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Properties

Label 18.12.3103584740...5247.2
Degree $18$
Signature $[12, 3]$
Discriminant $-\,3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 167$
Root discriminant $38.27$
Ramified primes $3, 7, 29, 167$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 51, 165, -368, -678, 2220, -489, -4233, 4836, 60, -3741, 3477, -1730, 441, 45, -84, 36, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 84*x^15 + 45*x^14 + 441*x^13 - 1730*x^12 + 3477*x^11 - 3741*x^10 + 60*x^9 + 4836*x^8 - 4233*x^7 - 489*x^6 + 2220*x^5 - 678*x^4 - 368*x^3 + 165*x^2 + 51*x + 1)
 
gp: K = bnfinit(x^18 - 9*x^17 + 36*x^16 - 84*x^15 + 45*x^14 + 441*x^13 - 1730*x^12 + 3477*x^11 - 3741*x^10 + 60*x^9 + 4836*x^8 - 4233*x^7 - 489*x^6 + 2220*x^5 - 678*x^4 - 368*x^3 + 165*x^2 + 51*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 36 x^{16} - 84 x^{15} + 45 x^{14} + 441 x^{13} - 1730 x^{12} + 3477 x^{11} - 3741 x^{10} + 60 x^{9} + 4836 x^{8} - 4233 x^{7} - 489 x^{6} + 2220 x^{5} - 678 x^{4} - 368 x^{3} + 165 x^{2} + 51 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-31035847405427991068883845247=-\,3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 167\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.27$
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:  $3, 7, 29, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{918638209} a^{16} - \frac{8}{918638209} a^{15} - \frac{160559326}{918638209} a^{14} + \frac{205277213}{918638209} a^{13} - \frac{327675809}{918638209} a^{12} + \frac{216088930}{918638209} a^{11} + \frac{292285731}{918638209} a^{10} - \frac{233972700}{918638209} a^{9} - \frac{412470381}{918638209} a^{8} - \frac{69197522}{918638209} a^{7} - \frac{118037833}{918638209} a^{6} + \frac{255614797}{918638209} a^{5} - \frac{93620017}{918638209} a^{4} + \frac{257585675}{918638209} a^{3} + \frac{175377468}{918638209} a^{2} + \frac{13303781}{918638209} a + \frac{441766144}{918638209}$, $\frac{1}{180971727173} a^{17} + \frac{90}{180971727173} a^{16} + \frac{2595354517}{180971727173} a^{15} + \frac{55205605358}{180971727173} a^{14} + \frac{47348637335}{180971727173} a^{13} + \frac{86608188609}{180971727173} a^{12} + \frac{40760403260}{180971727173} a^{11} - \frac{90094300023}{180971727173} a^{10} - \frac{20585880354}{180971727173} a^{9} + \frac{76175757683}{180971727173} a^{8} + \frac{88638978747}{180971727173} a^{7} + \frac{62178963883}{180971727173} a^{6} - \frac{24649833197}{180971727173} a^{5} - \frac{47499980769}{180971727173} a^{4} - \frac{33374071758}{180971727173} a^{3} - \frac{62721228538}{180971727173} a^{2} + \frac{10931918772}{180971727173} a - \frac{52245291624}{180971727173}$

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:  $14$
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:  \( 89878226.1543 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ $18$

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
$3$3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$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}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$167$$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
167.2.1.2$x^{2} + 334$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.3.0.1$x^{3} - x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$
167.3.0.1$x^{3} - x + 11$$1$$3$$0$$C_3$$[\ ]^{3}$