↑

Properties

Label 18.0.65701728236...8567.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{17}$
Root discriminant $27.18$
Ramified primes $3, 7$
Class number $4$
Class group $[2, 2]$
Galois group $C_2\times C_9:C_3$ (as 18T14)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1183, 0, 15435, 0, 61257, 0, 90587, 0, 65772, 0, 26040, 0, 5761, 0, 693, 0, 42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 42*x^16 + 693*x^14 + 5761*x^12 + 26040*x^10 + 65772*x^8 + 90587*x^6 + 61257*x^4 + 15435*x^2 + 1183)
 
gp: K = bnfinit(x^18 + 42*x^16 + 693*x^14 + 5761*x^12 + 26040*x^10 + 65772*x^8 + 90587*x^6 + 61257*x^4 + 15435*x^2 + 1183, 1)
 

Normalized defining polynomial

\( x^{18} + 42 x^{16} + 693 x^{14} + 5761 x^{12} + 26040 x^{10} + 65772 x^{8} + 90587 x^{6} + 61257 x^{4} + 15435 x^{2} + 1183 \)

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:  \(-65701728236743660173798567=-\,3^{24}\cdot 7^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.18$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{26} a^{15} - \frac{5}{26} a^{13} + \frac{5}{26} a^{11} + \frac{1}{26} a^{9} - \frac{1}{2} a^{8} + \frac{3}{13} a^{7} - \frac{2}{13} a^{5} - \frac{1}{2} a^{4} + \frac{9}{26} a^{3} + \frac{7}{26} a$, $\frac{1}{299554907626} a^{16} + \frac{19103418715}{149777453813} a^{14} - \frac{8841585484}{149777453813} a^{12} + \frac{4719374602}{149777453813} a^{10} - \frac{120369253375}{299554907626} a^{8} - \frac{25112408871}{299554907626} a^{6} - \frac{1}{2} a^{5} + \frac{18848889523}{299554907626} a^{4} - \frac{69146842147}{299554907626} a^{2} - \frac{1}{2} a + \frac{3260055272}{11521342601}$, $\frac{1}{299554907626} a^{17} + \frac{3642809627}{299554907626} a^{15} + \frac{2679757117}{149777453813} a^{13} - \frac{6801967999}{149777453813} a^{11} - \frac{396602105}{23042685202} a^{9} - \frac{41359560938}{149777453813} a^{7} - \frac{1}{2} a^{6} + \frac{3663773461}{149777453813} a^{5} - \frac{1}{2} a^{4} - \frac{40334092374}{149777453813} a^{3} - \frac{1}{2} a^{2} + \frac{142368150077}{299554907626} a - \frac{1}{2}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( -\frac{226378539}{299554907626} a^{17} - \frac{103001955}{23042685202} a^{16} - \frac{5229520677}{149777453813} a^{15} - \frac{4036960503}{23042685202} a^{14} - \frac{96653910129}{149777453813} a^{13} - \frac{30009282041}{11521342601} a^{12} - \frac{913315688136}{149777453813} a^{11} - \frac{423630165495}{23042685202} a^{10} - \frac{4684035413298}{149777453813} a^{9} - \frac{1472063919165}{23042685202} a^{8} - \frac{25738693040667}{299554907626} a^{7} - \frac{2481979166057}{23042685202} a^{6} - \frac{35760365449641}{299554907626} a^{5} - \frac{1765594056261}{23042685202} a^{4} - \frac{10699311536794}{149777453813} a^{3} - \frac{146579042862}{11521342601} a^{2} - \frac{1568013443199}{149777453813} a - \frac{7622000237}{23042685202} \) (order $14$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 457078.455661 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_9:C_3$ (as 18T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 22 conjugacy class representatives for $C_2\times C_9:C_3$
Character table for $C_2\times C_9:C_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 9.9.3063651608241.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ $18$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
3Data not computed
7Data not computed