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Properties

Label 18.2.90838095035...6864.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{20}\cdot 59^{9}$
Root discriminant $16.59$
Ramified primes $2, 59$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^2:D_9$ (as 18T67)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -14, -26, 19, 45, 4, -14, -61, 85, -45, 61, -57, 9, 15, -3, 0, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 3*x^14 + 15*x^13 + 9*x^12 - 57*x^11 + 61*x^10 - 45*x^9 + 85*x^8 - 61*x^7 - 14*x^6 + 4*x^5 + 45*x^4 + 19*x^3 - 26*x^2 - 14*x - 2)
 
gp: K = bnfinit(x^18 - x^17 - 3*x^14 + 15*x^13 + 9*x^12 - 57*x^11 + 61*x^10 - 45*x^9 + 85*x^8 - 61*x^7 - 14*x^6 + 4*x^5 + 45*x^4 + 19*x^3 - 26*x^2 - 14*x - 2, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 3 x^{14} + 15 x^{13} + 9 x^{12} - 57 x^{11} + 61 x^{10} - 45 x^{9} + 85 x^{8} - 61 x^{7} - 14 x^{6} + 4 x^{5} + 45 x^{4} + 19 x^{3} - 26 x^{2} - 14 x - 2 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9083809503541921316864=2^{20}\cdot 59^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.59$
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:  $2, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{18727794399627356} a^{17} - \frac{690968217060193}{4681948599906839} a^{16} - \frac{552058246590284}{4681948599906839} a^{15} - \frac{2162962204253396}{4681948599906839} a^{14} - \frac{5883080667912539}{18727794399627356} a^{13} - \frac{1132118861656679}{4681948599906839} a^{12} - \frac{3288846600232183}{18727794399627356} a^{11} - \frac{13049258903474}{4681948599906839} a^{10} + \frac{3619563954177577}{18727794399627356} a^{9} + \frac{1617699278913059}{4681948599906839} a^{8} + \frac{579874913993641}{18727794399627356} a^{7} + \frac{593779653106633}{4681948599906839} a^{6} + \frac{3314367777109289}{9363897199813678} a^{5} + \frac{4172288840307843}{9363897199813678} a^{4} + \frac{7299918092025111}{18727794399627356} a^{3} + \frac{1054438654108807}{9363897199813678} a^{2} + \frac{1605920304783297}{4681948599906839} a + \frac{1264294569185695}{9363897199813678}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 17860.4330384 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:D_9$ (as 18T67):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times C_2^2:D_9$
Character table for $C_2\times C_2^2:D_9$

Intermediate fields

3.1.59.1, 6.2.3286064.1, 9.1.775511104.1

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

Sibling fields

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 $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ R

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.16.9$x^{12} + 7 x^{10} + 4 x^{8} + 3 x^{6} - 4 x^{4} - x^{2} - 5$$6$$2$$16$$(C_6\times C_2):C_2$$[2, 2]_{3}^{2}$
59Data not computed