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Properties

Label 18.0.12009299766...7744.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{9}\cdot 7^{9}\cdot 37^{14}$
Root discriminant $191.51$
Ramified primes $2, 3, 7, 37$
Class number $21835008$ (GRH)
Class group $[2, 2, 18, 18, 36, 468]$ (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![2378782567897, 64419294176, 747079319085, 3474292326, 107199772771, -1404347650, 9318786102, -241380330, 545228110, -18246558, 22417227, -798220, 651907, -21092, 13029, -314, 164, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 164*x^16 - 314*x^15 + 13029*x^14 - 21092*x^13 + 651907*x^12 - 798220*x^11 + 22417227*x^10 - 18246558*x^9 + 545228110*x^8 - 241380330*x^7 + 9318786102*x^6 - 1404347650*x^5 + 107199772771*x^4 + 3474292326*x^3 + 747079319085*x^2 + 64419294176*x + 2378782567897)
 
gp: K = bnfinit(x^18 - 2*x^17 + 164*x^16 - 314*x^15 + 13029*x^14 - 21092*x^13 + 651907*x^12 - 798220*x^11 + 22417227*x^10 - 18246558*x^9 + 545228110*x^8 - 241380330*x^7 + 9318786102*x^6 - 1404347650*x^5 + 107199772771*x^4 + 3474292326*x^3 + 747079319085*x^2 + 64419294176*x + 2378782567897, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 164 x^{16} - 314 x^{15} + 13029 x^{14} - 21092 x^{13} + 651907 x^{12} - 798220 x^{11} + 22417227 x^{10} - 18246558 x^{9} + 545228110 x^{8} - 241380330 x^{7} + 9318786102 x^{6} - 1404347650 x^{5} + 107199772771 x^{4} + 3474292326 x^{3} + 747079319085 x^{2} + 64419294176 x + 2378782567897 \)

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:  \(-120092997667342126925823206694597892767744=-\,2^{24}\cdot 3^{9}\cdot 7^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $191.51$
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:  $2, 3, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{821083593224951651505642937337753673923088107831174305786061384433674} a^{17} + \frac{11071169826771956007024158729961293647151477753372224638509796543150}{410541796612475825752821468668876836961544053915587152893030692216837} a^{16} - \frac{70398465982325409586877502989683803406918820514430222170320790059859}{410541796612475825752821468668876836961544053915587152893030692216837} a^{15} - \frac{94736969064031614562057320068340639403235115784862741735899690107590}{410541796612475825752821468668876836961544053915587152893030692216837} a^{14} - \frac{80441158842025688482753361791839066819082629185722725368085464774819}{821083593224951651505642937337753673923088107831174305786061384433674} a^{13} - \frac{68278732006596904449502288387950961063797541771538066973330868070113}{821083593224951651505642937337753673923088107831174305786061384433674} a^{12} - \frac{36971735439596275693034477786790465364537958617837782755458077249921}{410541796612475825752821468668876836961544053915587152893030692216837} a^{11} + \frac{373185000795942873382254131591981400508010779087261925842074561192}{1526177682574259575289299140032999393909085702288428077669259078873} a^{10} + \frac{8833752882228987894179974953433305815651335192448959785823190417629}{821083593224951651505642937337753673923088107831174305786061384433674} a^{9} + \frac{63456713843573941607638936654997049790935407058322557758413070138693}{410541796612475825752821468668876836961544053915587152893030692216837} a^{8} + \frac{168635428730316458650122207437474056467269947809589182938585559184321}{410541796612475825752821468668876836961544053915587152893030692216837} a^{7} - \frac{165033055035742437051040559053239567140051758483231155083817644428041}{410541796612475825752821468668876836961544053915587152893030692216837} a^{6} - \frac{95738302619734175650477427200747205994297219237248033557965239679476}{410541796612475825752821468668876836961544053915587152893030692216837} a^{5} + \frac{363290101129159846203821477404290265010006903997177712232778108656777}{821083593224951651505642937337753673923088107831174305786061384433674} a^{4} - \frac{2450901046962828415250647448847343776586420224687327971606340550957}{9547483642150600598902824852764577603756838463153189602163504470159} a^{3} - \frac{77405917423739182349591717337671919966457576173870741434377797673328}{410541796612475825752821468668876836961544053915587152893030692216837} a^{2} + \frac{108055463496181693869916903650320758561285104401585729092971291133747}{821083593224951651505642937337753673923088107831174305786061384433674} a - \frac{51353947031198656120960188367296742135893409185623749486725182144023}{821083593224951651505642937337753673923088107831174305786061384433674}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{18}\times C_{18}\times C_{36}\times C_{468}$, which has order $21835008$ (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

\(\Q(\sqrt{-21}) \), 3.3.148.1, 3.3.1369.1, 6.0.3245647104.8, 6.0.1110822721344.1, 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$