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Properties

Label 18.0.15490206293...5167.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 47^{9}$
Root discriminant $25.09$
Ramified primes $7, 47$
Class number $5$
Class group $[5]$
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![13411, -3917, 6206, -4333, 1340, 3460, -3469, 4368, -2237, -743, 1430, -399, 27, -101, 83, -13, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 13*x^15 + 83*x^14 - 101*x^13 + 27*x^12 - 399*x^11 + 1430*x^10 - 743*x^9 - 2237*x^8 + 4368*x^7 - 3469*x^6 + 3460*x^5 + 1340*x^4 - 4333*x^3 + 6206*x^2 - 3917*x + 13411)
 
gp: K = bnfinit(x^18 - 2*x^17 - 13*x^15 + 83*x^14 - 101*x^13 + 27*x^12 - 399*x^11 + 1430*x^10 - 743*x^9 - 2237*x^8 + 4368*x^7 - 3469*x^6 + 3460*x^5 + 1340*x^4 - 4333*x^3 + 6206*x^2 - 3917*x + 13411, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 13 x^{15} + 83 x^{14} - 101 x^{13} + 27 x^{12} - 399 x^{11} + 1430 x^{10} - 743 x^{9} - 2237 x^{8} + 4368 x^{7} - 3469 x^{6} + 3460 x^{5} + 1340 x^{4} - 4333 x^{3} + 6206 x^{2} - 3917 x + 13411 \)

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:  \(-15490206293606403634785167=-\,7^{12}\cdot 47^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.09$
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:  $7, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{2} - \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{96} a^{15} + \frac{5}{96} a^{14} + \frac{1}{96} a^{13} - \frac{1}{32} a^{12} + \frac{5}{96} a^{11} - \frac{7}{96} a^{10} + \frac{3}{32} a^{9} - \frac{1}{96} a^{8} + \frac{1}{48} a^{7} + \frac{11}{48} a^{6} - \frac{5}{24} a^{5} + \frac{5}{48} a^{4} - \frac{17}{96} a^{3} + \frac{15}{32} a^{2} + \frac{9}{32} a - \frac{23}{96}$, $\frac{1}{96} a^{16} + \frac{1}{24} a^{13} - \frac{1}{24} a^{12} + \frac{1}{24} a^{11} + \frac{1}{12} a^{10} + \frac{1}{48} a^{9} + \frac{7}{96} a^{8} - \frac{1}{8} a^{7} - \frac{11}{48} a^{6} + \frac{1}{48} a^{5} + \frac{17}{96} a^{4} + \frac{17}{48} a^{3} + \frac{1}{16} a^{2} - \frac{13}{48} a + \frac{43}{96}$, $\frac{1}{15878064509515129916033184} a^{17} + \frac{72259420285506360506923}{15878064509515129916033184} a^{16} + \frac{1870119569525126293871}{3969516127378782479008296} a^{15} - \frac{151195353284345199357211}{3969516127378782479008296} a^{14} + \frac{96860800643745033314467}{1984758063689391239504148} a^{13} + \frac{10162678978529486114615}{330793010614898539917358} a^{12} - \frac{29621519386699698038851}{496189515922347809876037} a^{11} - \frac{633930900455966424769259}{7939032254757564958016592} a^{10} + \frac{155157720643753442104203}{5292688169838376638677728} a^{9} + \frac{231549553225818218455853}{15878064509515129916033184} a^{8} - \frac{191294799842731384988213}{2646344084919188319338864} a^{7} + \frac{12892852283170062197814}{165396505307449269958679} a^{6} + \frac{534594085264031216746141}{5292688169838376638677728} a^{5} + \frac{853503795898381044704129}{15878064509515129916033184} a^{4} - \frac{340601947502905989173453}{1323172042459594159669432} a^{3} + \frac{1885771433115103499430211}{3969516127378782479008296} a^{2} - \frac{5791872503019477128505331}{15878064509515129916033184} a - \frac{1842360464010747254462513}{5292688169838376638677728}$

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

Class group and class number

$C_{5}$, which has order $5$

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:  \( 29368.4164899 \)
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

\(\Q(\sqrt{-47}) \), 3.1.2303.1 x3, \(\Q(\zeta_{7})^+\), 6.0.249279023.1, 6.0.5087327.1 x2, 6.0.249279023.2, 9.3.12214672127.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.5087327.1
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$47$47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
47.6.3.2$x^{6} - 2209 x^{2} + 207646$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$