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Properties

Label 18.18.1226761765...7648.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{24}\cdot 13^{9}$
Root discriminant $24.76$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 30, -37, -204, 294, 604, -858, -984, 1201, 984, -858, -604, 294, 204, -37, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 30*x^16 - 37*x^15 + 204*x^14 + 294*x^13 - 604*x^12 - 858*x^11 + 984*x^10 + 1201*x^9 - 984*x^8 - 858*x^7 + 604*x^6 + 294*x^5 - 204*x^4 - 37*x^3 + 30*x^2 - 1)
 
gp: K = bnfinit(x^18 - 30*x^16 - 37*x^15 + 204*x^14 + 294*x^13 - 604*x^12 - 858*x^11 + 984*x^10 + 1201*x^9 - 984*x^8 - 858*x^7 + 604*x^6 + 294*x^5 - 204*x^4 - 37*x^3 + 30*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 30 x^{16} - 37 x^{15} + 204 x^{14} + 294 x^{13} - 604 x^{12} - 858 x^{11} + 984 x^{10} + 1201 x^{9} - 984 x^{8} - 858 x^{7} + 604 x^{6} + 294 x^{5} - 204 x^{4} - 37 x^{3} + 30 x^{2} - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12267617659000607237787648=2^{12}\cdot 3^{24}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.76$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{6}{13} a^{13} - \frac{1}{13} a^{12} + \frac{5}{13} a^{10} - \frac{1}{13} a^{9} + \frac{6}{13} a^{8} - \frac{6}{13} a^{6} - \frac{1}{13} a^{5} - \frac{5}{13} a^{4} + \frac{1}{13} a^{2} + \frac{6}{13} a - \frac{1}{13}$, $\frac{1}{13} a^{15} + \frac{2}{13} a^{13} + \frac{6}{13} a^{12} + \frac{5}{13} a^{11} - \frac{5}{13} a^{10} - \frac{1}{13} a^{9} + \frac{3}{13} a^{8} - \frac{6}{13} a^{7} - \frac{4}{13} a^{6} + \frac{1}{13} a^{5} + \frac{4}{13} a^{4} + \frac{1}{13} a^{3} + \frac{2}{13} a + \frac{6}{13}$, $\frac{1}{169} a^{16} + \frac{3}{169} a^{15} + \frac{6}{169} a^{14} + \frac{62}{169} a^{13} + \frac{45}{169} a^{12} + \frac{75}{169} a^{11} - \frac{9}{169} a^{10} - \frac{43}{169} a^{9} - \frac{64}{169} a^{8} + \frac{17}{169} a^{7} + \frac{43}{169} a^{6} + \frac{68}{169} a^{5} - \frac{59}{169} a^{4} - \frac{75}{169} a^{3} + \frac{6}{169} a^{2} - \frac{16}{169} a - \frac{25}{169}$, $\frac{1}{18421} a^{17} - \frac{33}{18421} a^{16} + \frac{405}{18421} a^{15} + \frac{223}{18421} a^{14} + \frac{7342}{18421} a^{13} + \frac{6528}{18421} a^{12} - \frac{4568}{18421} a^{11} - \frac{5439}{18421} a^{10} - \frac{2104}{18421} a^{9} - \frac{8937}{18421} a^{8} - \frac{4287}{18421} a^{7} - \frac{869}{18421} a^{6} - \frac{5926}{18421} a^{5} + \frac{1178}{18421} a^{4} + \frac{6593}{18421} a^{3} - \frac{5601}{18421} a^{2} - \frac{3271}{18421} a + \frac{2213}{18421}$

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:  $17$
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:  \( 3530519.19402 \) (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

\(\Q(\sqrt{13}) \), 3.3.4212.1 x3, \(\Q(\zeta_{9})^+\), 6.6.230632272.1, 6.6.2847312.1 x2, 6.6.14414517.1, 9.9.74724856128.1 x3

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

Sibling fields

Degree 6 sibling: 6.6.2847312.1
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$