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Properties

Label 13.13.3536296682...0881.1
Degree $13$
Signature $[13, 0]$
Discriminant $3^{12}\cdot 13^{16}$
Root discriminant $64.78$
Ramified primes $3, 13$
Class number $1$
Class group Trivial
Galois group $C_{13}:C_3$ (as 13T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-306, -2691, 10062, -6435, -7488, 7605, 1872, -2925, -156, 507, 0, -39, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 39*x^11 + 507*x^9 - 156*x^8 - 2925*x^7 + 1872*x^6 + 7605*x^5 - 7488*x^4 - 6435*x^3 + 10062*x^2 - 2691*x - 306)
 
gp: K = bnfinit(x^13 - 39*x^11 + 507*x^9 - 156*x^8 - 2925*x^7 + 1872*x^6 + 7605*x^5 - 7488*x^4 - 6435*x^3 + 10062*x^2 - 2691*x - 306, 1)
 

Normalized defining polynomial

\( x^{13} - 39 x^{11} + 507 x^{9} - 156 x^{8} - 2925 x^{7} + 1872 x^{6} + 7605 x^{5} - 7488 x^{4} - 6435 x^{3} + 10062 x^{2} - 2691 x - 306 \)

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:  \(353629668200918277880881=3^{12}\cdot 13^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.78$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{8} - \frac{1}{2} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{42} a^{10} + \frac{1}{21} a^{9} + \frac{1}{42} a^{8} - \frac{3}{7} a^{6} + \frac{3}{7} a^{4} - \frac{5}{14} a^{3} + \frac{2}{7} a^{2} + \frac{3}{14} a + \frac{3}{7}$, $\frac{1}{42} a^{11} - \frac{1}{14} a^{9} - \frac{1}{21} a^{8} + \frac{1}{14} a^{7} + \frac{5}{14} a^{6} - \frac{1}{14} a^{5} + \frac{2}{7} a^{4} - \frac{1}{2} a^{3} + \frac{1}{7} a^{2} - \frac{1}{2} a + \frac{1}{7}$, $\frac{1}{189966} a^{12} + \frac{914}{94983} a^{11} - \frac{68}{13569} a^{10} + \frac{937}{31661} a^{9} - \frac{6151}{94983} a^{8} + \frac{24}{31661} a^{7} - \frac{1879}{4523} a^{6} + \frac{15361}{63322} a^{5} + \frac{8614}{31661} a^{4} - \frac{12988}{31661} a^{3} - \frac{1242}{4523} a^{2} - \frac{6179}{31661} a + \frac{14101}{31661}$

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:  \( 166789217.838 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{13}:C_3$ (as 13T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 39
The 7 conjugacy class representatives for $C_{13}:C_3$
Character table for $C_{13}:C_3$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$3.13.12.1$x^{13} - 3$$13$$1$$12$$C_{13}:C_3$$[\ ]_{13}^{3}$
$13$13.13.16.2$x^{13} + 130 x^{4} + 13$$13$$1$$16$$C_{13}:C_3$$[4/3]_{3}$