↑

Properties

Label 18.0.73732319655...0464.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 109^{12}$
Root discriminant $45.64$
Ramified primes $2, 109$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, 0, 7232, 0, -14240, 0, -8076, 0, 24616, 0, 5497, 0, -3680, 0, 594, 0, -40, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 40*x^16 + 594*x^14 - 3680*x^12 + 5497*x^10 + 24616*x^8 - 8076*x^6 - 14240*x^4 + 7232*x^2 + 1024)
 
gp: K = bnfinit(x^18 - 40*x^16 + 594*x^14 - 3680*x^12 + 5497*x^10 + 24616*x^8 - 8076*x^6 - 14240*x^4 + 7232*x^2 + 1024, 1)
 

Normalized defining polynomial

\( x^{18} - 40 x^{16} + 594 x^{14} - 3680 x^{12} + 5497 x^{10} + 24616 x^{8} - 8076 x^{6} - 14240 x^{4} + 7232 x^{2} + 1024 \)

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:  \(-737323196555695092066492350464=-\,2^{18}\cdot 109^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.64$
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:  $2, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} - \frac{3}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{80} a^{12} + \frac{1}{40} a^{10} - \frac{1}{16} a^{9} - \frac{3}{80} a^{8} - \frac{1}{8} a^{7} + \frac{3}{40} a^{6} + \frac{1}{16} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{160} a^{13} - \frac{3}{160} a^{11} - \frac{3}{160} a^{9} + \frac{11}{160} a^{7} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{3}{10} a^{3} + \frac{1}{4} a^{2} + \frac{2}{5} a$, $\frac{1}{160} a^{14} - \frac{1}{160} a^{12} + \frac{1}{160} a^{10} - \frac{1}{16} a^{9} + \frac{1}{32} a^{8} - \frac{1}{8} a^{7} - \frac{9}{80} a^{6} + \frac{1}{16} a^{5} + \frac{7}{40} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a - \frac{1}{5}$, $\frac{1}{320} a^{15} - \frac{1}{160} a^{11} + \frac{3}{80} a^{9} + \frac{13}{320} a^{7} + \frac{17}{80} a^{5} + \frac{37}{80} a^{3} - \frac{3}{20} a$, $\frac{1}{36437286686720} a^{16} + \frac{3448086553}{1821864334336} a^{14} - \frac{31790269823}{18218643343360} a^{12} + \frac{100288727643}{4554660835840} a^{10} - \frac{1}{16} a^{9} - \frac{8766802683}{7287457337344} a^{8} - \frac{1}{8} a^{7} - \frac{438402802439}{9109321671680} a^{6} - \frac{3}{16} a^{5} - \frac{760059210663}{9109321671680} a^{4} - \frac{1}{8} a^{3} - \frac{97701565007}{455466083584} a^{2} + \frac{56172277689}{142333151120}$, $\frac{1}{72874573373440} a^{17} + \frac{3448086553}{3643728668672} a^{15} - \frac{31790269823}{36437286686720} a^{13} - \frac{184377574597}{9109321671680} a^{11} - \frac{8766802683}{14574914674688} a^{9} - \frac{2146400615879}{18218643343360} a^{7} - \frac{3037389628583}{18218643343360} a^{5} + \frac{357764518577}{910932167168} a^{3} + \frac{56172277689}{284666302240} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{11881}{90528320} a^{17} - \frac{96619}{18105664} a^{15} + \frac{736173}{9052832} a^{13} - \frac{23996537}{45264160} a^{11} + \frac{88269597}{90528320} a^{9} + \frac{57438081}{18105664} a^{7} - \frac{44402119}{11316040} a^{5} - \frac{77347503}{22632080} a^{3} + \frac{14607663}{5658020} a \) (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:  \( 3674078577.57 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.47524.1 x3, 3.3.11881.1, 6.0.9034122304.1, 6.0.760384.2 x2, 6.0.9034122304.2, 9.3.107334407093824.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.760384.2
Degree 9 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$109$109.9.6.1$x^{9} + 3270 x^{6} + 3552419 x^{3} + 1295029000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
109.9.6.1$x^{9} + 3270 x^{6} + 3552419 x^{3} + 1295029000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$