↑

Properties

Label 18.0.19432680584...7584.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{21}\cdot 7^{12}$
Root discriminant $37.29$
Ramified primes $2, 3, 7$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24576, -73728, 63744, -5376, -192, -36096, 49264, -18192, -8532, 12472, -3756, -1116, 1563, -702, 207, -56, 21, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 21*x^16 - 56*x^15 + 207*x^14 - 702*x^13 + 1563*x^12 - 1116*x^11 - 3756*x^10 + 12472*x^9 - 8532*x^8 - 18192*x^7 + 49264*x^6 - 36096*x^5 - 192*x^4 - 5376*x^3 + 63744*x^2 - 73728*x + 24576)
 
gp: K = bnfinit(x^18 - 6*x^17 + 21*x^16 - 56*x^15 + 207*x^14 - 702*x^13 + 1563*x^12 - 1116*x^11 - 3756*x^10 + 12472*x^9 - 8532*x^8 - 18192*x^7 + 49264*x^6 - 36096*x^5 - 192*x^4 - 5376*x^3 + 63744*x^2 - 73728*x + 24576, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 21 x^{16} - 56 x^{15} + 207 x^{14} - 702 x^{13} + 1563 x^{12} - 1116 x^{11} - 3756 x^{10} + 12472 x^{9} - 8532 x^{8} - 18192 x^{7} + 49264 x^{6} - 36096 x^{5} - 192 x^{4} - 5376 x^{3} + 63744 x^{2} - 73728 x + 24576 \)

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:  \(-19432680584168369411623747584=-\,2^{27}\cdot 3^{21}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.29$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{9} - \frac{1}{16} a^{7} + \frac{3}{32} a^{5} + \frac{1}{8} a^{4} + \frac{1}{16} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{10} - \frac{1}{16} a^{8} + \frac{3}{32} a^{6} + \frac{1}{8} a^{5} + \frac{1}{16} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{7} - \frac{1}{8} a^{6} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{192} a^{12} - \frac{1}{64} a^{11} + \frac{1}{96} a^{9} - \frac{3}{64} a^{8} + \frac{1}{64} a^{7} - \frac{5}{48} a^{6} + \frac{1}{32} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{384} a^{13} - \frac{1}{384} a^{12} - \frac{1}{96} a^{10} - \frac{5}{384} a^{9} - \frac{1}{128} a^{8} - \frac{5}{96} a^{7} - \frac{7}{96} a^{6} + \frac{1}{16} a^{5} + \frac{1}{32} a^{4} + \frac{3}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{768} a^{14} - \frac{1}{768} a^{13} - \frac{1}{192} a^{11} + \frac{7}{768} a^{10} - \frac{1}{256} a^{9} + \frac{1}{192} a^{8} + \frac{5}{192} a^{7} + \frac{1}{64} a^{6} - \frac{15}{64} a^{5} - \frac{1}{4} a^{4} - \frac{1}{16} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{27648} a^{15} - \frac{1}{2304} a^{14} + \frac{11}{9216} a^{13} - \frac{1}{512} a^{12} - \frac{131}{9216} a^{11} - \frac{7}{1152} a^{10} - \frac{67}{9216} a^{9} - \frac{245}{4608} a^{8} + \frac{67}{2304} a^{7} + \frac{103}{1728} a^{6} + \frac{473}{2304} a^{5} + \frac{259}{1152} a^{4} - \frac{61}{192} a^{3} + \frac{67}{288} a^{2} - \frac{1}{2} a - \frac{4}{9}$, $\frac{1}{7520256} a^{16} + \frac{19}{3760128} a^{15} + \frac{433}{835584} a^{14} - \frac{587}{626688} a^{13} - \frac{503}{2506752} a^{12} + \frac{547}{417792} a^{11} - \frac{36707}{2506752} a^{10} + \frac{83}{13056} a^{9} + \frac{18721}{313344} a^{8} - \frac{26629}{940032} a^{7} - \frac{104485}{1880064} a^{6} + \frac{857}{26112} a^{5} + \frac{281}{2304} a^{4} - \frac{83}{19584} a^{3} + \frac{18739}{39168} a^{2} - \frac{127}{612} a + \frac{233}{1224}$, $\frac{1}{1031692305273716736} a^{17} - \frac{3212351885}{128961538159214592} a^{16} + \frac{335512504615}{54299595014406144} a^{15} + \frac{27661909789783}{57316239181873152} a^{14} + \frac{13391589181539}{12736942040416256} a^{13} + \frac{66826276623271}{28658119590936576} a^{12} + \frac{107063721558821}{26453648853172224} a^{11} - \frac{86136648116329}{9049932502401024} a^{10} + \frac{174252413177105}{42987179386404864} a^{9} + \frac{326782546772885}{9920118319939584} a^{8} - \frac{590070880878281}{15171945665789952} a^{7} - \frac{4651895260305625}{128961538159214592} a^{6} - \frac{37222115383303}{447783118608384} a^{5} - \frac{186795035149733}{1343349355825152} a^{4} + \frac{2462970103911079}{5373397423300608} a^{3} - \frac{152345642210821}{2686698711650304} a^{2} + \frac{68821997129657}{167918669478144} a - \frac{28163232783535}{83959334739072}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( -1 \) (order $2$)
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:  \( 674206540.477 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-6}) \), 3.1.1176.1 x3, 3.1.10584.2 x3, 3.1.10584.1 x3, 3.1.216.1 x3, 6.0.33191424.2, 6.0.2688505344.3, 6.0.2688505344.2, 6.0.1119744.1, 9.1.28455140560896.4 x9

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

Sibling fields

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 R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
$3$3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$