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Properties

Label 18.18.1488294338...3712.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{9}\cdot 19^{16}$
Root discriminant $47.45$
Ramified primes $2, 3, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -40, 53, 2054, -4165, -19276, 26450, 38696, -38251, -26642, 20953, 7992, -5337, -1150, 680, 78, -42, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 42*x^16 + 78*x^15 + 680*x^14 - 1150*x^13 - 5337*x^12 + 7992*x^11 + 20953*x^10 - 26642*x^9 - 38251*x^8 + 38696*x^7 + 26450*x^6 - 19276*x^5 - 4165*x^4 + 2054*x^3 + 53*x^2 - 40*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 42*x^16 + 78*x^15 + 680*x^14 - 1150*x^13 - 5337*x^12 + 7992*x^11 + 20953*x^10 - 26642*x^9 - 38251*x^8 + 38696*x^7 + 26450*x^6 - 19276*x^5 - 4165*x^4 + 2054*x^3 + 53*x^2 - 40*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 42 x^{16} + 78 x^{15} + 680 x^{14} - 1150 x^{13} - 5337 x^{12} + 7992 x^{11} + 20953 x^{10} - 26642 x^{9} - 38251 x^{8} + 38696 x^{7} + 26450 x^{6} - 19276 x^{5} - 4165 x^{4} + 2054 x^{3} + 53 x^{2} - 40 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1488294338429317924379721203712=2^{18}\cdot 3^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.45$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(228=2^{2}\cdot 3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{228}(1,·)$, $\chi_{228}(131,·)$, $\chi_{228}(73,·)$, $\chi_{228}(11,·)$, $\chi_{228}(83,·)$, $\chi_{228}(85,·)$, $\chi_{228}(23,·)$, $\chi_{228}(25,·)$, $\chi_{228}(157,·)$, $\chi_{228}(35,·)$, $\chi_{228}(169,·)$, $\chi_{228}(47,·)$, $\chi_{228}(49,·)$, $\chi_{228}(119,·)$, $\chi_{228}(121,·)$, $\chi_{228}(61,·)$, $\chi_{228}(215,·)$, $\chi_{228}(191,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{37} a^{16} - \frac{15}{37} a^{15} + \frac{6}{37} a^{14} - \frac{15}{37} a^{13} - \frac{7}{37} a^{12} - \frac{1}{37} a^{11} - \frac{3}{37} a^{10} + \frac{1}{37} a^{9} - \frac{5}{37} a^{8} - \frac{10}{37} a^{7} - \frac{16}{37} a^{6} + \frac{7}{37} a^{5} - \frac{1}{37} a^{4} - \frac{16}{37} a^{3} + \frac{1}{37} a^{2} - \frac{10}{37} a - \frac{1}{37}$, $\frac{1}{5895184715274613315301251739} a^{17} + \frac{11364961519041214338917871}{5895184715274613315301251739} a^{16} + \frac{524462888200035116250848850}{5895184715274613315301251739} a^{15} - \frac{1038413124976726648240608816}{5895184715274613315301251739} a^{14} + \frac{2553049152655949071621836494}{5895184715274613315301251739} a^{13} - \frac{2725950507008945780243101596}{5895184715274613315301251739} a^{12} - \frac{993889325932283711369031646}{5895184715274613315301251739} a^{11} + \frac{1175767669748679325120218566}{5895184715274613315301251739} a^{10} - \frac{527845586311427606343403031}{5895184715274613315301251739} a^{9} - \frac{100495570222810866008428191}{5895184715274613315301251739} a^{8} + \frac{2672462168653905844282874823}{5895184715274613315301251739} a^{7} - \frac{551506653931246034594202430}{5895184715274613315301251739} a^{6} + \frac{351013468754412786345824347}{5895184715274613315301251739} a^{5} + \frac{2425957254026241082414890027}{5895184715274613315301251739} a^{4} - \frac{819685142869367623467201276}{5895184715274613315301251739} a^{3} + \frac{1141567858137289380088739412}{5895184715274613315301251739} a^{2} + \frac{2347186764967134712754781572}{5895184715274613315301251739} a - \frac{2259716292336653450103366149}{5895184715274613315301251739}$

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:  \( 1824767825.56 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{3}) \), 3.3.361.1, 6.6.225194688.1, \(\Q(\zeta_{19})^+\)

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 R $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
3Data not computed
19Data not computed