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Properties

Label 18.0.80269720285...2848.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{9}\cdot 7^{15}$
Root discriminant $31.24$
Ramified primes $2, 3, 7$
Class number $28$ (GRH)
Class group $[2, 14]$ (GRH)
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![74088, 0, 148176, 0, 123480, 0, 65464, 0, 27440, 0, 9212, 0, 2226, 0, 336, 0, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 28*x^16 + 336*x^14 + 2226*x^12 + 9212*x^10 + 27440*x^8 + 65464*x^6 + 123480*x^4 + 148176*x^2 + 74088)
 
gp: K = bnfinit(x^18 + 28*x^16 + 336*x^14 + 2226*x^12 + 9212*x^10 + 27440*x^8 + 65464*x^6 + 123480*x^4 + 148176*x^2 + 74088, 1)
 

Normalized defining polynomial

\( x^{18} + 28 x^{16} + 336 x^{14} + 2226 x^{12} + 9212 x^{10} + 27440 x^{8} + 65464 x^{6} + 123480 x^{4} + 148176 x^{2} + 74088 \)

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:  \(-802697202857257993500622848=-\,2^{33}\cdot 3^{9}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{14} a^{6}$, $\frac{1}{14} a^{7}$, $\frac{1}{14} a^{8}$, $\frac{1}{14} a^{9}$, $\frac{1}{14} a^{10}$, $\frac{1}{14} a^{11}$, $\frac{1}{196} a^{12}$, $\frac{1}{196} a^{13}$, $\frac{1}{1764} a^{14} - \frac{1}{882} a^{12} - \frac{1}{42} a^{8} + \frac{1}{126} a^{6} - \frac{1}{9} a^{4} + \frac{4}{9} a^{2} - \frac{1}{3}$, $\frac{1}{1764} a^{15} - \frac{1}{882} a^{13} - \frac{1}{42} a^{9} + \frac{1}{126} a^{7} - \frac{1}{9} a^{5} + \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{24382376676} a^{16} - \frac{3313717}{12191188338} a^{14} - \frac{3582553}{2031864723} a^{12} + \frac{6359509}{290266389} a^{10} + \frac{6676345}{1741598334} a^{8} - \frac{51110945}{1741598334} a^{6} - \frac{3870821}{124399881} a^{4} - \frac{8124296}{41466627} a^{2} + \frac{529975}{13822209}$, $\frac{1}{24382376676} a^{17} - \frac{3313717}{12191188338} a^{15} - \frac{3582553}{2031864723} a^{13} + \frac{6359509}{290266389} a^{11} + \frac{6676345}{1741598334} a^{9} - \frac{51110945}{1741598334} a^{7} - \frac{3870821}{124399881} a^{5} - \frac{8124296}{41466627} a^{3} + \frac{529975}{13822209} a$

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

Class group and class number

$C_{2}\times C_{14}$, which has order $28$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 31684.521927615173 \) (assuming GRH)
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

\(\Q(\sqrt{-42}) \), \(\Q(\zeta_{7})^+\), 3.1.588.1, 6.0.929359872.2, 6.0.232339968.1, 9.3.203297472.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed