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Properties

Label 18.6.11480578918...8688.2
Degree $18$
Signature $[6, 6]$
Discriminant $2^{6}\cdot 3^{33}\cdot 19^{9}$
Root discriminant $41.16$
Ramified primes $2, 3, 19$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T585

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-71, 1428, -7329, 16638, -17625, 4713, 7119, -7284, 4803, -4952, 3963, -2130, 912, -177, 3, -9, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 12*x^16 - 9*x^15 + 3*x^14 - 177*x^13 + 912*x^12 - 2130*x^11 + 3963*x^10 - 4952*x^9 + 4803*x^8 - 7284*x^7 + 7119*x^6 + 4713*x^5 - 17625*x^4 + 16638*x^3 - 7329*x^2 + 1428*x - 71)
 
gp: K = bnfinit(x^18 - 6*x^17 + 12*x^16 - 9*x^15 + 3*x^14 - 177*x^13 + 912*x^12 - 2130*x^11 + 3963*x^10 - 4952*x^9 + 4803*x^8 - 7284*x^7 + 7119*x^6 + 4713*x^5 - 17625*x^4 + 16638*x^3 - 7329*x^2 + 1428*x - 71, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 12 x^{16} - 9 x^{15} + 3 x^{14} - 177 x^{13} + 912 x^{12} - 2130 x^{11} + 3963 x^{10} - 4952 x^{9} + 4803 x^{8} - 7284 x^{7} + 7119 x^{6} + 4713 x^{5} - 17625 x^{4} + 16638 x^{3} - 7329 x^{2} + 1428 x - 71 \)

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:  \(114805789186292807734226138688=2^{6}\cdot 3^{33}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.16$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{12} + \frac{1}{9} a^{9} - \frac{4}{9} a^{6} - \frac{4}{9} a^{3} - \frac{4}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{13} + \frac{1}{9} a^{10} - \frac{4}{9} a^{7} - \frac{4}{9} a^{4} - \frac{4}{9} a$, $\frac{1}{156285519263846781004827} a^{17} + \frac{109616496153251356735}{156285519263846781004827} a^{16} + \frac{465848756054042790584}{52095173087948927001609} a^{15} - \frac{713446340535598705943}{156285519263846781004827} a^{14} + \frac{11782650242171757718174}{156285519263846781004827} a^{13} + \frac{696425067834406304354}{52095173087948927001609} a^{12} + \frac{13971162266124558654751}{156285519263846781004827} a^{11} + \frac{13975937429182206111205}{156285519263846781004827} a^{10} + \frac{3816292762971983194324}{52095173087948927001609} a^{9} - \frac{29694134562267085285072}{156285519263846781004827} a^{8} + \frac{72931747741345676493566}{156285519263846781004827} a^{7} + \frac{16329594737026994421763}{52095173087948927001609} a^{6} + \frac{26698833764013304042328}{156285519263846781004827} a^{5} - \frac{43343256799868640851410}{156285519263846781004827} a^{4} + \frac{13891198173729755862295}{52095173087948927001609} a^{3} + \frac{73652603622684127294922}{156285519263846781004827} a^{2} + \frac{16727145576527749401656}{156285519263846781004827} a + \frac{19073334595727445954128}{52095173087948927001609}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 26560913.5363 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T585:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 13824
The 96 conjugacy class representatives for t18n585 are not computed
Character table for t18n585 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.5609891727441.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
3Data not computed
$19$19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.6.5.1$x^{6} - 304$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.4.2$x^{6} - 19 x^{3} + 722$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$