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Properties

Label 18.10.3683108409...8689.1
Degree $18$
Signature $[10, 4]$
Discriminant $19^{16}\cdot 113^{2}$
Root discriminant $23.16$
Ramified primes $19, 113$
Class number $1$
Class group Trivial
Galois group 18T177

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 21, -110, 151, 138, -692, 764, -309, -141, 624, -583, 84, 149, -103, 23, 5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 5*x^16 + 23*x^15 - 103*x^14 + 149*x^13 + 84*x^12 - 583*x^11 + 624*x^10 - 141*x^9 - 309*x^8 + 764*x^7 - 692*x^6 + 138*x^5 + 151*x^4 - 110*x^3 + 21*x^2 + 3*x - 1)
 
gp: K = bnfinit(x^18 - 5*x^17 + 5*x^16 + 23*x^15 - 103*x^14 + 149*x^13 + 84*x^12 - 583*x^11 + 624*x^10 - 141*x^9 - 309*x^8 + 764*x^7 - 692*x^6 + 138*x^5 + 151*x^4 - 110*x^3 + 21*x^2 + 3*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 5 x^{16} + 23 x^{15} - 103 x^{14} + 149 x^{13} + 84 x^{12} - 583 x^{11} + 624 x^{10} - 141 x^{9} - 309 x^{8} + 764 x^{7} - 692 x^{6} + 138 x^{5} + 151 x^{4} - 110 x^{3} + 21 x^{2} + 3 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3683108409844954690118689=19^{16}\cdot 113^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.16$
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:  $19, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{37} a^{14} + \frac{10}{37} a^{13} - \frac{15}{37} a^{12} + \frac{3}{37} a^{11} + \frac{13}{37} a^{10} - \frac{16}{37} a^{9} + \frac{18}{37} a^{8} - \frac{8}{37} a^{7} - \frac{4}{37} a^{6} - \frac{3}{37} a^{5} + \frac{13}{37} a^{4} - \frac{12}{37} a^{3} - \frac{11}{37} a^{2} + \frac{12}{37} a + \frac{15}{37}$, $\frac{1}{37} a^{15} - \frac{4}{37} a^{13} + \frac{5}{37} a^{12} - \frac{17}{37} a^{11} + \frac{2}{37} a^{10} - \frac{7}{37} a^{9} - \frac{3}{37} a^{8} + \frac{2}{37} a^{7} + \frac{6}{37} a^{5} + \frac{6}{37} a^{4} - \frac{2}{37} a^{3} + \frac{11}{37} a^{2} + \frac{6}{37} a - \frac{2}{37}$, $\frac{1}{37} a^{16} + \frac{8}{37} a^{13} - \frac{3}{37} a^{12} + \frac{14}{37} a^{11} + \frac{8}{37} a^{10} + \frac{7}{37} a^{9} + \frac{5}{37} a^{7} - \frac{10}{37} a^{6} - \frac{6}{37} a^{5} + \frac{13}{37} a^{4} - \frac{1}{37} a^{2} + \frac{9}{37} a - \frac{14}{37}$, $\frac{1}{76998823957625813} a^{17} - \frac{399469617863490}{76998823957625813} a^{16} + \frac{672300783143947}{76998823957625813} a^{15} - \frac{77792625282744}{76998823957625813} a^{14} - \frac{9890854384862165}{76998823957625813} a^{13} + \frac{34280262183499178}{76998823957625813} a^{12} - \frac{25063808114239185}{76998823957625813} a^{11} - \frac{2610560911655476}{76998823957625813} a^{10} - \frac{16259137226463119}{76998823957625813} a^{9} + \frac{30742381514346674}{76998823957625813} a^{8} + \frac{6833474516834911}{76998823957625813} a^{7} - \frac{19146165942159894}{76998823957625813} a^{6} + \frac{31332679431565844}{76998823957625813} a^{5} + \frac{712136708772604}{2081049296152049} a^{4} - \frac{29768887310303105}{76998823957625813} a^{3} + \frac{17250213710762724}{76998823957625813} a^{2} - \frac{7344012750553674}{76998823957625813} a + \frac{24820708427495937}{76998823957625813}$

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:  $13$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 441589.256764 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T177:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 16 conjugacy class representatives for t18n177
Character table for t18n177

Intermediate fields

3.3.361.1, \(\Q(\zeta_{19})^+\)

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$