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Properties

Label 13.1.10493449363...7921.1
Degree $13$
Signature $[1, 6]$
Discriminant $4679^{6}$
Root discriminant $49.42$
Ramified prime $4679$
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![-1093, -1591, 1646, -1085, -5653, 5688, -4050, 1159, -353, 227, -34, -1, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 4*x^12 - x^11 - 34*x^10 + 227*x^9 - 353*x^8 + 1159*x^7 - 4050*x^6 + 5688*x^5 - 5653*x^4 - 1085*x^3 + 1646*x^2 - 1591*x - 1093)
 
gp: K = bnfinit(x^13 - 4*x^12 - x^11 - 34*x^10 + 227*x^9 - 353*x^8 + 1159*x^7 - 4050*x^6 + 5688*x^5 - 5653*x^4 - 1085*x^3 + 1646*x^2 - 1591*x - 1093, 1)
 

Normalized defining polynomial

\( x^{13} - 4 x^{12} - x^{11} - 34 x^{10} + 227 x^{9} - 353 x^{8} + 1159 x^{7} - 4050 x^{6} + 5688 x^{5} - 5653 x^{4} - 1085 x^{3} + 1646 x^{2} - 1591 x - 1093 \)

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:  \(10493449363391633467921=4679^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.42$
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:  $4679$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{109} a^{11} - \frac{42}{109} a^{10} + \frac{35}{109} a^{9} - \frac{45}{109} a^{8} - \frac{16}{109} a^{7} + \frac{41}{109} a^{6} + \frac{36}{109} a^{5} - \frac{54}{109} a^{4} - \frac{24}{109} a^{3} + \frac{38}{109} a^{2} + \frac{31}{109} a + \frac{48}{109}$, $\frac{1}{1176806024222426459401} a^{12} + \frac{3935993228775168292}{1176806024222426459401} a^{11} + \frac{3911619001758509898}{10796385543325013389} a^{10} + \frac{40365479569393941560}{1176806024222426459401} a^{9} + \frac{533958109194720465900}{1176806024222426459401} a^{8} - \frac{25341524464265657464}{1176806024222426459401} a^{7} - \frac{149787530153314701447}{1176806024222426459401} a^{6} + \frac{189907864768974372948}{1176806024222426459401} a^{5} - \frac{61562049401273894546}{1176806024222426459401} a^{4} - \frac{2059770810789861456}{22203887249479744517} a^{3} + \frac{563145905325746759659}{1176806024222426459401} a^{2} - \frac{12053064472433178236}{1176806024222426459401} a - \frac{35545834150936618481}{1176806024222426459401}$

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:  \( 396390.937641 \) (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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{13}$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
4679Data not computed