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Properties

Label 18.0.32248627200...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{9}\cdot 5^{12}$
Root discriminant $13.78$
Ramified primes $2, 3, 5$
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![1, -3, 5, 0, -6, -2, 10, -2, 11, -33, 17, -6, 12, 0, 0, -4, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + x^16 - 4*x^15 + 12*x^12 - 6*x^11 + 17*x^10 - 33*x^9 + 11*x^8 - 2*x^7 + 10*x^6 - 2*x^5 - 6*x^4 + 5*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^18 - x^17 + x^16 - 4*x^15 + 12*x^12 - 6*x^11 + 17*x^10 - 33*x^9 + 11*x^8 - 2*x^7 + 10*x^6 - 2*x^5 - 6*x^4 + 5*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + x^{16} - 4 x^{15} + 12 x^{12} - 6 x^{11} + 17 x^{10} - 33 x^{9} + 11 x^{8} - 2 x^{7} + 10 x^{6} - 2 x^{5} - 6 x^{4} + 5 x^{2} - 3 x + 1 \)

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:  \(-322486272000000000000=-\,2^{26}\cdot 3^{9}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.78$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{11} - \frac{1}{2} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6}$, $\frac{1}{30} a^{15} - \frac{1}{15} a^{14} + \frac{1}{30} a^{13} + \frac{1}{30} a^{12} - \frac{2}{5} a^{11} + \frac{7}{30} a^{10} + \frac{1}{30} a^{9} - \frac{1}{3} a^{8} - \frac{1}{5} a^{7} - \frac{1}{6} a^{6} - \frac{7}{30} a^{5} - \frac{2}{5} a^{4} + \frac{11}{30} a^{3} - \frac{13}{30} a^{2} + \frac{1}{5} a - \frac{1}{10}$, $\frac{1}{150} a^{16} + \frac{2}{25} a^{14} - \frac{2}{25} a^{13} + \frac{1}{15} a^{12} + \frac{4}{75} a^{11} + \frac{2}{5} a^{10} - \frac{3}{25} a^{9} + \frac{23}{50} a^{8} + \frac{34}{75} a^{7} - \frac{31}{75} a^{6} - \frac{23}{75} a^{5} - \frac{3}{25} a^{4} - \frac{6}{25} a^{3} + \frac{1}{3} a^{2} + \frac{17}{75} a - \frac{1}{150}$, $\frac{1}{7950} a^{17} - \frac{1}{1590} a^{16} + \frac{127}{7950} a^{15} + \frac{274}{3975} a^{14} - \frac{13}{265} a^{13} + \frac{23}{7950} a^{12} - \frac{129}{530} a^{11} - \frac{2813}{7950} a^{10} - \frac{1388}{3975} a^{9} + \frac{2273}{7950} a^{8} + \frac{3533}{7950} a^{7} - \frac{1361}{7950} a^{6} - \frac{2443}{7950} a^{5} + \frac{1387}{3975} a^{4} - \frac{188}{795} a^{3} + \frac{3439}{7950} a^{2} - \frac{1178}{3975} a - \frac{49}{795}$

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:  \( -\frac{434}{1325} a^{17} + \frac{1369}{3975} a^{16} - \frac{528}{1325} a^{15} + \frac{4972}{3975} a^{14} - \frac{808}{3975} a^{13} + \frac{88}{1325} a^{12} - \frac{14348}{3975} a^{11} + \frac{11356}{3975} a^{10} - \frac{4438}{795} a^{9} + \frac{1682}{159} a^{8} - \frac{25324}{3975} a^{7} - \frac{912}{1325} a^{6} - \frac{5356}{1325} a^{5} + \frac{120}{53} a^{4} + \frac{11456}{3975} a^{3} + \frac{2512}{3975} a^{2} - \frac{9562}{3975} a + \frac{6061}{3975} \) (order $6$)
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:  \( 2294.8992992867607 \)
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

\(\Q(\sqrt{-3}) \), 3.1.200.1, 3.1.300.1 x3, 6.0.1080000.2, 6.0.270000.1, 9.1.3456000000.1 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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.22.79$x^{12} + 2 x^{10} + 4 x^{8} + 4 x^{6} + 4 x^{4} + 4$$6$$2$$22$$D_6$$[3]_{3}^{2}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$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}$