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Properties

Label 18.2.18420461686...3872.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{18}\cdot 7^{11}\cdot 1373^{4}$
Root discriminant $32.71$
Ramified primes $2, 7, 1373$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T888

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-823543, 0, -117649, 0, 16807, 0, 31213, 0, 4116, 0, -245, 0, -77, 0, -14, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 14*x^14 - 77*x^12 - 245*x^10 + 4116*x^8 + 31213*x^6 + 16807*x^4 - 117649*x^2 - 823543)
 
gp: K = bnfinit(x^18 - 14*x^14 - 77*x^12 - 245*x^10 + 4116*x^8 + 31213*x^6 + 16807*x^4 - 117649*x^2 - 823543, 1)
 

Normalized defining polynomial

\( x^{18} - 14 x^{14} - 77 x^{12} - 245 x^{10} + 4116 x^{8} + 31213 x^{6} + 16807 x^{4} - 117649 x^{2} - 823543 \)

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:  \(1842046168626834314431823872=2^{18}\cdot 7^{11}\cdot 1373^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.71$
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:  $2, 7, 1373$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{49} a^{8} - \frac{2}{7} a^{4} + \frac{3}{7} a^{2}$, $\frac{1}{49} a^{9} - \frac{2}{7} a^{5} + \frac{3}{7} a^{3}$, $\frac{1}{343} a^{10} - \frac{2}{49} a^{6} - \frac{11}{49} a^{4} + \frac{2}{7} a^{2}$, $\frac{1}{343} a^{11} - \frac{2}{49} a^{7} - \frac{11}{49} a^{5} + \frac{2}{7} a^{3}$, $\frac{1}{2401} a^{12} - \frac{2}{343} a^{8} - \frac{11}{343} a^{6} - \frac{5}{49} a^{4} - \frac{2}{7} a^{2}$, $\frac{1}{2401} a^{13} - \frac{2}{343} a^{9} - \frac{11}{343} a^{7} - \frac{5}{49} a^{5} - \frac{2}{7} a^{3}$, $\frac{1}{16807} a^{14} - \frac{2}{2401} a^{10} - \frac{11}{2401} a^{8} - \frac{5}{343} a^{6} + \frac{12}{49} a^{4} - \frac{1}{7} a^{2}$, $\frac{1}{16807} a^{15} - \frac{2}{2401} a^{11} - \frac{11}{2401} a^{9} - \frac{5}{343} a^{7} + \frac{12}{49} a^{5} - \frac{1}{7} a^{3}$, $\frac{1}{2034374625451} a^{16} + \frac{8293093}{290624946493} a^{14} - \frac{48524926}{290624946493} a^{12} + \frac{127432273}{290624946493} a^{10} + \frac{52393833}{5931121357} a^{8} - \frac{37435285}{5931121357} a^{6} + \frac{79809061}{847303051} a^{4} + \frac{51710927}{121043293} a^{2} + \frac{4801066}{17291899}$, $\frac{1}{2034374625451} a^{17} + \frac{8293093}{290624946493} a^{15} - \frac{48524926}{290624946493} a^{13} + \frac{127432273}{290624946493} a^{11} + \frac{52393833}{5931121357} a^{9} - \frac{37435285}{5931121357} a^{7} + \frac{79809061}{847303051} a^{5} + \frac{51710927}{121043293} a^{3} + \frac{4801066}{17291899} a$

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:  $9$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1747982.45921 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T888:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 362880
The 36 conjugacy class representatives for t18n888
Character table for t18n888 is not computed

Intermediate fields

\(\Q(\sqrt{7}) \), 9.1.92371321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ R $18$ $18$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
1373Data not computed