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Properties

Label 13.13.4580101374...5681.1
Degree $13$
Signature $[13, 0]$
Discriminant $937^{12}$
Root discriminant $553.53$
Ramified prime $937$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{13}$ (as 13T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-190078187, -520002704, -420910202, -38878963, 87214801, 27799988, -3276218, -2039413, -37046, 46006, 1203, -432, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 432*x^11 + 1203*x^10 + 46006*x^9 - 37046*x^8 - 2039413*x^7 - 3276218*x^6 + 27799988*x^5 + 87214801*x^4 - 38878963*x^3 - 420910202*x^2 - 520002704*x - 190078187)
 
gp: K = bnfinit(x^13 - x^12 - 432*x^11 + 1203*x^10 + 46006*x^9 - 37046*x^8 - 2039413*x^7 - 3276218*x^6 + 27799988*x^5 + 87214801*x^4 - 38878963*x^3 - 420910202*x^2 - 520002704*x - 190078187, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 432 x^{11} + 1203 x^{10} + 46006 x^{9} - 37046 x^{8} - 2039413 x^{7} - 3276218 x^{6} + 27799988 x^{5} + 87214801 x^{4} - 38878963 x^{3} - 420910202 x^{2} - 520002704 x - 190078187 \)

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:  \(458010137458255714802917980980035681=937^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $553.53$
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:  $937$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(937\)
Dirichlet character group:    $\lbrace$$\chi_{937}(512,·)$, $\chi_{937}(1,·)$, $\chi_{937}(227,·)$, $\chi_{937}(36,·)$, $\chi_{937}(743,·)$, $\chi_{937}(359,·)$, $\chi_{937}(911,·)$, $\chi_{937}(657,·)$, $\chi_{937}(931,·)$, $\chi_{937}(629,·)$, $\chi_{937}(676,·)$, $\chi_{937}(721,·)$, $\chi_{937}(156,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13} a^{8} + \frac{5}{13} a^{7} + \frac{5}{13} a^{6} - \frac{1}{13} a^{5} + \frac{6}{13} a^{4} + \frac{4}{13} a^{3} + \frac{1}{13} a^{2} + \frac{5}{13} a$, $\frac{1}{13} a^{9} + \frac{6}{13} a^{7} - \frac{2}{13} a^{5} - \frac{6}{13} a^{3} + \frac{1}{13} a$, $\frac{1}{13} a^{10} - \frac{4}{13} a^{7} - \frac{6}{13} a^{6} + \frac{6}{13} a^{5} - \frac{3}{13} a^{4} + \frac{2}{13} a^{3} - \frac{5}{13} a^{2} - \frac{4}{13} a$, $\frac{1}{13} a^{11} + \frac{1}{13} a^{7} + \frac{6}{13} a^{5} - \frac{2}{13} a^{3} - \frac{6}{13} a$, $\frac{1}{891029064251339138293469075369393406023} a^{12} + \frac{25997500921253074733374179507804347058}{891029064251339138293469075369393406023} a^{11} + \frac{16239845924160948835310363935386889786}{891029064251339138293469075369393406023} a^{10} + \frac{1309927085208737250460133215652118434}{38740394097884310360585611972582322001} a^{9} - \frac{3629600035017329823260285700981242998}{891029064251339138293469075369393406023} a^{8} - \frac{1916567507354831040257380595961447057}{13298941257482673705872672766707364269} a^{7} + \frac{3257361921381006560016330368471974421}{13298941257482673705872672766707364269} a^{6} - \frac{95555447167992325250066220014543450743}{891029064251339138293469075369393406023} a^{5} - \frac{22291742392583831975351108290281930340}{891029064251339138293469075369393406023} a^{4} - \frac{89593371559550982460780777093637670926}{891029064251339138293469075369393406023} a^{3} - \frac{369244200074925091280716345369977237096}{891029064251339138293469075369393406023} a^{2} + \frac{19675026752412437119380474408094210284}{68540697250103010637959159643799492771} a - \frac{8868616155529606128375103205365864}{37721902724327468705536136292679963}$

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:  \( 169264238822310.44 \) (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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
937Data not computed