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Properties

Label 18.6.86596161341...7136.2
Degree $18$
Signature $[6, 6]$
Discriminant $2^{18}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}$
Root discriminant $67.59$
Ramified primes $2, 3, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-283207, 512050, -233624, 200028, -204442, -57922, 66773, -15000, 37846, -18830, 4063, -668, -514, 130, -29, -8, 13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 13*x^16 - 8*x^15 - 29*x^14 + 130*x^13 - 514*x^12 - 668*x^11 + 4063*x^10 - 18830*x^9 + 37846*x^8 - 15000*x^7 + 66773*x^6 - 57922*x^5 - 204442*x^4 + 200028*x^3 - 233624*x^2 + 512050*x - 283207)
 
gp: K = bnfinit(x^18 - 4*x^17 + 13*x^16 - 8*x^15 - 29*x^14 + 130*x^13 - 514*x^12 - 668*x^11 + 4063*x^10 - 18830*x^9 + 37846*x^8 - 15000*x^7 + 66773*x^6 - 57922*x^5 - 204442*x^4 + 200028*x^3 - 233624*x^2 + 512050*x - 283207, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 13 x^{16} - 8 x^{15} - 29 x^{14} + 130 x^{13} - 514 x^{12} - 668 x^{11} + 4063 x^{10} - 18830 x^{9} + 37846 x^{8} - 15000 x^{7} + 66773 x^{6} - 57922 x^{5} - 204442 x^{4} + 200028 x^{3} - 233624 x^{2} + 512050 x - 283207 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(865961613414533621361938102747136=2^{18}\cdot 3^{6}\cdot 7^{12}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.59$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} - \frac{5}{16} a^{6} - \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{7}{16} a^{3} + \frac{5}{16} a^{2} + \frac{5}{16} a + \frac{1}{16}$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{7}{32} a^{9} - \frac{1}{64} a^{8} + \frac{1}{16} a^{7} - \frac{3}{64} a^{6} + \frac{3}{16} a^{5} + \frac{25}{64} a^{4} - \frac{3}{16} a^{3} - \frac{15}{32} a^{2} - \frac{13}{32} a + \frac{1}{64}$, $\frac{1}{128} a^{15} - \frac{1}{128} a^{14} - \frac{1}{128} a^{13} - \frac{7}{128} a^{12} - \frac{1}{8} a^{11} + \frac{3}{64} a^{10} + \frac{1}{128} a^{9} + \frac{21}{128} a^{8} - \frac{23}{128} a^{7} + \frac{31}{128} a^{6} - \frac{51}{128} a^{5} + \frac{59}{128} a^{4} - \frac{25}{64} a^{3} + \frac{1}{32} a^{2} - \frac{53}{128} a - \frac{49}{128}$, $\frac{1}{512} a^{16} - \frac{1}{256} a^{15} - \frac{1}{128} a^{14} + \frac{5}{256} a^{13} - \frac{21}{512} a^{12} - \frac{21}{256} a^{11} - \frac{5}{512} a^{10} + \frac{31}{128} a^{9} + \frac{3}{64} a^{8} - \frac{21}{256} a^{7} - \frac{59}{256} a^{6} + \frac{79}{256} a^{5} + \frac{207}{512} a^{4} + \frac{43}{256} a^{3} - \frac{177}{512} a^{2} + \frac{39}{128} a + \frac{29}{512}$, $\frac{1}{1244165818366487788148703073523597997056} a^{17} - \frac{764920880906183669167793716923443397}{1244165818366487788148703073523597997056} a^{16} - \frac{45465838369436495692055797775875899}{21451134799422203243943156440062034432} a^{15} - \frac{709361973329621110356343711279699261}{622082909183243894074351536761798998528} a^{14} - \frac{30020140462495669088852050236911213203}{1244165818366487788148703073523597997056} a^{13} - \frac{50751141235111686175968928350303290587}{1244165818366487788148703073523597997056} a^{12} - \frac{116735980029814010203797748611671140231}{1244165818366487788148703073523597997056} a^{11} + \frac{30356951403686714149498238364621277835}{1244165818366487788148703073523597997056} a^{10} + \frac{44276349909531477479832615588206350569}{311041454591621947037175768380899499264} a^{9} + \frac{145877964056233217503213803837566957455}{622082909183243894074351536761798998528} a^{8} - \frac{347424518806298943943383247092555761}{766111957122221544426541301430786944} a^{7} - \frac{29054119377672197245884518309488603097}{77760363647905486759293942095224874816} a^{6} - \frac{147592357247631283323805317321159523755}{1244165818366487788148703073523597997056} a^{5} + \frac{88244817995834937047235059537491371103}{177737974052355398306957581931942571008} a^{4} - \frac{9738803968538976268354425096530159669}{177737974052355398306957581931942571008} a^{3} + \frac{47457854917568580140134112668966140751}{1244165818366487788148703073523597997056} a^{2} - \frac{418649656017098154068698760573531609783}{1244165818366487788148703073523597997056} a + \frac{438525291433222223315242742769040604569}{1244165818366487788148703073523597997056}$

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:  $11$
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:  \( 6946204917.89 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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

Sibling fields

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 R R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
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}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.6.5.2$x^{6} + 246$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
41.6.4.1$x^{6} + 1435 x^{3} + 2904768$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$