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Properties

Label 13.13.5428007703...5361.1
Degree $13$
Signature $[13, 0]$
Discriminant $13^{24}$
Root discriminant $113.90$
Ramified prime $13$
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![-12167, 363883, -128206, -376064, 110214, 137683, -27027, -24128, 2457, 2080, -65, -78, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 78*x^11 - 65*x^10 + 2080*x^9 + 2457*x^8 - 24128*x^7 - 27027*x^6 + 137683*x^5 + 110214*x^4 - 376064*x^3 - 128206*x^2 + 363883*x - 12167)
 
gp: K = bnfinit(x^13 - 78*x^11 - 65*x^10 + 2080*x^9 + 2457*x^8 - 24128*x^7 - 27027*x^6 + 137683*x^5 + 110214*x^4 - 376064*x^3 - 128206*x^2 + 363883*x - 12167, 1)
 

Normalized defining polynomial

\( x^{13} - 78 x^{11} - 65 x^{10} + 2080 x^{9} + 2457 x^{8} - 24128 x^{7} - 27027 x^{6} + 137683 x^{5} + 110214 x^{4} - 376064 x^{3} - 128206 x^{2} + 363883 x - 12167 \)

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:  \(542800770374370512771595361=13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $113.90$
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:  $13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(169=13^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{169}(1,·)$, $\chi_{169}(66,·)$, $\chi_{169}(131,·)$, $\chi_{169}(40,·)$, $\chi_{169}(105,·)$, $\chi_{169}(14,·)$, $\chi_{169}(79,·)$, $\chi_{169}(144,·)$, $\chi_{169}(53,·)$, $\chi_{169}(118,·)$, $\chi_{169}(27,·)$, $\chi_{169}(92,·)$, $\chi_{169}(157,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{23} a^{9} + \frac{10}{23} a^{8} - \frac{1}{23} a^{7} + \frac{6}{23} a^{6} + \frac{6}{23} a^{5} - \frac{2}{23} a^{4} + \frac{5}{23} a^{3} + \frac{5}{23} a^{2} + \frac{8}{23} a$, $\frac{1}{437} a^{10} - \frac{1}{437} a^{9} + \frac{50}{437} a^{8} + \frac{109}{437} a^{7} + \frac{9}{437} a^{6} + \frac{116}{437} a^{5} - \frac{42}{437} a^{4} - \frac{50}{437} a^{3} - \frac{70}{437} a^{2} - \frac{203}{437} a + \frac{6}{19}$, $\frac{1}{437} a^{11} - \frac{8}{437} a^{9} + \frac{26}{437} a^{8} + \frac{175}{437} a^{7} - \frac{217}{437} a^{6} + \frac{169}{437} a^{5} + \frac{22}{437} a^{4} + \frac{32}{437} a^{3} - \frac{121}{437} a^{2} - \frac{84}{437} a + \frac{6}{19}$, $\frac{1}{423161433854797657} a^{12} + \frac{304941979446326}{423161433854797657} a^{11} + \frac{130257164717920}{423161433854797657} a^{10} - \frac{6998038476714584}{423161433854797657} a^{9} + \frac{30498274535732460}{423161433854797657} a^{8} + \frac{2036350708613748}{22271654413410403} a^{7} - \frac{142556340917656304}{423161433854797657} a^{6} + \frac{51542855244125771}{423161433854797657} a^{5} + \frac{3019358022124234}{22271654413410403} a^{4} - \frac{5885617833605573}{22271654413410403} a^{3} - \frac{144979391882990390}{423161433854797657} a^{2} - \frac{2981980706926316}{18398323211078159} a - \frac{328928563213444}{799927096133833}$

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:  \( 2733056590.62 \) (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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ R ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
$13$13.13.24.1$x^{13} - 13 x^{12} + 13$$13$$1$$24$$C_{13}$$[2]$