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Properties

Label 13.1.96857448932...4849.1
Degree $13$
Signature $[1, 6]$
Discriminant $2143^{6}$
Root discriminant $34.47$
Ramified prime $2143$
Class number $1$
Class group Trivial
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![-657, 828, 983, -1784, 304, 942, -876, 336, -46, -2, -8, 11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 5*x^12 + 11*x^11 - 8*x^10 - 2*x^9 - 46*x^8 + 336*x^7 - 876*x^6 + 942*x^5 + 304*x^4 - 1784*x^3 + 983*x^2 + 828*x - 657)
 
gp: K = bnfinit(x^13 - 5*x^12 + 11*x^11 - 8*x^10 - 2*x^9 - 46*x^8 + 336*x^7 - 876*x^6 + 942*x^5 + 304*x^4 - 1784*x^3 + 983*x^2 + 828*x - 657, 1)
 

Normalized defining polynomial

\( x^{13} - 5 x^{12} + 11 x^{11} - 8 x^{10} - 2 x^{9} - 46 x^{8} + 336 x^{7} - 876 x^{6} + 942 x^{5} + 304 x^{4} - 1784 x^{3} + 983 x^{2} + 828 x - 657 \)

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:  \(96857448932353894849=2143^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.47$
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:  $2143$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{117} a^{10} + \frac{5}{39} a^{9} + \frac{14}{117} a^{8} - \frac{8}{117} a^{7} + \frac{19}{39} a^{5} + \frac{56}{117} a^{4} + \frac{7}{117} a^{3} + \frac{8}{117} a^{2} + \frac{19}{39} a + \frac{4}{13}$, $\frac{1}{4095} a^{11} - \frac{2}{4095} a^{10} - \frac{124}{4095} a^{9} - \frac{134}{1365} a^{8} - \frac{332}{4095} a^{7} + \frac{54}{455} a^{6} + \frac{452}{4095} a^{5} + \frac{647}{1365} a^{4} - \frac{661}{1365} a^{3} + \frac{1286}{4095} a^{2} + \frac{74}{455} a + \frac{153}{455}$, $\frac{1}{125548605} a^{12} + \frac{1987}{25109721} a^{11} - \frac{23486}{9657585} a^{10} + \frac{39808}{1931517} a^{9} - \frac{2987122}{41849535} a^{8} - \frac{3703723}{125548605} a^{7} + \frac{16171619}{125548605} a^{6} + \frac{5571508}{25109721} a^{5} - \frac{42385561}{125548605} a^{4} - \frac{954061}{8369907} a^{3} + \frac{3919534}{17935515} a^{2} - \frac{237467}{1349985} a - \frac{3039334}{13949845}$

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:  $6$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 175904.527309 \)
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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2143Data not computed