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Properties

Label 18.0.97577143684...9456.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{12}\cdot 769^{5}$
Root discriminant $46.35$
Ramified primes $2, 7, 769$
Class number $792$ (GRH)
Class group $[792]$ (GRH)
Galois group 18T696

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![769, 0, 5529, 0, 15974, 0, 24144, 0, 20724, 0, 10264, 0, 2878, 0, 442, 0, 34, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 34*x^16 + 442*x^14 + 2878*x^12 + 10264*x^10 + 20724*x^8 + 24144*x^6 + 15974*x^4 + 5529*x^2 + 769)
 
gp: K = bnfinit(x^18 + 34*x^16 + 442*x^14 + 2878*x^12 + 10264*x^10 + 20724*x^8 + 24144*x^6 + 15974*x^4 + 5529*x^2 + 769, 1)
 

Normalized defining polynomial

\( x^{18} + 34 x^{16} + 442 x^{14} + 2878 x^{12} + 10264 x^{10} + 20724 x^{8} + 24144 x^{6} + 15974 x^{4} + 5529 x^{2} + 769 \)

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:  \(-975771436849265818910759059456=-\,2^{18}\cdot 7^{12}\cdot 769^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.35$
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:  $2, 7, 769$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{10} - \frac{2}{7} a^{8} + \frac{3}{7} a^{4} - \frac{1}{7} a^{2} + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{11} - \frac{2}{7} a^{9} + \frac{3}{7} a^{5} - \frac{1}{7} a^{3} + \frac{1}{7} a$, $\frac{1}{49} a^{14} + \frac{3}{49} a^{12} + \frac{23}{49} a^{10} + \frac{11}{49} a^{8} + \frac{24}{49} a^{6} - \frac{3}{7} a^{4} + \frac{17}{49} a^{2} - \frac{2}{49}$, $\frac{1}{49} a^{15} + \frac{3}{49} a^{13} + \frac{23}{49} a^{11} + \frac{11}{49} a^{9} + \frac{24}{49} a^{7} - \frac{3}{7} a^{5} + \frac{17}{49} a^{3} - \frac{2}{49} a$, $\frac{1}{24353} a^{16} - \frac{69}{24353} a^{14} + \frac{591}{24353} a^{12} - \frac{830}{3479} a^{10} + \frac{10306}{24353} a^{8} + \frac{2906}{24353} a^{6} - \frac{3812}{24353} a^{4} - \frac{5391}{24353} a^{2} + \frac{4162}{24353}$, $\frac{1}{24353} a^{17} - \frac{69}{24353} a^{15} + \frac{591}{24353} a^{13} - \frac{830}{3479} a^{11} + \frac{10306}{24353} a^{9} + \frac{2906}{24353} a^{7} - \frac{3812}{24353} a^{5} - \frac{5391}{24353} a^{3} + \frac{4162}{24353} a$

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

Class group and class number

$C_{792}$, which has order $792$ (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:  \( 60132.7264179 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T696:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 192 conjugacy class representatives for t18n696 are not computed
Character table for t18n696 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.69573030289.1

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 R $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
$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}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
769Data not computed