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Properties

Label 18.0.13407337425...6048.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 23^{9}\cdot 127^{14}$
Root discriminant $415.13$
Ramified primes $2, 23, 127$
Class number $5710787712$ (GRH)
Class group $[12, 252, 1888488]$ (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![14816731136, -15402059776, 10681441664, -4306452928, 1353700784, -348451720, 122868164, -42389938, 10462064, -883965, -103469, -31030, 40510, -8692, -104, 296, -14, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 14*x^16 + 296*x^15 - 104*x^14 - 8692*x^13 + 40510*x^12 - 31030*x^11 - 103469*x^10 - 883965*x^9 + 10462064*x^8 - 42389938*x^7 + 122868164*x^6 - 348451720*x^5 + 1353700784*x^4 - 4306452928*x^3 + 10681441664*x^2 - 15402059776*x + 14816731136)
 
gp: K = bnfinit(x^18 - 7*x^17 - 14*x^16 + 296*x^15 - 104*x^14 - 8692*x^13 + 40510*x^12 - 31030*x^11 - 103469*x^10 - 883965*x^9 + 10462064*x^8 - 42389938*x^7 + 122868164*x^6 - 348451720*x^5 + 1353700784*x^4 - 4306452928*x^3 + 10681441664*x^2 - 15402059776*x + 14816731136, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} - 14 x^{16} + 296 x^{15} - 104 x^{14} - 8692 x^{13} + 40510 x^{12} - 31030 x^{11} - 103469 x^{10} - 883965 x^{9} + 10462064 x^{8} - 42389938 x^{7} + 122868164 x^{6} - 348451720 x^{5} + 1353700784 x^{4} - 4306452928 x^{3} + 10681441664 x^{2} - 15402059776 x + 14816731136 \)

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:  \(-134073374251346122646523762386906440653122306048=-\,2^{18}\cdot 23^{9}\cdot 127^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $415.13$
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:  $2, 23, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} + \frac{1}{16} a^{5} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{1}{64} a^{10} - \frac{3}{64} a^{8} + \frac{3}{32} a^{7} + \frac{5}{64} a^{6} - \frac{3}{16} a^{5} + \frac{3}{32} a^{4} - \frac{3}{8} a^{3} + \frac{3}{8} a^{2}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{1}{32} a^{10} + \frac{1}{64} a^{9} + \frac{5}{64} a^{7} - \frac{1}{32} a^{6} + \frac{5}{32} a^{5} - \frac{3}{16} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{13} - \frac{1}{128} a^{12} + \frac{3}{128} a^{11} + \frac{3}{128} a^{10} + \frac{3}{128} a^{9} + \frac{5}{128} a^{8} - \frac{3}{128} a^{7} - \frac{1}{32} a^{6} + \frac{11}{64} a^{5} - \frac{5}{32} a^{4} + \frac{5}{16} a^{3} - \frac{3}{8} a^{2}$, $\frac{1}{9344} a^{15} - \frac{9}{4672} a^{14} + \frac{1}{2336} a^{13} + \frac{3}{2336} a^{12} - \frac{55}{2336} a^{11} - \frac{5}{292} a^{10} + \frac{199}{4672} a^{9} - \frac{13}{584} a^{8} - \frac{293}{9344} a^{7} - \frac{483}{4672} a^{6} + \frac{715}{4672} a^{5} + \frac{475}{2336} a^{4} - \frac{405}{1168} a^{3} + \frac{229}{584} a^{2} - \frac{16}{73} a + \frac{4}{73}$, $\frac{1}{299008} a^{16} - \frac{1}{74752} a^{15} - \frac{489}{149504} a^{14} + \frac{1129}{149504} a^{13} - \frac{1121}{149504} a^{12} + \frac{1227}{149504} a^{11} - \frac{325}{18688} a^{10} - \frac{5275}{149504} a^{9} + \frac{11249}{299008} a^{8} - \frac{1293}{149504} a^{7} + \frac{9429}{149504} a^{6} - \frac{1601}{74752} a^{5} - \frac{7}{512} a^{4} + \frac{7687}{18688} a^{3} + \frac{1317}{4672} a^{2} - \frac{877}{2336} a - \frac{139}{292}$, $\frac{1}{187053819202271225179713893144930110658660391657472} a^{17} - \frac{262914538340703314915712405454371618401932027}{187053819202271225179713893144930110658660391657472} a^{16} - \frac{1035130992645646940283306078509434380981568731}{93526909601135612589856946572465055329330195828736} a^{15} - \frac{5437925428325510133950690796427872775913031329}{11690863700141951573732118321558131916166274478592} a^{14} + \frac{27464344727837513394171010180647939178908681185}{5845431850070975786866059160779065958083137239296} a^{13} - \frac{144560525943466309655660804619296228642842184943}{46763454800567806294928473286232527664665097914368} a^{12} + \frac{1770206212945586868387011829996888764880922851131}{93526909601135612589856946572465055329330195828736} a^{11} - \frac{385258862852770409813747657214545991085397112515}{93526909601135612589856946572465055329330195828736} a^{10} - \frac{2968220765732072327590071064410043292558401202165}{187053819202271225179713893144930110658660391657472} a^{9} - \frac{9091716876331451915878825679266183574337952222497}{187053819202271225179713893144930110658660391657472} a^{8} - \frac{41130436932920617413061808891715412287226132897}{365339490629435986679128697548691622380196077456} a^{7} - \frac{3818702870861018806372741553298974278791824426245}{93526909601135612589856946572465055329330195828736} a^{6} - \frac{3809364620327737799791313839499537175660550407143}{46763454800567806294928473286232527664665097914368} a^{5} + \frac{2048174224579188856634505380090278376338817620823}{23381727400283903147464236643116263832332548957184} a^{4} - \frac{778856016106423354871798451172591796447787173613}{11690863700141951573732118321558131916166274478592} a^{3} - \frac{621134373117176197314301248176395924596805914205}{2922715925035487893433029580389532979041568619648} a^{2} + \frac{195320142920815560460312584913799832742773031091}{1461357962517743946716514790194766489520784309824} a + \frac{34424878501393203038904488997949427542193293405}{182669745314717993339564348774345811190098038728}$

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

Class group and class number

$C_{12}\times C_{252}\times C_{1888488}$, which has order $5710787712$ (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:  \( 5546046730.2947445 \) (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

\(\Q(\sqrt{-23}) \), 3.3.16129.1, 3.3.1016.1, 6.0.3165179847047.1, 6.0.12559458752.3, 9.9.272832440404737536.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$127$127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.1$x^{3} - 127$$3$$1$$2$$C_3$$[\ ]_{3}$
127.6.5.1$x^{6} - 127$$6$$1$$5$$C_6$$[\ ]_{6}$
127.6.5.1$x^{6} - 127$$6$$1$$5$$C_6$$[\ ]_{6}$