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Properties

Label 18.0.18075490334...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 5^{12}$
Root discriminant $25.30$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3^2$ (as 18T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20, -120, 120, 72, 453, -90, 48, -516, 1134, -1198, 1029, -732, 465, -282, 162, -72, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 24*x^16 - 72*x^15 + 162*x^14 - 282*x^13 + 465*x^12 - 732*x^11 + 1029*x^10 - 1198*x^9 + 1134*x^8 - 516*x^7 + 48*x^6 - 90*x^5 + 453*x^4 + 72*x^3 + 120*x^2 - 120*x + 20)
 
gp: K = bnfinit(x^18 - 6*x^17 + 24*x^16 - 72*x^15 + 162*x^14 - 282*x^13 + 465*x^12 - 732*x^11 + 1029*x^10 - 1198*x^9 + 1134*x^8 - 516*x^7 + 48*x^6 - 90*x^5 + 453*x^4 + 72*x^3 + 120*x^2 - 120*x + 20, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 24 x^{16} - 72 x^{15} + 162 x^{14} - 282 x^{13} + 465 x^{12} - 732 x^{11} + 1029 x^{10} - 1198 x^{9} + 1134 x^{8} - 516 x^{7} + 48 x^{6} - 90 x^{5} + 453 x^{4} + 72 x^{3} + 120 x^{2} - 120 x + 20 \)

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:  \(-18075490334784000000000000=-\,2^{18}\cdot 3^{24}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.30$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{3}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{80} a^{14} + \frac{1}{20} a^{13} - \frac{1}{40} a^{12} + \frac{1}{80} a^{10} + \frac{9}{40} a^{9} + \frac{13}{80} a^{8} + \frac{3}{40} a^{7} + \frac{1}{40} a^{6} - \frac{11}{40} a^{5} - \frac{23}{80} a^{4} + \frac{3}{40} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{160} a^{15} - \frac{1}{160} a^{14} + \frac{1}{20} a^{13} - \frac{1}{16} a^{12} + \frac{1}{160} a^{11} + \frac{13}{160} a^{10} + \frac{33}{160} a^{9} + \frac{21}{160} a^{8} + \frac{11}{80} a^{7} + \frac{1}{20} a^{6} - \frac{73}{160} a^{5} + \frac{21}{160} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{1600} a^{16} - \frac{1}{800} a^{15} - \frac{3}{1600} a^{14} + \frac{17}{800} a^{13} - \frac{1}{320} a^{12} - \frac{7}{400} a^{11} - \frac{3}{25} a^{10} - \frac{11}{200} a^{9} + \frac{57}{320} a^{8} - \frac{93}{800} a^{7} + \frac{43}{320} a^{6} - \frac{321}{800} a^{5} - \frac{5}{64} a^{4} + \frac{91}{200} a^{3} + \frac{33}{80} a^{2} + \frac{3}{8} a - \frac{21}{80}$, $\frac{1}{4140256374400} a^{17} + \frac{1281184119}{4140256374400} a^{16} - \frac{536414349}{828051274880} a^{15} + \frac{10538938931}{4140256374400} a^{14} - \frac{9882325451}{4140256374400} a^{13} - \frac{198546744353}{4140256374400} a^{12} + \frac{8575271883}{103506409360} a^{11} + \frac{17731350787}{207012818720} a^{10} + \frac{12689530589}{78118044800} a^{9} + \frac{685278657279}{4140256374400} a^{8} + \frac{239460883869}{4140256374400} a^{7} + \frac{897525705293}{4140256374400} a^{6} + \frac{1878412251773}{4140256374400} a^{5} + \frac{1914262278023}{4140256374400} a^{4} - \frac{72176433423}{1035064093600} a^{3} - \frac{43215921237}{207012818720} a^{2} + \frac{2971053049}{207012818720} a + \frac{45348441879}{207012818720}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1315307643}{1562360896} a^{17} + \frac{7521227991}{1562360896} a^{16} - \frac{29446607019}{1562360896} a^{15} + \frac{86395513041}{1562360896} a^{14} - \frac{188700989199}{1562360896} a^{13} + \frac{317650854795}{1562360896} a^{12} - \frac{260962590783}{781180448} a^{11} + \frac{407756093727}{781180448} a^{10} - \frac{1123421842417}{1562360896} a^{9} + \frac{1258874723913}{1562360896} a^{8} - \frac{1136607763323}{1562360896} a^{7} + \frac{358785288213}{1562360896} a^{6} + \frac{36834390063}{1562360896} a^{5} + \frac{131551297425}{1562360896} a^{4} - \frac{140551176435}{390590224} a^{3} - \frac{62931269205}{390590224} a^{2} - \frac{56936543625}{390590224} a + \frac{23595617093}{390590224} \) (order $4$)
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:  \( 2143781.2150954185 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.675.1, 3.3.2700.1, 6.0.1166400.3 x2, 6.0.29160000.1, 6.0.29160000.2, 9.3.531441000000.1

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
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 R R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
3Data not computed
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$