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Properties

Label 18.0.17513643935...9696.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{12}\cdot 13^{6}$
Root discriminant $17.21$
Ramified primes $2, 7, 13$
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![8, 0, 0, -4, -14, 42, 1, -140, 280, -290, 182, -70, 32, -28, 21, -8, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^15 + 21*x^14 - 28*x^13 + 32*x^12 - 70*x^11 + 182*x^10 - 290*x^9 + 280*x^8 - 140*x^7 + x^6 + 42*x^5 - 14*x^4 - 4*x^3 + 8)
 
gp: K = bnfinit(x^18 - 8*x^15 + 21*x^14 - 28*x^13 + 32*x^12 - 70*x^11 + 182*x^10 - 290*x^9 + 280*x^8 - 140*x^7 + x^6 + 42*x^5 - 14*x^4 - 4*x^3 + 8, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{15} + 21 x^{14} - 28 x^{13} + 32 x^{12} - 70 x^{11} + 182 x^{10} - 290 x^{9} + 280 x^{8} - 140 x^{7} + x^{6} + 42 x^{5} - 14 x^{4} - 4 x^{3} + 8 \)

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:  \(-17513643935890567069696=-\,2^{18}\cdot 7^{12}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.21$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{26} a^{14} + \frac{2}{13} a^{13} + \frac{2}{13} a^{12} + \frac{3}{13} a^{11} - \frac{3}{13} a^{10} + \frac{1}{26} a^{9} - \frac{11}{26} a^{7} - \frac{7}{26} a^{6} + \frac{2}{13} a^{5} - \frac{9}{26} a^{4} + \frac{3}{26} a^{3} + \frac{1}{13} a^{2} - \frac{2}{13} a + \frac{6}{13}$, $\frac{1}{78} a^{15} + \frac{1}{78} a^{14} - \frac{4}{39} a^{13} + \frac{7}{78} a^{12} + \frac{5}{26} a^{11} + \frac{19}{78} a^{10} - \frac{8}{39} a^{9} - \frac{11}{78} a^{8} + \frac{1}{3} a^{7} + \frac{25}{78} a^{6} - \frac{4}{39} a^{5} - \frac{3}{26} a^{4} - \frac{11}{26} a^{3} + \frac{29}{78} a^{2} + \frac{4}{13} a - \frac{5}{39}$, $\frac{1}{156} a^{16} + \frac{1}{13} a^{13} + \frac{5}{156} a^{12} - \frac{5}{39} a^{11} + \frac{7}{39} a^{10} + \frac{7}{78} a^{9} - \frac{1}{78} a^{8} - \frac{11}{78} a^{7} + \frac{5}{13} a^{6} - \frac{1}{39} a^{5} - \frac{9}{52} a^{4} + \frac{25}{78} a^{3} - \frac{1}{6} a^{2} + \frac{2}{39} a + \frac{10}{39}$, $\frac{1}{1181247132} a^{17} + \frac{1193741}{1181247132} a^{16} + \frac{481090}{295311783} a^{15} + \frac{5037737}{590623566} a^{14} + \frac{244243}{2329876} a^{13} - \frac{87502407}{393749044} a^{12} + \frac{1436807}{15144194} a^{11} + \frac{44059199}{295311783} a^{10} + \frac{48260783}{590623566} a^{9} - \frac{30843991}{295311783} a^{8} + \frac{18646609}{196874522} a^{7} + \frac{22774987}{98437261} a^{6} - \frac{521836837}{1181247132} a^{5} - \frac{53075467}{1181247132} a^{4} + \frac{133027847}{295311783} a^{3} + \frac{50290841}{196874522} a^{2} - \frac{104431060}{295311783} a - \frac{108579983}{295311783}$

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:  \( \frac{4435837}{393749044} a^{17} + \frac{10029719}{196874522} a^{16} + \frac{9353599}{590623566} a^{15} - \frac{47409803}{590623566} a^{14} - \frac{14460943}{90865164} a^{13} + \frac{190146023}{295311783} a^{12} - \frac{12125463}{15144194} a^{11} + \frac{308958517}{590623566} a^{10} - \frac{339319667}{295311783} a^{9} + \frac{1519779947}{295311783} a^{8} - \frac{5517177841}{590623566} a^{7} + \frac{2780530559}{295311783} a^{6} - \frac{5035012507}{1181247132} a^{5} - \frac{1077790}{98437261} a^{4} + \frac{67158857}{98437261} a^{3} + \frac{357489790}{295311783} a^{2} - \frac{50360289}{98437261} a - \frac{36693923}{295311783} \) (order $4$)
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:  \( 13709.522062672431 \)
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

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{7})^+\), 3.1.2548.1, 6.0.25969216.1, 6.0.153664.1, 9.3.16542390592.2

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$