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Properties

Label 13.1.25285076456...3641.1
Degree $13$
Signature $[1, 6]$
Discriminant $3691^{6}$
Root discriminant $44.30$
Ramified prime $3691$
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![576, -3936, 10576, -13920, 9140, -3034, 849, -62, -65, 51, -25, 17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 6*x^12 + 17*x^11 - 25*x^10 + 51*x^9 - 65*x^8 - 62*x^7 + 849*x^6 - 3034*x^5 + 9140*x^4 - 13920*x^3 + 10576*x^2 - 3936*x + 576)
 
gp: K = bnfinit(x^13 - 6*x^12 + 17*x^11 - 25*x^10 + 51*x^9 - 65*x^8 - 62*x^7 + 849*x^6 - 3034*x^5 + 9140*x^4 - 13920*x^3 + 10576*x^2 - 3936*x + 576, 1)
 

Normalized defining polynomial

\( x^{13} - 6 x^{12} + 17 x^{11} - 25 x^{10} + 51 x^{9} - 65 x^{8} - 62 x^{7} + 849 x^{6} - 3034 x^{5} + 9140 x^{4} - 13920 x^{3} + 10576 x^{2} - 3936 x + 576 \)

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:  \(2528507645606750993641=3691^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.30$
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:  $3691$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{5} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{6} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{12} a^{5} - \frac{1}{4} a^{4} - \frac{1}{12} a^{3} - \frac{5}{12} a^{2} + \frac{1}{3} a$, $\frac{1}{36} a^{8} - \frac{1}{36} a^{7} + \frac{1}{36} a^{6} - \frac{1}{36} a^{5} - \frac{7}{36} a^{4} + \frac{13}{36} a^{3} + \frac{7}{18} a^{2} - \frac{7}{18} a + \frac{1}{3}$, $\frac{1}{36} a^{9} - \frac{1}{18} a^{5} + \frac{1}{6} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{5}{18} a + \frac{1}{3}$, $\frac{1}{144} a^{10} + \frac{1}{144} a^{8} + \frac{5}{144} a^{7} + \frac{5}{144} a^{6} - \frac{7}{144} a^{5} + \frac{5}{36} a^{4} - \frac{11}{144} a^{3} - \frac{1}{18} a - \frac{1}{6}$, $\frac{1}{1728} a^{11} - \frac{1}{864} a^{10} + \frac{7}{576} a^{9} - \frac{13}{1728} a^{8} + \frac{59}{1728} a^{7} - \frac{3}{64} a^{6} + \frac{65}{864} a^{5} + \frac{397}{1728} a^{4} - \frac{31}{96} a^{3} + \frac{49}{216} a^{2} + \frac{91}{216} a - \frac{1}{36}$, $\frac{1}{1285632} a^{12} - \frac{7}{160704} a^{11} + \frac{313}{142848} a^{10} + \frac{1973}{1285632} a^{9} + \frac{8537}{1285632} a^{8} + \frac{461}{15872} a^{7} + \frac{3203}{160704} a^{6} - \frac{30431}{1285632} a^{5} - \frac{475}{35712} a^{4} + \frac{46403}{321408} a^{3} - \frac{19031}{160704} a^{2} + \frac{1759}{6696} a + \frac{1369}{4464}$

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:  \( 4940532.71808 \) (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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
3691Data not computed