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Properties

Label 13.1.27277017024...0625.1
Degree $13$
Signature $[1, 6]$
Discriminant $5^{6}\cdot 347^{6}$
Root discriminant $31.27$
Ramified primes $5, 347$
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![-395, -220, -803, 168, 140, 147, -137, -67, 28, 17, 0, -1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 3*x^12 - x^11 + 17*x^9 + 28*x^8 - 67*x^7 - 137*x^6 + 147*x^5 + 140*x^4 + 168*x^3 - 803*x^2 - 220*x - 395)
 
gp: K = bnfinit(x^13 - 3*x^12 - x^11 + 17*x^9 + 28*x^8 - 67*x^7 - 137*x^6 + 147*x^5 + 140*x^4 + 168*x^3 - 803*x^2 - 220*x - 395, 1)
 

Normalized defining polynomial

\( x^{13} - 3 x^{12} - x^{11} + 17 x^{9} + 28 x^{8} - 67 x^{7} - 137 x^{6} + 147 x^{5} + 140 x^{4} + 168 x^{3} - 803 x^{2} - 220 x - 395 \)

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:  \(27277017024655140625=5^{6}\cdot 347^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.27$
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:  $5, 347$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{45} a^{9} - \frac{7}{45} a^{8} - \frac{1}{45} a^{7} - \frac{1}{15} a^{6} - \frac{11}{45} a^{5} - \frac{2}{5} a^{4} - \frac{19}{45} a^{3} - \frac{4}{15} a^{2} + \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{45} a^{10} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{2}{45} a^{6} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{2}{15} a^{2} - \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{225} a^{11} + \frac{2}{225} a^{9} + \frac{31}{225} a^{8} - \frac{1}{25} a^{7} + \frac{19}{225} a^{6} + \frac{7}{25} a^{5} - \frac{91}{225} a^{4} - \frac{37}{225} a^{3} - \frac{44}{225} a^{2} + \frac{1}{9} a + \frac{8}{45}$, $\frac{1}{3910669425} a^{12} + \frac{1626238}{782133885} a^{11} - \frac{1639271}{1303556475} a^{10} + \frac{37978661}{3910669425} a^{9} + \frac{60551034}{434518825} a^{8} - \frac{131243072}{1303556475} a^{7} + \frac{15122002}{100273575} a^{6} - \frac{174387819}{434518825} a^{5} - \frac{921205927}{3910669425} a^{4} + \frac{1704459976}{3910669425} a^{3} - \frac{8313493}{17380753} a^{2} + \frac{17992241}{60164145} a - \frac{7659005}{17380753}$

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:  \( 179464.45726 \)
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 ${\href{/LocalNumberField/2.13.0.1}{13} }$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.13.0.1}{13} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.13.0.1}{13} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.13.0.1}{13} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
347Data not computed