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Properties

Label 18.4.31529670483...5248.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,2^{39}\cdot 3^{18}\cdot 23^{6}$
Root discriminant $38.31$
Ramified primes $2, 3, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T903

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, 384, 1728, 3568, 2340, 1908, -559, 216, -360, -92, -186, -180, -46, -48, 9, -8, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 6*x^16 - 8*x^15 + 9*x^14 - 48*x^13 - 46*x^12 - 180*x^11 - 186*x^10 - 92*x^9 - 360*x^8 + 216*x^7 - 559*x^6 + 1908*x^5 + 2340*x^4 + 3568*x^3 + 1728*x^2 + 384*x - 64)
 
gp: K = bnfinit(x^18 + 6*x^16 - 8*x^15 + 9*x^14 - 48*x^13 - 46*x^12 - 180*x^11 - 186*x^10 - 92*x^9 - 360*x^8 + 216*x^7 - 559*x^6 + 1908*x^5 + 2340*x^4 + 3568*x^3 + 1728*x^2 + 384*x - 64, 1)
 

Normalized defining polynomial

\( x^{18} + 6 x^{16} - 8 x^{15} + 9 x^{14} - 48 x^{13} - 46 x^{12} - 180 x^{11} - 186 x^{10} - 92 x^{9} - 360 x^{8} + 216 x^{7} - 559 x^{6} + 1908 x^{5} + 2340 x^{4} + 3568 x^{3} + 1728 x^{2} + 384 x - 64 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-31529670483033070306483765248=-\,2^{39}\cdot 3^{18}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.31$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{14} + \frac{1}{16} a^{12} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{16} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1221289650276132479804960} a^{17} + \frac{8262347046035953241373}{305322412569033119951240} a^{16} + \frac{2739429587253525000633}{122128965027613247980496} a^{15} + \frac{9128719784264410014747}{76330603142258279987810} a^{14} - \frac{57329062697123299629247}{1221289650276132479804960} a^{13} - \frac{8580051655283490854423}{305322412569033119951240} a^{12} + \frac{11923507563860555009035}{122128965027613247980496} a^{11} + \frac{5490833394032167727895}{61064482513806623990248} a^{10} - \frac{116024425346294868064533}{610644825138066239902480} a^{9} - \frac{5827586016166268051821}{305322412569033119951240} a^{8} - \frac{11924820080007073232993}{76330603142258279987810} a^{7} - \frac{2682156425065601162491}{15266120628451655997562} a^{6} - \frac{239896442380541183280919}{1221289650276132479804960} a^{5} - \frac{7099771696108021115939}{15266120628451655997562} a^{4} + \frac{5876519785039893058851}{30532241256903311995124} a^{3} - \frac{491165017619949207753}{6637443751500719998940} a^{2} + \frac{26447936067062609018739}{76330603142258279987810} a + \frac{15293228164687920658786}{38165301571129139993905}$

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:  $10$
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:  \( 18057994.161 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T903:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 559872
The 174 conjugacy class representatives for t18n903 are not computed
Character table for t18n903 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 3.1.23.1, 6.2.270848.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.12.30.107$x^{12} - 6 x^{10} - 3 x^{8} + 20 x^{6} + 3 x^{4} - 14 x^{2} + 7$$4$$3$$30$12T134$[2, 2, 2, 3, 7/2, 7/2, 7/2]^{3}$
3Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$