↑

Properties

Label 18.0.28360588773...2403.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 331^{8}$
Root discriminant $22.83$
Ramified primes $3, 331$
Class number $4$
Class group $[4]$
Galois group $S_3\times S_4$ (as 18T69)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![117, 18, -33, -78, -203, 306, 23, -317, 295, -38, -123, 149, -67, 9, 25, -21, 13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 13*x^16 - 21*x^15 + 25*x^14 + 9*x^13 - 67*x^12 + 149*x^11 - 123*x^10 - 38*x^9 + 295*x^8 - 317*x^7 + 23*x^6 + 306*x^5 - 203*x^4 - 78*x^3 - 33*x^2 + 18*x + 117)
 
gp: K = bnfinit(x^18 - 4*x^17 + 13*x^16 - 21*x^15 + 25*x^14 + 9*x^13 - 67*x^12 + 149*x^11 - 123*x^10 - 38*x^9 + 295*x^8 - 317*x^7 + 23*x^6 + 306*x^5 - 203*x^4 - 78*x^3 - 33*x^2 + 18*x + 117, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 13 x^{16} - 21 x^{15} + 25 x^{14} + 9 x^{13} - 67 x^{12} + 149 x^{11} - 123 x^{10} - 38 x^{9} + 295 x^{8} - 317 x^{7} + 23 x^{6} + 306 x^{5} - 203 x^{4} - 78 x^{3} - 33 x^{2} + 18 x + 117 \)

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:  \(-2836058877396286768262403=-\,3^{9}\cdot 331^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.83$
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:  $3, 331$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{77454} a^{16} + \frac{3175}{38727} a^{15} + \frac{2909}{38727} a^{14} + \frac{11203}{25818} a^{13} + \frac{30049}{77454} a^{12} + \frac{61}{993} a^{11} - \frac{18386}{38727} a^{10} - \frac{19639}{77454} a^{9} - \frac{5329}{25818} a^{8} + \frac{30109}{77454} a^{7} - \frac{2449}{38727} a^{6} - \frac{3394}{38727} a^{5} + \frac{599}{5958} a^{4} - \frac{265}{12909} a^{3} - \frac{15325}{38727} a^{2} + \frac{5749}{12909} a - \frac{433}{1986}$, $\frac{1}{68283809349444} a^{17} - \frac{72674227}{68283809349444} a^{16} + \frac{4098169345841}{34141904674722} a^{15} - \frac{87267220283}{583622302132} a^{14} - \frac{345710772421}{2626300359594} a^{13} - \frac{8340925433987}{22761269783148} a^{12} - \frac{1163548365670}{17070952337361} a^{11} - \frac{11788159148347}{68283809349444} a^{10} - \frac{4772616542623}{11380634891574} a^{9} + \frac{948393688831}{17070952337361} a^{8} - \frac{33311840621753}{68283809349444} a^{7} + \frac{642717662423}{2626300359594} a^{6} - \frac{24589479602971}{68283809349444} a^{5} - \frac{1939804774099}{22761269783148} a^{4} + \frac{7487753669938}{17070952337361} a^{3} + \frac{516308490505}{11380634891574} a^{2} - \frac{1279316905213}{22761269783148} a + \frac{100708400235}{583622302132}$

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

Class group and class number

$C_{4}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 47915.6179638 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times S_4$ (as 18T69):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 15 conjugacy class representatives for $S_3\times S_4$
Character table for $S_3\times S_4$

Intermediate fields

3.3.993.1, 3.1.331.1, 6.0.2958147.1, 9.3.324097543467.1

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
331Data not computed