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Properties

Label 13.1.48095468356...7449.1
Degree $13$
Signature $[1, 6]$
Discriminant $1907^{6}$
Root discriminant $32.66$
Ramified prime $1907$
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![374, -271, 543, -509, 62, 245, -405, 394, -251, 93, -34, 15, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 + 15*x^11 - 34*x^10 + 93*x^9 - 251*x^8 + 394*x^7 - 405*x^6 + 245*x^5 + 62*x^4 - 509*x^3 + 543*x^2 - 271*x + 374)
 
gp: K = bnfinit(x^13 - x^12 + 15*x^11 - 34*x^10 + 93*x^9 - 251*x^8 + 394*x^7 - 405*x^6 + 245*x^5 + 62*x^4 - 509*x^3 + 543*x^2 - 271*x + 374, 1)
 

Normalized defining polynomial

\( x^{13} - x^{12} + 15 x^{11} - 34 x^{10} + 93 x^{9} - 251 x^{8} + 394 x^{7} - 405 x^{6} + 245 x^{5} + 62 x^{4} - 509 x^{3} + 543 x^{2} - 271 x + 374 \)

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:  \(48095468356445867449=1907^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.66$
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:  $1907$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{7}{16} a + \frac{3}{8}$, $\frac{1}{800} a^{11} + \frac{3}{800} a^{10} + \frac{9}{200} a^{9} + \frac{37}{800} a^{8} + \frac{13}{160} a^{7} + \frac{21}{400} a^{6} + \frac{47}{800} a^{5} - \frac{39}{800} a^{4} + \frac{39}{100} a^{3} + \frac{59}{800} a^{2} + \frac{47}{160} a - \frac{143}{400}$, $\frac{1}{6335420800} a^{12} - \frac{476131}{791927600} a^{11} + \frac{32328983}{6335420800} a^{10} + \frac{35401901}{6335420800} a^{9} - \frac{97718111}{3167710400} a^{8} - \frac{95148473}{6335420800} a^{7} + \frac{311378581}{1267084160} a^{6} + \frac{283402191}{1583855200} a^{5} + \frac{845594001}{6335420800} a^{4} - \frac{67292787}{333443200} a^{3} + \frac{1119002513}{3167710400} a^{2} + \frac{2746542529}{6335420800} a + \frac{123143293}{3167710400}$

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:  \( 333391.286353 \) (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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.13.0.1}{13} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.13.0.1}{13} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.13.0.1}{13} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.13.0.1}{13} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.13.0.1}{13} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1907Data not computed