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Properties

Label 18.6.26856034367...4592.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{16}\cdot 3^{9}\cdot 113^{6}$
Root discriminant $15.51$
Ramified primes $2, 3, 113$
Class number $1$
Class group Trivial
Galois group $C_2\times (C_3\times A_4):S_3$ (as 18T156)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -3, 14, -30, 27, -22, -9, -5, 15, 29, -37, 20, -12, 0, 7, 1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + x^16 + 7*x^15 - 12*x^13 + 20*x^12 - 37*x^11 + 29*x^10 + 15*x^9 - 5*x^8 - 9*x^7 - 22*x^6 + 27*x^5 - 30*x^4 + 14*x^3 - 3*x^2 - x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + x^16 + 7*x^15 - 12*x^13 + 20*x^12 - 37*x^11 + 29*x^10 + 15*x^9 - 5*x^8 - 9*x^7 - 22*x^6 + 27*x^5 - 30*x^4 + 14*x^3 - 3*x^2 - x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + x^{16} + 7 x^{15} - 12 x^{13} + 20 x^{12} - 37 x^{11} + 29 x^{10} + 15 x^{9} - 5 x^{8} - 9 x^{7} - 22 x^{6} + 27 x^{5} - 30 x^{4} + 14 x^{3} - 3 x^{2} - x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2685603436730970734592=2^{16}\cdot 3^{9}\cdot 113^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.51$
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:  $2, 3, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{132248335648652} a^{17} + \frac{7044528725167}{132248335648652} a^{16} + \frac{14033304335357}{66124167824326} a^{15} - \frac{38356813921379}{132248335648652} a^{14} + \frac{55821817943851}{132248335648652} a^{13} + \frac{10626346311061}{132248335648652} a^{12} + \frac{31215053949715}{132248335648652} a^{11} - \frac{8802085880438}{33062083912163} a^{10} - \frac{54489923204427}{132248335648652} a^{9} - \frac{25336940360719}{66124167824326} a^{8} - \frac{20382138277163}{132248335648652} a^{7} + \frac{19070596739997}{66124167824326} a^{6} + \frac{10160184970002}{33062083912163} a^{5} - \frac{29085940963345}{132248335648652} a^{4} - \frac{59324454296105}{132248335648652} a^{3} + \frac{61138506016343}{132248335648652} a^{2} + \frac{7420626727555}{66124167824326} a + \frac{43614157439501}{132248335648652}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6093.44526767 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times (C_3\times A_4):S_3$ (as 18T156):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 38 conjugacy class representatives for $C_2\times (C_3\times A_4):S_3$
Character table for $C_2\times (C_3\times A_4):S_3$ is not computed

Intermediate fields

3.1.339.1, 6.2.5516208.2, 9.3.623331504.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.16.7$x^{12} + 5 x^{10} + 4 x^{8} + x^{6} + 4 x^{4} + x^{2} + 3$$6$$2$$16$12T42$[2, 2]_{3}^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$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$$[\ ]_{2}^{6}$
$113$113.6.3.2$x^{6} - 12769 x^{2} + 7214485$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
113.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
113.6.3.2$x^{6} - 12769 x^{2} + 7214485$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$