↑

Properties

Label 18.0.13368127977...1536.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{12}\cdot 11^{9}$
Root discriminant $19.27$
Ramified primes $2, 7, 11$
Class number $1$
Class group Trivial
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 11, -34, 56, -87, 166, -26, 160, 85, 61, 22, 19, -27, 21, -13, 8, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 8*x^16 - 13*x^15 + 21*x^14 - 27*x^13 + 19*x^12 + 22*x^11 + 61*x^10 + 85*x^9 + 160*x^8 - 26*x^7 + 166*x^6 - 87*x^5 + 56*x^4 - 34*x^3 + 11*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^18 - x^17 + 8*x^16 - 13*x^15 + 21*x^14 - 27*x^13 + 19*x^12 + 22*x^11 + 61*x^10 + 85*x^9 + 160*x^8 - 26*x^7 + 166*x^6 - 87*x^5 + 56*x^4 - 34*x^3 + 11*x^2 - 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 8 x^{16} - 13 x^{15} + 21 x^{14} - 27 x^{13} + 19 x^{12} + 22 x^{11} + 61 x^{10} + 85 x^{9} + 160 x^{8} - 26 x^{7} + 166 x^{6} - 87 x^{5} + 56 x^{4} - 34 x^{3} + 11 x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-133681279779085528641536=-\,2^{12}\cdot 7^{12}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.27$
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:  $2, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} + \frac{4}{15} a^{11} + \frac{1}{15} a^{10} + \frac{4}{15} a^{9} + \frac{4}{15} a^{8} - \frac{2}{15} a^{7} + \frac{7}{15} a^{6} - \frac{1}{3} a^{5} - \frac{2}{15} a^{4} - \frac{2}{15} a^{3} + \frac{4}{15} a^{2} + \frac{1}{15} a + \frac{1}{15}$, $\frac{1}{15} a^{13} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a - \frac{4}{15}$, $\frac{1}{15} a^{14} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{2} - \frac{4}{15} a$, $\frac{1}{15} a^{15} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{3} - \frac{4}{15} a^{2}$, $\frac{1}{103935} a^{16} - \frac{71}{7995} a^{15} + \frac{989}{34645} a^{14} + \frac{406}{103935} a^{13} - \frac{36}{34645} a^{12} - \frac{7772}{34645} a^{11} - \frac{2897}{34645} a^{10} - \frac{2211}{6929} a^{9} + \frac{2751}{6929} a^{8} - \frac{5774}{34645} a^{7} + \frac{2663}{6929} a^{6} + \frac{11}{2665} a^{5} + \frac{11138}{34645} a^{4} + \frac{7379}{103935} a^{3} + \frac{30542}{103935} a^{2} - \frac{2259}{34645} a - \frac{2900}{20787}$, $\frac{1}{64127895} a^{17} + \frac{51}{21375965} a^{16} - \frac{450163}{21375965} a^{15} + \frac{33274}{64127895} a^{14} - \frac{26540}{986583} a^{13} + \frac{1578868}{64127895} a^{12} - \frac{23149658}{64127895} a^{11} - \frac{30199397}{64127895} a^{10} + \frac{14861206}{64127895} a^{9} - \frac{22869056}{64127895} a^{8} + \frac{2264692}{64127895} a^{7} - \frac{30279068}{64127895} a^{6} - \frac{17700536}{64127895} a^{5} - \frac{7818444}{21375965} a^{4} + \frac{5766713}{12825579} a^{3} + \frac{4295077}{64127895} a^{2} - \frac{1016941}{4275193} a - \frac{22801231}{64127895}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6989.09727608 \)
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

\(\Q(\sqrt{-11}) \), 3.1.44.1 x3, \(\Q(\zeta_{7})^+\), 6.0.21296.1, 6.0.51131696.1 x2, 6.0.3195731.1, 9.3.10021812416.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.51131696.1
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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$