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Properties

Label 18.0.58116758997...6247.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,13^{9}\cdot 19^{17}$
Root discriminant $58.17$
Ramified primes $13, 19$
Class number $9774$ (GRH)
Class group $[9774]$ (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![7634353, -7260376, 7260376, -5390491, 5390491, -2647993, 2647993, -819661, 819661, -159430, 159430, -19381, 19381, -1426, 1426, -58, 58, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 58*x^16 - 58*x^15 + 1426*x^14 - 1426*x^13 + 19381*x^12 - 19381*x^11 + 159430*x^10 - 159430*x^9 + 819661*x^8 - 819661*x^7 + 2647993*x^6 - 2647993*x^5 + 5390491*x^4 - 5390491*x^3 + 7260376*x^2 - 7260376*x + 7634353)
 
gp: K = bnfinit(x^18 - x^17 + 58*x^16 - 58*x^15 + 1426*x^14 - 1426*x^13 + 19381*x^12 - 19381*x^11 + 159430*x^10 - 159430*x^9 + 819661*x^8 - 819661*x^7 + 2647993*x^6 - 2647993*x^5 + 5390491*x^4 - 5390491*x^3 + 7260376*x^2 - 7260376*x + 7634353, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 58 x^{16} - 58 x^{15} + 1426 x^{14} - 1426 x^{13} + 19381 x^{12} - 19381 x^{11} + 159430 x^{10} - 159430 x^{9} + 819661 x^{8} - 819661 x^{7} + 2647993 x^{6} - 2647993 x^{5} + 5390491 x^{4} - 5390491 x^{3} + 7260376 x^{2} - 7260376 x + 7634353 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-58116758997176374949719154916247=-\,13^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.17$
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:  $13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(247=13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{247}(1,·)$, $\chi_{247}(66,·)$, $\chi_{247}(131,·)$, $\chi_{247}(196,·)$, $\chi_{247}(118,·)$, $\chi_{247}(129,·)$, $\chi_{247}(12,·)$, $\chi_{247}(144,·)$, $\chi_{247}(90,·)$, $\chi_{247}(155,·)$, $\chi_{247}(92,·)$, $\chi_{247}(157,·)$, $\chi_{247}(103,·)$, $\chi_{247}(235,·)$, $\chi_{247}(51,·)$, $\chi_{247}(116,·)$, $\chi_{247}(181,·)$, $\chi_{247}(246,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2117473} a^{10} + \frac{562712}{2117473} a^{9} + \frac{30}{2117473} a^{8} + \frac{370913}{2117473} a^{7} + \frac{315}{2117473} a^{6} - \frac{896729}{2117473} a^{5} + \frac{1350}{2117473} a^{4} + \frac{540025}{2117473} a^{3} + \frac{2025}{2117473} a^{2} - \frac{572714}{2117473} a + \frac{486}{2117473}$, $\frac{1}{2117473} a^{11} + \frac{33}{2117473} a^{9} + \frac{429337}{2117473} a^{8} + \frac{396}{2117473} a^{7} - \frac{283277}{2117473} a^{6} + \frac{2079}{2117473} a^{5} + \frac{1051632}{2117473} a^{4} + \frac{4455}{2117473} a^{3} - \frac{864040}{2117473} a^{2} + \frac{2673}{2117473} a - \frac{324015}{2117473}$, $\frac{1}{2117473} a^{12} + \frac{917098}{2117473} a^{9} - \frac{594}{2117473} a^{8} + \frac{181432}{2117473} a^{7} - \frac{8316}{2117473} a^{6} + \frac{999067}{2117473} a^{5} - \frac{40095}{2117473} a^{4} + \frac{372392}{2117473} a^{3} - \frac{64152}{2117473} a^{2} - \frac{481710}{2117473} a - \frac{16038}{2117473}$, $\frac{1}{2117473} a^{13} - \frac{702}{2117473} a^{9} + \frac{195641}{2117473} a^{8} - \frac{11232}{2117473} a^{7} + \frac{89525}{2117473} a^{6} - \frac{66339}{2117473} a^{5} + \frac{1011797}{2117473} a^{4} - \frac{151632}{2117473} a^{3} - \frac{581339}{2117473} a^{2} - \frac{94770}{2117473} a - \frac{1040298}{2117473}$, $\frac{1}{2117473} a^{14} - \frac{747986}{2117473} a^{9} + \frac{9828}{2117473} a^{8} + \frac{21272}{2117473} a^{7} + \frac{154791}{2117473} a^{6} + \frac{397520}{2117473} a^{5} + \frac{796068}{2117473} a^{4} - \frac{511456}{2117473} a^{3} - \frac{790693}{2117473} a^{2} - \frac{765656}{2117473} a + \frac{341172}{2117473}$, $\frac{1}{2117473} a^{15} + \frac{12285}{2117473} a^{9} - \frac{831351}{2117473} a^{8} + \frac{221130}{2117473} a^{7} + \frac{973607}{2117473} a^{6} - \frac{724354}{2117473} a^{5} - \frac{764977}{2117473} a^{4} - \frac{917996}{2117473} a^{3} - \frac{87201}{2117473} a^{2} + \frac{14852}{2117473} a - \frac{684160}{2117473}$, $\frac{1}{2117473} a^{16} - \frac{198926}{2117473} a^{9} - \frac{147420}{2117473} a^{8} - \frac{1008175}{2117473} a^{7} - \frac{359183}{2117473} a^{6} + \frac{456242}{2117473} a^{5} - \frac{562962}{2117473} a^{4} - \frac{251417}{2117473} a^{3} + \frac{547403}{2117473} a^{2} + \frac{862024}{2117473} a + \frac{381909}{2117473}$, $\frac{1}{2117473} a^{17} - \frac{192780}{2117473} a^{9} + \frac{724659}{2117473} a^{8} + \frac{533570}{2117473} a^{7} - \frac{406258}{2117473} a^{6} - \frac{998077}{2117473} a^{5} - \frac{620388}{2117473} a^{4} - \frac{197156}{2117473} a^{3} - \frac{750169}{2117473} a^{2} - \frac{923436}{2117473} a - \frac{725722}{2117473}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{9774}$, which has order $9774$ (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:  $8$
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:  \( 22305.8950792 \) (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

\(\Q(\sqrt{-247}) \), 3.3.361.1, 6.0.5439989503.1, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
13Data not computed
19Data not computed