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Properties

Label 18.6.58065480530...3552.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{24}\cdot 3^{6}\cdot 7^{15}$
Root discriminant $18.39$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (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![1, 10, -25, -28, -215, -184, -74, 372, 324, 90, -195, -162, 3, 24, 51, -20, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 4*x^16 - 20*x^15 + 51*x^14 + 24*x^13 + 3*x^12 - 162*x^11 - 195*x^10 + 90*x^9 + 324*x^8 + 372*x^7 - 74*x^6 - 184*x^5 - 215*x^4 - 28*x^3 - 25*x^2 + 10*x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + 4*x^16 - 20*x^15 + 51*x^14 + 24*x^13 + 3*x^12 - 162*x^11 - 195*x^10 + 90*x^9 + 324*x^8 + 372*x^7 - 74*x^6 - 184*x^5 - 215*x^4 - 28*x^3 - 25*x^2 + 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 4 x^{16} - 20 x^{15} + 51 x^{14} + 24 x^{13} + 3 x^{12} - 162 x^{11} - 195 x^{10} + 90 x^{9} + 324 x^{8} + 372 x^{7} - 74 x^{6} - 184 x^{5} - 215 x^{4} - 28 x^{3} - 25 x^{2} + 10 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(58065480530762296983552=2^{24}\cdot 3^{6}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.39$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{4}{9} a^{10} + \frac{2}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{3} a^{6} - \frac{4}{9} a^{5} + \frac{2}{9} a^{4} - \frac{2}{9} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{33032165480828523} a^{17} - \frac{1164509908917818}{33032165480828523} a^{16} - \frac{4685132636094514}{33032165480828523} a^{15} - \frac{1758562514158343}{33032165480828523} a^{14} - \frac{9051154876189}{155080589111871} a^{13} - \frac{747709987257082}{33032165480828523} a^{12} + \frac{672540464981263}{33032165480828523} a^{11} - \frac{548752139087864}{11010721826942841} a^{10} + \frac{11367959890241567}{33032165480828523} a^{9} + \frac{8479863929952103}{33032165480828523} a^{8} - \frac{308054502925574}{3670240608980947} a^{7} - \frac{12277481389071490}{33032165480828523} a^{6} + \frac{6753136413168266}{33032165480828523} a^{5} + \frac{15076666989931243}{33032165480828523} a^{4} - \frac{8765155211443964}{33032165480828523} a^{3} - \frac{1184429772996371}{11010721826942841} a^{2} + \frac{3141698278504579}{33032165480828523} a + \frac{3987518871428384}{11010721826942841}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 25505.791722934875 \) (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

\(\Q(\sqrt{7}) \), \(\Q(\zeta_{7})^+\), 3.1.588.1, \(\Q(\zeta_{28})^+\), 6.2.38723328.1, 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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed