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Properties

Label 13.13.9269664678...3521.1
Degree $13$
Signature $[13, 0]$
Discriminant $677^{12}$
Root discriminant $410.06$
Ramified prime $677$
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![-1052321, -140970591, 238916059, -43862553, -61633266, 13888428, 4472586, -1071164, -114643, 31073, 765, -312, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 312*x^11 + 765*x^10 + 31073*x^9 - 114643*x^8 - 1071164*x^7 + 4472586*x^6 + 13888428*x^5 - 61633266*x^4 - 43862553*x^3 + 238916059*x^2 - 140970591*x - 1052321)
 
gp: K = bnfinit(x^13 - x^12 - 312*x^11 + 765*x^10 + 31073*x^9 - 114643*x^8 - 1071164*x^7 + 4472586*x^6 + 13888428*x^5 - 61633266*x^4 - 43862553*x^3 + 238916059*x^2 - 140970591*x - 1052321, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 312 x^{11} + 765 x^{10} + 31073 x^{9} - 114643 x^{8} - 1071164 x^{7} + 4472586 x^{6} + 13888428 x^{5} - 61633266 x^{4} - 43862553 x^{3} + 238916059 x^{2} - 140970591 x - 1052321 \)

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:  \(9269664678331989431355838883693521=677^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $410.06$
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:  $677$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(677\)
Dirichlet character group:    $\lbrace$$\chi_{677}(1,·)$, $\chi_{677}(263,·)$, $\chi_{677}(40,·)$, $\chi_{677}(457,·)$, $\chi_{677}(362,·)$, $\chi_{677}(333,·)$, $\chi_{677}(365,·)$, $\chi_{677}(115,·)$, $\chi_{677}(533,·)$, $\chi_{677}(246,·)$, $\chi_{677}(538,·)$, $\chi_{677}(426,·)$, $\chi_{677}(383,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{251} a^{11} - \frac{120}{251} a^{10} - \frac{104}{251} a^{9} + \frac{1}{251} a^{8} - \frac{12}{251} a^{7} - \frac{30}{251} a^{6} + \frac{101}{251} a^{5} + \frac{24}{251} a^{4} - \frac{109}{251} a^{3} + \frac{72}{251} a^{2} - \frac{99}{251} a + \frac{39}{251}$, $\frac{1}{247991603590071823323312008680790511125929} a^{12} - \frac{173841080938944349711428293033003149984}{247991603590071823323312008680790511125929} a^{11} - \frac{244037454696820741331980859250437301416}{247991603590071823323312008680790511125929} a^{10} + \frac{4836768563075012632581195297654652874744}{247991603590071823323312008680790511125929} a^{9} - \frac{2110954914885104158062441012147337476653}{247991603590071823323312008680790511125929} a^{8} - \frac{42320734222505387798445750702518953638722}{247991603590071823323312008680790511125929} a^{7} - \frac{82542071030247337364090651235791583823022}{247991603590071823323312008680790511125929} a^{6} - \frac{123132428440511594133830528635691380864511}{247991603590071823323312008680790511125929} a^{5} - \frac{90481097186908731358539796652060698663923}{247991603590071823323312008680790511125929} a^{4} - \frac{101356725092329349913138192730394852488826}{247991603590071823323312008680790511125929} a^{3} - \frac{82074285220565530803700852812698843769960}{247991603590071823323312008680790511125929} a^{2} + \frac{23860956993005583662825549883550703823636}{247991603590071823323312008680790511125929} a + \frac{11388584444425828158165716368050814706566}{247991603590071823323312008680790511125929}$

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:  \( 3454123871256.3633 \) (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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
677Data not computed