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Properties

Label 18.0.20903572991...7872.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{8}\cdot 7^{15}$
Root discriminant $22.45$
Ramified primes $2, 3, 7$
Class number $3$
Class group $[3]$
Galois group $C_2\times S_3^2$ (as 18T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, -144, 376, -218, -714, 852, 278, -680, 132, 191, -105, 10, 36, -16, -4, 6, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 6*x^15 - 4*x^14 - 16*x^13 + 36*x^12 + 10*x^11 - 105*x^10 + 191*x^9 + 132*x^8 - 680*x^7 + 278*x^6 + 852*x^5 - 714*x^4 - 218*x^3 + 376*x^2 - 144*x + 36)
 
gp: K = bnfinit(x^18 - x^17 + 6*x^15 - 4*x^14 - 16*x^13 + 36*x^12 + 10*x^11 - 105*x^10 + 191*x^9 + 132*x^8 - 680*x^7 + 278*x^6 + 852*x^5 - 714*x^4 - 218*x^3 + 376*x^2 - 144*x + 36, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 6 x^{15} - 4 x^{14} - 16 x^{13} + 36 x^{12} + 10 x^{11} - 105 x^{10} + 191 x^{9} + 132 x^{8} - 680 x^{7} + 278 x^{6} + 852 x^{5} - 714 x^{4} - 218 x^{3} + 376 x^{2} - 144 x + 36 \)

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:  \(-2090357299107442691407872=-\,2^{26}\cdot 3^{8}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.45$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{12} - \frac{1}{2} a^{10} - \frac{1}{3} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{12} - \frac{1}{3} a^{11} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{2346} a^{16} - \frac{167}{2346} a^{15} + \frac{43}{1173} a^{14} + \frac{15}{782} a^{13} + \frac{539}{2346} a^{12} - \frac{973}{2346} a^{11} - \frac{197}{2346} a^{10} + \frac{339}{782} a^{9} + \frac{583}{1173} a^{8} + \frac{5}{51} a^{7} - \frac{605}{2346} a^{6} + \frac{42}{391} a^{5} - \frac{526}{1173} a^{4} - \frac{514}{1173} a^{3} + \frac{58}{1173} a^{2} + \frac{101}{391} a + \frac{50}{391}$, $\frac{1}{15290181420448014} a^{17} - \frac{58534138807}{899422436496942} a^{16} + \frac{275527366271332}{7645090710224007} a^{15} + \frac{605398250007350}{7645090710224007} a^{14} + \frac{103682984977759}{15290181420448014} a^{13} + \frac{544653982108339}{15290181420448014} a^{12} - \frac{4232734173409255}{15290181420448014} a^{11} + \frac{1556603941651937}{15290181420448014} a^{10} - \frac{3351349805724226}{7645090710224007} a^{9} + \frac{3104246182885268}{7645090710224007} a^{8} + \frac{2104270238347475}{15290181420448014} a^{7} + \frac{2620436141168927}{15290181420448014} a^{6} - \frac{638353557603703}{2548363570074669} a^{5} - \frac{217677977141319}{849454523358223} a^{4} + \frac{920566503642410}{2548363570074669} a^{3} - \frac{2476469211107545}{7645090710224007} a^{2} + \frac{336304062497644}{2548363570074669} a + \frac{268202448904957}{849454523358223}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 256078.49869658783 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 18T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.1176.1, 3.1.588.1, 6.0.9680832.1, 6.0.2420208.1, 9.1.78066229248.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.11.5$x^{6} + 6$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.5$x^{6} + 6$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7Data not computed