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Properties

Label 18.0.74422866183...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{30}\cdot 5^{12}\cdot 7^{12}\cdot 29^{5}$
Root discriminant $86.56$
Ramified primes $2, 5, 7, 29$
Class number $36864$ (GRH)
Class group $[2, 2, 6, 1536]$ (GRH)
Galois group 18T176

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20511149, 0, 55167918, 0, 56338590, 0, 29008613, 0, 8207609, 0, 1290015, 0, 109769, 0, 4986, 0, 113, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 113*x^16 + 4986*x^14 + 109769*x^12 + 1290015*x^10 + 8207609*x^8 + 29008613*x^6 + 56338590*x^4 + 55167918*x^2 + 20511149)
 
gp: K = bnfinit(x^18 + 113*x^16 + 4986*x^14 + 109769*x^12 + 1290015*x^10 + 8207609*x^8 + 29008613*x^6 + 56338590*x^4 + 55167918*x^2 + 20511149, 1)
 

Normalized defining polynomial

\( x^{18} + 113 x^{16} + 4986 x^{14} + 109769 x^{12} + 1290015 x^{10} + 8207609 x^{8} + 29008613 x^{6} + 56338590 x^{4} + 55167918 x^{2} + 20511149 \)

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:  \(-74422866183848971206656000000000000=-\,2^{30}\cdot 5^{12}\cdot 7^{12}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.56$
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:  $2, 5, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{29} a^{8} - \frac{9}{29} a^{6} - \frac{6}{29} a^{4} + \frac{11}{29} a^{2}$, $\frac{1}{29} a^{9} - \frac{9}{29} a^{7} - \frac{6}{29} a^{5} + \frac{11}{29} a^{3}$, $\frac{1}{29} a^{10} - \frac{14}{29} a^{4} + \frac{12}{29} a^{2}$, $\frac{1}{29} a^{11} - \frac{14}{29} a^{5} + \frac{12}{29} a^{3}$, $\frac{1}{1682} a^{12} - \frac{3}{1682} a^{10} + \frac{27}{1682} a^{8} + \frac{497}{1682} a^{6} - \frac{659}{1682} a^{4} - \frac{5}{58} a^{2} - \frac{1}{2}$, $\frac{1}{1682} a^{13} - \frac{3}{1682} a^{11} + \frac{27}{1682} a^{9} + \frac{497}{1682} a^{7} - \frac{659}{1682} a^{5} - \frac{5}{58} a^{3} - \frac{1}{2} a$, $\frac{1}{829226} a^{14} - \frac{3}{829226} a^{12} - \frac{12327}{829226} a^{10} + \frac{5717}{829226} a^{8} - \frac{395813}{829226} a^{6} + \frac{13985}{28594} a^{4} + \frac{287}{986} a^{2} - \frac{2}{17}$, $\frac{1}{1658452} a^{15} + \frac{245}{829226} a^{13} - \frac{1}{3364} a^{12} + \frac{3697}{414613} a^{11} + \frac{3}{3364} a^{10} - \frac{4783}{829226} a^{9} - \frac{27}{3364} a^{8} + \frac{53277}{829226} a^{7} + \frac{1185}{3364} a^{6} - \frac{2553}{28594} a^{5} - \frac{1023}{3364} a^{4} - \frac{375}{986} a^{3} + \frac{5}{116} a^{2} - \frac{21}{68} a + \frac{1}{4}$, $\frac{1}{319327613851324} a^{16} - \frac{775923}{12281831301974} a^{14} - \frac{1}{3364} a^{13} - \frac{11453804461}{79831903462831} a^{12} + \frac{3}{3364} a^{11} + \frac{68501541569}{159663806925662} a^{10} - \frac{27}{3364} a^{9} + \frac{2362976794039}{159663806925662} a^{8} + \frac{1185}{3364} a^{7} - \frac{12026101209}{423511424206} a^{6} - \frac{1023}{3364} a^{5} - \frac{85344224847}{189849948782} a^{4} + \frac{5}{116} a^{3} - \frac{2901771369}{13093099916} a^{2} + \frac{1}{4} a - \frac{24233808}{112871551}$, $\frac{1}{319327613851324} a^{17} - \frac{775923}{12281831301974} a^{15} - \frac{1}{1658452} a^{14} - \frac{11453804461}{79831903462831} a^{13} + \frac{3}{1658452} a^{12} + \frac{68501541569}{159663806925662} a^{11} - \frac{16267}{1658452} a^{10} + \frac{2362976794039}{159663806925662} a^{9} + \frac{22877}{1658452} a^{8} - \frac{12026101209}{423511424206} a^{7} - \frac{690759}{1658452} a^{6} - \frac{85344224847}{189849948782} a^{5} + \frac{22497}{57188} a^{4} - \frac{2901771369}{13093099916} a^{3} + \frac{665}{1972} a^{2} - \frac{24233808}{112871551} a + \frac{1}{17}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{1536}$, which has order $36864$ (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:  \( 252178.434124 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T176:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 40 conjugacy class representatives for t18n176
Character table for t18n176 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.9800.1, 9.9.941192000000.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.24.244$x^{12} - 8 x^{11} + 4 x^{10} - 8 x^{9} - 14 x^{8} + 8 x^{7} + 8 x^{5} + 16 x^{3} - 8 x^{2} + 16 x + 8$$4$$3$$24$$C_2^2 \times A_4$$[2, 2, 3]^{6}$
$5$5.9.6.1$x^{9} - 25 x^{3} + 250$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
5.9.6.1$x^{9} - 25 x^{3} + 250$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
7Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$