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Properties

Label 18.12.1614537673...3639.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{12}\cdot 22679^{3}$
Root discriminant $19.47$
Ramified primes $7, 22679$
Class number $1$
Class group Trivial
Galois group 18T207

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -11, 44, -41, -242, 858, -980, 18, 704, -308, 55, -266, 115, 170, -141, 9, 24, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 24*x^16 + 9*x^15 - 141*x^14 + 170*x^13 + 115*x^12 - 266*x^11 + 55*x^10 - 308*x^9 + 704*x^8 + 18*x^7 - 980*x^6 + 858*x^5 - 242*x^4 - 41*x^3 + 44*x^2 - 11*x + 1)
 
gp: K = bnfinit(x^18 - 9*x^17 + 24*x^16 + 9*x^15 - 141*x^14 + 170*x^13 + 115*x^12 - 266*x^11 + 55*x^10 - 308*x^9 + 704*x^8 + 18*x^7 - 980*x^6 + 858*x^5 - 242*x^4 - 41*x^3 + 44*x^2 - 11*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 24 x^{16} + 9 x^{15} - 141 x^{14} + 170 x^{13} + 115 x^{12} - 266 x^{11} + 55 x^{10} - 308 x^{9} + 704 x^{8} + 18 x^{7} - 980 x^{6} + 858 x^{5} - 242 x^{4} - 41 x^{3} + 44 x^{2} - 11 x + 1 \)

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:  \(-161453767328118264113639=-\,7^{12}\cdot 22679^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.47$
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:  $7, 22679$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7312335294151} a^{17} - \frac{1135076181209}{7312335294151} a^{16} + \frac{1829963536136}{7312335294151} a^{15} + \frac{3315956732765}{7312335294151} a^{14} + \frac{796880584715}{7312335294151} a^{13} + \frac{1184438009091}{7312335294151} a^{12} + \frac{1237088366408}{7312335294151} a^{11} + \frac{3296448809479}{7312335294151} a^{10} - \frac{2871482297997}{7312335294151} a^{9} + \frac{1380888636732}{7312335294151} a^{8} + \frac{746826996328}{7312335294151} a^{7} + \frac{3166217574969}{7312335294151} a^{6} - \frac{402545410840}{7312335294151} a^{5} - \frac{13306920224}{7312335294151} a^{4} - \frac{1392925285092}{7312335294151} a^{3} + \frac{1981780889925}{7312335294151} a^{2} + \frac{1108548468370}{7312335294151} a + \frac{1724166228059}{7312335294151}$

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:  $14$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 110975.912853 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T207:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 17 conjugacy class representatives for t18n207
Character table for t18n207

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.54452279.1, 9.7.2668161671.1 x3

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

Sibling fields

Degree 9 sibling: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
22679Data not computed