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Properties

Label 18.10.6838720957...8977.1
Degree $18$
Signature $[10, 4]$
Discriminant $7^{12}\cdot 83^{4}\cdot 97\cdot 181^{4}$
Root discriminant $39.99$
Ramified primes $7, 83, 97, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-461, 3092, 12122, 6924, 11423, 21162, 1095, 988, 4397, -5122, 2, 445, -819, 448, -40, -1, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 7*x^16 - x^15 - 40*x^14 + 448*x^13 - 819*x^12 + 445*x^11 + 2*x^10 - 5122*x^9 + 4397*x^8 + 988*x^7 + 1095*x^6 + 21162*x^5 + 11423*x^4 + 6924*x^3 + 12122*x^2 + 3092*x - 461)
 
gp: K = bnfinit(x^18 - 6*x^17 + 7*x^16 - x^15 - 40*x^14 + 448*x^13 - 819*x^12 + 445*x^11 + 2*x^10 - 5122*x^9 + 4397*x^8 + 988*x^7 + 1095*x^6 + 21162*x^5 + 11423*x^4 + 6924*x^3 + 12122*x^2 + 3092*x - 461, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 7 x^{16} - x^{15} - 40 x^{14} + 448 x^{13} - 819 x^{12} + 445 x^{11} + 2 x^{10} - 5122 x^{9} + 4397 x^{8} + 988 x^{7} + 1095 x^{6} + 21162 x^{5} + 11423 x^{4} + 6924 x^{3} + 12122 x^{2} + 3092 x - 461 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(68387209571737753818736598977=7^{12}\cdot 83^{4}\cdot 97\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.99$
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:  $7, 83, 97, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} - \frac{2}{7} a^{15} + \frac{2}{7} a^{14} + \frac{1}{7} a^{13} - \frac{2}{7} a^{12} + \frac{2}{7} a^{11} + \frac{2}{7} a^{10} - \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{10407749863742300931235200915342480191} a^{17} - \frac{482552858811319973933516092024417537}{10407749863742300931235200915342480191} a^{16} + \frac{3894610926992222341779797008682604378}{10407749863742300931235200915342480191} a^{15} + \frac{1473407492967898415287259314413441881}{10407749863742300931235200915342480191} a^{14} - \frac{2492063279734197468858168388058349493}{10407749863742300931235200915342480191} a^{13} + \frac{898829149419138066206948754946998690}{10407749863742300931235200915342480191} a^{12} + \frac{4715758744789528593039835620889961654}{10407749863742300931235200915342480191} a^{11} + \frac{32256052034691990013467599319702192}{212403058443720427168065324802907759} a^{10} - \frac{4867266068494049292546689107419326854}{10407749863742300931235200915342480191} a^{9} - \frac{609124370270724116655437061499336834}{10407749863742300931235200915342480191} a^{8} - \frac{4592612629623271446817223674838138579}{10407749863742300931235200915342480191} a^{7} - \frac{2799499038541550845778262133203717881}{10407749863742300931235200915342480191} a^{6} - \frac{2752147405249299803012963188437647240}{10407749863742300931235200915342480191} a^{5} + \frac{3806611977668563657854320914488996063}{10407749863742300931235200915342480191} a^{4} + \frac{2908695100281884984357958236417355495}{10407749863742300931235200915342480191} a^{3} - \frac{1328591021674995568317002266520834107}{10407749863742300931235200915342480191} a^{2} + \frac{124749456544424525693869260174805075}{10407749863742300931235200915342480191} a + \frac{4060183944785818119622590030303932249}{10407749863742300931235200915342480191}$

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:  $13$
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:  \( 117573040.211 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 192 conjugacy class representatives for t18n839 are not computed
Character table for t18n839 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ $18$ $18$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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
$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}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.3.2.1$x^{3} - 83$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
83.3.2.1$x^{3} - 83$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
97Data not computed
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$