↑

Properties

Label 13.13.5712743366...0001.1
Degree $13$
Signature $[13, 0]$
Discriminant $443^{12}$
Root discriminant $277.23$
Ramified prime $443$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{13}$ (as 13T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![144209, 896956, -3051848, -4904338, 1207646, 1686466, -161679, -208418, 9116, 10752, -181, -204, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 204*x^11 - 181*x^10 + 10752*x^9 + 9116*x^8 - 208418*x^7 - 161679*x^6 + 1686466*x^5 + 1207646*x^4 - 4904338*x^3 - 3051848*x^2 + 896956*x + 144209)
 
gp: K = bnfinit(x^13 - x^12 - 204*x^11 - 181*x^10 + 10752*x^9 + 9116*x^8 - 208418*x^7 - 161679*x^6 + 1686466*x^5 + 1207646*x^4 - 4904338*x^3 - 3051848*x^2 + 896956*x + 144209, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 204 x^{11} - 181 x^{10} + 10752 x^{9} + 9116 x^{8} - 208418 x^{7} - 161679 x^{6} + 1686466 x^{5} + 1207646 x^{4} - 4904338 x^{3} - 3051848 x^{2} + 896956 x + 144209 \)

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

Invariants

Degree:  $13$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[13, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57127433662862356193722241010001=443^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $277.23$
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:  $443$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(443\)
Dirichlet character group:    $\lbrace$$\chi_{443}(1,·)$, $\chi_{443}(35,·)$, $\chi_{443}(356,·)$, $\chi_{443}(38,·)$, $\chi_{443}(238,·)$, $\chi_{443}(115,·)$, $\chi_{443}(56,·)$, $\chi_{443}(339,·)$, $\chi_{443}(184,·)$, $\chi_{443}(378,·)$, $\chi_{443}(347,·)$, $\chi_{443}(188,·)$, $\chi_{443}(383,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{13} a^{9} + \frac{1}{13} a^{8} + \frac{1}{13} a^{6} - \frac{5}{13} a^{5} - \frac{1}{13} a^{4} + \frac{6}{13} a^{3} + \frac{2}{13} a^{2} - \frac{5}{13} a$, $\frac{1}{13} a^{10} - \frac{1}{13} a^{8} + \frac{1}{13} a^{7} - \frac{6}{13} a^{6} + \frac{4}{13} a^{5} - \frac{6}{13} a^{4} - \frac{4}{13} a^{3} + \frac{6}{13} a^{2} + \frac{5}{13} a$, $\frac{1}{13} a^{11} + \frac{2}{13} a^{8} - \frac{6}{13} a^{7} + \frac{5}{13} a^{6} + \frac{2}{13} a^{5} - \frac{5}{13} a^{4} - \frac{1}{13} a^{3} - \frac{6}{13} a^{2} - \frac{5}{13} a$, $\frac{1}{432722201776761950983227367994717} a^{12} + \frac{6132693084373423328017707127835}{432722201776761950983227367994717} a^{11} + \frac{8458462850465764937788002340981}{432722201776761950983227367994717} a^{10} - \frac{493414414981056712922091778245}{33286323213597073152555951384209} a^{9} - \frac{65935521193329922084561157670291}{432722201776761950983227367994717} a^{8} - \frac{16297117949662467759009940308987}{432722201776761950983227367994717} a^{7} + \frac{20666775024114935539912340017631}{432722201776761950983227367994717} a^{6} + \frac{59482574856588687674043583015858}{432722201776761950983227367994717} a^{5} + \frac{139038000495827761297681830144415}{432722201776761950983227367994717} a^{4} + \frac{67702185105869952054654645782006}{432722201776761950983227367994717} a^{3} - \frac{1517923257468752235281606723322}{5927701394202218506619552986229} a^{2} - \frac{53958423060799896815628012672323}{432722201776761950983227367994717} a - \frac{11704460423143117442598855999542}{33286323213597073152555951384209}$

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:  $12$
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:  \( 983737918371.0616 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{13}$ (as 13T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 13
The 13 conjugacy class representatives for $C_{13}$
Character table for $C_{13}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{13}$

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
443Data not computed