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Properties

Label 18.0.16191937479...2368.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{31}$
Root discriminant $13.27$
Ramified primes $2, 3$
Class number $1$
Class group Trivial
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -120, 381, -819, 1401, -2067, 2793, -3447, 3861, -3832, 3336, -2517, 1635, -903, 417, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 417*x^14 - 903*x^13 + 1635*x^12 - 2517*x^11 + 3336*x^10 - 3832*x^9 + 3861*x^8 - 3447*x^7 + 2793*x^6 - 2067*x^5 + 1401*x^4 - 819*x^3 + 381*x^2 - 120*x + 19)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 417*x^14 - 903*x^13 + 1635*x^12 - 2517*x^11 + 3336*x^10 - 3832*x^9 + 3861*x^8 - 3447*x^7 + 2793*x^6 - 2067*x^5 + 1401*x^4 - 819*x^3 + 381*x^2 - 120*x + 19, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 417 x^{14} - 903 x^{13} + 1635 x^{12} - 2517 x^{11} + 3336 x^{10} - 3832 x^{9} + 3861 x^{8} - 3447 x^{7} + 2793 x^{6} - 2067 x^{5} + 1401 x^{4} - 819 x^{3} + 381 x^{2} - 120 x + 19 \)

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:  \(-161919374795459002368=-\,2^{18}\cdot 3^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.27$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{19618} a^{16} - \frac{4}{9809} a^{15} + \frac{449}{9809} a^{14} + \frac{3663}{19618} a^{13} + \frac{2347}{19618} a^{12} - \frac{3211}{19618} a^{11} - \frac{1463}{19618} a^{10} + \frac{3648}{9809} a^{9} - \frac{1634}{9809} a^{8} + \frac{6805}{19618} a^{7} - \frac{7517}{19618} a^{6} + \frac{1977}{19618} a^{5} + \frac{6573}{19618} a^{4} + \frac{4873}{9809} a^{3} - \frac{8911}{19618} a^{2} - \frac{5119}{19618} a + \frac{1669}{9809}$, $\frac{1}{7082098} a^{17} + \frac{86}{3541049} a^{16} - \frac{775182}{3541049} a^{15} + \frac{578006}{3541049} a^{14} + \frac{53529}{7082098} a^{13} - \frac{481910}{3541049} a^{12} - \frac{1795759}{7082098} a^{11} - \frac{1167776}{3541049} a^{10} - \frac{1699154}{3541049} a^{9} - \frac{570274}{3541049} a^{8} - \frac{2510037}{7082098} a^{7} - \frac{1102233}{3541049} a^{6} - \frac{1265861}{7082098} a^{5} - \frac{1228031}{3541049} a^{4} - \frac{863825}{7082098} a^{3} + \frac{1740886}{3541049} a^{2} - \frac{184389}{3541049} a + \frac{42141}{186371}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{69113}{7082098} a^{17} + \frac{751535}{7082098} a^{16} - \frac{111706}{208297} a^{15} + \frac{12588291}{7082098} a^{14} - \frac{15073433}{3541049} a^{13} + \frac{27616060}{3541049} a^{12} - \frac{38805502}{3541049} a^{11} + \frac{80274143}{7082098} a^{10} - \frac{23997749}{3541049} a^{9} - \frac{16123753}{7082098} a^{8} + \frac{44190695}{3541049} a^{7} - \frac{65867383}{3541049} a^{6} + \frac{69229242}{3541049} a^{5} - \frac{115360681}{7082098} a^{4} + \frac{99081309}{7082098} a^{3} - \frac{40446303}{3541049} a^{2} + \frac{58694775}{7082098} a - \frac{608655}{186371} \) (order $18$)
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:  \( 4471.846205846836 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 3.1.216.1, \(\Q(\zeta_{9})\), 6.0.139968.1, 9.3.7346640384.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
3Data not computed