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Properties

Label 13.1.93648472563...6281.1
Degree $13$
Signature $[1, 6]$
Discriminant $2131^{6}$
Root discriminant $34.38$
Ramified prime $2131$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{13}$ (as 13T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2048, -1456, 40, 856, -1126, 257, 399, -384, 175, -50, 5, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 5*x^12 + 8*x^11 + 5*x^10 - 50*x^9 + 175*x^8 - 384*x^7 + 399*x^6 + 257*x^5 - 1126*x^4 + 856*x^3 + 40*x^2 - 1456*x + 2048)
 
gp: K = bnfinit(x^13 - 5*x^12 + 8*x^11 + 5*x^10 - 50*x^9 + 175*x^8 - 384*x^7 + 399*x^6 + 257*x^5 - 1126*x^4 + 856*x^3 + 40*x^2 - 1456*x + 2048, 1)
 

Normalized defining polynomial

\( x^{13} - 5 x^{12} + 8 x^{11} + 5 x^{10} - 50 x^{9} + 175 x^{8} - 384 x^{7} + 399 x^{6} + 257 x^{5} - 1126 x^{4} + 856 x^{3} + 40 x^{2} - 1456 x + 2048 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $13$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(93648472563048956281=2131^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.38$
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:  $2131$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{24} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{3}$, $\frac{1}{24} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{3} a$, $\frac{1}{96} a^{10} + \frac{1}{96} a^{9} - \frac{1}{96} a^{8} - \frac{1}{16} a^{7} + \frac{1}{32} a^{6} - \frac{5}{32} a^{5} - \frac{1}{32} a^{4} - \frac{3}{8} a^{3} + \frac{1}{3} a^{2} - \frac{5}{12} a - \frac{1}{3}$, $\frac{1}{99072} a^{11} - \frac{269}{99072} a^{10} - \frac{1991}{99072} a^{9} - \frac{107}{6192} a^{8} - \frac{4099}{33024} a^{7} + \frac{3637}{33024} a^{6} + \frac{7949}{33024} a^{5} - \frac{277}{5504} a^{4} + \frac{3403}{12384} a^{3} - \frac{4133}{12384} a^{2} - \frac{2665}{6192} a + \frac{137}{387}$, $\frac{1}{11690496} a^{12} + \frac{11}{2922624} a^{11} + \frac{9155}{2922624} a^{10} - \frac{167719}{11690496} a^{9} + \frac{142255}{11690496} a^{8} + \frac{32735}{649472} a^{7} + \frac{1113}{11008} a^{6} + \frac{927023}{3896832} a^{5} + \frac{1123411}{5845248} a^{4} + \frac{162079}{730656} a^{3} - \frac{389767}{1461312} a^{2} - \frac{197789}{730656} a - \frac{6230}{22833}$

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:  $6$
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:  \( 746752.77063 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{13}$ (as 13T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 26
The 8 conjugacy class representatives for $D_{13}$
Character table for $D_{13}$

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2131Data not computed