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Properties

Label 18.0.33733203082...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{22}\cdot 3^{30}\cdot 5^{8}$
Root discriminant $29.77$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, 0, 2304, 0, 3456, 0, 3888, 0, 2592, 0, 1008, 0, 177, 0, 18, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 + 18*x^14 + 177*x^12 + 1008*x^10 + 2592*x^8 + 3888*x^6 + 3456*x^4 + 2304*x^2 + 256)
 
gp: K = bnfinit(x^18 + 9*x^16 + 18*x^14 + 177*x^12 + 1008*x^10 + 2592*x^8 + 3888*x^6 + 3456*x^4 + 2304*x^2 + 256, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} + 18 x^{14} + 177 x^{12} + 1008 x^{10} + 2592 x^{8} + 3888 x^{6} + 3456 x^{4} + 2304 x^{2} + 256 \)

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:  \(-337332030823872921600000000=-\,2^{22}\cdot 3^{30}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.77$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{6} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{12} a^{9} + \frac{1}{12} a^{7} - \frac{1}{2} a^{5} + \frac{1}{12} a^{3} + \frac{1}{3} a$, $\frac{1}{36} a^{10} - \frac{1}{36} a^{8} + \frac{1}{9} a^{6} + \frac{13}{36} a^{4} + \frac{7}{18} a^{2} + \frac{4}{9}$, $\frac{1}{72} a^{11} - \frac{1}{72} a^{9} - \frac{1}{9} a^{7} + \frac{13}{72} a^{5} + \frac{7}{36} a^{3} - \frac{4}{9} a$, $\frac{1}{2160} a^{12} - \frac{1}{80} a^{10} - \frac{7}{360} a^{8} + \frac{61}{432} a^{6} + \frac{1}{20} a^{4} + \frac{7}{18} a^{2} + \frac{37}{135}$, $\frac{1}{4320} a^{13} - \frac{1}{160} a^{11} - \frac{1}{72} a^{10} - \frac{7}{720} a^{9} - \frac{5}{72} a^{8} + \frac{61}{864} a^{7} + \frac{1}{36} a^{6} - \frac{19}{40} a^{5} + \frac{23}{72} a^{4} + \frac{7}{36} a^{3} - \frac{5}{18} a^{2} + \frac{37}{270} a + \frac{1}{9}$, $\frac{1}{12960} a^{14} + \frac{1}{12960} a^{12} - \frac{13}{2160} a^{10} - \frac{151}{12960} a^{8} - \frac{269}{1620} a^{6} + \frac{11}{540} a^{4} - \frac{203}{810} a^{2} - \frac{157}{405}$, $\frac{1}{25920} a^{15} + \frac{1}{25920} a^{13} - \frac{13}{4320} a^{11} - \frac{151}{25920} a^{9} - \frac{269}{3240} a^{7} + \frac{11}{1080} a^{5} - \frac{203}{1620} a^{3} + \frac{124}{405} a$, $\frac{1}{155520} a^{16} + \frac{1}{31104} a^{14} - \frac{1}{77760} a^{12} - \frac{247}{155520} a^{10} - \frac{613}{7776} a^{8} - \frac{1}{6} a^{7} - \frac{499}{9720} a^{6} - \frac{1}{2} a^{5} + \frac{121}{2430} a^{4} + \frac{433}{1215} a^{2} - \frac{1}{6} a + \frac{311}{1215}$, $\frac{1}{155520} a^{17} - \frac{1}{155520} a^{15} - \frac{1}{19440} a^{13} + \frac{221}{155520} a^{11} + \frac{803}{77760} a^{9} + \frac{559}{4860} a^{7} - \frac{895}{1944} a^{5} - \frac{1057}{2430} a^{3} + \frac{344}{1215} a$

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:  \( -\frac{1}{576} a^{17} - \frac{31}{2160} a^{15} - \frac{61}{2880} a^{13} - \frac{19}{64} a^{11} - \frac{13373}{8640} a^{9} - \frac{827}{240} a^{7} - \frac{871}{180} a^{5} - \frac{1199}{270} a^{3} - \frac{146}{45} a \) (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:  \( 1629603.9494643004 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.243.1, 6.0.3779136.2, 9.3.143489070000.1

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

Sibling fields

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.12.16.3$x^{12} - 30 x^{10} - 5 x^{8} + 19 x^{4} + 30 x^{2} + 1$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
$3$3.6.10.2$x^{6} + 9$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.6.10.2$x^{6} + 9$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
3.6.10.2$x^{6} + 9$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$