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Properties

Label 13.13.4912589042...4641.1
Degree $13$
Signature $[13, 0]$
Discriminant $53^{12}$
Root discriminant $39.05$
Ramified prime $53$
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![-1, 10, 68, -291, -215, 738, 246, -601, -116, 190, 19, -24, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 24*x^11 + 19*x^10 + 190*x^9 - 116*x^8 - 601*x^7 + 246*x^6 + 738*x^5 - 215*x^4 - 291*x^3 + 68*x^2 + 10*x - 1)
 
gp: K = bnfinit(x^13 - x^12 - 24*x^11 + 19*x^10 + 190*x^9 - 116*x^8 - 601*x^7 + 246*x^6 + 738*x^5 - 215*x^4 - 291*x^3 + 68*x^2 + 10*x - 1, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} - 24 x^{11} + 19 x^{10} + 190 x^{9} - 116 x^{8} - 601 x^{7} + 246 x^{6} + 738 x^{5} - 215 x^{4} - 291 x^{3} + 68 x^{2} + 10 x - 1 \)

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:  \(491258904256726154641=53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.05$
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:  $53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(53\)
Dirichlet character group:    $\lbrace$$\chi_{53}(1,·)$, $\chi_{53}(36,·)$, $\chi_{53}(10,·)$, $\chi_{53}(44,·)$, $\chi_{53}(13,·)$, $\chi_{53}(46,·)$, $\chi_{53}(15,·)$, $\chi_{53}(16,·)$, $\chi_{53}(49,·)$, $\chi_{53}(24,·)$, $\chi_{53}(47,·)$, $\chi_{53}(28,·)$, $\chi_{53}(42,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{23} a^{11} - \frac{3}{23} a^{10} - \frac{6}{23} a^{9} - \frac{5}{23} a^{8} - \frac{10}{23} a^{7} + \frac{5}{23} a^{6} + \frac{5}{23} a^{5} - \frac{3}{23} a^{4} - \frac{1}{23} a^{3} + \frac{4}{23} a^{2} + \frac{11}{23} a + \frac{2}{23}$, $\frac{1}{435105007} a^{12} - \frac{6511450}{435105007} a^{11} - \frac{211168269}{435105007} a^{10} - \frac{202582962}{435105007} a^{9} - \frac{11741}{435105007} a^{8} + \frac{23782233}{435105007} a^{7} - \frac{11983132}{435105007} a^{6} + \frac{20632332}{435105007} a^{5} - \frac{26107826}{435105007} a^{4} + \frac{165061215}{435105007} a^{3} - \frac{53912393}{435105007} a^{2} + \frac{83591149}{435105007} a + \frac{80272926}{435105007}$

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:  \( 1314145.36669 \)
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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ R ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
$53$53.13.12.1$x^{13} - 53$$13$$1$$12$$C_{13}$$[\ ]_{13}$