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Properties

Label 18.18.3830034706...5792.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 19^{16}\cdot 37^{3}$
Root discriminant $50.01$
Ramified primes $2, 19, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![647, -12592, -16553, 212908, -223063, -313520, 412283, 202114, -281919, -72124, 95078, 15264, -16889, -1850, 1544, 108, -66, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 66*x^16 + 108*x^15 + 1544*x^14 - 1850*x^13 - 16889*x^12 + 15264*x^11 + 95078*x^10 - 72124*x^9 - 281919*x^8 + 202114*x^7 + 412283*x^6 - 313520*x^5 - 223063*x^4 + 212908*x^3 - 16553*x^2 - 12592*x + 647)
 
gp: K = bnfinit(x^18 - 2*x^17 - 66*x^16 + 108*x^15 + 1544*x^14 - 1850*x^13 - 16889*x^12 + 15264*x^11 + 95078*x^10 - 72124*x^9 - 281919*x^8 + 202114*x^7 + 412283*x^6 - 313520*x^5 - 223063*x^4 + 212908*x^3 - 16553*x^2 - 12592*x + 647, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 66 x^{16} + 108 x^{15} + 1544 x^{14} - 1850 x^{13} - 16889 x^{12} + 15264 x^{11} + 95078 x^{10} - 72124 x^{9} - 281919 x^{8} + 202114 x^{7} + 412283 x^{6} - 313520 x^{5} - 223063 x^{4} + 212908 x^{3} - 16553 x^{2} - 12592 x + 647 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3830034706318154794675914145792=2^{18}\cdot 19^{16}\cdot 37^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.01$
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:  $2, 19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{113} a^{16} + \frac{9}{113} a^{15} + \frac{3}{113} a^{14} - \frac{16}{113} a^{13} + \frac{35}{113} a^{12} + \frac{32}{113} a^{11} + \frac{41}{113} a^{10} - \frac{48}{113} a^{9} - \frac{18}{113} a^{8} - \frac{31}{113} a^{7} - \frac{11}{113} a^{6} - \frac{25}{113} a^{5} + \frac{1}{113} a^{4} + \frac{25}{113} a^{3} + \frac{18}{113} a^{2} + \frac{29}{113} a - \frac{50}{113}$, $\frac{1}{84278131884111759842344849397472623} a^{17} - \frac{45432673966796734633667223797118}{84278131884111759842344849397472623} a^{16} + \frac{40125117206342112770728760451310166}{84278131884111759842344849397472623} a^{15} - \frac{31129519030202013051370287710566188}{84278131884111759842344849397472623} a^{14} + \frac{16216027771631122308936313850201701}{84278131884111759842344849397472623} a^{13} - \frac{33418341946964599794051543774970024}{84278131884111759842344849397472623} a^{12} + \frac{36729398170315193629989502975122520}{84278131884111759842344849397472623} a^{11} + \frac{28593058978085596824089003368612016}{84278131884111759842344849397472623} a^{10} + \frac{3880754659711246131619233062803454}{84278131884111759842344849397472623} a^{9} - \frac{2032496288778112420718130610791073}{84278131884111759842344849397472623} a^{8} + \frac{9781820093998736252266621561751894}{84278131884111759842344849397472623} a^{7} + \frac{31679275574951597366039533206801623}{84278131884111759842344849397472623} a^{6} + \frac{37249015883645039568733011777273502}{84278131884111759842344849397472623} a^{5} + \frac{24840418087377783499947945197237349}{84278131884111759842344849397472623} a^{4} + \frac{40390017586134492851932860644978995}{84278131884111759842344849397472623} a^{3} - \frac{35952663429739539019151499073816208}{84278131884111759842344849397472623} a^{2} - \frac{16243270373569445314143840271309042}{84278131884111759842344849397472623} a + \frac{29969122906989144954974027017339013}{84278131884111759842344849397472623}$

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:  $17$
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:  \( 2587028370.07 \) (assuming GRH)
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, 6.6.308600128.1, \(\Q(\zeta_{19})^+\)

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 R ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $18$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
2Data not computed
19Data not computed
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$