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Properties

Label 13.1.39258873741...6656.1
Degree $13$
Signature $[1, 6]$
Discriminant $2^{18}\cdot 157^{6}$
Root discriminant $26.93$
Ramified primes $2, 157$
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![-8, 20, 136, 168, -394, 211, 96, -162, 64, 16, -32, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 6*x^12 + 18*x^11 - 32*x^10 + 16*x^9 + 64*x^8 - 162*x^7 + 96*x^6 + 211*x^5 - 394*x^4 + 168*x^3 + 136*x^2 + 20*x - 8)
 
gp: K = bnfinit(x^13 - 6*x^12 + 18*x^11 - 32*x^10 + 16*x^9 + 64*x^8 - 162*x^7 + 96*x^6 + 211*x^5 - 394*x^4 + 168*x^3 + 136*x^2 + 20*x - 8, 1)
 

Normalized defining polynomial

\( x^{13} - 6 x^{12} + 18 x^{11} - 32 x^{10} + 16 x^{9} + 64 x^{8} - 162 x^{7} + 96 x^{6} + 211 x^{5} - 394 x^{4} + 168 x^{3} + 136 x^{2} + 20 x - 8 \)

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:  \(3925887374183366656=2^{18}\cdot 157^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.93$
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:  $2, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{832} a^{11} + \frac{1}{104} a^{10} - \frac{9}{832} a^{9} - \frac{11}{416} a^{8} - \frac{81}{832} a^{7} - \frac{23}{208} a^{6} + \frac{33}{832} a^{5} + \frac{17}{416} a^{4} + \frac{1}{16} a^{3} + \frac{23}{104} a^{2} + \frac{75}{208} a - \frac{3}{104}$, $\frac{1}{21632} a^{12} - \frac{11}{21632} a^{11} + \frac{671}{21632} a^{10} - \frac{1307}{21632} a^{9} + \frac{1169}{21632} a^{8} - \frac{9}{21632} a^{7} + \frac{9}{1664} a^{6} - \frac{4129}{21632} a^{5} - \frac{1337}{10816} a^{4} - \frac{1813}{5408} a^{3} + \frac{2009}{5408} a^{2} - \frac{79}{5408} a - \frac{359}{2704}$

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:  \( 656720.852793 \) (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 R ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.13.0.1}{13} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.13.0.1}{13} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. 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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
$157$$\Q_{157}$$x + 5$$1$$1$$0$Trivial$[\ ]$
157.2.1.2$x^{2} + 785$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.2$x^{2} + 785$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.2$x^{2} + 785$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.2$x^{2} + 785$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.2$x^{2} + 785$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.2$x^{2} + 785$$2$$1$$1$$C_2$$[\ ]_{2}$