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Properties

Label 18.0.10540406459...3888.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 29^{9}\cdot 37^{14}$
Root discriminant $245.56$
Ramified primes $2, 3, 29, 37$
Class number $156266496$ (GRH)
Class group $[2, 2, 2, 2, 2, 12, 12, 33912]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3443926733743, -1110688906253, 1179077566010, -296201831072, 174264924081, -35293480367, 15004567540, -2489903023, 846514320, -115487055, 33000244, -3666123, 901320, -79111, 16771, -1074, 192, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 192*x^16 - 1074*x^15 + 16771*x^14 - 79111*x^13 + 901320*x^12 - 3666123*x^11 + 33000244*x^10 - 115487055*x^9 + 846514320*x^8 - 2489903023*x^7 + 15004567540*x^6 - 35293480367*x^5 + 174264924081*x^4 - 296201831072*x^3 + 1179077566010*x^2 - 1110688906253*x + 3443926733743)
 
gp: K = bnfinit(x^18 - 7*x^17 + 192*x^16 - 1074*x^15 + 16771*x^14 - 79111*x^13 + 901320*x^12 - 3666123*x^11 + 33000244*x^10 - 115487055*x^9 + 846514320*x^8 - 2489903023*x^7 + 15004567540*x^6 - 35293480367*x^5 + 174264924081*x^4 - 296201831072*x^3 + 1179077566010*x^2 - 1110688906253*x + 3443926733743, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 192 x^{16} - 1074 x^{15} + 16771 x^{14} - 79111 x^{13} + 901320 x^{12} - 3666123 x^{11} + 33000244 x^{10} - 115487055 x^{9} + 846514320 x^{8} - 2489903023 x^{7} + 15004567540 x^{6} - 35293480367 x^{5} + 174264924081 x^{4} - 296201831072 x^{3} + 1179077566010 x^{2} - 1110688906253 x + 3443926733743 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-10540406459397059490686352709797854892453888=-\,2^{12}\cdot 3^{9}\cdot 29^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $245.56$
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:  $2, 3, 29, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{17} + \frac{36960851712875848150644681889226771138804807149763195352180819580353993}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{16} + \frac{23537040766547260463653266475139795887298178631062442830164926837489412}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{15} + \frac{16891864113024611354469734949697862811957517912214613049471063821659947}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{14} - \frac{19406403080230510286842292921460591868657451825885476878654615020764167}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{13} + \frac{26496947669114982098249846293169400973131239084454560198753502980876339}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{12} + \frac{11916201188623108159717394751865764202809466004338913968405870899435400}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{11} + \frac{37942173547995342456437187714195917144323099022164544248250182848867320}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{10} + \frac{13217057596755533238228566528619694201770294916992669209025250045451735}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{9} - \frac{38633540016015304700603540617842262079659452391590656692823926654968692}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{8} - \frac{25369526658383899194234865489289260433370551906961111027722832357757418}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{7} - \frac{20971006952174468813643704093655288372730644687223681663100975404264603}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{6} + \frac{29441132287069328898081831220431137861735699534932468554737412936871808}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{5} + \frac{30581503552603920904339491118217371051308899031355599266923099819278186}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{4} - \frac{27126169816354846758416530463254711465841287374348522713355223356916048}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{3} + \frac{34430175194295918435489840073445072068123622952905514813888673119126949}{85452178137760780425437518781617748956149022387394234125249469687781691} a^{2} - \frac{2984601233138252070980314300932021880517923746611477338717243832043951}{85452178137760780425437518781617748956149022387394234125249469687781691} a - \frac{19608231867186806184878908329948056636644683909257849263989179015189756}{85452178137760780425437518781617748956149022387394234125249469687781691}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{12}\times C_{12}\times C_{33912}$, which has order $156266496$ (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:  $8$
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:  \( 615797.1340659427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-87}) \), 3.3.148.1, 3.3.1369.1, 6.0.14423849712.4, 6.0.1234140640983.2, 9.9.6075640136512.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$