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Properties

Label 18.6.54226471004...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{18}\cdot 3^{25}\cdot 5^{12}$
Root discriminant $26.90$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -96, 24, -288, 240, 516, -888, 1044, -1347, 1090, -618, 324, -147, 36, -15, 12, 6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 6*x^16 + 12*x^15 - 15*x^14 + 36*x^13 - 147*x^12 + 324*x^11 - 618*x^10 + 1090*x^9 - 1347*x^8 + 1044*x^7 - 888*x^6 + 516*x^5 + 240*x^4 - 288*x^3 + 24*x^2 - 96*x + 64)
 
gp: K = bnfinit(x^18 - 6*x^17 + 6*x^16 + 12*x^15 - 15*x^14 + 36*x^13 - 147*x^12 + 324*x^11 - 618*x^10 + 1090*x^9 - 1347*x^8 + 1044*x^7 - 888*x^6 + 516*x^5 + 240*x^4 - 288*x^3 + 24*x^2 - 96*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 6 x^{16} + 12 x^{15} - 15 x^{14} + 36 x^{13} - 147 x^{12} + 324 x^{11} - 618 x^{10} + 1090 x^{9} - 1347 x^{8} + 1044 x^{7} - 888 x^{6} + 516 x^{5} + 240 x^{4} - 288 x^{3} + 24 x^{2} - 96 x + 64 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(54226471004352000000000000=2^{18}\cdot 3^{25}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.90$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{20} a^{12} - \frac{3}{20} a^{10} - \frac{1}{5} a^{8} + \frac{3}{20} a^{6} - \frac{9}{20} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{20} a^{13} - \frac{3}{20} a^{11} - \frac{1}{5} a^{9} + \frac{3}{20} a^{7} - \frac{9}{20} a^{5} - \frac{2}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{20} a^{14} - \frac{3}{20} a^{10} + \frac{1}{20} a^{8} - \frac{1}{4} a^{4} + \frac{1}{10} a^{2} + \frac{2}{5}$, $\frac{1}{40} a^{15} - \frac{1}{40} a^{14} - \frac{1}{40} a^{13} - \frac{1}{40} a^{12} - \frac{1}{10} a^{10} - \frac{1}{8} a^{9} + \frac{3}{40} a^{8} + \frac{7}{40} a^{7} - \frac{3}{40} a^{6} + \frac{7}{20} a^{5} + \frac{7}{20} a^{4} - \frac{1}{2} a^{3} - \frac{1}{10} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{80} a^{16} - \frac{1}{80} a^{15} - \frac{1}{80} a^{14} + \frac{1}{80} a^{13} + \frac{1}{8} a^{11} + \frac{3}{16} a^{10} - \frac{1}{16} a^{9} + \frac{7}{80} a^{8} - \frac{17}{80} a^{7} - \frac{3}{40} a^{6} - \frac{3}{10} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{2}{5} a^{2} + \frac{1}{10} a$, $\frac{1}{978423513875572640} a^{17} - \frac{5595808353566917}{978423513875572640} a^{16} - \frac{736352582274521}{978423513875572640} a^{15} - \frac{10394694656543667}{978423513875572640} a^{14} - \frac{380437584677928}{30575734808611645} a^{13} - \frac{6000537655107463}{489211756937786320} a^{12} - \frac{186491373124184577}{978423513875572640} a^{11} + \frac{18056452618417787}{195684702775114528} a^{10} + \frac{107522084272602111}{978423513875572640} a^{9} - \frac{229653119070913309}{978423513875572640} a^{8} + \frac{14458852455434433}{489211756937786320} a^{7} + \frac{21668089681363033}{122302939234446580} a^{6} - \frac{40327198637103849}{244605878468893160} a^{5} + \frac{59455305338498643}{122302939234446580} a^{4} - \frac{4889029537938375}{24460587846889316} a^{3} - \frac{17233030004832603}{122302939234446580} a^{2} + \frac{105550035880253}{353476702989730} a + \frac{13142313009412571}{30575734808611645}$

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:  $11$
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:  \( 14377045.320330542 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

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

\(\Q(\sqrt{3}) \), 3.1.675.1, 3.3.2700.1 x3, 6.2.87480000.1, 6.6.87480000.1, 9.3.531441000000.1 x3

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$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}$
3Data not computed
$5$5.6.4.1$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}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$