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Properties

Label 18.2.23898616113...8125.1
Degree $18$
Signature $[2, 8]$
Discriminant $5^{15}\cdot 23^{8}$
Root discriminant $15.41$
Ramified primes $5, 23$
Class number $1$
Class group Trivial
Galois group $D_{18}$ (as 18T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 20, 64, 64, 0, 0, 25, 60, 0, 0, -1, 15, 6, -4, 0, -1, 5, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 5*x^16 - x^15 - 4*x^13 + 6*x^12 + 15*x^11 - x^10 + 60*x^7 + 25*x^6 + 64*x^3 + 64*x^2 + 20*x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + 5*x^16 - x^15 - 4*x^13 + 6*x^12 + 15*x^11 - x^10 + 60*x^7 + 25*x^6 + 64*x^3 + 64*x^2 + 20*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 5 x^{16} - x^{15} - 4 x^{13} + 6 x^{12} + 15 x^{11} - x^{10} + 60 x^{7} + 25 x^{6} + 64 x^{3} + 64 x^{2} + 20 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2389861611358642578125=5^{15}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.41$
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:  $5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{14} + \frac{3}{7} a^{13} + \frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{133} a^{16} - \frac{2}{133} a^{15} - \frac{2}{19} a^{14} + \frac{20}{133} a^{13} + \frac{3}{133} a^{12} + \frac{25}{133} a^{11} + \frac{30}{133} a^{10} - \frac{53}{133} a^{9} - \frac{6}{133} a^{8} - \frac{53}{133} a^{7} + \frac{60}{133} a^{6} + \frac{25}{133} a^{5} - \frac{27}{133} a^{4} + \frac{46}{133} a^{3} + \frac{3}{133} a^{2} + \frac{45}{133} a + \frac{52}{133}$, $\frac{1}{3325036900717} a^{17} + \frac{11778030749}{3325036900717} a^{16} + \frac{32287422905}{3325036900717} a^{15} + \frac{974745058759}{3325036900717} a^{14} - \frac{200134429162}{3325036900717} a^{13} - \frac{225102781499}{3325036900717} a^{12} + \frac{150412542461}{475005271531} a^{11} + \frac{1010422044717}{3325036900717} a^{10} + \frac{50033342028}{3325036900717} a^{9} - \frac{1585202501755}{3325036900717} a^{8} - \frac{257057708056}{3325036900717} a^{7} + \frac{429332549701}{3325036900717} a^{6} + \frac{1074834199997}{3325036900717} a^{5} + \frac{1224340117828}{3325036900717} a^{4} + \frac{1075622607015}{3325036900717} a^{3} - \frac{675000409416}{3325036900717} a^{2} - \frac{1487841397450}{3325036900717} a - \frac{991283206661}{3325036900717}$

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:  $9$
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:  \( 2645.14483524 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{18}$ (as 18T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $D_{18}$
Character table for $D_{18}$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.23.1, 6.2.66125.1, 9.1.4372515625.1

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

Sibling fields

Galois closure: data not computed
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 $18$ $18$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.5.1$x^{6} - 5$$6$$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}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$