962584787814681}{14185433968108682098223} a^{4} + \frac{11946715535538512825613}{28370867936217364196446} a^{3} + \frac{9511028178554866882991}{28370867936217364196446} a^{2} - \frac{5686516697534158101135}{14185433968108682098223} a + \frac{35063055123650928005}{535299395022969135782}$

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:  \( 68786041.1916 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\wr S_3$ (as 18T86):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 162
The 22 conjugacy class representatives for $C_3\wr S_3$
Character table for $C_3\wr S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{229}) \), 3.3.229.1 x3, 6.6.12008989.1, 9.9.16440305941.1

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$37$37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.2$x^{3} + 74$$3$$1$$2$$C_3$$[\ ]_{3}$
229Data not computed