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Properties

Label 18.0.40310784000...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{9}\cdot 5^{15}$
Root discriminant $18.02$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![4, 20, 44, 38, -10, 0, 70, 60, -60, -65, 149, -40, -86, 88, -20, -22, 20, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 20*x^16 - 22*x^15 - 20*x^14 + 88*x^13 - 86*x^12 - 40*x^11 + 149*x^10 - 65*x^9 - 60*x^8 + 60*x^7 + 70*x^6 - 10*x^4 + 38*x^3 + 44*x^2 + 20*x + 4)
 
gp: K = bnfinit(x^18 - 7*x^17 + 20*x^16 - 22*x^15 - 20*x^14 + 88*x^13 - 86*x^12 - 40*x^11 + 149*x^10 - 65*x^9 - 60*x^8 + 60*x^7 + 70*x^6 - 10*x^4 + 38*x^3 + 44*x^2 + 20*x + 4, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 20 x^{16} - 22 x^{15} - 20 x^{14} + 88 x^{13} - 86 x^{12} - 40 x^{11} + 149 x^{10} - 65 x^{9} - 60 x^{8} + 60 x^{7} + 70 x^{6} - 10 x^{4} + 38 x^{3} + 44 x^{2} + 20 x + 4 \)

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:  \(-40310784000000000000000=-\,2^{26}\cdot 3^{9}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.02$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{10672} a^{16} - \frac{595}{10672} a^{15} + \frac{937}{10672} a^{14} - \frac{43}{10672} a^{13} - \frac{275}{10672} a^{12} - \frac{2577}{10672} a^{11} + \frac{1803}{10672} a^{10} - \frac{1619}{10672} a^{9} - \frac{209}{2668} a^{8} - \frac{1511}{5336} a^{7} + \frac{11}{46} a^{6} + \frac{2163}{5336} a^{5} + \frac{73}{5336} a^{4} - \frac{1419}{5336} a^{3} + \frac{127}{1334} a^{2} - \frac{1269}{2668} a + \frac{1249}{2668}$, $\frac{1}{7886608} a^{17} + \frac{277}{7886608} a^{16} + \frac{233139}{7886608} a^{15} - \frac{55415}{7886608} a^{14} + \frac{825327}{7886608} a^{13} + \frac{1011583}{7886608} a^{12} + \frac{330613}{7886608} a^{11} + \frac{1479885}{7886608} a^{10} - \frac{586909}{3943304} a^{9} - \frac{891603}{3943304} a^{8} - \frac{84957}{985826} a^{7} - \frac{620699}{3943304} a^{6} - \frac{887177}{3943304} a^{5} + \frac{1281513}{3943304} a^{4} - \frac{680461}{1971652} a^{3} - \frac{185317}{1971652} a^{2} + \frac{402101}{1971652} a - \frac{481949}{985826}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 41069.839937116616 \) (assuming GRH)
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

\(\Q(\sqrt{-15}) \), 3.1.200.1, 3.1.300.1, 6.0.1350000.1, 6.0.5400000.2, 9.1.3456000000.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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}$
2.6.11.9$x^{6} + 6 x^{4} + 2$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.6.11.9$x^{6} + 6 x^{4} + 2$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
$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}$
$5$5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$