↑

Properties

Label 18.0.68313613530...4848.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 163^{10}$
Root discriminant $58.69$
Ramified primes $2, 3, 163$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19363, -68391, 121521, -280422, 739367, -1365187, 1685627, -1438355, 844354, -311864, 45543, 16857, -10303, 2007, -123, 5, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 7*x^16 + 5*x^15 - 123*x^14 + 2007*x^13 - 10303*x^12 + 16857*x^11 + 45543*x^10 - 311864*x^9 + 844354*x^8 - 1438355*x^7 + 1685627*x^6 - 1365187*x^5 + 739367*x^4 - 280422*x^3 + 121521*x^2 - 68391*x + 19363)
 
gp: K = bnfinit(x^18 - 6*x^17 + 7*x^16 + 5*x^15 - 123*x^14 + 2007*x^13 - 10303*x^12 + 16857*x^11 + 45543*x^10 - 311864*x^9 + 844354*x^8 - 1438355*x^7 + 1685627*x^6 - 1365187*x^5 + 739367*x^4 - 280422*x^3 + 121521*x^2 - 68391*x + 19363, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 7 x^{16} + 5 x^{15} - 123 x^{14} + 2007 x^{13} - 10303 x^{12} + 16857 x^{11} + 45543 x^{10} - 311864 x^{9} + 844354 x^{8} - 1438355 x^{7} + 1685627 x^{6} - 1365187 x^{5} + 739367 x^{4} - 280422 x^{3} + 121521 x^{2} - 68391 x + 19363 \)

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:  \(-68313613530252822324639135694848=-\,2^{18}\cdot 3^{9}\cdot 163^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.69$
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:  $2, 3, 163$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{7232936414078024364800687284486} a^{17} - \frac{209139205740835534228658344389}{3616468207039012182400343642243} a^{16} + \frac{372189180383925961730725025457}{3616468207039012182400343642243} a^{15} - \frac{898916844377703630161447119555}{7232936414078024364800687284486} a^{14} + \frac{987683303995524676045174913859}{7232936414078024364800687284486} a^{13} - \frac{484413010097609683360012933639}{7232936414078024364800687284486} a^{12} + \frac{2298299638365223827599912370353}{7232936414078024364800687284486} a^{11} + \frac{3167128980904388948183522091701}{7232936414078024364800687284486} a^{10} + \frac{59013929104936916749071822884}{212733423943471304847079037779} a^{9} - \frac{1407056054345553382770936824159}{3616468207039012182400343642243} a^{8} + \frac{3588836602838893971466662761613}{7232936414078024364800687284486} a^{7} - \frac{1820129902386627410600658566543}{7232936414078024364800687284486} a^{6} - \frac{852319428233451477531857865381}{7232936414078024364800687284486} a^{5} + \frac{1675680272002358129442780149417}{7232936414078024364800687284486} a^{4} + \frac{933581945408891616098117551723}{3616468207039012182400343642243} a^{3} - \frac{18358007557638527091705181706}{3616468207039012182400343642243} a^{2} - \frac{1049631898959352882720089273134}{3616468207039012182400343642243} a + \frac{53212083817262512660411910725}{425466847886942609694158075558}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{106162196638896905020}{15580983648847492259} a^{17} + \frac{526714789706723237266}{15580983648847492259} a^{16} - \frac{196078708992008114970}{15580983648847492259} a^{15} - \frac{734564677077314019126}{15580983648847492259} a^{14} + \frac{12295069454705809723744}{15580983648847492259} a^{13} - \frac{200297752502919658061611}{15580983648847492259} a^{12} + \frac{885761170821310038397934}{15580983648847492259} a^{11} - \frac{869600584979944663204310}{15580983648847492259} a^{10} - \frac{337545615683961262781150}{916528449932205427} a^{9} + \frac{27148633111277920927860791}{15580983648847492259} a^{8} - \frac{61441076957859074312888696}{15580983648847492259} a^{7} + \frac{88881859084804636681073582}{15580983648847492259} a^{6} - \frac{86627705791182808539114610}{15580983648847492259} a^{5} + \frac{54947258915451773425186249}{15580983648847492259} a^{4} - \frac{21414207147504007514840398}{15580983648847492259} a^{3} + \frac{7524609636783242255089740}{15580983648847492259} a^{2} - \frac{5084799422095966362762640}{15580983648847492259} a + \frac{116391921529521596088022}{916528449932205427} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 272590163.0199287 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.1304.1, 6.0.717363.1, 6.0.45911232.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$163$$\Q_{163}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{163}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{163}$$x + 4$$1$$1$$0$Trivial$[\ ]$
163.2.1.2$x^{2} + 652$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.2$x^{2} + 652$$2$$1$$1$$C_2$$[\ ]_{2}$
163.2.1.2$x^{2} + 652$$2$$1$$1$$C_2$$[\ ]_{2}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.6.5.5$x^{6} + 10432$$6$$1$$5$$C_6$$[\ ]_{6}$