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Properties

Label 13.13.2554203806...6961.1
Degree $13$
Signature $[13, 0]$
Discriminant $131^{12}$
Root discriminant $90.03$
Ramified prime $131$
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![15667, 21271, -34869, -47723, 20328, 33548, -3352, -9610, -33, 1199, 27, -60, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 60*x^11 + 27*x^10 + 1199*x^9 - 33*x^8 - 9610*x^7 - 3352*x^6 + 33548*x^5 + 20328*x^4 - 47723*x^3 - 34869*x^2 + 21271*x + 15667)
 
gp: K = bnfinit(x^13 - x^12 - 60*x^11 + 27*x^10 + 1199*x^9 - 33*x^8 - 9610*x^7 - 3352*x^6 + 33548*x^5 + 20328*x^4 - 47723*x^3 - 34869*x^2 + 21271*x + 15667, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 60 x^{11} + 27 x^{10} + 1199 x^{9} - 33 x^{8} - 9610 x^{7} - 3352 x^{6} + 33548 x^{5} + 20328 x^{4} - 47723 x^{3} - 34869 x^{2} + 21271 x + 15667 \)

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:  \(25542038069936263923006961=131^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.03$
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:  $131$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(131\)
Dirichlet character group:    $\lbrace$$\chi_{131}(1,·)$, $\chi_{131}(99,·)$, $\chi_{131}(39,·)$, $\chi_{131}(80,·)$, $\chi_{131}(107,·)$, $\chi_{131}(45,·)$, $\chi_{131}(112,·)$, $\chi_{131}(113,·)$, $\chi_{131}(52,·)$, $\chi_{131}(84,·)$, $\chi_{131}(60,·)$, $\chi_{131}(62,·)$, $\chi_{131}(63,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3869} a^{11} + \frac{495}{3869} a^{10} + \frac{1213}{3869} a^{9} - \frac{1773}{3869} a^{8} + \frac{987}{3869} a^{7} - \frac{1345}{3869} a^{6} - \frac{1792}{3869} a^{5} + \frac{849}{3869} a^{4} + \frac{7}{73} a^{3} + \frac{377}{3869} a^{2} + \frac{197}{3869} a - \frac{1849}{3869}$, $\frac{1}{29655505980893777} a^{12} + \frac{2786585719338}{29655505980893777} a^{11} + \frac{8307218062751569}{29655505980893777} a^{10} - \frac{5608908248182115}{29655505980893777} a^{9} - \frac{7315472545156425}{29655505980893777} a^{8} - \frac{5769496573751300}{29655505980893777} a^{7} - \frac{11587114687714914}{29655505980893777} a^{6} + \frac{6380121130974331}{29655505980893777} a^{5} - \frac{5444767598768430}{29655505980893777} a^{4} - \frac{3678758867361965}{29655505980893777} a^{3} + \frac{2424995795257099}{29655505980893777} a^{2} - \frac{223606030901704}{29655505980893777} a - \frac{9357701767805873}{29655505980893777}$

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:  \( 292369424.433 \) (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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
$131$131.13.12.1$x^{13} - 131$$13$$1$$12$$C_{13}$$[\ ]_{13}$