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Properties

Label 13.13.5909151103...1441.1
Degree $13$
Signature $[13, 0]$
Discriminant $79^{12}$
Root discriminant $56.45$
Ramified prime $79$
Class number $1$
Class group Trivial
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![293, -2196, 411, 6321, -4575, -4414, 5541, -617, -1193, 365, 77, -36, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 36*x^11 + 77*x^10 + 365*x^9 - 1193*x^8 - 617*x^7 + 5541*x^6 - 4414*x^5 - 4575*x^4 + 6321*x^3 + 411*x^2 - 2196*x + 293)
 
gp: K = bnfinit(x^13 - x^12 - 36*x^11 + 77*x^10 + 365*x^9 - 1193*x^8 - 617*x^7 + 5541*x^6 - 4414*x^5 - 4575*x^4 + 6321*x^3 + 411*x^2 - 2196*x + 293, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 36 x^{11} + 77 x^{10} + 365 x^{9} - 1193 x^{8} - 617 x^{7} + 5541 x^{6} - 4414 x^{5} - 4575 x^{4} + 6321 x^{3} + 411 x^{2} - 2196 x + 293 \)

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:  \(59091511031674153381441=79^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.45$
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:  $79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(79\)
Dirichlet character group:    $\lbrace$$\chi_{79}(64,·)$, $\chi_{79}(1,·)$, $\chi_{79}(67,·)$, $\chi_{79}(38,·)$, $\chi_{79}(65,·)$, $\chi_{79}(8,·)$, $\chi_{79}(10,·)$, $\chi_{79}(46,·)$, $\chi_{79}(18,·)$, $\chi_{79}(52,·)$, $\chi_{79}(21,·)$, $\chi_{79}(22,·)$, $\chi_{79}(62,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{23} a^{10} - \frac{11}{23} a^{9} - \frac{10}{23} a^{8} + \frac{8}{23} a^{7} - \frac{8}{23} a^{6} - \frac{9}{23} a^{5} + \frac{3}{23} a^{4} - \frac{8}{23} a^{3} + \frac{7}{23} a^{2} - \frac{7}{23} a + \frac{11}{23}$, $\frac{1}{23} a^{11} + \frac{7}{23} a^{9} - \frac{10}{23} a^{8} + \frac{11}{23} a^{7} - \frac{5}{23} a^{6} - \frac{4}{23} a^{5} + \frac{2}{23} a^{4} + \frac{11}{23} a^{3} + \frac{1}{23} a^{2} + \frac{3}{23} a + \frac{6}{23}$, $\frac{1}{1253201} a^{12} + \frac{16227}{1253201} a^{11} - \frac{3951}{1253201} a^{10} - \frac{40039}{1253201} a^{9} + \frac{222896}{1253201} a^{8} + \frac{417009}{1253201} a^{7} - \frac{554348}{1253201} a^{6} + \frac{285793}{1253201} a^{5} + \frac{183385}{1253201} a^{4} - \frac{221709}{1253201} a^{3} - \frac{546617}{1253201} a^{2} - \frac{616022}{1253201} a - \frac{404244}{1253201}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 14045368.3938 \)
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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
$79$79.13.12.1$x^{13} - 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$