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Properties

Label 18.6.41619125031...1857.1
Degree $18$
Signature $[6, 6]$
Discriminant $19^{16}\cdot 113^{3}$
Root discriminant $30.12$
Ramified primes $19, 113$
Class number $1$
Class group Trivial
Galois group $C_2\times C_2^2:C_9$ (as 18T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4181, -20001, 41301, -53722, 46651, -22543, -1829, 14901, -15321, 10127, -5196, 2227, -833, 242, -12, -42, 27, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 27*x^16 - 42*x^15 - 12*x^14 + 242*x^13 - 833*x^12 + 2227*x^11 - 5196*x^10 + 10127*x^9 - 15321*x^8 + 14901*x^7 - 1829*x^6 - 22543*x^5 + 46651*x^4 - 53722*x^3 + 41301*x^2 - 20001*x + 4181)
 
gp: K = bnfinit(x^18 - 8*x^17 + 27*x^16 - 42*x^15 - 12*x^14 + 242*x^13 - 833*x^12 + 2227*x^11 - 5196*x^10 + 10127*x^9 - 15321*x^8 + 14901*x^7 - 1829*x^6 - 22543*x^5 + 46651*x^4 - 53722*x^3 + 41301*x^2 - 20001*x + 4181, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 27 x^{16} - 42 x^{15} - 12 x^{14} + 242 x^{13} - 833 x^{12} + 2227 x^{11} - 5196 x^{10} + 10127 x^{9} - 15321 x^{8} + 14901 x^{7} - 1829 x^{6} - 22543 x^{5} + 46651 x^{4} - 53722 x^{3} + 41301 x^{2} - 20001 x + 4181 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(416191250312479879983411857=19^{16}\cdot 113^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.12$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{37} a^{16} + \frac{12}{37} a^{15} + \frac{9}{37} a^{14} + \frac{2}{37} a^{13} - \frac{15}{37} a^{11} + \frac{14}{37} a^{10} + \frac{13}{37} a^{9} - \frac{1}{37} a^{8} - \frac{18}{37} a^{7} + \frac{6}{37} a^{6} + \frac{18}{37} a^{5} + \frac{17}{37} a^{4} + \frac{15}{37} a^{3} + \frac{15}{37} a^{2} - \frac{16}{37} a$, $\frac{1}{1050783468577241060475023973507319} a^{17} + \frac{605884770921904589739570193698}{1050783468577241060475023973507319} a^{16} - \frac{3482825510387472346876476275372}{9298968748471159827212601535463} a^{15} + \frac{353248948772792903217203096736280}{1050783468577241060475023973507319} a^{14} - \frac{240226509118543563301391388283661}{1050783468577241060475023973507319} a^{13} - \frac{375696491105143213609354749809281}{1050783468577241060475023973507319} a^{12} - \frac{519136461352475103187276223006485}{1050783468577241060475023973507319} a^{11} + \frac{299987703954357935523575924648917}{1050783468577241060475023973507319} a^{10} + \frac{3392685522396871596246459318687}{6958830917730073248178966711969} a^{9} + \frac{219521312072241862719240754196751}{1050783468577241060475023973507319} a^{8} + \frac{520730953193334949915039133502851}{1050783468577241060475023973507319} a^{7} + \frac{361807738175692740803730366219181}{1050783468577241060475023973507319} a^{6} + \frac{136539987409730029887833115964115}{1050783468577241060475023973507319} a^{5} + \frac{14227947368760612477717304998228}{1050783468577241060475023973507319} a^{4} + \frac{63115891836919933834226248433825}{1050783468577241060475023973507319} a^{3} + \frac{201460577389768151105962498601}{6958830917730073248178966711969} a^{2} - \frac{3075134133296082638182530395990}{9298968748471159827212601535463} a - \frac{16930900466123048534112646322}{251323479688409725059800041499}$

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:  $11$
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:  \( 1701924.22156 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2:C_9$ (as 18T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 24 conjugacy class representatives for $C_2\times C_2^2:C_9$
Character table for $C_2\times C_2^2:C_9$ is not computed

Intermediate fields

3.3.361.1, 6.2.14726273.1, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
19Data not computed
$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$$0$Trivial$[\ ]$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.2$x^{2} + 339$$2$$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}$