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Properties

Label 13.1.83815694592...6649.1
Degree $13$
Signature $[1, 6]$
Discriminant $4507^{6}$
Root discriminant $48.57$
Ramified prime $4507$
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![-9216, 12288, -15040, 9024, -4692, 1034, -225, -6, -132, 99, -65, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 6*x^12 + 24*x^11 - 65*x^10 + 99*x^9 - 132*x^8 - 6*x^7 - 225*x^6 + 1034*x^5 - 4692*x^4 + 9024*x^3 - 15040*x^2 + 12288*x - 9216)
 
gp: K = bnfinit(x^13 - 6*x^12 + 24*x^11 - 65*x^10 + 99*x^9 - 132*x^8 - 6*x^7 - 225*x^6 + 1034*x^5 - 4692*x^4 + 9024*x^3 - 15040*x^2 + 12288*x - 9216, 1)
 

Normalized defining polynomial

\( x^{13} - 6 x^{12} + 24 x^{11} - 65 x^{10} + 99 x^{9} - 132 x^{8} - 6 x^{7} - 225 x^{6} + 1034 x^{5} - 4692 x^{4} + 9024 x^{3} - 15040 x^{2} + 12288 x - 9216 \)

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:  \(8381569459284757656649=4507^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.57$
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:  $4507$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{48} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{3}{16} a^{5} - \frac{1}{16} a^{4} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} + \frac{5}{12} a$, $\frac{1}{288} a^{10} + \frac{1}{288} a^{9} - \frac{1}{32} a^{8} - \frac{7}{96} a^{6} - \frac{7}{32} a^{5} + \frac{11}{96} a^{4} - \frac{11}{48} a^{3} + \frac{11}{72} a^{2} - \frac{1}{18} a + \frac{1}{3}$, $\frac{1}{131328} a^{11} - \frac{17}{65664} a^{10} + \frac{85}{8208} a^{9} - \frac{521}{14592} a^{8} - \frac{1915}{43776} a^{7} + \frac{415}{5472} a^{6} - \frac{1025}{21888} a^{5} - \frac{5699}{43776} a^{4} + \frac{23515}{65664} a^{3} + \frac{277}{32832} a^{2} + \frac{3875}{8208} a + \frac{335}{684}$, $\frac{1}{74068992} a^{12} + \frac{109}{37034496} a^{11} - \frac{15607}{9258624} a^{10} + \frac{34823}{8229888} a^{9} - \frac{568159}{24689664} a^{8} - \frac{9007}{6172416} a^{7} - \frac{1274921}{12344832} a^{6} + \frac{952405}{24689664} a^{5} - \frac{3407051}{37034496} a^{4} + \frac{180665}{393984} a^{3} - \frac{19369}{49248} a^{2} - \frac{637}{4104} a + \frac{3485}{10716}$

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:  \( 10670572.6643 \) (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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
4507Data not computed