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Properties

Label 18.0.34542799956...7184.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{30}$
Root discriminant $15.72$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
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![2, 12, 36, 78, 90, 0, -63, 48, 234, 272, 153, 18, -24, -12, 9, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 + 9*x^14 - 12*x^13 - 24*x^12 + 18*x^11 + 153*x^10 + 272*x^9 + 234*x^8 + 48*x^7 - 63*x^6 + 90*x^4 + 78*x^3 + 36*x^2 + 12*x + 2)
 
gp: K = bnfinit(x^18 - 6*x^16 + 9*x^14 - 12*x^13 - 24*x^12 + 18*x^11 + 153*x^10 + 272*x^9 + 234*x^8 + 48*x^7 - 63*x^6 + 90*x^4 + 78*x^3 + 36*x^2 + 12*x + 2, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} + 9 x^{14} - 12 x^{13} - 24 x^{12} + 18 x^{11} + 153 x^{10} + 272 x^{9} + 234 x^{8} + 48 x^{7} - 63 x^{6} + 90 x^{4} + 78 x^{3} + 36 x^{2} + 12 x + 2 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-3454279995636458717184=-\,2^{24}\cdot 3^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.72$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{968} a^{16} + \frac{3}{484} a^{15} + \frac{39}{968} a^{14} - \frac{49}{242} a^{13} + \frac{19}{121} a^{12} + \frac{13}{121} a^{11} + \frac{4}{121} a^{10} + \frac{89}{484} a^{9} - \frac{427}{968} a^{8} + \frac{35}{121} a^{7} + \frac{7}{968} a^{6} + \frac{95}{484} a^{5} + \frac{43}{242} a^{4} - \frac{81}{484} a^{3} - \frac{151}{484} a^{2} - \frac{36}{121} a + \frac{215}{484}$, $\frac{1}{7198395680432} a^{17} - \frac{2410935369}{7198395680432} a^{16} + \frac{733397874885}{7198395680432} a^{15} - \frac{388285872589}{7198395680432} a^{14} - \frac{24876186311}{899799460054} a^{13} + \frac{218595852447}{1799598920108} a^{12} + \frac{27985266449}{163599901828} a^{11} + \frac{405805403687}{3599197840216} a^{10} + \frac{1383515802667}{7198395680432} a^{9} + \frac{2794148616073}{7198395680432} a^{8} + \frac{472039070115}{7198395680432} a^{7} - \frac{248479716789}{654399607312} a^{6} + \frac{901487758661}{3599197840216} a^{5} + \frac{5307868075}{327199803656} a^{4} + \frac{583575493}{7436359174} a^{3} - \frac{527597243059}{3599197840216} a^{2} + \frac{975830843743}{3599197840216} a + \frac{595908011731}{3599197840216}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{2856751839}{725498456} a^{17} + \frac{164335428}{90687307} a^{16} + \frac{16556575941}{725498456} a^{15} - \frac{1906031817}{181374614} a^{14} - \frac{5581393125}{181374614} a^{13} + \frac{5571791808}{90687307} a^{12} + \frac{12060288981}{181374614} a^{11} - \frac{36941792367}{362749228} a^{10} - \frac{403613619179}{725498456} a^{9} - \frac{295391866617}{362749228} a^{8} - \frac{393338878167}{725498456} a^{7} + \frac{24369573303}{362749228} a^{6} + \frac{20184518457}{90687307} a^{5} - \frac{37171894299}{362749228} a^{4} - \frac{112006295847}{362749228} a^{3} - \frac{29703193515}{181374614} a^{2} - \frac{22846859823}{362749228} a - \frac{1505308729}{90687307} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 21530.848473016555 \)
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

\(\Q(\sqrt{-1}) \), 3.1.324.1 x3, 3.1.108.1, 6.0.186624.1, 6.0.419904.2, 9.1.14693280768.1 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
3Data not computed