↑

Properties

Label 18.2.50181356668...6768.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{26}\cdot 3^{32}\cdot 7^{9}$
Root discriminant $50.77$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-39366, 0, 0, 0, 0, 0, -47952, 0, 0, -7106, 0, 0, -486, 0, 0, -66, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 66*x^15 - 486*x^12 - 7106*x^9 - 47952*x^6 - 39366)
 
gp: K = bnfinit(x^18 - 66*x^15 - 486*x^12 - 7106*x^9 - 47952*x^6 - 39366, 1)
 

Normalized defining polynomial

\( x^{18} - 66 x^{15} - 486 x^{12} - 7106 x^{9} - 47952 x^{6} - 39366 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5018135666827513318018822176768=2^{26}\cdot 3^{32}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.77$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{15} a^{9} + \frac{2}{15} a^{6} - \frac{1}{3} a^{3} - \frac{1}{5}$, $\frac{1}{15} a^{10} + \frac{2}{15} a^{7} - \frac{1}{3} a^{4} - \frac{1}{5} a$, $\frac{1}{45} a^{11} - \frac{1}{15} a^{8} + \frac{22}{45} a^{2}$, $\frac{1}{135} a^{12} - \frac{1}{45} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{67}{135} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{405} a^{13} - \frac{4}{135} a^{10} + \frac{1}{15} a^{7} + \frac{157}{405} a^{4} + \frac{2}{5} a$, $\frac{1}{3645} a^{14} - \frac{1}{1215} a^{13} + \frac{1}{405} a^{12} - \frac{13}{1215} a^{11} + \frac{13}{405} a^{10} - \frac{4}{135} a^{9} - \frac{7}{45} a^{8} + \frac{2}{15} a^{7} + \frac{1}{15} a^{6} + \frac{1642}{3645} a^{5} + \frac{383}{1215} a^{4} + \frac{22}{405} a^{3} + \frac{19}{135} a^{2} - \frac{19}{45} a + \frac{1}{15}$, $\frac{1}{317024031735} a^{15} + \frac{322033964}{105674677245} a^{12} + \frac{60126457}{3913876935} a^{9} + \frac{26899001581}{317024031735} a^{6} - \frac{261593342}{2348326161} a^{3} + \frac{177464221}{434875215}$, $\frac{1}{951072095205} a^{16} + \frac{322033964}{317024031735} a^{13} + \frac{60126457}{11741630805} a^{10} + \frac{26899001581}{951072095205} a^{7} + \frac{2086732819}{7044978483} a^{4} - \frac{257410994}{1304625645} a$, $\frac{1}{2853216285615} a^{17} + \frac{12221767}{190214419041} a^{14} + \frac{1}{1215} a^{13} - \frac{1}{405} a^{12} - \frac{38417453}{3913876935} a^{11} - \frac{13}{405} a^{10} + \frac{4}{135} a^{9} - \frac{290125030154}{2853216285615} a^{8} - \frac{2}{15} a^{7} - \frac{1}{15} a^{6} + \frac{3697678747}{11741630805} a^{5} - \frac{383}{1215} a^{4} - \frac{22}{405} a^{3} + \frac{1192173056}{3913876935} a^{2} + \frac{19}{45} a - \frac{1}{15}$

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:  $9$
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:  \( 422411479.76656485 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

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 $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{7}) \), 3.1.108.1, 6.2.64012032.1, 9.1.11019960576.1

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

Sibling fields

Degree 18 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$3$3.9.16.6$x^{9} + 6 x^{8} + 3 x^{3} + 18 x + 6$$9$$1$$16$$(C_9:C_3):C_2$$[3/2, 2, 13/6]_{2}$
3.9.16.6$x^{9} + 6 x^{8} + 3 x^{3} + 18 x + 6$$9$$1$$16$$(C_9:C_3):C_2$$[3/2, 2, 13/6]_{2}$
7Data not computed